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+{
+    "url": "http://arxiv.org/abs/2405.03962v1",
+    "title": "AdsorbDiff: Adsorbate Placement via Conditional Denoising Diffusion",
+    "abstract": "Determining the optimal configuration of adsorbates on a slab (adslab) is\npivotal in the exploration of novel catalysts across diverse applications.\nTraditionally, the quest for the lowest energy adslab configuration involves\nplacing the adsorbate onto the slab followed by an optimization process. Prior\nmethodologies have relied on heuristics, problem-specific intuitions, or\nbrute-force approaches to guide adsorbate placement. In this work, we propose a\nnovel framework for adsorbate placement using denoising diffusion. The model is\ndesigned to predict the optimal adsorbate site and orientation corresponding to\nthe lowest energy configuration. Further, we have an end-to-end evaluation\nframework where diffusion-predicted adslab configuration is optimized with a\npretrained machine learning force field and finally evaluated with Density\nFunctional Theory (DFT). Our findings demonstrate an acceleration of up to 5x\nor 3.5x improvement in accuracy compared to the previous best approach. Given\nthe novelty of this framework and application, we provide insights into the\nimpact of pre-training, model architectures, and conduct extensive experiments\nto underscore the significance of this approach.",
+    "authors": "Adeesh Kolluru, John R Kitchin",
+    "published": "2024-05-07",
+    "updated": "2024-05-07",
+    "primary_cat": "cs.LG",
+    "cats": [
+        "cs.LG",
+        "physics.chem-ph"
+    ],
+    "label": "Original Paper",
+    "paper_cat": "Diffusion AND Model",
+    "gt": "Determining the optimal configuration of adsorbates on a slab (adslab) is\npivotal in the exploration of novel catalysts across diverse applications.\nTraditionally, the quest for the lowest energy adslab configuration involves\nplacing the adsorbate onto the slab followed by an optimization process. Prior\nmethodologies have relied on heuristics, problem-specific intuitions, or\nbrute-force approaches to guide adsorbate placement. In this work, we propose a\nnovel framework for adsorbate placement using denoising diffusion. The model is\ndesigned to predict the optimal adsorbate site and orientation corresponding to\nthe lowest energy configuration. Further, we have an end-to-end evaluation\nframework where diffusion-predicted adslab configuration is optimized with a\npretrained machine learning force field and finally evaluated with Density\nFunctional Theory (DFT). Our findings demonstrate an acceleration of up to 5x\nor 3.5x improvement in accuracy compared to the previous best approach. Given\nthe novelty of this framework and application, we provide insights into the\nimpact of pre-training, model architectures, and conduct extensive experiments\nto underscore the significance of this approach.",
+    "main_content": "Introduction Heterogenous catalysis plays an important role in developing chemicals in industries, environmental protection through converters, and the synthesis of alternative fuels (Liu & Li, 2017; Zitnick et al., 2020). Modeling these chemical reactions involve an intermediate adsorbate on a catalyst slab which determines the efficacy of the catalyst for that particular reaction. Discovering a novel catalyst computationally involves screening through billions of candidates and finding the lowest energy configuration. 1Department of Chemical Engineering, Carnegie Mellon University. Correspondence to: Adeesh Kolluru <akolluru@andrew.cmu.edu>, John R. Kitchin <jkitchin@andrew.cmu.edu>. Finding the lowest energy configuration for an adsorbate and slab requires a global optimum (which is non-convex) search across different sites on the slab. Conventional approaches solve this in two steps (1) heuristically place the adsorbate on certain important sites and (2) perform optimization with quantum mechanical calculators like Density Functional Theory (DFT) on each of these sites. The lowest energy site out of these is considered for calculating adsorption energy, which is a thermodynamic descriptor for how good that catalyst is. With recent advances in machine learning methods for predicting forces, it has become possible to perform optimization with ML force fields (MLFFs) instead of Density Functional Theory (DFT) making this process faster and easier to test many sites and find better minima. These ML force fields are trained on DFT data to predict energies and forces corresponding to different adslab configurations. The recent release of the OC20-Dense dataset (Lan et al., 2023) signifies a significant advancement in the computation of the lowest energy adslab configuration. This work employs a blend of heuristic and random adsorbate placements across 100 sites, with subsequent optimizations across each site using Density Functional Theory (DFT) to calculate adsorption energy. The study further introduces AdsorbML, a paradigm characterized by a brute-force exploration of initial adsorbate placements. Employing pre-trained machine learning (ML) force fields from OC20, AdsorbML streamlines the optimization process, culminating in the determination of the lowest energy adsorbate-slab (adslab) configuration. The predictive accuracy of these configurations is rigorously validated against DFT single-points or complete DFT optimization. This hybrid approach results in a computational acceleration of 2000-fold in adsorption energy calculations compared to the sole reliance on DFT calculations. Recent developments in graph neural network (GNN) based ML architectures have increased the accuracies of adsorption energy prediction significantly by encoding geometric information of atoms in more explicit ways. However, there\u2019s little to no work done on improving the adsorption site prediction which could help us get away with the currently used brute-force approach. In this work, we develop a novel conditional denoising diffu1 arXiv:2405.03962v1  [cs.LG]  7 May 2024 \fAdsorbate placement via conditional denoising diffusion sion framework for adsorbate placement. We first formulate a diffusion framework over the space of the 2D translation and 3D rigid rotation of an adsorbate molecule over the slab considering periodic boundary conditions (PBC) of the slab. Through the learned diffusion process, we sample the most stable site by iteratively updating the center of mass of adsorbate and rigid orientation. Performing a naive unconditional diffusion framework on the most optimal adsorbate site and orientation \u2014 corresponding to the lowest energy adslab configuration out of 100 densely sampled calculations in OC20-Dense \u2014 leads to throwing away 99% of DFT optimal energy data. Therefore, we modify the diffusion training to be conditional on relative energies (relative across densely sampled sites of an adslab combination). This leads to significant improvements in accuracies and sample efficiency during diffusion training. After sampling for the optimal site and orientation of adsorbate on the slab, we perform ML force field (MLFF) optimization and DFT single-point verification similar to AdsorbML. This comprehensive end-to-end evaluation helps in robust assessment of the practical impact of the learned diffusion model. There have been significant advances in diffusion generative models in molecular and material discovery, and analogous problems in molecular docking on proteins. However, this is the first work to frame the adsorbate placement problem considering all its symmetries with the slab in a diffusion framework. Intuitively, the reverse diffusion process of AdsorbDiff helps in skipping multiple minima sites due to its energy-based conditional sampling which is followed by a local optimization with a DFT-learned MLFF to find a global optimum. To facilitate further research on this problem, we provide comprehensive results on the importance of GNN architectures for the diffusion task, show the importance of pretraining, and demonstrate the success of our approach to in-distribution (ID) and out-of-distribution (OOD) splits. The summary of contributions of this work are \u2022 We propose AdsorbDiff, a novel conditional denoising diffusion framework designed to leverage the translation, rotation, and periodic symmetries inherent in adsorbate and slab interactions. Additionally, this framework is adept at efficiently predicting the lowest energy site by conditional training on relative energies. \u2022 We present our results in a comprehensive end-to-end evaluation framework, integrated with DFT, to accurately gauge the true capability of our approach in predicting optimal adsorption energies. \u2022 We achieve a 31.8% success rate, 3.5x higher than the naive AdsorbML baseline of 9.1% with a single site prediction. Alternatively, we demonstrate that a comparable level of accuracy could be achieved by AdsorbML by employing 5x more placements. \u2022 We demonstrate that pretraining on large-scale local optimization data can significantly improve the results on the search for global optima. \u2022 We show that diffusion results exhibit insignificant dependence on GNN architectures, in contrast to the notable differences observed for the same architectures when trained on DFT forces. \u2022 We highlight the model\u2019s generalization capabilities to previously unseen adsorbates and slabs. 2. Background and Related Work Force-fields: Energy and forces (as a gradient of energy with respect to positions) are calculated using ab initio quantum mechanical methods like Density Functional Theory (DFT). ML models can be trained to predict these energies and forces, and are called ML force-fields (MLFFs). These force fields can be utilized to perform structure optimization to get the lowest energy structures. Optimization: For adsorption energy prediction, we start with an optimized adsorbate and slab, place the adsorbate on a slab, and perform optimization to get an adslab configuration with the lowest energy. Usually, second-order optimizers like BFGS, L-BFGS, Conjugate gradient descent, etc are used to solve this optimization problem. Since this is non-convex, the initial guess of adsorbate placement or the strategy of optimization is critical to finding an adslab configuration corresponding to the global optimum. AdsorbML (Lan et al., 2023) method starts with combining heuristic and random initial placements which is a brute-force approach to finding better minima. \u201dEasy Potential\u201d from (Schaarschmidt et al., 2022) trains a simple harmonic potential to guess this initial placement. Learn2Hop (Merchant et al., 2021) also learns the optimization landscape to navigate through better and hop through local minima. There are approaches like minima hopping that help in navigating through the entire optimization landscape with a force-field (Jung et al., 2023) and help in finding better minima, but these could be computationally expensive. GNNs: Message-Passing Neural Networks (MPNN) are a class of graph neural networks (GNN) that are utilized across material property prediction tasks. Different architectures encode the geometric information in different ways. SchNet (Sch\u00a8 utt et al., 2018) only encodes the distance information. Including more explicit geometric features have improved the model prediction as DimeNet (Gasteiger et al., 2020b;a) incorporates triplets. SphereNet (Liu et al., 2021), GemNet (Gasteiger et al., 2021; 2022) incorporates complete geometric information explicitly by giving triplets and quadruplets information. PaiNN (Sch\u00a8 utt et al., 2021) incorporates directional information and applies only linear operations on those features. Equivariant models like NequIP (Batzner et al., 2022), Allegro (Musaelian et al., 2023), MACE (Batatia et al., 2022), SCN (Zitnick et al., 2 \fAdsorbate placement via conditional denoising diffusion Figure 1. Overview of AdsorbDiff: Random initial site and orientation for the adsorbate are selected, followed by sampling over 2D translation, 3D rigid rotations, and considering periodic boundary conditions (PBC) to predict the optimal site and orientation. MLFF optimization is then conducted from the predicted site with a fixed interstitial gap until convergence. The final prediction undergoes constraint verification, and DFT verification is performed on valid structures to calculate success rates. 2022), Equiformer (Liao & Smidt, 2022; Liao et al., 2023) utilize spherical harmonics in representing the geometric features. Diffusion Models: Diffusion models are a class of generative models that have shown impressive results across different domains starting from computer vision (Dhariwal & Nichol, 2021; Croitoru et al., 2023), language models (Gong et al., 2022), temporal data modeling, to applications in molecules (Xu et al., 2022; 2023; Arts et al., 2023; Hoogeboom et al., 2022; Jing et al., 2022), proteins (Wu et al., 2022; Trippe et al., 2022; Watson et al., 2022; 2023) and materials (Xie et al., 2021; Fu et al., 2023; Zeni et al., 2023; Merchant et al., 2023; Yang et al., 2023b). There are different kinds of formulations proposed for diffusion models like denoising diffusion probabilistic models (DDPMs), score-based generative models (SGMs), and stochastic differential equations (Score SDEs) (Yang et al., 2023a). Many of these formulations have been adapted to problems in molecular and material discovery. For example, CDVAE (Xie et al., 2021) adapts concepts from noise-conditioned score networks (NCSN) for bulk discovery. Conditional diffusion has also been recently utilized across proteins (Krishna et al., 2024), catalyst and materials (Zheng et al., 2023) for generating structures with required properties. Diffusion models have also been recently utilized for molecular docking on proteins (Corso et al., 2022). Although this problem is somewhat analogous to placing adsorbate on a slab, as far as we know there hasn\u2019t been previous work on formulating adsorbate placement in a diffusion framework. AdsorbDiff also differs from molecular docking in several key aspects \u2013 2D translation formulation, periodic boundary conditions, conditional denoising formulation, and the requirement of DFT level accuracy as opposed to simple force-fields for proteins making our end-to-end evaluation with DFT critical. 3. AdsorbDiff 3.1. Overview The objective of this research is to enhance the efficiency of adsorption energy calculation, representing the lowest energy configuration of an adsorbate on a slab. The methodology of this work involves the initial placement of an adsorbate on a random site within the 2D surface of the slab, followed by reverse diffusion to predict the optimal adsorption site and orientation. Employing machine learning force field optimization, the structure undergoes iterative updates with an optimizer until forces converge close to 0. Subsequently, the final structure is verified for compliance with constraints essential for defining adsorption energy. On the optimized structure, a single Density Functional Theory (DFT) calculation is conducted to obtain the predicted energy (EP red). A successful outcome is determined by the predicted energy being within 0.1 eV or lower than the DFT baseline of adsorption energy in OC20-Dense data, indicating the model\u2019s ability to provide a comparable or superior estimate of adsorption energy (shown in Figure 1). 3 \fAdsorbate placement via conditional denoising diffusion The code is open-sourced with MIT License1. 3.2. Adsorbate placement Various adsorbate placement strategies were explored for the OC20-Dense dataset, incorporating a combination of heuristic and random approaches. Specifically, 100 sites were selected for each adslab configuration, utilizing a blend of heuristic and random placements. The heuristic placement involved strategically situating the adsorbate\u2019s binding site on either an on-top site, hollow site, or bridge site, with a specified interstitial gap denoting the distance between the connecting atom of the slab and the corresponding adsorbate atom. Additional random sites are introduced through the random rotation of the adsorbate along the normal of the slab, accompanied by a slight translational wobble along the surface from the heuristic site. 3.3. Diffusion for adsorbate placement In this work, our objective is to develop a diffusion model aimed at predicting the adsorbate orientation and site corresponding to the lowest energy, as established through benchmarking with the OC20-Dense dataset. The adsorbate motion is constrained within a manifold (Mc) and utilizes the combined action group (A), as described in DiffDock (Corso et al., 2022). This manifold permits the adsorbate to navigate towards configurations with lowenergy adslab states through a combination of translations, rotations, and torsion angle adjustments. Note, for fair comparisons with our baselines, torsion angle alterations are disregarded in our analysis due to the smaller size of the adsorbate employed in this study. This approach aligns with the methodology of AdsorbML, which does not introduce randomness in torsion angles as part of its benchmark. In our framework, we specifically consider translations in the 2D plane parallel to the slab while accounting for periodic boundary conditions (PBC). The z-coordinate is meticulously aligned to denote the normal direction of the slab and the diffusion process is executed across the xycoordinates. Therefore, the adsorbate movements are associated with the 2D translation group T(2), and rigid rotations are modeled using the SO(3) group. The translation operation, denoted as Atr : T(2) \u00d7 R2n \u2192R2n, is defined as Atr(r, x)i = xi + r, employing the isomorphism T(2) \u223c = R2, where xi \u2208R2 represents the position of the i-th adsorbate atom. Similarly, the rotation operation, denoted as Arot : SO(3) \u00d7 R3n \u2192R3n, is defined by Arot(R, x)i = R(xi \u2212\u00af x) + \u00af x, where \u00af x = 1 n P i xi, signifying rotations around the center-of-mass of the adsorbate. For the initial coordinates of adsorbate, we select a random 1https://github.com/AdeeshKolluru/ AdsorbDiff point on the slab. This point is considered as the center-ofmass of the adsorbate in fractional coordinates. We then convert from fractional coordinates to real coordinates and perform a reverse diffusion process to get to the lowest energy site (as shown in Algorithm 1). The work conducted by De et al. (De Bortoli et al., 2022) and Corso et al. (Corso et al., 2022) has demonstrated the applicability of the diffusion framework to Riemannian manifolds. In this context, the score model constitutes the tangent space, and a geodesic random walk serves as the reverse stochastic differential equation (SDE) solver. The score model is trained using denoising score matching (Song & Ermon, 2019), wherein a score function s\u03b8(x) is learned to approximate the gradient of the probability density \u2207xp(x) at varying noise levels (as shown in Algorithm 2). The learned scores for translations and rotations are treated as independent entities, assuming the tangent space is a direct sum of individual tangent spaces, with contributions from torsion being neglected. The forward SDE for both translation and rotation is defined as dx = q d\u03c32(t) dt dw, 4 \fAdsorbate placement via conditional denoising diffusion where w represents the corresponding Wiener process. In the translational scenario within T(2), the model learns a score for a standard Gaussian distribution with variance \u03c32(t). For rotations in SO(3), the diffusion kernel is governed by the IGSO(3) distribution, which can be sampled in the axis-angle parameterization. This involves sampling a unit vector \u03c9\u2032 \u2208so(3) uniformly and a random angle \u03c9 from the interval [0, \u03c0], as outlined by Equations 1 and 2. The score of diffusion kernel is defined in Equation 3. The computation of R\u2032 = R(\u03c9\u02c6 \u03c9)R, where R is the result of applying the Euler vector \u03c9\u02c6 \u03c9 to R, has been established in prior work by Yim et al. (Yim et al., 2023). To efficiently carry out the score computation and sampling processes, it is feasible to precompute the truncated infinite series and interpolate the cumulative distribution function (CDF) of p(\u03c9). p(\u03c9) = 1 \u2212cos(\u03c9) \u03c0 f(\u03c9) (1) f(\u03c9) = \u221e X l=0 (2l + 1) exp \u0012 \u2212l(l + 1)\u03c32 2 \u0013 \u00d7 sin \u0012\u0012 l + 1 2 \u0013 \u03c9 \u0013 sin \u0010\u03c9 2 \u0011 (2) \u2207ln pt(R\u2032|R) = \u0012 d d\u03c9 log f(\u03c9) \u0013 \u02c6 \u03c9 (3) 3.4. Conditional denoising diffusion for adsorbate placement While the OC Challenge set provides densely calculated adsorption energies for 244 systems, a total of 244 * 100 DFT optimization benchmarks were conducted. This involved performing 100 different random placements for each configuration. Notably, the naive denoising diffusion setup was exclusively trained on the 244 lowest energy configurations. To leverage the entirety of the DFT optimization data, a conditional diffusion model is employed. In this model, the optimized position is conditioned on the relative energy, specifically relative to the energy of the lowest energy configuration (Ec rel-i = Ec min \u2212Ec i ). This approach allows for a more comprehensive utilization of the available DFT optimization data. 3.5. Graph Neural Network (GNN) architecture The inputs to the ML model are the 3D positions of all input atoms from the adslab configuration and their corresponding atomic numbers. The outputs predict per-atom 3D vectors. These vectors are forces in the case of force fields and the score function in the case of diffusion. To predict multiple score functions (for translation and rotation), multiple output heads are trained each predicting independent score functions. All architectures used in this work come under the messagepassing neural network (MPNN) framework of graph neural networks (GNNs). MPNNs operate by passing messages between nodes in the graph, allowing information to be exchanged and aggregated iteratively. The key components of an MPNN include message passing, updating node states, and global readout. In the message-passing step, nodes exchange information based on their local context, and this information is then used to update the states of the nodes (as shown in Equation 4). h(t+1) v = Update \u0010 h(t) v , Aggregate \u0010 {m(t) u\u2192v | u \u2208N(v)} \u0011\u0011 (4) Here, h(t) v represents embeddings of node v at iteration t, m(t) u\u2192v denotes the message from node u to v at iteration t, N(v) represents the neighborhood of node v, and Update and Aggregate are differentiable functions for updating node states and aggregating messages, respectively. In our study, we systematically investigate diverse architectures employed in the training of diffusion models to discern the significance of architectural decisions in this context. Specifically, we have chosen to assess the performance of PaiNN, GemNet-OC, and EquiformerV2, each distinguished by its treatment of explicit geometric information and rotational symmetries (Duval et al., 2023). This selection is grounded in the diverse characteristics they bring to the table. Furthermore, we employ these architectures in benchmarking against OC20 force-field evaluation, thereby facilitating comparative analysis of architectural significance in the realms of force-fields and diffusion. 4. Results In this section, we present results demonstrating the impact of AdsorbDiff in accelerating the search for adsorption energy or better global optima. Specifically, we demonstrate the impact of conditional denoising training over unconditional training and a randomly placed adsorbate baseline. This random baseline is equivalent to performing AdsorbML on a single site (Nsite=1). Additionally, we demonstrate the impact of pretraining, model architectures, and the generalization of this approach to new adsorbates and slabs. 4.1. Datasets We utilize two publicly available datasets for this work OC20-Dense (Lan et al., 2023) and OC20 (Chanussot et al., 2021). OC20: Open Catalyst 2020 (OC20) is a large-scale dataset that contains converged DFT optimization trajectories of 5 \fAdsorbate placement via conditional denoising diffusion 460k unique adslab configurations, encompassing 55 unique elements and 74 adsorbates. Note that these optimizations are local optimizations performed with a single heuristic placement. ML force field models are trained on the forces derived from these DFT trajectories. Additionally, the optimized structure from OC20 is utilized for pre-training the diffusion model. OC20-Dense: The OC20-Dense dataset serves as a DFT benchmark for adsorption energies, employing dense placement on 100 random sites per adslab configuration, followed by DFT optimization. This dataset releases both in-distribution (ID) and out-of-distribution (OOD) data, relative to OC20. The ID data incorporates adsorbates and slabs from OC20\u2019s training set but presents different combinations and configurations, while OOD introduces new adsorbates and/or slabs not found in the OC20 training set. A subset of OC20-Dense ID and OOD was utilized in the Open Catalyst Challenge 2023, hosted at the AI for Science Workshop during NeurIPS 2023 2. We split the ID data into 80/20 ratios for training the diffusion model and validating the sampling process. These smaller subsets make it computationally cheaper to perform end-to-end iterations. 4.2. Metric and constraints Our success metric is defined by the final energy calculated through DFT. For real-world applications, this energy (DDF T T otal) is used in calculating the adsorption energy EDF T Ads as EDF T Adsorption = EDF T T otal \u2212EDF T Slab \u2212EDF T Adsorbate, where EDF T Slab and EDF T Adsorbate are the independent energies of slab and adsorbate respectively. This adsorption energy acts as a thermodynamic description of how good a catalyst is for downstream application. The DFT Success Rate (SR) is defined as the percentage of valid structures within 0.1 eV or lower of the DFT computed adsorption energy benchmark in the OC20-Dense data (as described in AdsorbML). This is computationally expensive to calculate but is accurate. Metrics calculated from ML predictions are inexpensive but are also inaccurate, discussed further in Appendix C. Since we calculate adsorption energies, the adsorbate and slab must not change during optimization. Therefore, the structures are considered an anomaly due to (1) adsorbate desorption: adsorbate moves far away from the slab, (2) adsorbate dissociation: atoms in adsorbate dissociate into multiple adsorbates, (3) slab mismatch/reconstruction: slab reconstructs into a completely different structure during optimization (4) adsorbate intercalation: when any of the adsorbate atoms detaches and get into the slab. Experimental setup: All presented results are based on the DFT success rate metric as defined in the preceding 2https://opencatalystproject.org/ challenge.html section. Throughout the diffusion process, we employ the EquiformerV2 architecture, unless explicitly stated otherwise, owing to its state-of-the-art performance in AdsorbML. Additionally, for MLFF optimization, we utilize GemNetOC pre-trained on OC20, chosen for its lower inference cost. Further specifics regarding model and training hyperparameters are available in Appendix D. All results are shown on the val ID split apart from the OOD section. 4.3. Conditional vs Unconditional diffusion 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) Random Unconditional Conditional 9.1% 11.4% 31.8% Conditional vs Unconditional Diffusion (Nsite=1) Figure 2. Comparison of conditional and unconditional diffusion with a baseline of random placement. Conditional diffusion training on relative energies of configurations of adslab significantly improves success rates over unconditional training and AdsorbML baseline. We demonstrate the importance of conditional training on relative energies (as shown in Section 3.4) over unconditional diffusion training in Figure 2. We compare both of these approaches to a naive baseline of AdsorbML with a single site (Nsite=1) where MLFF optimization is performed on a random adsorbate placement. It is noteworthy that the performance of unconditional training is suboptimal, this may be ascribed to the unexploited potential of additional data made available through conditional training. 4.4. AdsorbDiff vs AdsorbML AdsorbML conducts MLFF optimization and DFT evaluations on adsorption sites randomly placed within the system. A comparative analysis is drawn with AdsorbDiff, where the prediction of adsorption sites is facilitated through the utilization of diffusion models. As depicted in Figure 3, it is evident that AdsorbDiff exhibits notably superior performance, particularly at lower Nsites. However, as the number of adsorption sites (Nsites) increases, AdsorbDiff tends to either converge to or underperform in comparison to the brute force approach employed by AdsorbML. Adsorbate sites sampled from AdsorbDiff have less diversity by design as it\u2019s trained to predict the global optima. We calculate the average across the standard deviation of the points sampled at 10 Nsites and get 8.1 \u02da A for AdsorbML and 2.7 \u02da A for AdsorbDiff. AdsorbML\u2019s brute force placements have more randomness which leads to fewer anomalies post the MLFF 6 \fAdsorbate placement via conditional denoising diffusion 2 4 6 8 10 Number of Sites 10 15 20 25 30 35 40 45 DFT Success Rate (%) 9.1% 31.8% 20.5% 34.1% 34.1% 36.3% 47.7% 41.0% AdsorbDiff vs AdsorbML AdsorbML AdsorbDiff AdsorbDiff (Nsite=1) Figure 3. DFT Success Rates (%) for AdsorbDiff and AdsorbML across a varying number of site predictions. AdsorbDiff performs 3.5x better than AdsorbML utilizing a single site prediction. At higher sites, AdsorbML performs better due to the brute-force nature of site prediction that reduces anomalies. 2 4 6 8 10 Number of Sites 10 15 20 25 30 Anomalies 31.8% 25.0% 18.2% 20.5% 11.4% 22.7% 6.8% 13.6% AdsorbML AdsorbDiff Figure 4. Anomalies in AdsorbDiff and AdsorbML with respect to Nsites. A system is labeled as anomalous if all its predicted sites result in anomalies. AdsorbML has fewer anomalies than AdsorbDiff at higher Nsites due to more randomness in initial sites. optimization process shown in Figure 4. 4.5. Impact of pretraining Conditional diffusion benefits from training on a dataset that is 100 times more extensive than the unconditional approach, a consequence of leveraging multiple local optima within a unique adslab configuration. The substantial increase in training data size manifests in a notable enhancement in the success rate for the conditional approach. The OC20 IS2RE dataset, containing optimization data for 460,000 distinct adslab combinations, serves as a valuable resource for pretraining the diffusion model. It is important to acknowledge that this pretraining process results in a model that learns the local optima of an adslab combination, with the caveat that the model may not capture global optima for an adslab combination. 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) Random PT Zero-shot PT Conditional 9.1% 29.6% 31.8% Impact of Pre-training (Nsite=1) Figure 5. Impact of pretraining on 460k OC20 local optima data on DFT Success Rate. PT Zero-shot measures zero-shot generalization of OC20 pre-trained model to OC20-Dense data. PT Conditional is finetuned on OC20 Dense data conditionally on relative energies of adslab configurations. Random baseline corresponds to randomly placed adsorbate. IS2RS Pretraining (PT) Zero-shot: Taking advantage of the diffusion model pre-trained on OC20 IS2RE data, we conduct a zero-shot validation on the OC20-Dense ID val split. This experimental setup allows us to assess the model\u2019s ability to predict better global optima having trained on a large dataset of local optima. Notably, we observe a substantial increase in DFT success rate in the zero-shot setting (as shown in Figure 5). IS2RS Pretraining (PT) Conditional: In this approach, we utilize the pre-trained model using the OC20-Dense data as described in Section 3.4. We observe that although this gives a 2% improvement over zero-shot, it converges to the same results as just training conditionally on OC20-Dense (shown in Figure 5). 4.6. Impact of architectures Architectures characterized by richer geometric information and extensive many-body interaction capabilities, such as eSCN and EquiformerV2, have demonstrated superior performance in force evaluations within the OC20 dataset compared to simpler models like PaiNN, which primarily encode directional information and apply linear transformations. Our benchmarking involves the evaluation of three architectures that exhibit progressively improved performance in OC20 Force MAE, revealing significant differences among them. This evaluation is specifically conducted in the context of the zero-shot assessment following pretraining (PT zeroshot) on an extensive dataset encompassing 460,000 OC20 instances. This choice is inspired by insights from the GemNet-OC paper (Gasteiger et al., 2022), suggesting that certain architectural choices manifest optimal performance only at higher data scales. 7 \fAdsorbate placement via conditional denoising diffusion 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) PaiNN GemNet-OC EquiformerV2 27.3% 27.3% 29.6% Impact of GNN architectures on diffusion Figure 6. Impact of Graph Neural Network (GNN) architectures on the diffusion process for DFT Success Rate keeping other parts of the framework same. Different architectures perform similarly on the task of diffusion sampling. Interestingly, in the realm of the diffusion task, we note that the disparity in success rates among these architectures is marginal (as shown in Figure 6) which has been recently demonstrated in applications of molecular generation tasks as well (Wang et al., 2023). The intuition behind this result is that the diffusion model\u2019s score function can be thought of as learning a harmonic potential (Xie et al., 2021). Harmonic potentials are simpler force-fields than ab-initio DFT calculations involved in OC20 forces. This could result in simpler architectures being able to capture the underlying complexity of the diffusion task defined in our work. 4.7. OOD generalization We measure the success of AdsorbDiff in out-of-distribution (OOD) cases where the model hasn\u2019t seen the adsorbate or the slab even during the pre-training on OC20. We pick a random 50 samples out of 200 validation OOD split defined in Open Catalyst Challenge 2023. We observe a marginal decrease of only 3.8% in results for the OOD case compared to the ID scenario and consistently observe significant improvement over the AdsorbML (Nsite=1) baseline. 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) Random AdsorbDiff 8.4% 28% OOD Results Figure 7. Comparison of DFT Success Rate for In-Distribution (ID) and Out-of-Distribution (OOD) splits using the AdsorbDiff method. Random baseline corresponds to randomly placed adsorbate. 4.8. Inference cost In the case of conditional diffusion, our approach maintains a maximum step limit of 100, with adsorbate placement converging, on average, within 98 steps. In contrast, for MLFF optimization with a maximum step limit of 300 and Fmax criteria of 0.01 eV/A (consistent with AdsorbML), the convergence occurs in approximately 286 steps. Consequently, for scenarios with a single adsorption site (Nsite 1), AdsorbDiff incurs approximately 34% more inference cost than AdsorbML, given the GNN architecture for diffusion and MLFF optimization is the same. This end-to-end ML framework is O(104) times faster than the conventional DFT pipelines (Lan et al., 2023). In Section 4.6, we illustrate that simpler and faster models such as PaiNN yield comparable performance to more intricate and slower models like EquiformerV2. This enhances the efficiency of our diffusion-based approach, as its computational burden becomes negligible in comparison to MLFF optimization, which would require more computationally intensive ML architectures (details in Appendix B). 5.",
+    "additional_graph_info": {
+        "graph": [
+            [
+                "Adeesh Kolluru",
+                "Muhammed Shuaibi"
+            ],
+            [
+                "Adeesh Kolluru",
+                "Nima Shoghi"
+            ],
+            [
+                "Adeesh Kolluru",
+                "Abhishek Das"
+            ],
+            [
+                "Muhammed Shuaibi",
+                "Abhishek Das"
+            ],
+            [
+                "Muhammed Shuaibi",
+                "Aditya Grover"
+            ],
+            [
+                "Muhammed Shuaibi",
+                "Anuroop Sriram"
+            ],
+            [
+                "Nima Shoghi",
+                "C. Lawrence Zitnick"
+            ],
+            [
+                "Abhishek Das",
+                "Devi Parikh"
+            ],
+            [
+                "Abhishek Das",
+                "Dhruv Batra"
+            ]
+        ],
+        "node_feat": {
+            "Adeesh Kolluru": [
+                {
+                    "url": "http://arxiv.org/abs/2405.03962v1",
+                    "title": "AdsorbDiff: Adsorbate Placement via Conditional Denoising Diffusion",
+                    "abstract": "Determining the optimal configuration of adsorbates on a slab (adslab) is\npivotal in the exploration of novel catalysts across diverse applications.\nTraditionally, the quest for the lowest energy adslab configuration involves\nplacing the adsorbate onto the slab followed by an optimization process. Prior\nmethodologies have relied on heuristics, problem-specific intuitions, or\nbrute-force approaches to guide adsorbate placement. In this work, we propose a\nnovel framework for adsorbate placement using denoising diffusion. The model is\ndesigned to predict the optimal adsorbate site and orientation corresponding to\nthe lowest energy configuration. Further, we have an end-to-end evaluation\nframework where diffusion-predicted adslab configuration is optimized with a\npretrained machine learning force field and finally evaluated with Density\nFunctional Theory (DFT). Our findings demonstrate an acceleration of up to 5x\nor 3.5x improvement in accuracy compared to the previous best approach. Given\nthe novelty of this framework and application, we provide insights into the\nimpact of pre-training, model architectures, and conduct extensive experiments\nto underscore the significance of this approach.",
+                    "authors": "Adeesh Kolluru, John R Kitchin",
+                    "published": "2024-05-07",
+                    "updated": "2024-05-07",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "physics.chem-ph"
+                    ],
+                    "main_content": "Introduction Heterogenous catalysis plays an important role in developing chemicals in industries, environmental protection through converters, and the synthesis of alternative fuels (Liu & Li, 2017; Zitnick et al., 2020). Modeling these chemical reactions involve an intermediate adsorbate on a catalyst slab which determines the efficacy of the catalyst for that particular reaction. Discovering a novel catalyst computationally involves screening through billions of candidates and finding the lowest energy configuration. 1Department of Chemical Engineering, Carnegie Mellon University. Correspondence to: Adeesh Kolluru <akolluru@andrew.cmu.edu>, John R. Kitchin <jkitchin@andrew.cmu.edu>. Finding the lowest energy configuration for an adsorbate and slab requires a global optimum (which is non-convex) search across different sites on the slab. Conventional approaches solve this in two steps (1) heuristically place the adsorbate on certain important sites and (2) perform optimization with quantum mechanical calculators like Density Functional Theory (DFT) on each of these sites. The lowest energy site out of these is considered for calculating adsorption energy, which is a thermodynamic descriptor for how good that catalyst is. With recent advances in machine learning methods for predicting forces, it has become possible to perform optimization with ML force fields (MLFFs) instead of Density Functional Theory (DFT) making this process faster and easier to test many sites and find better minima. These ML force fields are trained on DFT data to predict energies and forces corresponding to different adslab configurations. The recent release of the OC20-Dense dataset (Lan et al., 2023) signifies a significant advancement in the computation of the lowest energy adslab configuration. This work employs a blend of heuristic and random adsorbate placements across 100 sites, with subsequent optimizations across each site using Density Functional Theory (DFT) to calculate adsorption energy. The study further introduces AdsorbML, a paradigm characterized by a brute-force exploration of initial adsorbate placements. Employing pre-trained machine learning (ML) force fields from OC20, AdsorbML streamlines the optimization process, culminating in the determination of the lowest energy adsorbate-slab (adslab) configuration. The predictive accuracy of these configurations is rigorously validated against DFT single-points or complete DFT optimization. This hybrid approach results in a computational acceleration of 2000-fold in adsorption energy calculations compared to the sole reliance on DFT calculations. Recent developments in graph neural network (GNN) based ML architectures have increased the accuracies of adsorption energy prediction significantly by encoding geometric information of atoms in more explicit ways. However, there\u2019s little to no work done on improving the adsorption site prediction which could help us get away with the currently used brute-force approach. In this work, we develop a novel conditional denoising diffu1 arXiv:2405.03962v1  [cs.LG]  7 May 2024 \fAdsorbate placement via conditional denoising diffusion sion framework for adsorbate placement. We first formulate a diffusion framework over the space of the 2D translation and 3D rigid rotation of an adsorbate molecule over the slab considering periodic boundary conditions (PBC) of the slab. Through the learned diffusion process, we sample the most stable site by iteratively updating the center of mass of adsorbate and rigid orientation. Performing a naive unconditional diffusion framework on the most optimal adsorbate site and orientation \u2014 corresponding to the lowest energy adslab configuration out of 100 densely sampled calculations in OC20-Dense \u2014 leads to throwing away 99% of DFT optimal energy data. Therefore, we modify the diffusion training to be conditional on relative energies (relative across densely sampled sites of an adslab combination). This leads to significant improvements in accuracies and sample efficiency during diffusion training. After sampling for the optimal site and orientation of adsorbate on the slab, we perform ML force field (MLFF) optimization and DFT single-point verification similar to AdsorbML. This comprehensive end-to-end evaluation helps in robust assessment of the practical impact of the learned diffusion model. There have been significant advances in diffusion generative models in molecular and material discovery, and analogous problems in molecular docking on proteins. However, this is the first work to frame the adsorbate placement problem considering all its symmetries with the slab in a diffusion framework. Intuitively, the reverse diffusion process of AdsorbDiff helps in skipping multiple minima sites due to its energy-based conditional sampling which is followed by a local optimization with a DFT-learned MLFF to find a global optimum. To facilitate further research on this problem, we provide comprehensive results on the importance of GNN architectures for the diffusion task, show the importance of pretraining, and demonstrate the success of our approach to in-distribution (ID) and out-of-distribution (OOD) splits. The summary of contributions of this work are \u2022 We propose AdsorbDiff, a novel conditional denoising diffusion framework designed to leverage the translation, rotation, and periodic symmetries inherent in adsorbate and slab interactions. Additionally, this framework is adept at efficiently predicting the lowest energy site by conditional training on relative energies. \u2022 We present our results in a comprehensive end-to-end evaluation framework, integrated with DFT, to accurately gauge the true capability of our approach in predicting optimal adsorption energies. \u2022 We achieve a 31.8% success rate, 3.5x higher than the naive AdsorbML baseline of 9.1% with a single site prediction. Alternatively, we demonstrate that a comparable level of accuracy could be achieved by AdsorbML by employing 5x more placements. \u2022 We demonstrate that pretraining on large-scale local optimization data can significantly improve the results on the search for global optima. \u2022 We show that diffusion results exhibit insignificant dependence on GNN architectures, in contrast to the notable differences observed for the same architectures when trained on DFT forces. \u2022 We highlight the model\u2019s generalization capabilities to previously unseen adsorbates and slabs. 2. Background and Related Work Force-fields: Energy and forces (as a gradient of energy with respect to positions) are calculated using ab initio quantum mechanical methods like Density Functional Theory (DFT). ML models can be trained to predict these energies and forces, and are called ML force-fields (MLFFs). These force fields can be utilized to perform structure optimization to get the lowest energy structures. Optimization: For adsorption energy prediction, we start with an optimized adsorbate and slab, place the adsorbate on a slab, and perform optimization to get an adslab configuration with the lowest energy. Usually, second-order optimizers like BFGS, L-BFGS, Conjugate gradient descent, etc are used to solve this optimization problem. Since this is non-convex, the initial guess of adsorbate placement or the strategy of optimization is critical to finding an adslab configuration corresponding to the global optimum. AdsorbML (Lan et al., 2023) method starts with combining heuristic and random initial placements which is a brute-force approach to finding better minima. \u201dEasy Potential\u201d from (Schaarschmidt et al., 2022) trains a simple harmonic potential to guess this initial placement. Learn2Hop (Merchant et al., 2021) also learns the optimization landscape to navigate through better and hop through local minima. There are approaches like minima hopping that help in navigating through the entire optimization landscape with a force-field (Jung et al., 2023) and help in finding better minima, but these could be computationally expensive. GNNs: Message-Passing Neural Networks (MPNN) are a class of graph neural networks (GNN) that are utilized across material property prediction tasks. Different architectures encode the geometric information in different ways. SchNet (Sch\u00a8 utt et al., 2018) only encodes the distance information. Including more explicit geometric features have improved the model prediction as DimeNet (Gasteiger et al., 2020b;a) incorporates triplets. SphereNet (Liu et al., 2021), GemNet (Gasteiger et al., 2021; 2022) incorporates complete geometric information explicitly by giving triplets and quadruplets information. PaiNN (Sch\u00a8 utt et al., 2021) incorporates directional information and applies only linear operations on those features. Equivariant models like NequIP (Batzner et al., 2022), Allegro (Musaelian et al., 2023), MACE (Batatia et al., 2022), SCN (Zitnick et al., 2 \fAdsorbate placement via conditional denoising diffusion Figure 1. Overview of AdsorbDiff: Random initial site and orientation for the adsorbate are selected, followed by sampling over 2D translation, 3D rigid rotations, and considering periodic boundary conditions (PBC) to predict the optimal site and orientation. MLFF optimization is then conducted from the predicted site with a fixed interstitial gap until convergence. The final prediction undergoes constraint verification, and DFT verification is performed on valid structures to calculate success rates. 2022), Equiformer (Liao & Smidt, 2022; Liao et al., 2023) utilize spherical harmonics in representing the geometric features. Diffusion Models: Diffusion models are a class of generative models that have shown impressive results across different domains starting from computer vision (Dhariwal & Nichol, 2021; Croitoru et al., 2023), language models (Gong et al., 2022), temporal data modeling, to applications in molecules (Xu et al., 2022; 2023; Arts et al., 2023; Hoogeboom et al., 2022; Jing et al., 2022), proteins (Wu et al., 2022; Trippe et al., 2022; Watson et al., 2022; 2023) and materials (Xie et al., 2021; Fu et al., 2023; Zeni et al., 2023; Merchant et al., 2023; Yang et al., 2023b). There are different kinds of formulations proposed for diffusion models like denoising diffusion probabilistic models (DDPMs), score-based generative models (SGMs), and stochastic differential equations (Score SDEs) (Yang et al., 2023a). Many of these formulations have been adapted to problems in molecular and material discovery. For example, CDVAE (Xie et al., 2021) adapts concepts from noise-conditioned score networks (NCSN) for bulk discovery. Conditional diffusion has also been recently utilized across proteins (Krishna et al., 2024), catalyst and materials (Zheng et al., 2023) for generating structures with required properties. Diffusion models have also been recently utilized for molecular docking on proteins (Corso et al., 2022). Although this problem is somewhat analogous to placing adsorbate on a slab, as far as we know there hasn\u2019t been previous work on formulating adsorbate placement in a diffusion framework. AdsorbDiff also differs from molecular docking in several key aspects \u2013 2D translation formulation, periodic boundary conditions, conditional denoising formulation, and the requirement of DFT level accuracy as opposed to simple force-fields for proteins making our end-to-end evaluation with DFT critical. 3. AdsorbDiff 3.1. Overview The objective of this research is to enhance the efficiency of adsorption energy calculation, representing the lowest energy configuration of an adsorbate on a slab. The methodology of this work involves the initial placement of an adsorbate on a random site within the 2D surface of the slab, followed by reverse diffusion to predict the optimal adsorption site and orientation. Employing machine learning force field optimization, the structure undergoes iterative updates with an optimizer until forces converge close to 0. Subsequently, the final structure is verified for compliance with constraints essential for defining adsorption energy. On the optimized structure, a single Density Functional Theory (DFT) calculation is conducted to obtain the predicted energy (EP red). A successful outcome is determined by the predicted energy being within 0.1 eV or lower than the DFT baseline of adsorption energy in OC20-Dense data, indicating the model\u2019s ability to provide a comparable or superior estimate of adsorption energy (shown in Figure 1). 3 \fAdsorbate placement via conditional denoising diffusion The code is open-sourced with MIT License1. 3.2. Adsorbate placement Various adsorbate placement strategies were explored for the OC20-Dense dataset, incorporating a combination of heuristic and random approaches. Specifically, 100 sites were selected for each adslab configuration, utilizing a blend of heuristic and random placements. The heuristic placement involved strategically situating the adsorbate\u2019s binding site on either an on-top site, hollow site, or bridge site, with a specified interstitial gap denoting the distance between the connecting atom of the slab and the corresponding adsorbate atom. Additional random sites are introduced through the random rotation of the adsorbate along the normal of the slab, accompanied by a slight translational wobble along the surface from the heuristic site. 3.3. Diffusion for adsorbate placement In this work, our objective is to develop a diffusion model aimed at predicting the adsorbate orientation and site corresponding to the lowest energy, as established through benchmarking with the OC20-Dense dataset. The adsorbate motion is constrained within a manifold (Mc) and utilizes the combined action group (A), as described in DiffDock (Corso et al., 2022). This manifold permits the adsorbate to navigate towards configurations with lowenergy adslab states through a combination of translations, rotations, and torsion angle adjustments. Note, for fair comparisons with our baselines, torsion angle alterations are disregarded in our analysis due to the smaller size of the adsorbate employed in this study. This approach aligns with the methodology of AdsorbML, which does not introduce randomness in torsion angles as part of its benchmark. In our framework, we specifically consider translations in the 2D plane parallel to the slab while accounting for periodic boundary conditions (PBC). The z-coordinate is meticulously aligned to denote the normal direction of the slab and the diffusion process is executed across the xycoordinates. Therefore, the adsorbate movements are associated with the 2D translation group T(2), and rigid rotations are modeled using the SO(3) group. The translation operation, denoted as Atr : T(2) \u00d7 R2n \u2192R2n, is defined as Atr(r, x)i = xi + r, employing the isomorphism T(2) \u223c = R2, where xi \u2208R2 represents the position of the i-th adsorbate atom. Similarly, the rotation operation, denoted as Arot : SO(3) \u00d7 R3n \u2192R3n, is defined by Arot(R, x)i = R(xi \u2212\u00af x) + \u00af x, where \u00af x = 1 n P i xi, signifying rotations around the center-of-mass of the adsorbate. For the initial coordinates of adsorbate, we select a random 1https://github.com/AdeeshKolluru/ AdsorbDiff point on the slab. This point is considered as the center-ofmass of the adsorbate in fractional coordinates. We then convert from fractional coordinates to real coordinates and perform a reverse diffusion process to get to the lowest energy site (as shown in Algorithm 1). The work conducted by De et al. (De Bortoli et al., 2022) and Corso et al. (Corso et al., 2022) has demonstrated the applicability of the diffusion framework to Riemannian manifolds. In this context, the score model constitutes the tangent space, and a geodesic random walk serves as the reverse stochastic differential equation (SDE) solver. The score model is trained using denoising score matching (Song & Ermon, 2019), wherein a score function s\u03b8(x) is learned to approximate the gradient of the probability density \u2207xp(x) at varying noise levels (as shown in Algorithm 2). The learned scores for translations and rotations are treated as independent entities, assuming the tangent space is a direct sum of individual tangent spaces, with contributions from torsion being neglected. The forward SDE for both translation and rotation is defined as dx = q d\u03c32(t) dt dw, 4 \fAdsorbate placement via conditional denoising diffusion where w represents the corresponding Wiener process. In the translational scenario within T(2), the model learns a score for a standard Gaussian distribution with variance \u03c32(t). For rotations in SO(3), the diffusion kernel is governed by the IGSO(3) distribution, which can be sampled in the axis-angle parameterization. This involves sampling a unit vector \u03c9\u2032 \u2208so(3) uniformly and a random angle \u03c9 from the interval [0, \u03c0], as outlined by Equations 1 and 2. The score of diffusion kernel is defined in Equation 3. The computation of R\u2032 = R(\u03c9\u02c6 \u03c9)R, where R is the result of applying the Euler vector \u03c9\u02c6 \u03c9 to R, has been established in prior work by Yim et al. (Yim et al., 2023). To efficiently carry out the score computation and sampling processes, it is feasible to precompute the truncated infinite series and interpolate the cumulative distribution function (CDF) of p(\u03c9). p(\u03c9) = 1 \u2212cos(\u03c9) \u03c0 f(\u03c9) (1) f(\u03c9) = \u221e X l=0 (2l + 1) exp \u0012 \u2212l(l + 1)\u03c32 2 \u0013 \u00d7 sin \u0012\u0012 l + 1 2 \u0013 \u03c9 \u0013 sin \u0010\u03c9 2 \u0011 (2) \u2207ln pt(R\u2032|R) = \u0012 d d\u03c9 log f(\u03c9) \u0013 \u02c6 \u03c9 (3) 3.4. Conditional denoising diffusion for adsorbate placement While the OC Challenge set provides densely calculated adsorption energies for 244 systems, a total of 244 * 100 DFT optimization benchmarks were conducted. This involved performing 100 different random placements for each configuration. Notably, the naive denoising diffusion setup was exclusively trained on the 244 lowest energy configurations. To leverage the entirety of the DFT optimization data, a conditional diffusion model is employed. In this model, the optimized position is conditioned on the relative energy, specifically relative to the energy of the lowest energy configuration (Ec rel-i = Ec min \u2212Ec i ). This approach allows for a more comprehensive utilization of the available DFT optimization data. 3.5. Graph Neural Network (GNN) architecture The inputs to the ML model are the 3D positions of all input atoms from the adslab configuration and their corresponding atomic numbers. The outputs predict per-atom 3D vectors. These vectors are forces in the case of force fields and the score function in the case of diffusion. To predict multiple score functions (for translation and rotation), multiple output heads are trained each predicting independent score functions. All architectures used in this work come under the messagepassing neural network (MPNN) framework of graph neural networks (GNNs). MPNNs operate by passing messages between nodes in the graph, allowing information to be exchanged and aggregated iteratively. The key components of an MPNN include message passing, updating node states, and global readout. In the message-passing step, nodes exchange information based on their local context, and this information is then used to update the states of the nodes (as shown in Equation 4). h(t+1) v = Update \u0010 h(t) v , Aggregate \u0010 {m(t) u\u2192v | u \u2208N(v)} \u0011\u0011 (4) Here, h(t) v represents embeddings of node v at iteration t, m(t) u\u2192v denotes the message from node u to v at iteration t, N(v) represents the neighborhood of node v, and Update and Aggregate are differentiable functions for updating node states and aggregating messages, respectively. In our study, we systematically investigate diverse architectures employed in the training of diffusion models to discern the significance of architectural decisions in this context. Specifically, we have chosen to assess the performance of PaiNN, GemNet-OC, and EquiformerV2, each distinguished by its treatment of explicit geometric information and rotational symmetries (Duval et al., 2023). This selection is grounded in the diverse characteristics they bring to the table. Furthermore, we employ these architectures in benchmarking against OC20 force-field evaluation, thereby facilitating comparative analysis of architectural significance in the realms of force-fields and diffusion. 4. Results In this section, we present results demonstrating the impact of AdsorbDiff in accelerating the search for adsorption energy or better global optima. Specifically, we demonstrate the impact of conditional denoising training over unconditional training and a randomly placed adsorbate baseline. This random baseline is equivalent to performing AdsorbML on a single site (Nsite=1). Additionally, we demonstrate the impact of pretraining, model architectures, and the generalization of this approach to new adsorbates and slabs. 4.1. Datasets We utilize two publicly available datasets for this work OC20-Dense (Lan et al., 2023) and OC20 (Chanussot et al., 2021). OC20: Open Catalyst 2020 (OC20) is a large-scale dataset that contains converged DFT optimization trajectories of 5 \fAdsorbate placement via conditional denoising diffusion 460k unique adslab configurations, encompassing 55 unique elements and 74 adsorbates. Note that these optimizations are local optimizations performed with a single heuristic placement. ML force field models are trained on the forces derived from these DFT trajectories. Additionally, the optimized structure from OC20 is utilized for pre-training the diffusion model. OC20-Dense: The OC20-Dense dataset serves as a DFT benchmark for adsorption energies, employing dense placement on 100 random sites per adslab configuration, followed by DFT optimization. This dataset releases both in-distribution (ID) and out-of-distribution (OOD) data, relative to OC20. The ID data incorporates adsorbates and slabs from OC20\u2019s training set but presents different combinations and configurations, while OOD introduces new adsorbates and/or slabs not found in the OC20 training set. A subset of OC20-Dense ID and OOD was utilized in the Open Catalyst Challenge 2023, hosted at the AI for Science Workshop during NeurIPS 2023 2. We split the ID data into 80/20 ratios for training the diffusion model and validating the sampling process. These smaller subsets make it computationally cheaper to perform end-to-end iterations. 4.2. Metric and constraints Our success metric is defined by the final energy calculated through DFT. For real-world applications, this energy (DDF T T otal) is used in calculating the adsorption energy EDF T Ads as EDF T Adsorption = EDF T T otal \u2212EDF T Slab \u2212EDF T Adsorbate, where EDF T Slab and EDF T Adsorbate are the independent energies of slab and adsorbate respectively. This adsorption energy acts as a thermodynamic description of how good a catalyst is for downstream application. The DFT Success Rate (SR) is defined as the percentage of valid structures within 0.1 eV or lower of the DFT computed adsorption energy benchmark in the OC20-Dense data (as described in AdsorbML). This is computationally expensive to calculate but is accurate. Metrics calculated from ML predictions are inexpensive but are also inaccurate, discussed further in Appendix C. Since we calculate adsorption energies, the adsorbate and slab must not change during optimization. Therefore, the structures are considered an anomaly due to (1) adsorbate desorption: adsorbate moves far away from the slab, (2) adsorbate dissociation: atoms in adsorbate dissociate into multiple adsorbates, (3) slab mismatch/reconstruction: slab reconstructs into a completely different structure during optimization (4) adsorbate intercalation: when any of the adsorbate atoms detaches and get into the slab. Experimental setup: All presented results are based on the DFT success rate metric as defined in the preceding 2https://opencatalystproject.org/ challenge.html section. Throughout the diffusion process, we employ the EquiformerV2 architecture, unless explicitly stated otherwise, owing to its state-of-the-art performance in AdsorbML. Additionally, for MLFF optimization, we utilize GemNetOC pre-trained on OC20, chosen for its lower inference cost. Further specifics regarding model and training hyperparameters are available in Appendix D. All results are shown on the val ID split apart from the OOD section. 4.3. Conditional vs Unconditional diffusion 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) Random Unconditional Conditional 9.1% 11.4% 31.8% Conditional vs Unconditional Diffusion (Nsite=1) Figure 2. Comparison of conditional and unconditional diffusion with a baseline of random placement. Conditional diffusion training on relative energies of configurations of adslab significantly improves success rates over unconditional training and AdsorbML baseline. We demonstrate the importance of conditional training on relative energies (as shown in Section 3.4) over unconditional diffusion training in Figure 2. We compare both of these approaches to a naive baseline of AdsorbML with a single site (Nsite=1) where MLFF optimization is performed on a random adsorbate placement. It is noteworthy that the performance of unconditional training is suboptimal, this may be ascribed to the unexploited potential of additional data made available through conditional training. 4.4. AdsorbDiff vs AdsorbML AdsorbML conducts MLFF optimization and DFT evaluations on adsorption sites randomly placed within the system. A comparative analysis is drawn with AdsorbDiff, where the prediction of adsorption sites is facilitated through the utilization of diffusion models. As depicted in Figure 3, it is evident that AdsorbDiff exhibits notably superior performance, particularly at lower Nsites. However, as the number of adsorption sites (Nsites) increases, AdsorbDiff tends to either converge to or underperform in comparison to the brute force approach employed by AdsorbML. Adsorbate sites sampled from AdsorbDiff have less diversity by design as it\u2019s trained to predict the global optima. We calculate the average across the standard deviation of the points sampled at 10 Nsites and get 8.1 \u02da A for AdsorbML and 2.7 \u02da A for AdsorbDiff. AdsorbML\u2019s brute force placements have more randomness which leads to fewer anomalies post the MLFF 6 \fAdsorbate placement via conditional denoising diffusion 2 4 6 8 10 Number of Sites 10 15 20 25 30 35 40 45 DFT Success Rate (%) 9.1% 31.8% 20.5% 34.1% 34.1% 36.3% 47.7% 41.0% AdsorbDiff vs AdsorbML AdsorbML AdsorbDiff AdsorbDiff (Nsite=1) Figure 3. DFT Success Rates (%) for AdsorbDiff and AdsorbML across a varying number of site predictions. AdsorbDiff performs 3.5x better than AdsorbML utilizing a single site prediction. At higher sites, AdsorbML performs better due to the brute-force nature of site prediction that reduces anomalies. 2 4 6 8 10 Number of Sites 10 15 20 25 30 Anomalies 31.8% 25.0% 18.2% 20.5% 11.4% 22.7% 6.8% 13.6% AdsorbML AdsorbDiff Figure 4. Anomalies in AdsorbDiff and AdsorbML with respect to Nsites. A system is labeled as anomalous if all its predicted sites result in anomalies. AdsorbML has fewer anomalies than AdsorbDiff at higher Nsites due to more randomness in initial sites. optimization process shown in Figure 4. 4.5. Impact of pretraining Conditional diffusion benefits from training on a dataset that is 100 times more extensive than the unconditional approach, a consequence of leveraging multiple local optima within a unique adslab configuration. The substantial increase in training data size manifests in a notable enhancement in the success rate for the conditional approach. The OC20 IS2RE dataset, containing optimization data for 460,000 distinct adslab combinations, serves as a valuable resource for pretraining the diffusion model. It is important to acknowledge that this pretraining process results in a model that learns the local optima of an adslab combination, with the caveat that the model may not capture global optima for an adslab combination. 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) Random PT Zero-shot PT Conditional 9.1% 29.6% 31.8% Impact of Pre-training (Nsite=1) Figure 5. Impact of pretraining on 460k OC20 local optima data on DFT Success Rate. PT Zero-shot measures zero-shot generalization of OC20 pre-trained model to OC20-Dense data. PT Conditional is finetuned on OC20 Dense data conditionally on relative energies of adslab configurations. Random baseline corresponds to randomly placed adsorbate. IS2RS Pretraining (PT) Zero-shot: Taking advantage of the diffusion model pre-trained on OC20 IS2RE data, we conduct a zero-shot validation on the OC20-Dense ID val split. This experimental setup allows us to assess the model\u2019s ability to predict better global optima having trained on a large dataset of local optima. Notably, we observe a substantial increase in DFT success rate in the zero-shot setting (as shown in Figure 5). IS2RS Pretraining (PT) Conditional: In this approach, we utilize the pre-trained model using the OC20-Dense data as described in Section 3.4. We observe that although this gives a 2% improvement over zero-shot, it converges to the same results as just training conditionally on OC20-Dense (shown in Figure 5). 4.6. Impact of architectures Architectures characterized by richer geometric information and extensive many-body interaction capabilities, such as eSCN and EquiformerV2, have demonstrated superior performance in force evaluations within the OC20 dataset compared to simpler models like PaiNN, which primarily encode directional information and apply linear transformations. Our benchmarking involves the evaluation of three architectures that exhibit progressively improved performance in OC20 Force MAE, revealing significant differences among them. This evaluation is specifically conducted in the context of the zero-shot assessment following pretraining (PT zeroshot) on an extensive dataset encompassing 460,000 OC20 instances. This choice is inspired by insights from the GemNet-OC paper (Gasteiger et al., 2022), suggesting that certain architectural choices manifest optimal performance only at higher data scales. 7 \fAdsorbate placement via conditional denoising diffusion 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) PaiNN GemNet-OC EquiformerV2 27.3% 27.3% 29.6% Impact of GNN architectures on diffusion Figure 6. Impact of Graph Neural Network (GNN) architectures on the diffusion process for DFT Success Rate keeping other parts of the framework same. Different architectures perform similarly on the task of diffusion sampling. Interestingly, in the realm of the diffusion task, we note that the disparity in success rates among these architectures is marginal (as shown in Figure 6) which has been recently demonstrated in applications of molecular generation tasks as well (Wang et al., 2023). The intuition behind this result is that the diffusion model\u2019s score function can be thought of as learning a harmonic potential (Xie et al., 2021). Harmonic potentials are simpler force-fields than ab-initio DFT calculations involved in OC20 forces. This could result in simpler architectures being able to capture the underlying complexity of the diffusion task defined in our work. 4.7. OOD generalization We measure the success of AdsorbDiff in out-of-distribution (OOD) cases where the model hasn\u2019t seen the adsorbate or the slab even during the pre-training on OC20. We pick a random 50 samples out of 200 validation OOD split defined in Open Catalyst Challenge 2023. We observe a marginal decrease of only 3.8% in results for the OOD case compared to the ID scenario and consistently observe significant improvement over the AdsorbML (Nsite=1) baseline. 0 5 10 15 20 25 30 35 40 DFT Success Rate (%) Random AdsorbDiff 8.4% 28% OOD Results Figure 7. Comparison of DFT Success Rate for In-Distribution (ID) and Out-of-Distribution (OOD) splits using the AdsorbDiff method. Random baseline corresponds to randomly placed adsorbate. 4.8. Inference cost In the case of conditional diffusion, our approach maintains a maximum step limit of 100, with adsorbate placement converging, on average, within 98 steps. In contrast, for MLFF optimization with a maximum step limit of 300 and Fmax criteria of 0.01 eV/A (consistent with AdsorbML), the convergence occurs in approximately 286 steps. Consequently, for scenarios with a single adsorption site (Nsite 1), AdsorbDiff incurs approximately 34% more inference cost than AdsorbML, given the GNN architecture for diffusion and MLFF optimization is the same. This end-to-end ML framework is O(104) times faster than the conventional DFT pipelines (Lan et al., 2023). In Section 4.6, we illustrate that simpler and faster models such as PaiNN yield comparable performance to more intricate and slower models like EquiformerV2. This enhances the efficiency of our diffusion-based approach, as its computational burden becomes negligible in comparison to MLFF optimization, which would require more computationally intensive ML architectures (details in Appendix B). 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2206.02005v2",
+                    "title": "Open Challenges in Developing Generalizable Large Scale Machine Learning Models for Catalyst Discovery",
+                    "abstract": "The development of machine learned potentials for catalyst discovery has\npredominantly been focused on very specific chemistries and material\ncompositions. While effective in interpolating between available materials,\nthese approaches struggle to generalize across chemical space. The recent\ncuration of large-scale catalyst datasets has offered the opportunity to build\na universal machine learning potential, spanning chemical and composition\nspace. If accomplished, said potential could accelerate the catalyst discovery\nprocess across a variety of applications (CO2 reduction, NH3 production, etc.)\nwithout additional specialized training efforts that are currently required.\nThe release of the Open Catalyst 2020 (OC20) has begun just that, pushing the\nheterogeneous catalysis and machine learning communities towards building more\naccurate and robust models. In this perspective, we discuss some of the\nchallenges and findings of recent developments on OC20. We examine the\nperformance of current models across different materials and adsorbates to\nidentify notably underperforming subsets. We then discuss some of the modeling\nefforts surrounding energy-conservation, approaches to finding and evaluating\nthe local minima, and augmentation of off-equilibrium data. To complement the\ncommunity's ongoing developments, we end with an outlook to some of the\nimportant challenges that have yet to be thoroughly explored for large-scale\ncatalyst discovery.",
+                    "authors": "Adeesh Kolluru, Muhammed Shuaibi, Aini Palizhati, Nima Shoghi, Abhishek Das, Brandon Wood, C. Lawrence Zitnick, John R Kitchin, Zachary W Ulissi",
+                    "published": "2022-06-04",
+                    "updated": "2022-06-13",
+                    "primary_cat": "physics.chem-ph",
+                    "cats": [
+                        "physics.chem-ph",
+                        "cond-mat.mtrl-sci"
+                    ],
+                    "main_content": "Introduction Catalysts have played a key role in the synthesis of everyday chemicals and fuels necessary for a 21st century society. As renewable energy prices continue to decrease, traditional chemical synthesis processes are being revisited for more sustainable alternatives. At the center of this, catalyst discovery plays a key role in the advancement of renewable energy processes and sustainable chemical production, i.e. ammonia for fertilizer and hydrogen production. Unfortunately, the search space for catalyst materials is enormous for even high-throughput experiments.2 This presents a need for computational tools to simulate systems through quantum mechanical (QM) models like Density Functional Theory (DFT). QM approaches have made notable advancements in bridging computational results to experimental \ufb01ndings.3\u20138 While effective, QM tools scale very poorly, O(N 3) or 1 arXiv:2206.02005v2  [physics.chem-ph]  13 Jun 2022 \fFigure 1: Summary of challenges associated with training on large dataset with large ML potentials discussed in the paper. Top left Trade o\ufb00s in direct and gradient GNN force predictions. Top right An example system for a case where the distance metrics are relatively good for the direct approach but the force metrics are worse. Bottom left Demonstration of inconsistent error across a metallic surface and a non-metal through an example. Bottom right Augmenting existing relaxation datasets with o\ufb00-equlibrium data can aid in relaxation performance. worse in the number of electrons. The computational cost associated with QM tools render them infeasible to the scale of the systems and search space desired for catalyst discovery. As a result, the catalysis community has moved towards a more data driven approach.9\u201313 With the QM data available, researchers are often interested in building machine learning surrogates for a particular chemical property.14\u201317 Such e\ufb00orts, however, were limited to the \ufb01nite data available, often for a very speci\ufb01c chemistry or system, limiting the generalizability ability of such models.10,18 Fortunately, as the community continues to curate larger, and more diverse datasets, machine learning models will continue to improve as they move towards larger, and more sophisticated architectures. In the \ufb01eld of small molecules, a vast collection of datasets have been developed for varying use cases, including molecular dynamics simulations (MD17,19 ANI-1,20 COLL21) and quantum mechanical properties (QM922 Alchemy23). These datasets are often limited to a few (5-10) unique elements, on average 10-20 atoms per system, and training set sizes in the range of 10k-1M samples. In the \ufb01eld of heterogeneous catalysis, datasets are often much more limited with training set sizes between 100 50k.24\u201327 These datasets were often created for very speci\ufb01c applications involving a handful of small adsorbates (i.e. hydrogen containing adsorbates on transition metal surfaces, CO2 re2 \fduction catalysts, etc.). The release of OC20 marks a push towards a large, sparse collection of the material space. OC20 spans 55 unique elements, 82 adsorbates and includes a collection of unary, binary and ternary materials. A total of 1.28 million DFT relaxations were performed, comprising \u223c260M single point evaluations of system energy and per-atom forces. OC20 presented several practical tasks for the community to work towards. The most general of the tasks, Structure to Energy and Forces (S2EF) evaluates a model\u2019s ability to serve as a surrogate to DFT predicting a con\ufb01guration\u2019s energy and per-atom forces. Initial Structure to Relaxed Energy (IS2RE) asks to predict the relaxed state energy, given only the initial structure. Initial Structure to Relaxed Structure (IS2RS) explores how well the relaxed structure can be predicted given only the initial con\ufb01guration. In the scope of OC20, all energies were referenced to represent adsorption energy. For more details, we refer readers to the original manuscript.1 In this perspective we shed light on the challenges of training Graph Neural Networks (GNNs) on large-scale datasets spanning material and composition space, illustrated in Figure 1. We begin with a quick overview on the current state of the community\u2019s progress and share some takeaways from what we have observed. We then discuss some telling trends on the performance of models across di\ufb00erent adsorbates and material types. We discuss how di\ufb00erent approaches and modeling decisions impact the prediction tasks and highlight the challenges associated with each. Further, we explain what the accuracies in various proposed metrics mean and some of the challenges in analyzing them. Finally, we share our outlook on the direction the community is headed and what still remains to achieve a large scale, generalizable potential for catalyst discovery. Community progress in developing ML models for catalysis Molecular modeling has progressed at an incredible rate over the past few decades. Simple linear models, neural networks, and kernel methods were originally developed relying on hand-crafted atomic representations, or descriptors28\u201332 as inputs to the models. Descriptors capture invariant geometric information in the form of bonds and angles of the local environment of an atom. While e\ufb00ective, the parameterization of such descriptors has been a challenging and non-trivial task. The past few years has seen a shift towards deep learning approaches. Rather than relying on hand crafted representations, models are being developed to learn similar or more expressive representations, speci\ufb01cally by exploiting the graphical nature of molecules using Graph Neural Networks (GNNs).33\u201338 Such models only take in 3D atomic coordinates and atomic numbers. A graph is then generated, where atoms are treated as nodes, and the distance between them as edges. Once a graph has been constructed, GNNs will undergo several rounds of message passing in which node representations are updated based o\ufb00messages sent between neighboring nodes. While models may di\ufb00er in their exact architecture, the update and message functions often include a series of multilayer perceptrons and nonlinearities. Unlike traditional descriptor based models, GNNs end up learning node representations as part of the training process. Learned representations proceed through a \ufb01nal output block where a \ufb01nal prediction is made. In recent years, GNNs have come to surpass traditional descriptor based models.33\u201338 While typically data hungry, recent models like NequIP36 are demonstrating great performance with as little as 100 samples. GNNs continue to gain traction as models continue to demonstrate state of the art performance on molecular datasets. Since the release of OC20, the community has been rapidly developing new approaches to improve existing baselines. Models being de3 \fFigure 2: Community progress on the OC20 dataset since release. Left: IS2RE performance for both direct and relaxation based approaches. The current error target of 0.10 eV would make these models more practically useful for researchers\u2019 applications. Right: S2EF performance as evaluated by mean absolute error of the forces. IS2RE and S2EF MAEs for their median baselines are 1.756 eV and 0.084 eV/\u02da A , respectively. veloped range from traditional descriptor-style models39 to complex and large GNN architectures.35,40\u201343 Godwin, et al. present a simple, but e\ufb00ective GNN regularization technique to improve graph-level predictions, namely IS2RE. Liu, et al. use a similar technique in addition to a graph-based transformer to win 1st place in the NeurIPS 2021 Open Catalyst Challenge44 for direct IS2RE predictions. Klicpera, et al.35,45 and Shuaibi, et al.40 explore various higher order representations (i.e., triplets and quadruplets) and leverage training on the entire OC20 to achieve impressive performance on the S2EF task, with GemNet-OC45 holding the current state of the art across all tasks. Sriram, et al.41 introduces Graph Parallelism, allowing them to scale GemNet to nearly a billion parameters across multiple GPUs. The scale and diversity of OC20 has additionally enabled transfer learning approaches to smaller datasets. Kolluru, et al.46 propose a transfer learning technique to use OC20 pretrained models to improve performance on smaller, out-of-distribution datasets. Similar work has also been demonstrated for other big material datasets.47 As the community continues to improve performance (Figure 2), it\u2019s important to understand some of the challenges, trends, and pitfalls in developing a generalizable potential. Where are molecular GNNs still erroneous? Most of the independent work done in developing ML potentials has been con\ufb01ned to datasets built for certain applications. For example, ML potentials for the applications of CO2RR are usually just trained with CO and H adsorbates.27,48\u201350 While this approach might interpolate well across materials, extrapolation to di\ufb00erent adsorbates or more complicated materials will likely su\ufb00er in performance. A universal ML potential, if possible, would \ufb01rst require a large, diverse dataset that spans material and chemical space. OC20 dataset was created to build ML potentials that cover a large and diverse space of heterogeneous catalysts. Errors across material types: With over 300k unique surfaces, OC20 spans a vast range of material compositions. When training large GNNs on the entire OC20 dataset, we observe that the accuracies are not uniform across element and adsorbate types. To analyze this, we divide the validation set into four di\ufb00erent material types: intermetallics, metalloids, nonmetals and halides, Figure 3(a). The distribution of data across these classes of materials is not the same, we have signi\ufb01cantly more intermetallics and relatively fewer halides. We observe that the performance on non-metals is signi\ufb01cantly worse, although both nonmetals and 4 \f(a) (c) (d) (b) Figure 3: Analysis of GemNet-dT errors on the OC20 validation sets. (a) The categorization of OC20 elements into intermetallics, nonmetals, metalloids and halides for analysis. (b) Model performance across the di\ufb00erent distributions and material types. (c) Errors averaged across all validation splits for speci\ufb01c adsorbate containing systems. (d) Errors averaged across all validation splits for adsorabtes containing certain elements. metalloids contribute to similar percentage of training data (Figure 3(b)). On the other hand, models tend to do much better across the board for intermetallics. Inaccuracies coming from non-metals disproportionately contribute to the overall errors, leading to worse performance for both force and energy predictions. Errors across adsorbates: Large adsorbates are inherently more complicated as the degrees of freedom increases with the number of atoms. However, we observe no correlation with our model\u2019s performance and the size of the adsorbate. Model accuracies are poor for bidentate adsorbates like *CH*COH, *N*NO, *CH2*O, shown in Figure 3(c). Figure 3(d) also shows that adsorbates with N and O are generally more erroneous. Modeling trade-o\ufb00s Energy-conserving forces Force predictions play an important role in the applications of ML models for catalyst discovery. While some tasks may only be interested in property predictions like adsorption or formation energy,48,52,53 forces are necessary to study dynamics such as structural relaxations, molecular dynamics, and transition state calculations.1,33,36,54 Physically, energy-conserving forces are derived as the gradient of energy with respect to atomic positions: Fi = \u2212dE dxi (1) Energy-conservation is critical in studying molecular dynamics accurately. ML models estimating energy-conserving forces must ensure the architecture is continuous and di\ufb00erentiable, often satis\ufb01ed by appropriate non-linear 5 \fTable 1: Results on the OC20 S2EF task via gradient-derived or direct force predictions. All models were trained on the OC20 S2EF All dataset. Results reported for the validation set. Energy metrics are unavailable for the gradient based SpinConv model due to being optimized only on forces. Model Energy MAE (eV) \u2193 Force MAE (eV/\u02da A ) \u2193 ID OOD Ads. OOD Cat. OOD Both ID OOD Ads. OOD Cat. OOD Both Median 2.04 2.42 1.99 2.58 0.081 0.080 0.079 0.098 Gradient forces SpinConv40 0.031 0.035 0.032 0.042 GemNet-dT35 0.36 0.39 0.48 0.58 0.030 0.034 0.033 0.042 Direct forces SpinConv40 0.26 0.29 0.38 0.47 0.027 0.030 0.029 0.037 GemNet-dT35 0.23 0.25 0.35 0.41 0.021 0.024 0.025 0.032 Table 2: Results on the OC20 IS2RE task using one of two approaches. Direct Directly predicting the relaxed state energy and Relaxation Training a model for energy and force predictions, followed by an iterative ML-based geometry optimization to arrive at a relaxed structure and energy. Relaxation results on the 2M subset suggest that competitive results are still possible with a limited compute budget. Results reported for the test set. Energy MAE [eV] \u2193 Energy within Threshold (EwT) \u2191 Model Approach Dataset Size ID OOD Ads OOD Cat OOD Both ID OOD Ads OOD Cat OOD Both Median baseline 1.75 1.88 1.71 1.66 0.71% 0.72% 0.89% 0.74% DimeNet++34 Direct 460,328 0.56 0.73 0.58 0.66 4.25% 2.07% 4.10% 2.41% SpinConv40 Direct 460,328 0.56 0.72 0.57 0.67 4.08% 2.26% 3.82% 2.33% NoisyNodes43 Direct 460,328 0.42 0.57 0.44 0.47 9.12% 3.49% 8.01% 4.64% Graphormer42 Direct 460,328 0.40 0.57 0.42 0.50 8.97% 3.45% 8.18% 3.79% DimeNet++ \u2013 LF + LE1,34,51 Relaxation 2,000,000 0.53 0.57 0.56 0.52 6.79% 4.71% 6.49% 4.54% SpinConv40,51 Relaxation 2,000,000 0.46 0.51 0.47 0.44 7.38% 4.82% 7.05% 5.31% GemNet-dT35 Relaxation 2,000,000 0.44 0.44 0.45 0.42 9.37% 6.59% 8.42% 6.40% GemNet-OC45 Relaxation 2,000,000 0.41 0.42 0.42 0.39 11.02% 8.68% 10.10% 7.82% DimeNet++ \u2013 LF + LE1,34 Relaxation 133,934,018 0.50 0.54 0.58 0.61 6.57% 4.34% 5.09% 3.93% SpinConv40 Relaxation 133,934,018 0.42 0.44 0.46 0.42 9.37% 7.47% 8.16% 6.56% GemNet-dT35 Relaxation 133,934,018 0.39 0.39 0.43 0.38 12.37% 9.11% 10.09% 7.87% GemNet-OC45 Relaxation 133,934,018 0.35 0.35 0.38 0.34 16.06% 12.62% 13.17% 11.06% activation functions.33\u201335 Geometrically, forces derived in an energy-conserving manner ensures forces are rotationally equivariant, a necessary physical relation of molecular systems.55 Unfortunately, a gradient calculation increases model overhead in both memory usage and computational time by a factor of 2-4.40,56 For datasets like MD17, calculating forces as a gradient is known to help in model accuracies as that is an important physical prior to the model.35,36,40 Models trained on MD17 are often used to run molecular dynamics, further necessitating the need for energy-conservation.36 However, for the OC20 dataset, particularly in the task of geometric optimization, we observe that the gradient approach for calculating forces to perform worse than direct prediction of forces for GemNet-dT35 and Spinconv.40 Dimenet ++34 and ForceNet56 were built for gradient and direct approach respectively. The gradient approach could also make the training unstable in certain cases, which has been observed for ForceNet56 and GemNet-OC.45 Table 1 compares performance on the S2EF task for two recent top performing models, GemNet-dT35 and SpinConv.40 Not only are the force accuracies worse for the gradient approach, but the corresponding relaxed structure and relaxed energy 6 \fmetrics calculated via optimization are also signi\ufb01cantly worse.40 While energy-conservation plays a critical role in many molecular applications, we observe that direct force computations brings e\ufb03ciency and performance advantages.40,56 Models trained for direct force predictions are limited to applications where strict enforcement of energy-conservation can reasonably be ignored, i.e. OC20\u2019s structural relaxations. Here, atomic positions are updated solely from force estimates.1,57 If necessary, DFT, or a subsequent ML model, can then be used to make reliable energy predictions on the ML optimized structure. Similarly, transition states or saddle points can be derived in a similar manner with direct-force models. We want to emphasize that although unorthodox, direct-force models still prove to be useful in certain catalyst applications, i.e. OC20-like tasks. Prediction of relaxed energy and structure Adsorption energy is one of many properties that helps inform catalyst performance.58 Computationally, this is computed via a series of QM structural relaxations. The relaxed energy is then referenced to represent the adsorption energy, see Chanussot et al.,1 Garc\u00b4 \u0131a-Muelas et al.59 for more details. From a data-driven approach, we can predict the relaxed energy or the relaxed structure of an atomic system usually via two methods. First, we can build a surrogate to DFT, approximating system energy and per-atom structures, and running ML optimizations to \ufb01nd the minimum energy, a common approach within the \ufb01eld. Alternatively, given a large enough dataset of relaxed structures and energies, we can try to predict these properties directly using a ML model instead of optimizing via an iterative loop. The advantage of the direct method over the relaxation approach is that it requires only a single call to the ML model, whereas the relaxation approach could require on average 200-300 calls for a single relaxation. Direct approaches are particularly advantageous when we talk about the computational cost of approaching large scale inference on the order of hundreds of millions to billions of systems. The community has made tremendous progress in predicting adsorption energy as evaluated by the OC20 IS2RE task (Figure 2). Direct approaches, despite using 300x less data, are approaching the competitive relaxation based approaches of GemNet-XL and GemNet-OC. Inference time aside, models trained on the full 133M dataset for the relaxation based approaches are typically compute intensive, using between 128-512 GPUs.1,35,40,56 While this is certainly a small price to pay if the models developed accelerate the discovery process, it does make it di\ufb03cult for the community to engage in and aid in development. This has been particularly observed in the NeurIPS 2021 Open Catalyst Challenge,44 where of the 30 submissions, 0 were made via the relaxation approach. Here, we show that models trained on a 2M subset of the full dataset are still able to provide competitive results and even, averaged across all splits, out perform direct approaches. Given the trends in the 2M dataset correlate well with the full 133M dataset,45 this should help incentivize the community to explore other approaches even with resource limitations. Although the relaxation approach is computationally expensive for both training and inference, we have observed that the models trained through this approach tend to generalize better on out-of-distribution (OOD) data, Table 2. Direct relaxed energy predictions are an easier ML problem than direct structure predictions. For a system of size N, energy predictions require a single scalar output, while structure predictions require 3N components. We \ufb01nd that for relaxed energy prediction tasks, metrics are closer for direct and relaxation approach whereas for structure prediction task the metrics are worse. The OC20 paper provides a baseline for relaxed structure prediction only via the relaxation approach.1 In Table 3 we provide baselines for direct relaxed structure prediction. A considerable gap exists between the direct and relaxation based approaches (especially in the DFT based metrics). 7 \fMetrics for \ufb01nding local minima Relaxed structure prediction is less straightforward than some of the other common energy and force prediction tasks. Given a dataset like OC20 where relaxed structures are not necessarily global minima, a model trained on such a dataset could either (1) predict and arrive at the same local minima, (2) arrive at a di\ufb00erent, but still suitable minima, or (3) fail to arrive at any sort of minima. To account for this, two main metrics have been presented in the OC20 paper. Average Distance within Threshold (ADwT) is a distance based metric and measures how close the predicted structure compares to the actual structure. This is similar to the Global Distance Test (GDT) metric in the protein folding task.60,61 ADwT takes an average across different thresholds varying from 0.1 to 0.5\u02da A to ensure a signal is captured. For the OC20 dataset, we evaluate this metric for the input initial structures for an accuracy of 21.18% on the in-domain validation set.1 Models, at the bare minimum, should perform better than this baseline. To ensure invariance to arbitrary coordinate reference frames, we predict the di\ufb00erence between initial and \ufb01nal positions instead of the \ufb01nal position Cartesian coordinates. Predicting the delta di\ufb00erence helps simplify this task and results in improved ADwT accuracies. Table 3: Baseline metrics for IS2RS direct task in comparison with the relaxation approach. Metrics are reported on a 2k subset of the validation set, across all splits. DwT is evaluated at a threshold of 0.04 \u02da A. For compute reasons, DFT-based metrics were evaluated on a 200 system subset of the 2k, 50 systems from each split. Model DwT (at 0.04 \u02da A ) \u2191ADwT \u2191FbT* \u2191AFbT* \u2191 Direct ForceNet56 0.70 45.69% 0.00% 0.00% SpinConv40 1.05 47.76% 0.00% 0.00% GemNet-dT35 1.75 45.87% 0.00% 0.08% Relaxation ForceNet56 1.45 46.51% 0.00% 7.64% SpinConv40 8.20 55.81% 0.00% 12.55% GemNet-dT35 13.95 60.88% 0.00% 20.35% A model that predicts a relaxed structure that is not identical to its DFT reference may still be considered successful for two reasons. (1) the model could have predicted a symmetrically identical site on the surface and (2) the model predicted a di\ufb00erent, but still suitable local minima. The former is more a concern surrounding the distance-based metric, as ADwT, although accounts for periodic-boundary conditions, does not consider symmetrically identical sites. While it is rather unlikely an adsorbate initialized over a particular site will hop several sites over to a symmetrically identical site, it is worth raising awareness to the possibility. On the other hand, a model that arrives at a di\ufb00erent relaxed structure entirely will fail according to ADwT. However, to verify whether the model has predicted a di\ufb00erent suitable minima, we can evaluate the DFT forces corresponding to the ML predicted structures. This metric is called Average Force below Threshold (AFbT) and it measures the percent of structures having their forces close to zero.1 Since models are expected to predict relaxed structures, DFT forces should be close to zero. This is a stricter metric as compared to ADwT. However, this is far more expensive due to the additional DFT calculations. A more practically useful metric would be number of DFT calculations required to \ufb01nd the relaxed structure starting from the ML relaxed structure. This would give us an idea of the percent of DFT calculations that the current ML models can reduce. Although useful, this is a signi\ufb01cantly more expensive metric than AFbT calculations. While it is not something Open Catalyst Project\u2019s (OCP) tracks on their public leaderboard, we bring awareness to it as there could be instances where models do poorly on ADwT and AFbT but resulting structures are only a few DFT steps away from the relaxed structure. In Table 3 we compare relaxed structure prediction via a direct and relaxation approach. We observe that direct methods, although having competitive ADwT metrics, have AFbT metrics that are signi\ufb01cantly worse. This suggests that direct models do a reasonable job at getting close to the relaxed structure but are in high-force con\ufb01gurations, failing to capture repulsive physical interactions.43 We speculate models struggle with this since small perturbations distances can have large consequences on 8 \fforces, e.g. moving two atoms at an equilibrium bond length fractions of an angstrom towards each other. Relaxed structure prediction via the relaxation approach avoids this issue by using ML forces to drive a geometric optimizer. We observe that distance metrics at tighter thresholds correlate better with force based metrics, however, going below 0.04 \u02da A does not give su\ufb03cient signal and the accuracies for most systems fall to zero. Moreover, the Distance within Threshold (DwT) at 0.04 \u02da A isn\u2019t a good enough signal that can replace AFbT. For example, DwT (at 0.04\u02da A ) for ForceNet relaxation approach and GemNet-dT direct approach are similar, however, the AFbT metrics still di\ufb00er by 7.56% (as shown in Table 3). We believe that \ufb01nding non DFT-based metrics that correlate well with DFT-based metrics is still an open and important question in the community which would make model evaluation computationally less expensive. Additional data The OC20 paper1 released two additional data subsets generated with ab-initio molecular dynamics (\u2018MD\u2019) and structural perturbations (\u2018Rattled\u2019). These provide 38M and 17M additional S2EF training data points respectively. Table 4 presents results for GemNet-OC45 models trained on S2EF, Rattled, and MD data compared against similar analysis from the OC20 paper for DimeNet++.21,34 First, on the force MAE metric, addition of MD data hurts DimeNet++ while it improves GemNetOC. We speculate this to be another artifact of modeling forces as negative gradients of energy (as in DimeNet++) vs. direct prediction (as in GemNet-OC). Second, consistent with the OC20 paper, adding MD data to the training set provides a useful signal for IS2RS structure relaxations as per the AFbT metric. Finally, adding Rattled data helps with IS2RS metrics, but did not help or marginally hurt the S2EF force MAE. This could be due to a variety of reasons \u2013 random perturbations being too large / small to be useful, intermediate structures along a trajectory being less useful compared to closer to the local minimum (as in MD initial structures), etc. A promising direction here could be active learning approaches to optimally query additional training data points. Table 4: Results with DimeNet++ (DN++) and GemNet-OC (GN-OC) trained on MD and Rattled. S2EF results reported for the validation in-distribution set. IS2RS results reported on the test set. S2EF Val ID IS2RS Test Training Data (# samples) Force MAE \u2193ADwT \u2191AFbT \u2191 DN++ ( 20M (20M) 0.0511 34.37% 2.67% 20M + MD (58M) 0.0594 47.69% 17.09% 20M + Rattled (37M) 0.0614 43.94% 12.51% GN-OC ( All (133M) 0.0179 60.33% 35.27% All + MD (172M) 0.0173 60.77% 38.05% All + MD + Rattled (189M) 0.0174 Summary and Outlook The development of generalizable or universal ML models has only recently been seriously considered with the emergence of largescale datasets like OC20.1 Since its release, the catalysis and ML communities have both made tremendous progressive in developing models for catalyst applications. As the community continues to grow and as more datasets emerge that span material and composition space, the prospect of large-scale generalizable models is within reason. Progress thus far has demonstrated several challenges in accomplishing this feat: classes of materials and adsorbates with inconsistent errors, energy-conserving forces, relaxed vs direct approaches, DFT metrics, and data augmentation strategies. In this perspective, we discussed these challenges in detail and provided some insights as to how and why they are important. Although these challenges were discussed in the context of OC20, we anticipate similar challenges to future datasets of its kind. Datasets like OC20 has o\ufb00ered new ways to how we think about building large, generalizable, and reliable models. While model performance has been the focal point of community progress thus far, we provide an outlook of other important challenges that we hope the community to engage in. 9 \fTraining strategies. OC20 was released with prede\ufb01ned training, validation, and test sets. Its splits were curated in a manner to tackle the problem of building a single generalizable model for catalysis. However, it could be the case that multiple models for di\ufb00erent subsets of the data, e.g. adsorbates, compositions, materials, do better. In the case of nonmetals, for instance, we have shown that this actually hurts performance a possible consequence of the reduced dataset size. Uncertainty and active learning. While model performance is a necessary step for the discovery process, it is not always su\ufb03cient. A practical ML-aided catalyst discovery pipeline will ultimately turn to experiments to validate whether the ML predicted \u201cgreat\u201d catalyst is at all e\ufb00ective. Having con\ufb01dence in these predictions is particularly important to avoid wasted expensive experiments. Uncertainty quanti\ufb01cation has been a particularly popular topic within the catalysis community, often focused on the small data regime and active learning.62\u201368 The e\ufb00ectiveness of traditional uncertainty estimation techniques on large datasets like OC20 is a necessary and important step for the future of this work. Similarly, how to best leverage active learning for either dataset generation and/or augmentation69 or online active learning64,65 at the scale of OC20 will be an exciting future direction. Model e\ufb03ciency. In addition to model performance and reliability, model e\ufb03ciency will continue to be critical for all applications. For training, faster, more data e\ufb03cient models can help attract the community to tackle some of the bigger challenges like a surrogate to DFT, i.e. OC20\u2019s S2EF task. Progress so far has shown that the best models are also the largest models. From an inference perspective, this poses obvious challenges of slower speeds and ultimately reduced screening throughput. While models still remain orders of magnitude faster than DFT, when considering the possibility of screening billions of systems, computational costs add up. Recent models encoding equivariant representations36,70 have shown incredible scaling and e\ufb03ciency gains that could be promising to explore. Moving forward, ef\ufb01cient architectures and model distillation71 will be an important contribution to reduce the computational cost of large-scale inference, even if it means sacri\ufb01cing some accuracy. Data augmentation. The scale of OC20 makes data augmentation a non-trivial challenge. With 130M+ training data points, randomly adding 10-100k data points will likely have negligible impact on the models. We observed that models using the additional MD data are able to perform the best, while the rattled data has little impact. Identifying strategies to combine and train large molecular and material datasets like ANI-120 and OQMD72 with OC20 could help improve models even further. The biggest challenge surrounding this comes from combining datasets of varying levels of DFT theory. Energy-conserving forces. In the context of OC20, we have observed that the best performing models make a direct force-prediction. While this may be suitable for some applications, the more physically motivated gradient approach to force prediction is desired for other applications like MD. The same direct models applied to MD17 observe the opposite e\ufb00ect, better performance via the gradient method.45 It remains an open question why this is the case, and we encourage others to investigate this observation. Physics-based modeling. The majority of models submitted to OC20 have followed a purely data-driven approach, only taking in atomic numbers and positions as inputs. Exploring ways to leverage OC20 charge density or Bader charge data 1 could prove useful, particularly in the low data regime. Additionally, models like UNiTE73 or OrbNet74 that leverage tight binding DFT75 for featurization could be interesting to explore for catalyst applications. 1To be made publically available at https://github.com/Open-CatalystProject/ocp/blob/main/DATASET.md 10 \fGlossary ADwT Average Distance within Threshold. 8 AFbT Average Force below Threshold. 8, 9 DFT Density Functional Theory. 1, 8 DwT Distance within Threshold. 9 EwT Energy within Threshold. 6 GNNs Graph Neural Networks. 3, 4 IS2RE Initial Structure to Relaxed Energy. 3, 4, 6 IS2RS Initial Structure to Relaxed Structure. 3, 9 OC20 Open Catalyst 2020 Dataset. 1, 3\u201310 OCP Open Catalyst Project\u2019s. 8 S2EF Structure to Energy and Forces. 3, 4, 9 11"
+                }
+            ],
+            "Muhammed Shuaibi": [
+                {
+                    "url": "http://arxiv.org/abs/2106.09575v1",
+                    "title": "Rotation Invariant Graph Neural Networks using Spin Convolutions",
+                    "abstract": "Progress towards the energy breakthroughs needed to combat climate change can\nbe significantly accelerated through the efficient simulation of atomic\nsystems. Simulation techniques based on first principles, such as Density\nFunctional Theory (DFT), are limited in their practical use due to their high\ncomputational expense. Machine learning approaches have the potential to\napproximate DFT in a computationally efficient manner, which could dramatically\nincrease the impact of computational simulations on real-world problems.\nApproximating DFT poses several challenges. These include accurately modeling\nthe subtle changes in the relative positions and angles between atoms, and\nenforcing constraints such as rotation invariance or energy conservation. We\nintroduce a novel approach to modeling angular information between sets of\nneighboring atoms in a graph neural network. Rotation invariance is achieved\nfor the network's edge messages through the use of a per-edge local coordinate\nframe and a novel spin convolution over the remaining degree of freedom. Two\nmodel variants are proposed for the applications of structure relaxation and\nmolecular dynamics. State-of-the-art results are demonstrated on the\nlarge-scale Open Catalyst 2020 dataset. Comparisons are also performed on the\nMD17 and QM9 datasets.",
+                    "authors": "Muhammed Shuaibi, Adeesh Kolluru, Abhishek Das, Aditya Grover, Anuroop Sriram, Zachary Ulissi, C. Lawrence Zitnick",
+                    "published": "2021-06-17",
+                    "updated": "2021-06-17",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.CE",
+                        "I.2.6; J.2"
+                    ],
+                    "main_content": "Introduction Many of the world\u2019s challenges such as \ufb01nding energy solutions to address climate change [35, 3] and drug discovery [22, 28] are fundamentally problems of atomic-scale design. A notable example is the discovery of new catalyst materials to drive chemical reactions that are essential for addressing energy scarcity, renewable energy storage, and more broadly climate change [35, 23]. Potential catalyst materials are typically modeled using Density Functional Theory (DFT) that estimates the forces that are exerted on each atom and the energy of a system or structure of atoms. Unfortunately, the computational complexity of DFT limits the scale at which it can be applied. Ef\ufb01cient machine learning approximations to DFT calculations hold the potential to signi\ufb01cantly increase the discovery rate of new materials for these important global problems. Graph Neural Networks (GNNs) [10, 34] are a common approach to modeling atomic structures, where each node represents an atom and the edges represent the atom\u2019s neighbors [26, 9, 13, 25, 27, 33, 20, 15]. A signi\ufb01cant challenge in designing models is utilizing relative angular information between atoms, while maintaining a model\u2019s invariance to system rotations. Numerous approaches have been proposed, such as only using the distance between atoms [25, 27, 33], or limiting equivariant angular representations to linear transformations to maintain equivariance [31, 2, 1, 29]. One promising approach is the use of triplets of neighboring atoms to de\ufb01ne local coordinate frames that are invariant to system rotations [15, 14]. The relative angles between the three atoms may be used to update the GNN\u2019s messages while maintaining the network\u2019s invariance to rotations. It has been shown Preprint. Under review. arXiv:2106.09575v1  [cs.LG]  17 Jun 2021 \fFigure 1: Illustration of projecting an atom \u00b4 s in the neighborhood of s onto a sphere in a local coordinate frame de\ufb01ned by atom s and t (left). For each projected atom, a corresponding latitude \u03c6 (inclination) and longitude \u03b8 (azimuth) is computed for its projection onto a 2D reference frame (middle). The spin convolution is done in the longitudinal direction, corresponding to a roll is 3D space. (right) Example channel \ufb01lters that are learned using the grid-based approach for the \ufb01rst through third message blocks and the force block. that this additional angular information results in signi\ufb01cantly improved accuracies on several tasks [15, 14, 3]. We propose encoding angular information using a local reference frame de\ufb01ned by only two atoms; the source and target atoms for each edge in a GNN. Using this reference frame, a spherical representation of the incoming messages to the source atom is created, Figure 1. The representation has the bene\ufb01t of encoding all neighboring atom information, and not just information between atom triplets, which may result in higher-order information being captured. The complication is a reference frame de\ufb01ned by two atoms (or two 3D points) still has one remaining degree of freedom the roll rotation about the axis de\ufb01ned by the two 3D points. If this \ufb01nal degree of freedom is not accounted for, the model will not be invariant to system rotations. Our solution is to perform a convolution on the spherical representation across this \ufb01nal rotation, called a \u201cspin convolution\u201d. By globally pooling the convolution\u2019s features, the resulting SpinConv model maintains rotation invariance while enabling the capture of rich angular information. We describe two model variations that are used depending on the importance of energy conservation in the \ufb01nal application. We propose an energy-centric model that enforces energy conservation by calculating the forces using the negative partial derivative of the energy with respect to the atoms\u2019 positions [4]. Our second approach is a force-centric model that directly estimates the atom forces that is not energy conserving. While the force-centric model\u2019s energy estimation is rotation invariant, the model\u2019s \ufb01nal force estimation layer is not strictly rotation equivariant, but through its architectural design it is encouraged to learn rotation equivariance during training. Results are demonstrated on the Open Catalyst 2020 (OC20) dataset [3] aimed at simulating catalyst materials that are useful for climate change related applications. The OC20 dataset contains over 130M training examples for approximating the DFT-estimated forces and energies. Our SpinConv model achieves state-of-the-art performance for both energy and force estimation. Notably, the force-centric variant, which is not energy conserving, outperforms the energy-centric models. Signi\ufb01cant gains in accuracy are achieved for predicting relaxed energies from initial structures, by using the force-centric approach to predict the relaxed structure followed by its energy. Ablation studies are performed on numerous architectural choices, such as the choice of spherical representation and the size of the model. For completeness, we also evaluate our model on the MD17 [4, 5] and QM9 [22] datasets that measure accuracy for molecular dynamics and property prediction tasks respectively for small molecules. Results compare favorably with respect to state-of-the-art methods. 2 Approach We model a system or structure of atoms using a Graph Neural Network (GNN) [10, 16, 34], where the nodes represent atoms and the edges represent the atoms\u2019 neighbors. In this section, we describe both an energy-centric and force-centric model to estimating atomic forces, which vary in how they estimate forces and whether they are energy conserving. We begin by describing the components shared by each approach, followed by how these components are used. Code will be released upon acceptance under a permissive open-source license. 2 \fFigure 2: (left) Overall model diagram for energy-centric model taking atom positions x and atomic numbers a as input and estimating the energy E. (right) Diagram of the embedding and force blocks. The force block is only used in the force-centric model to estimate the per-atom forces after the message blocks. 2.1 Inputs and Outputs The inputs to the network are the 3D positions xi and the atomic numbers ai for all i \u2208n atoms. The outputs are the per atom forces fi \u2208R3 and the overall structure\u2019s energy E. The 3D distance offset between a pair of source and target atoms s and t respectively is xst = xs \u2212xt with a distance of dst = \u2225xst\u22252. Directional information is encoded using the normalized unit vector \u02c6 xst = xst/dst. The graph neural network is constructed with each atom t as a node and the edges representing the atom\u2019s neighbors s \u2208Nt, where Nt contains all atoms s with dst < \u03b4. Each edge has a corresponding message mst that passes information from atom s to t. The output forces and energy are computed as a function of edge messages mst that we describe next. 2.2 Energy and force estimation The energy-centric and force-centric models compute the structure\u2019s energy E as an output. Our GNN model updates for each edge an M-dimensional hidden message h(k) st \u2208RM for K iterations. The structure\u2019s energy E \u2208R is computed as a function of the \ufb01nal layer of the edge messages in the GNN: E(x, a) = X t Fe(at, X s h(K) st ), (1) where Fe is a single embedding block described later. As we also discuss later, the edge messages hst are invariant to system rotations, so the estimated energy E is also invariant. The estimation of the forces varies for the energy-centric and force-centric models. The energy-centric model estimates the forces using the negative partial derivative of the energy with respect to the atom positions. This approach to force estimation has the bene\ufb01t of enforcing energy conservation [4], i.e., the forces along any closed path sum to zero. The calculation of the partial derivative [4, 25, 27] requires an additional step similar to performing backpropagation when updating the network\u2019s weights: f = \u2212\u2202 \u2202xE(x, a) (2) The force-centric model estimates forces directly for an atom t using: ft = Ff(at, \u02c6 xt, h(K) t ), (3) where Ff is the force block we describe later, \u02c6 xt are all the normalized unit vectors for the neighbors of t and h(K) t are all incoming messages to atom t. This has the bene\ufb01t of improved ef\ufb01ciency since it does not require an extra backward pass to estimate the forces. The tradeoff is that it does not enforce energy conservation, i.e., the sum of the forces along a closed path may not equal zero. Depending on the application, an energy-centric or force-centric approach may be most suitable. In either model, losses may be applied to both the energy and force estimates with weights determined by the needs of the application. 3 \f2.3 Messages The edge messages are iteratively updated to allow information from increasingly distant atoms to be captured. Each message is represented by a tuple, mst = {\u02c6 xst, dst, hk st}, where hk st is the message\u2019s hidden state at iteration k. Both \u02c6 xst and dst are used to update the message\u2019s hidden state hst, which is itself rotation invariant due to the spin convolution that we describe later. The hidden state hst \u2208RM is updated using: h(k+1) st = h(k) st + Fh \u0010 as, at, m(k) st , m(k) s \u0011 , (4) where m(k) s is the set of messages coming into node s, i.e., all m\u00b4 ss with \u00b4 s \u2208Ns. The form of Fh is illustrated in Figure 2. It contains three parts; the spin convolution that transforms a spherical projection of the messages into a rotation invariant representation, the distance block that encodes the distance dst between atoms, and the embedding block that incorporates information about the atoms\u2019 atomic numbers. The output of the spin convolution is passed through an embedding block, added to the output of the distance block and \ufb01nally passed through another embedding block. We describe each of these parts in turn. The hidden messages are initialized using just a distance block followed by and embedding block, Figure 2. 2.3.1 Spin Convolution The spin convolution captures information about the neighbors \u00b4 s \u2208Ns of atom s when updating the message hidden state hst. The spin convolution has three stages that we describe in turn; projection, convolution and pooling. The convolution captures the relative angular information between the neighboring atoms, and the pooling ensures the output D-dimensional feature representation is invariant to system rotations. An important feature is the angular information of the neighboring atoms in Ns relative to s and t. This information is encoded by creating a local reference frame in which atom s is the center (0, 0, 0) and the z-axis points from atom s to atom t. As shown in Figure 1(left), this \ufb01xes all degrees of freedom except the roll rotation about the vector from s to t. The spin convolution is performed across a discretized set of rotations about the roll rotation axis. At each rotation, the atoms \u00b4 s are projected onto a sphere centered on s and used to create a spherical representation of the hidden states h\u00b4 ss. Each atom \u00b4 s \u2208Ns is projected using a polar coordinate frame (\u03c6, \u03b8) where \u03c6 may be viewed as the latitude (inclination) and \u03b8 as the longitude (azimuth). The polar coordinates are computed in the local edge coordinate frame using \u00af x\u00b4 ss = Rst \u02c6 x\u00b4 ss where Rst is a 3D rotation matrix that satis\ufb01es Rst \u02c6 xst = (0, 0, 1). To capture the rich information encoded in the relative angular information between atoms, a set of \ufb01lters is applied to the spherical representation (Figure 1(right)), similar to how a \ufb01lter is applied to an image patch with traditional CNNs. We explore two potential spherical representations: spherical harmonics and a grid-based approach. Spherical harmonics represent a spherical function using a set of basis functions that are equivariant to rotations. The degree \u2113indicates the number of basis functions L = (\u2113+ 1)2 used. The spherical representation of the incoming messages for each atom is RL \u00d7 RM, where M is the size of the message hidden states in h. The second approach uses the computed polar coordinates (\u03c6, \u03b8) for all \u00b4 s \u2208Ns to create a grid-based representation, Figure 1(middle). The polar coordinates are discretized creating a R\u03c6 \u00d7 R\u03b8 \u00d7 RM feature representation. Each message hidden state h(k) s\u00b4 s \u2208RM is added to the 3D feature representation using bilinear interpolation with its corresponding (\u03c6, \u03b8). A 1D convolution is performed with either spherical representation in the longitudinal direction. Filters have the same size as the feature representation, RL \u00d7 RM or R\u03c6 \u00d7 R\u03b8 \u00d7 RM for spherical harmonics and the grid-based approach respectively. Full coverage \ufb01lters are used since the angular relationship between atoms at distant angles is important, e.g., the forces of atoms at exactly 180\u25e6from each other may cancel out. Large \ufb01lters also enable the network to learn the complex relationships between numerous neighboring atoms. Rotations are performed using Wigner D-matrices for the spherical harmonic representation, while a simple translation is used for the grid-based representation. The result of the convolution is a R\u03b8 \u00d7 RD feature vector corresponding to D \ufb01lters applied to each longitudinal orientation. To make the representation invariant to rotations, average pooling is performed in the longitudinal direction resulting in a \ufb01nal RD feature vector. 4 \fFigure 3: Illustration of learned embeddings (weights on the one-hot embeddings) for the source as and target at atomic numbers plotted on a periodic table. A random sample of 12 values from each embedding are shown. Embeddings are from the \ufb01rst embedding block in the \ufb01rst message update. Note that neighboring atoms in the periodic table with similar properties have similar weights. Elements not in the OC20 dataset are marked with a light grey checkerboard pattern. 2.3.2 Distance Block The distance block encodes the distance between two atoms. The distance is encoded using a set of evenly distributed Gaussian basis functions G with means \u00b5i and standard deviation \u03c3. The means of the basis functions are evenly distributed from 0 to \u03b4 angstroms. Since the atomic radii of each element varies, the relative position of two atoms s and t is highly dependent on their atomic numbers as and at. To account for this, gain vasat and offset uasat scalars for the distance dst are learned for each potential pair of atomic numbers: bi = Gi(vasatdst + uasat \u2212\u00b5i, \u03c3) (5) The resulting feature b is passed through a linear transformation to create a D-dimensional feature vector that is passed to the next block. 2.3.3 Embedding Block The embedding block incorporates the atomic number information as and at into the update of the message\u2019s hidden state. The embedding operation may be interpreted as a mixture of experts [18] approach that computes B different variations of the input, which are weighted by an embedding computed from the atoms\u2019 atomic numbers. The block\u2019s inputs are used to compute B sets of hidden values Vst \u2208RD \u00d7 RB. A one-hot embedding for the atomic numbers as and at are concatenated and used to compute an B dimensional vector, vst \u2208RB, for weighting the B different sets of hidden values. An illustration of the learned embeddings are shown in Figure 3. vst is computed using a two layer network and softmax. The matrix Vst is multiplied by vector vst resulting in a vector of length D. As shown in Figure 2, the result is passed through an additional fully connected layer before being passed to the next block. The output of the block is either D if it is used in the message update. If the embedding block is used to compute the \ufb01nal energy, only the atomic number at embedding is used, the input dimension is M instead of D, and the output is size 1. 2.4 Force Block The force block computes the per-atom 3D forces f from at, \u02c6 xt, and h(K) t using Equation (3). The force block uses a similar spin convolution as the message block, except the sphere is centered on the target atom t and is orientated along the x, y and z axes to compute fx, fy and fz respectively. That is, the force block is used three times to compute the force magnitude in each orthogonal direction for each atom. The force block uses the same embedding blocks as message passing, Figure 2. The same weights are used to compute forces in each of the three directions, only the orientation of the sphere used to create the convolutional features changes. To add more robustness to the force estimation and encourage rotational equivariance, the overall structure may be randomly rotated several times and the forces estimated. The multiple estimates may then be rotated back to the original reference frame and averaged. For both training and testing, \ufb01ve random rotations are used. Empirically, this approach encourages the networks to learn an approximate rotation equivariant representation even though rotation equivariance is not strictly enforced. 5 \fModel Hidden #Msg #Params Train Inference OC20 Test dim layers time time Energy MAE [eV] \u2193 Force MAE [eV/\u00c5] \u2193 Force Cos \u2191 EFwT [%] \u2191 Median \u2013 \u2013 \u2013 2.258 0.08438 0.0156 0.005 SchNet[27, 3] 1024 5 9.1M 194d 0.8h \u2013 0.04903 0.3413 0 DimeNet++[14, 3] 192 3 1.8M 587d 8.5h 0.5343 0.04758 0.3560 0.05 DimeNet++ energy-only[14, 3] 192 3 1.8M 587d 8.5h 0.4802 0.3459 0.1021 0.0 DimeNet++ force-only[14, 3] 192 3 1.8M 587d 8.5h \u2013 0.03573 0.4785 \u2013 DimeNet++-large[14, 3] 512 3 10.7M 1600d 27.0h \u2013 0.03275 0.5408 \u2013 ForceNet[12] 512 5 11.3M 31d 1.3h \u2013 0.03432 0.4770 \u2013 ForceNet-large[12] 768 7 34.8M 194d 3.5h 0.03113 0.5195 SpinConv (energy-centric) 256 3 6.1M 275d 22.7h 0.4114 0.03888 0.4299 0.16 SpinConv (energy-centric) force-only 256 3 6.1M 380d 22.7h \u2013 0.03258 0.4976 \u2013 SpinConv (force-centric) 256 3 8.5M 275d 9.1h 0.3363 0.02966 0.5391 0.45 Table 1: Comparison of SpinConv to existing GNN models on the S2EF task. Average results across all four test splits are reported. We mark as bold the best performance and close ones, i.e., within 0.0005 MAE, which according to our preliminary experiments, is a good threshold to meaningfully distinguish model performance. Training time is in GPU days, and inference time is in GPU hours. Median represents the trivial baseline of always predicting the median training force across all the validation atoms. Model Energy MAE (eV) \u2193 Force MAE (eV/\u00c5) \u2193 ID OOD Ads. OOD Cat. OOD Both ID OOD Ads. OOD Cat. OOD Both Median 2.043 2.420 1.992 2.577 0.0809 0.0801 0.0787 0.0978 Energy Loss Only SchNet 0.395 0.446 0.551 0.703 DimeNet++ 0.359 0.402 0.506 0.654 Force Loss Only SchNet 0.0443 0.0469 0.0459 0.0590 DimeNet++ 0.0331 0.0341 0.0340 0.0417 DimeNet++-large 0.0281 0.0289 0.0312 0.0371 ForceNet 0.0313 0.0320 0.0331 0.0409 ForceNet-large 0.0278 0.0283 0.0309 0.0375 SpinConv (energy-centric) 0.0309 0.0321 0.0315 0.0393 Energy and Force Loss SchNet 0.443 0.491 0.529 0.716 0.0493 0.0527 0.0508 0.0652 DimeNet++ 0.486 0.470 0.533 0.648 0.0443 0.0458 0.0444 0.0558 SpinConv (energy-centric) 0.351 0.367 0.411 0.517 0.0358 0.0374 0.0364 0.0460 SpinConv (force-centric) 0.261 0.275 0.350 0.459 0.0269 0.0277 0.0285 0.0356 Table 2: Comparison of SpinConv to existing GNN models on different test splits. We mark as bold the best performance and close ones, i.e., within 0.0005 MAE, which according to our preliminary experiments, is a good threshold to meaningfully distinguish model performance. Training time is in GPU days, and inference time is in GPU hours. Median represents the trivial baseline of always predicting the median training force across all the validation atoms. 3 Experiments In this section, we begin by presenting our primary results on the Open Catalyst 2020 (OC20) dataset [3] and compare against state-of-the-art models. This is followed by results on the smaller datasets of MD17 [4, 5] and QM9 [22] for additional model comparison. Implementation details. For all models, the edge messages have size M = 32 with K = 3 layers, the hidden dimension D = 256 and embedding dimension B = 8. Unless otherwise stated, the convolutional \ufb01lters are of size 16x12 and 12x8 for the force-centric and energy-centric models respectively. A smaller \ufb01lter size was used for the energy-centric model due to memory constraints. GroupNorm [32] is applied after the spin convolution with group size 4. An L1 loss is used for all experiments. The force loss was weighed by 100 with respect to the energy loss, except for the force-only model where the energy loss is set to 0. All models were trained with Adam (amsgrad) to convergence with the learning rate multiplied by 0.8 when the validation error plateaus. Training was performed using batch sizes ranging from 64 to 96 samples across 32 Volta 32GB GPUs. The Swish [21] function is used for all non-linear activation functions. The neighbors s \u2208Nt of each atom t are found using a distance threshold of \u03b4 = 6\u00c5. If more than 30 atoms are within the distance threshold, only the closest 30 are used. The distance block uses 256 to 512 Gaussian basis functions with \u03c3\u2019s equal to three times the distance between Gaussian means. 6 \fModel Hidden #Msg #Params Train OC20 Val ID 30k dim layers time Energy MAE [eV] \u2193 Force MAE [eV/\u00c5] \u2193 Force Cos \u2191 EFwT [%] \u2191 Median Energy-Centric SpinConv (grid 12x8) 128 2 1.3M 54d \u2013 0.0417 0.401 \u2013 SpinConv (spherical harmonics, \u2113= 5) 256 3 6.4M 119d \u2013 0.0405 0.411 \u2013 SpinConv (grid 12x8) 256 3 6.1M 87d \u2013 0.0406 0.426 \u2013 Force-Centric SpinConv (grid 12x8) 128 2 1.8M 54d 0.376 0.0370 0.436 0.15% SpinConv (grid no conv 16x12) 256 3 8.5M 56d 0.341 0.0348 0.462 0.20% SpinConv (spherical harmonics, \u2113= 5) 256 3 8.1M 113d 0.321 0.0328 0.484 0.22% SpinConv (grid 16x12) 256 3 8.5M 76d 0.317 0.0326 0.484 0.20% Table 3: Ablation studies for SpinConv model variations trained for 560k steps (32-48 batch size, 0.2 epochs) with 16 Volta 32 GB GPUs. Training time is in GPU days and the validation set is a 30k random sample of the OC20 ID Validation set. Figure 4: Performance of SpinConv ablations on OC20 Val ID 30k (Table 3). All models trained for 560k steps and plotted against wall-clock training time. Note force-centric models and grid-based approaches converge more quickly than energy-centric models and those using spherical harmonics. 3.1 OC20 The OC20 dataset [3] contains over 130 million structures used to train models for predicting forces and energies during structure relaxations that is released under a CC Attribution 4.0 License. Since the goal of a structure relaxation is to \ufb01nd a local energy minimum, energy conservation in optional for this task. We report results for the Structure to Energy and Forces (S2EF), the Initial Structure to Relaxed Energy (IS2RE) and the Initial Structure to Relaxed Structure (IS2RS) tasks. 3.1.1 Structure to Energy and Forces (S2EF) There are four metrics for the S2EF task, the energy and force Mean Absolute Error (MAE), the Force Cosine similarity, and the Energy and Forces within a Threshold (EFwT). The EFwT metric is meant to indicate the percentage of energy and force predictions that would be useful in practice. Results for three model variants are shown in Table 1 on the test set. The SpinConv force-centric approach has the lowest energy MAE and force MAE of all models. While still low in absolute terms, the SpinConv models are improving over other models on the EFwT metric. DimeNet++-large slightly out performs SpinConv on the force cosine metric. The training time for the SpinConv is signi\ufb01cantly faster than DimeNet++, while being a little slower than ForceNet [12] or SchNet [27]. In Table 2 we examine the performance of SpinConv across different test splits. Note that the energy prediction of SpinConv is sign\ufb01cantly better than SchNet or DimeNet++. Across all models the accuracy for the in domain split are highest and decline for the three Out of Domain (OOD Adsorbate, OOD Catalyst, OOD Both) splits. SpinConv outperforms all models on each of the different domain splits. When comparing energy-centric approaches trained with both force and energy losses (bottom rows), the SpinConv model does signi\ufb01cantly better at predicting both. In fact, the energy-centric approach trained on forces and energy outperforms the DimeNet++ [14] model when trained on only energy, or energy and forces. We examine variations of the SpinConv model in Table 3 and Figure 4 through ablation studies. We trained three variants of the energy-centric model and four variants of the force-centric model. The grid-based and spherical harmonic approaches produced similar accuracies. However, the grid-based approach was signi\ufb01cantly faster to train, so it was used in the remaining experiments. Smaller models lead to reduced performance on the OC20 dataset, but we found for smaller datasets such as MD17 or QM9 smaller model sizes can be bene\ufb01cial to avoid over\ufb01tting. Finally, we test the impact of not performing the convolution (no conv) and only applying the \ufb01lter at a single rotation. Rotation invariance was maintained by orienting the \ufb01lter based on the mean angle of the neighboring atoms 7 \fEnergy MAE [eV] \u2193 EwT \u2191 Model Approach ID OOD Ads OOD Cat OOD Both ID OOD Ads OOD Cat OOD Both Median baseline 1.7499 1.8793 1.7090 1.6636 0.71% 0.72% 0.89% 0.74% CGCNN [33] Direct 0.6149 0.9155 0.6219 0.8511 3.40% 1.93% 3.10% 2.00% SchNet [25] Direct 0.6387 0.7342 0.6616 0.7037 2.96% 2.33% 2.94% 2.21% DimeNet++ [15] Direct 0.5620 0.7252 0.5756 0.6613 4.25% 2.07% 4.10% 2.41% SpinConv Direct 0.5583 0.7230 0.5687 0.6738 4.08% 2.26% 3.82% 2.33% DimeNet++ Relaxation 0.6908 0.6842 0.7027 0.6834 4.25% 3.36% 3.76% 3.52% DimeNet++ \u2013 force-only + energy-only Relaxation 0.5124 0.5744 0.5935 0.6126 6.12% 4.29% 5.07% 3.85% DimeNet++ \u2013 large force-only + energy-only Relaxation 0.5034 0.5430 0.5789 0.6113 6.57% 4.34% 5.09% 3.93% SpinConv (force-centric) Relaxation 0.4235 0.4415 0.4572 0.4245 9.37% 6.75% 8.49% 6.76% Table 4: Initial Structure to Relaxed Energy (IS2RE) results on the OC20 test split as evaluated by the Energy MAE (eV) and Energy within Threshold (EwT) [3] (see OC20 discussion board). Comparisons made for the direct and relaxation approaches using various models. Model Inference AFbT (%) \u2191 ADwT (%) \u2191 time \u2193 ID OOD Ads. OOD Cat. OOD Both Average ID OOD Ads. OOD Cat. OOD Both Average SchNet [25] 54.1h 5.28 2.82 2.62 2.73 3.36 32.49 28.59 30.99 35.08 31.79 DimeNet++ [14] 407.6h 17.52 14.67 14.32 14.43 15.23 48.76 45.19 48.59 53.14 48.92 DimeNet++-large [14] 814.6h 25.65 20.73 20.24 20.67 21.82 52.45 48.47 50.99 54.82 51.68 ForceNet [12] 75.1h 10.75 7.74 7.54 7.78 8.45 46.83 41.26 46.45 49.60 46.04 ForceNet-large [12] 186.9h 14.77 12.23 12.16 11.46 12.66 50.59 45.16 49.80 52.94 49.62 SpinConv (force-centric) 263.2h 21.10 15.70 15.86 14.01 16.67 53.68 48.87 53.92 58.03 53.62 Table 5: Relaxed structure from initial structure (IS2RS) results on the OC20 test split, as evaluated by Average Distance within Threshold (ADwT) and Average Forces below Threshold (AFbT). All values in percentages, higher is better. Results computed via the OCP evaluation server. Inference times are total across the 4 splits. weighted by distance. The result of not performing the convolution is signi\ufb01cantly reduced accuracy. However, its faster training time may make it suitable for some applications. Finally, for the force-centric SpinConv model we explore results when varying the number of random rotations used in the force block. The force MAE when using a single random rotation is 0.0276 and improves slightly to 0.0270 when using 5 random rotations. Increasing the number of rotations beyond 5 leads to negligible gains. The standard deviation of the force estimates at different random rotations is 0.004 eV/\u00c5. This is equal to 15% of the force MAE, which indicates the amount of error due to the model not being strictly rotation equivariant is small relative to the overall error of the model. 3.1.2 Initial Structure to Relaxed Energy (IS2RE) The Initial Structure to Relaxed Energy (IS2RE) task takes an initial atomic structure and attempts to predict the energy of the structure after it has been relaxed. Two approaches may be taken to address this problem, the direct and relaxation approaches [3]. The direct treats the task as a standard regression problem and directly estimates the relaxed energy from the initial structure. The relaxation approach computes the relaxed structure using the ML predicted forces to update the atom positions. Next, given the ML relaxed structure the energy is estimated. We show results for both approaches in the OC20 dataset using SpinConv in Table 4. The results of the SpinConv model signi\ufb01cantly outperform all previous approaches using the relaxation approach for both energy MAE and Energy within Threshold (EwT) metrics. DimeNet++ also shows improved results for the relaxation approach with the best approach using two models; DimeNet++-large for force estimation and DimeNet++ (energy-only) for the energy estimation. Note in contrast to other approaches, SpinConv shows good results across all test splits, including those with out of domain adsorbates and catalysts. Using the direct approach, SpinConv is comparable to DimeNet++\u2019s direct approach. 3.1.3 Initial Structure to Relaxed Structure (IS2RS) Our \ufb01nal results on the OC20 dataset are on the IS2RS task where predicted forces are used to relax an atom structure to a local energy minimum. The is performed by iteratively estimating the forces 8 \fMolecule GDML PhysNet PhysNet-ens5 SchNet DimeNet* SpinConv Aspirin 0.02 0.06 0.04 0.33 0.09 0.07 Benzene 0.24 0.15 0.14 0.17 0.15 0.17 Ethanol 0.09 0.03 0.02 0.05 0.03 0.02 Malonaldehyde 0.09 0.04 0.03 0.08 0.04 0.04 Naphthalene 0.03 0.04 0.03 0.11 0.06 0.04 Salicylic 0.03 0.04 0.03 0.19 0.09 0.05 Toluene 0.05 0.03 0.03 0.09 0.05 0.03 Uracil 0.03 0.03 0.03 0.11 0.04 0.03 Mean 0.073 0.053 0.044 0.141 0.069 0.058 Table 6: Forces MAE (kcal/mol\u00c5) on MD17 for models trained using 50k samples. Best results for models not using domain speci\ufb01c information are in bold. *The DimeNet results were trained in-house as the original authors did not use the 50k dataset. DimeNet was found to outperform DimeNet++ on this task. Task \u03b1 \u2206\u03f5 \u03f5HOMO \u03f5LUMO \u00b5 C\u03bd G H R2 U U0 ZPVE Units bohr3 meV meV meV D cal/mol K meV meV bohr3 meV meV meV NMP [9] .092 69 43 38 .030 .040 19 17 .180 20 20 1.50 Schnet [25] .235 63 41 34 .033 .033 14 14 .073 19 14 1.70 Cormorant [1] .085 61 34 38 .038 .026 20 21 .961 21 22 2.03 L1Net [19] .088 68 46 35 .043 .031 14 14 .354 14 13 1.56 LieConv [7] .084 49 30 25 .032 .038 22 24 .800 19 19 2.28 TFN [29] .223 58 40 38 .064 .101 SE(3)-Tr. [8] .142 53 35 33 .051 .054 EGNN [24] .071 48 29 25 .029 .031 12 12 .106 12 11 1.55 DimeNet++ [14] .044 33 25 20 .030 .023 8 7 .331 6 6 1.21 SphereNet [17] .047 32 24 19 .027 .022 8 6 .292 7 6 1.12 SpinConv .058 47 26 22 .027 .028 12 12 .156 12 12 1.50 Table 7: Mean absolute error results for QM9 dataset [22] on 12 properties for small molecules. that are in turn used to update the atoms positions. This process is repeated until convergence or 200 iterations. Results are shown in Table 5. The suggested metrics are Average Distance within Threshold (ADwT) metric, which measures whether the atom positions are close to those found using DFT and Average Forces below Threshold (AFbT), which measures whether a true energy minimum was found (i.e., forces are close to zero). On the ADwT metric, SpinConv outperforms other approaches (53.62% averaged across splits). On the AFbT metric, DimeNet++-large outperforms SpinConv (21.82% vs. 16.67%), but is more than \u223c3x slower (814.6h vs. 263.2h) during inference. SpinConv outperforms all other models. 3.2 MD17 The MD17 dataset [4, 5] contains molecular dynamic simulations for eight small molecules. Two training datasets are commonly used, one containing 1k examples and another containing 50k examples. We found the 1k training dataset to be too small for the SpinConv model, and may be more appropriate for approaches that incorporate prior chemistry knowledge, such as hand-coded features or force \ufb01elds [4, 30]. The 50k dataset provides signi\ufb01cantly more training data, but the remaining validation and test data are highly similar to those found in training, and may not guarantee independent samples in the test set[6]. Nevertheless, we report results on MD17 for comparison to prior work on the molecular dynamics task. Research in this domain would greatly bene\ufb01t from the generation of a larger dataset. Results are shown in Table 6. SpinConv is on par or better for 7 of the 8 molecules when compared to DimeNet [15]. Both SpinConv and DimeNet perform well with respect to the GDML [4] and PhysNet [30] models that take advantage of domain-speci\ufb01c information. Given the smaller dataset size, the SpinConv model uses a reduced 8x8 grid-based spherical representation. Other model parameters are the same as previously described. 3.3 QM9 Our \ufb01nal set of results are on the popular QM9 dataset [22] that tests the prediction of numerous properties for small molecules. While the SpinConv model was designed to estimate energies and per-atom forces, we may use the same model to predict other proprieties. Results are shown in Table 7 on a random test split for an energy-centric 8x8 grid-based SpinConv model. The results 9 \fof DimeNet++ and the recent SphereNet[17] outperform those of others. However, DimeNet++, SphereNet and SpinConv perform well with respect to other approaches across many properties. 4 Related work A common approach to estimating molecular and atomic properties is the use of GNNs [26, 9, 13, 25, 27, 33, 20, 15] where nodes represent atoms and edges connect neighboring atoms. One of the \ufb01rst approaches for force estimation was SchNet [25], which computed forces using only the distance between atoms without the use of angular information. Unlike previous approaches that used discrete distance \ufb01lters [33], SchNet proposed the used of differentiable edge \ufb01lters. This enabled the construction of an energy-conserving model for molecular dynamics that estimates forces by taking the negative gradient of the energy with respect to the atom positions [4]. DimeNet extended this approach to also represent the angular information between triplets of atoms [15, 14]. The more recent SphereNet further extends this by capturing dihedral angles [17]. SpinConv is able to model relative angular relationships between all neighboring atoms, and not just triplets of atoms, due to the use of the spin convolutional \ufb01lter. In parallel to invariant models, rotational equivariant networks are explored in depth by [31, 2, 1, 29, 24]. This was accomplished by decoupling the network-fed invariant information (distance), from the equivariant information (distance vector), followed by the careful combination via tensor products. The energy-centric SpinConv model is invariant to rotations due to the use of global pooling after the spin convolution. The \ufb01nal force block of the force-centric model is not strictly rotation equivariant, but is encouraged to learn rotation equivariance during training. Another approach to force estimation is to directly regress the forces as an output of the network. This doesn\u2019t enforce energy conservation or rotational equivariance, but as shown by ForceNet [12], such models can still produce accurate force estimates. Numerous approaches incorporate more domain speci\ufb01c information into machine learning models. These include GDML [4] and PhysNet [30] that use handcrafted features and force-\ufb01elds respectively. OrbNet [20] is a hybrid approach that utilizes proprietary orbital features that improves accuracy while achieving signi\ufb01cant ef\ufb01ciency gains over DFT. While these approaches can lead to improved accuracy, they typically result in increased computational expense over ML models. 5 Discussion While the SpinConv model demonstrates improved performance, it still has signi\ufb01cant limitations. Most notable is the force and energy estimates are still signi\ufb01cantly lower than desired for practical applications. Further research is needed to improve accuracies, so that machine learning models can be widely adopted. Currently, the SpinConv model does not take advantage of domain speci\ufb01c information. Results could be signi\ufb01cantly improved, especially for smaller datasets (e.g., MD17 1k), if more domain information was integrated into the model [4, 30, 20]. The use of the spin convolution becomes increasingly expensive as the size of the \ufb01lter increases, since the number of convolutions is equal to the longitudinal dimension of the \ufb01lter. If \ufb01lters of higher resolution are needed, more computationally ef\ufb01cient approaches may be required. In conclusion, we propose the SpinConv model that effectively captures the relative angular information of neighboring atoms, while maintaining the invariance of the energy estimation with respect to system rotations. This is enabled by utilizing a spin convolution over a spherical representation in a per-edge local reference frame, followed by global pooling. Two model variants are proposed based on whether energy conservation is enforced. Results demonstrate state-of-the-art results on the OC20 dataset, and strong results on both the MD17 and QM9 datasets. 6 Societal Impact This work is motivated by the problems we face due to climate change [35], many of which require innovative solutions to reduce energy usage and replace traditional chemical feedstocks with renewable alternatives. For example, one of the most energy intensive chemical processes is the development of new electrochemical catalysts for ammonia fertilizer production that helped to feed the world\u2019s growing population during the 20th century [11]. This is also an illustrative example of possible 10 \funintended consequences as advancements in chemistry and materials may be used for numerous purposes. As ammonia fertilization increased in use, its overuse in today\u2019s farming has led to ocean \u201cdead zones\u201d and its production is very carbon intensive. Knowledge and techniques used to create ammonia were also transferred to the creation of explosives during wartime. We hope to steer the use of ML for atomic simulations to societally-bene\ufb01cial uses by training and testing our approaches on datasets, such as OC20, that were speci\ufb01cally designed to address chemical reactions useful for addressing climate change."
+                }
+            ],
+            "Nima Shoghi": [
+                {
+                    "url": "http://arxiv.org/abs/2310.16802v2",
+                    "title": "From Molecules to Materials: Pre-training Large Generalizable Models for Atomic Property Prediction",
+                    "abstract": "Foundation models have been transformational in machine learning fields such\nas natural language processing and computer vision. Similar success in atomic\nproperty prediction has been limited due to the challenges of training\neffective models across multiple chemical domains. To address this, we\nintroduce Joint Multi-domain Pre-training (JMP), a supervised pre-training\nstrategy that simultaneously trains on multiple datasets from different\nchemical domains, treating each dataset as a unique pre-training task within a\nmulti-task framework. Our combined training dataset consists of $\\sim$120M\nsystems from OC20, OC22, ANI-1x, and Transition-1x. We evaluate performance and\ngeneralization by fine-tuning over a diverse set of downstream tasks and\ndatasets including: QM9, rMD17, MatBench, QMOF, SPICE, and MD22. JMP\ndemonstrates an average improvement of 59% over training from scratch, and\nmatches or sets state-of-the-art on 34 out of 40 tasks. Our work highlights the\npotential of pre-training strategies that utilize diverse data to advance\nproperty prediction across chemical domains, especially for low-data tasks.\nPlease visit https://nima.sh/jmp for further information.",
+                    "authors": "Nima Shoghi, Adeesh Kolluru, John R. Kitchin, Zachary W. Ulissi, C. Lawrence Zitnick, Brandon M. Wood",
+                    "published": "2023-10-25",
+                    "updated": "2024-05-06",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Computing atomic properties accurately and efficiently for a vast array of molecules and materials is crucial for a range of applications, from drug discovery (Chan et al., 2019; Deng et al., 2022) to catalyst design (Zitnick et al., 2020). Currently, the quantum chemistry method Density Functional Theory (DFT) is commonly employed for atomic property calculations. Unfortunately, DFT\u2019s use is limited by its significant computational expense, which can range from hours to days for certain calculations. Machine learning (ML) potentials, which approximate or augment DFT, are capable of reducing the computational cost by orders of magnitude (Behler, 2016). In recent years, much progress has been made towards this goal (Kolluru et al., 2022b), fueled in part by the release of large and diverse DFT-generated datasets for training ML models. While these datasets are incredibly useful, they are also extremely expensive to generate, e.g., \u223c400 million CPU hours for the Open Catalyst 2020 dataset (OC20) (Chanussot et al., 2021). As a consequence, it is impractical to create a large dataset for every specific chemistry problem of interest. Similarly, it is non-ideal to train a model from scratch for all use cases, which is common practice currently. Foundation models (FMs) \u2014 large pre-trained models that can be fine-tuned for various tasks \u2014 have achieved remarkable success in domains such as natural language processing (NLP) and computer vision (CV), especially when fine-tuned on low-resource downstream tasks. Several key factors have enabled this effectiveness: (1) the availability of massive datasets, (2) the development of widely adopted pre-training strategies, and (3) the establishment of diverse benchmarks to rigorously assess the performance of these fine-tuned models. Despite the availability of large DFT-labeled datasets (e.g., OC20) and the existence of a wide and diverse range of downstream tasks (e.g., QM9 (Ruddigkeit et al., 2012b), MatBench (Dunn et al., 2020)), the adoption of pre-training in ML 1 arXiv:2310.16802v2  [cs.LG]  6 May 2024 \fPublished as a conference paper at ICLR 2024 Data Normalization & Temperature Sampling Pre-train Backbone Model QM9 Multi-Task Learning SPICE MD17 MD22 QMOF Matbench Backbone Model MD22 E F QMOF EG QM9 Cv SPICE E F Matbench Eform MD17 E F Fine-tune Knowledge Transfer OC20 E F OC22 E F ANI1x E F Trans1x E F Small Molecules Large Molecules Materials Small Molecules Catalysis OC20 OC22 Trans1x ANI1x Learned Representations Figure 1: An overview of the Joint Multi-domain Pre-training (JMP) method. Left: JMP\u2019s pretraining setup, where a single model is simultaneously trained on set of diverse pre-training datasets using multi-task learning. Center: JMP\u2019s fine-tuning process, where the pre-trained JMP backbone is equipped with new prediction heads and trained on downstream tasks. Right: t-SNE visualizations of JMP\u2019s node-level (\u02dc h) embeddings for randomly selected structures from all datasets. for atomic property prediction has been noticeably less prevalent. This under-utilization becomes evident when noting that most of the state-of-the-art (SOTA) results on downstream tasks come from models trained from scratch. More specifically, prior to our work, all previous SOTA results on the rMD17, MD22, SPICE, and MatBench datasets come from models trained form scratch. For QM9, models trained from scratch hold SOTA status for 7 of the 12 targets. In total, out of the 40 total tasks explored in this work\u2019s evaluation benchmark, models trained from scratch hold the previous SOTA on 34 tasks. At its core, the challenge of pre-training for atomic property prediction lies in the complexity and diversity of the underlying chemical space. Target applications vary from drug design to catalysis, the data ranges from small molecules with only 4 atoms to periodic crystals with hundreds, and even the properties of interest for each application vary from various energies to forces to phonon peaks. Furthermore, the nature of atomic properties imposes a unique set of challenges. Unlike in NLP or CV, where the data is often discrete and finite, atomic properties are continuous and can span several orders of magnitude. This requires models to be robust to outliers and capable of predicting highly variable outputs. Further, existing pre-training strategies (e.g., Zaidi et al. (2022); Zhou et al. (2023b)) are designed with equilibrium systems in mind and are not directly applicable to non-equilibrium systems, which are common in DFT datasets (e.g., over 99.7% of OC20\u2019s training data comprises of non-equilibrium structures). These challenges motivate the need for a flexible and generalizable pre-training strategy that can be adapted to different applications and datasets. In this work, we introduce Joint Multi-domain Pre-training (JMP), a supervised pre-training strategy tailored to the challenges and opportunities of machine learning for atomic modeling. JMP concurrently trains over 120 million diverse equilibrium and non-equilibrium atomic structures by framing each chemical domain as a separate pre-training task in a multi-task framework. This large-scale pre-training enables learning generalizable representations of atomic interactions. The contributions of our work are summarized as follows: First, we introduce the JMP method, shown in Figure 1, and demonstrate its powerful generalization ability by evaluating its fine-tuning performance across a diverse benchmark suite spanning small molecules, large molecules, and materials. Our results show that JMP consistently outperforms training from scratch and sets or matches the state-of-the-art on 34 out of the 40 fine-tuning benchmarks. Second, we show that JMP enables efficient scaling to larger models that would normally overfit if trained from scratch on small datasets. Pre-training acts as a strong regularizer, allowing us to train a 235M parameter model that sets new state-of-the-art performance on multiple low-data benchmarks. Finally, we conduct a detailed analysis of JMP\u2019s 2 \fPublished as a conference paper at ICLR 2024 computational requirements. While expensive upfront, we show JMP\u2019s pre-training cost is recovered by enabling over 12x faster fine-tuning compared to training from scratch. By pre-training large models on diverse chemical data, we believe JMP represents an important step towards the goal of a universal ML potential, and that the continued growth of available data and compute power will only improve JMP\u2019s ability to learn transferable atomic representations. 2 RELATED WORK Machine learning potentials: There has been significant progress in developing ML models for atomic property prediction. Initial approaches focused on descriptor-based methods, where these descriptors were hand-fitted physically meaningful analytical functions (Gonz\u00e1lez, 2011; Sundius, 2002; Dinur and Hagler, 1991). These functions were incorporated into gaussian process models (Chmiela et al., 2017) or neural networks (Behler and Parrinello, 2007). Recent advances in graph neural networks (GNNs) have shown to be a promising approach for these tasks, surpassing descriptorbased methods (Gasteiger et al., 2020; Sch\u00fctt et al., 2017; Batzner et al., 2021; Batatia et al., 2022) on multiple benchmarks across the atomic domains of small molecules, catalysts, and bulk materials. While much progress has been made, it remains difficult for a single model to perform well across all chemical domains. Pretraining and transfer learning on 3D atomic systems: The concept of transfer learning, where representations are learned on one dataset and transferred to another, has been successfully applied to a number of atomic modeling tasks (Kolluru et al., 2022a; Cai et al., 2020; Tsubaki and Mizoguchi, 2021; Smith et al., 2018). However, most of the focus in this area has been on transferring representations within the same chemical domain with a limited amount of pre-training data (Smith et al., 2019; Yamada et al., 2019; Pesciullesi et al., 2020). There are beginning to be more dedicated works on pre-training (Zhu et al., 2022; Liu et al., 2021; Jiao et al., 2022; Zhou et al., 2023b), but most do not explore generalization across multiple chemical domains. Many of these works focus on self-supervised pre-training on molecular graphs and/or 3D atomic structures. Recent self-supervised methods have focused on denoising methods (Song et al., 2020) applied to equilibrium structures \u2014 i.e. the per atom forces are close to zero (Zaidi et al., 2022; Feng et al., 2023b; Liu et al., 2022). The original formulation of denoising equilibrium structures is applicable to less than 1% of our training data because most of the atomic properties data is non-equilibrium. This is an active area of research and since the beginning of our present work, alternative formulations that could apply to non-equilibrium data have started to emerge (Feng et al., 2023a; Zheng et al., 2023). 3 DATASETS We separate the atomic space into four domains for the purposes of this manuscript including, small molecules (1-20 atoms), large molecules (more than 20 atoms), materials, and catalysis (contains material surfaces with molecules). Each dataset sample contains a 3D atomic structure (positions and atomic numbers) and a set of atomic properties. The atomic properties can be either node-level (e.g., forces) or graph-level (e.g., energy). The datasets are summarized in Table 1, with additional information, including details on train, validation, and test splits, in Appendix H. To study the ability of pre-trained models to generalize across domains and tasks, we only pre-train on small molecule and catalysis datasets, and fine-tune on small molecule, large molecule, and materials datasets. Our pre-training datasets include the ANI-1x (Smith et al., 2020) and Transition1x (Schreiner et al., 2022) small molecule datasets and the OC20 (Chanussot et al., 2021) and OC22 (Tran et al., 2022) catalysis datasets. These datasets were chosen due to their diversity and large size. The combined pre-training dataset contains over 120M training examples with energy and force labels, with the majority of the data (> 99%) coming from non-equilibrium structures. Due to the difference in underlying DFT theory and software used across the datasets, we utilize different prediction heads for each dataset. We also use a per-dataset linear referencing scheme for the energies. For fine-tuning, we use smaller datasets from three domains to evaluate how pre-trained models perform in similar (small molecule) and unseen domains (large molecule and materials). These datasets may contain in-distribution (ID) (i.e., energies and forces) or out-of-distribution (OOD) labels (e.g., QM9\u2019s \u2206\u03f5). 3 \fPublished as a conference paper at ICLR 2024 Dataset Domain Labels Elements Avg size Train Set Description Pretraining Datasets OC20 Catalyst E, F 55 \u223c73 (7-225) 100M Catalyst relaxations OC22 Catalyst E, F 51 \u223c80 (17-228) 8M Oxide catalyst relaxations ANI-1x Small Molecule E, F H, C, N, O \u223c15 (4-63) 2M MD simulations Transition-1x Small Molecule E, F H, C, N, O \u223c14 (4-23) 10M Reactions database Finetuning Datasets Matbench Materials (OOD) ID / OOD 84 \u223c30 (4-444) \u223c600\u2013130k Material properties QMOF Materials (OOD) OOD 77 \u223c109 (17, 500) 10k MOF properties MD17 Small Mols. (ID) ID H, C, N, O \u223c13 (9-21) 1k MD simulation QM9 Small Mols. (ID) ID / OOD H, C, N, O \u223c18 (3-29) \u223c130k QM properties SPICE Large Mols. (OOD) ID H, C, N, O, S \u223c46 (26-96) 1300, \u223c34k MD simulations MD22 Large Mols. (OOD) ID H, C, N, O \u223c67 (42-370) \u223c600\u20138k MD simulations Table 1: Summary of datasets and their properties, including the domain, target labels, atomic elements present, their sizes and a brief description. 4 JOINT MULTI-DOMAIN PRE-TRAINING Joint Multi-domain Pre-training (JMP), shown in Figure 1, is based on the intuition that pre-training on a diverse set of chemical domains should lead to better representation learning and thus better generalization through fine-tuning. The pre-training task is framed as a multi-task supervised learning problem, where each label of each pre-training dataset is treated as a separate task. This allows us to pre-train a single backbone model on multiple chemical domains and labels simultaneously. Notation: We use the following notation throughout this section. Let D = {D1, . . . , DM} be the set of M datasets that we pre-train on. Each dataset, Di, is a set systems (e.g., molecules or crystals), where each system is a tuple of atomic numbers (Z), atomic positions (R), and target (i.e., ground-truth) energy ( \u02c6 E) and forces ( \u02c6 F). For a given mini-batch of B systems, Wb is the index of the dataset that system b \u2208B belongs to, and Nb is the number of atoms in system b. Model Architecture: Our goal in this work is to design model-agnostic strategies for supervised pre-training. For our backbone model architecture, we chose GemNet-OC (Gasteiger et al., 2022) for its effectiveness across a wide spectrum of chemical domains as well as at large scales (Sriram et al., 2022). GemNet-OC is a message-passing neural network that computes a node representation hi for each atom i and an edge representation mij for pairs of nearby atoms, i and j. Using these representations, prediction heads compute desired target properties. System-level scalar predictions, such as energy, are computed by summing the node representations, E = PN i=1 MLP(hi). Nodelevel vector predictions, such as forces, are computed by summing the edge direction unit vectors, weighted by the edge representations, Fi = PN j=1 (MLP(mij) \u00b7 \u02c6 rij). During pre-training, we compute forces using a direct equivariant block, similar to Klicpera et al. (2021)\u2019s model setup for the OC20 dataset. This is for two reasons: (1) direct force prediction is much more computationally efficient than gradient-based force prediction, as the latter needs to perform a secondary backward pass to compute the gradient of the energy with respect to the atomic positions and (2) previous works (Gasteiger et al., 2022) have shown that for larger datasets, direct force prediction shows much faster convergence while producing similar converged accuracies to gradient-based force prediction. 4.1 MULTI-TASK PRE-TRAINING In the multi-task setting, each dataset has its own energy and force prediction heads, as shown in Figure 1 (left). This allows us to train a single model on multiple datasets simultaneously. In the following sections, we describe each of these imbalances and our proposed solutions in detail. Data Normalization: When pre-training on multiple datasets, we first need to normalize the targets to make sure they are on a common scale across datasets. Since our pre-training task is energy and force prediction, for each dataset we first linearly reference the total energies and then normalize them to unit Gaussian distributions. We normalize the forces by dividing them by component-wise RMS force. This puts the energy and forces for each dataset on a common scale. Dataset Size Imbalance: Our pre-training datasets vary greatly in size, from 2 million to 100 million training samples, for a total of 120M samples. To maintain a proper balance between the total contribution of large, high-resource and small, low-resource pre-training datasets and to prevent overfitting to high-resource datasets and underfitting on low-resource datasets, we use temperature 4 \fPublished as a conference paper at ICLR 2024 sampling (Devlin et al., 2018) during batch construction. Specifically, we sample each dataset i with probability pi \u221d( |Di| P j |Dj|)1/T , where |Di| is the number of samples in dataset i and T is the temperature hyperparameter. Inspired by Shaham et al. (2023), which shows that T = 2 optimizes model performance on high and low-resource languages for large models, we use T = 2. System Size Imbalance: The number of atoms per system varies greatly across our pre-training datasets. For example, Transition-1x has 14 atoms per system on average, while OC22 has 80 atoms per system on average. The naive loss reduction method shown in the non-teal terms of Equation (1), which is the default behavior of most machine learning libraries, computes an atom-level force loss and then averages the force loss across all atoms in the batch. This leads to datasets with more atoms per system dominating the force loss. To address this issue, we propose a structure-wise loss reduction strategy which first computes the average force loss for each system and then computes the average force loss across all systems. This ensures that the relative importance of the force loss is roughly equal across datasets, regardless of the number of nodes per system. In Equation (1), the updates to the naive formulation of the loss function are shown in teal and removed terms are red. This simple change leads to a significant improvement in model performance, as shown in Section 5.1. L = 1 B B X b=0 h \u03bb(Wb) E \f \f \f \u02c6 Eb \u2212Eb \f \f \f i | {z } Energy Loss (LE) + 1 B 1 P b Nb B X b=0 \" 1 Nb \u03bb(Wb) F Nb X i=0 \r \r \r \u02c6 Fb,i \u2212Fb,i \r \r \r 2 # | {z } Force Loss (LF ) (1) Loss Imbalance Within a Single Dataset: In the single-dataset setting, \u03bbE and \u03bbF are typically tuned by grid search, but this approach is not feasible in the multi-dataset setting, as there 2 \u00b7 M hyperparameters to tune, and changing one hyperparameter affects the optimal values of the others. Therefore, we need a simple heuristic to determine the loss coefficients for each dataset that provides a reasonable balance between the energy and force losses. Inspired by Tran et al. (2022)\u2019s size invariant force loss, which computes a dynamic \u03bbF based on the number of atoms in each system of the input batch, we fix \u03bb(i) E = 1 and \u03bb(i) F = \u27e8N\u27e9Di, where \u27e8N\u27e9Di is the average number of atoms per system in the ith dataset, Di. This provides a reasonable balance between energy and force loss within each dataset. Fine-Tuning: Once we have a fully pre-trained model, we can fine-tune it on downstream tasks. Our fine-tuning procedure is very similar to other fine-tuning procedures in the machine learning literature (Devlin et al., 2018; Zaidi et al., 2022): We discard the pre-training prediction heads, add new randomly initialized prediction heads for the downstream task, and fine-tune the entire model on the downstream task. This procedure is illustrated in Figure 1 (middle). For fine-tuning tasks with force labels, we have the option of using the directly computed forces (i.e., using the direct equivariant block) or computing the forces by taking the gradient of the energy with respect to the atomic positions. Our initial experiments showed that JMP works well with both methods. In our evaluations, however, we chose to compute forces conservatively by taking the gradient of the energy with respect to the atomic positions, as this is the standard approach in the literature. 5 EXPERIMENTS We benchmark our pre-trained models on a diverse set of atomic ML tasks. In previous related works, evaluations are commonly restricted to downstream datasets that align closely with the pre-training dataset\u2019s domain (Zaidi et al., 2022; Zhou et al., 2023a). We posit that true success in pre-training models for atomic machine learning tasks requires adeptness at out-of-domain extrapolation. To test this hypothesis, our benchmark uniquely spans across diverse domains including small molecules (QM9 and rMD17), large molecules (MD22 and SPICE), and materials (MatBench and QMOF). We compare our fine-tuned models (JMP) to randomly initialized models trained from scratch (GNOC) to demonstrate the effectiveness of JMP. We also compare to previous state-of-the-art (SOTA) models where available. For each task, we present results for both a small (\u223c30M parameters, labeled with the -S suffix) and large (\u223c230M parameters, labeled with the -L suffix) pre-trained model to probe the impact of the model size. These two variants utilize the GN-OC Base and Large backbone architectures from Gasteiger et al. (2022), respectively. Finally, we conduct ablation studies to understand the impact of various components of JMP. More information on the datasets used for 5 \fPublished as a conference paper at ICLR 2024 (c) Fine-tuned Large vs Scratch Large (b) Fine-tuned Large vs Fine-tuned Small 75 50 25 0 25 50 75 Relative Improvement (%) Fine-tuning T asks (a) Scrach Large vs Scratch Small QM9 MD17 MD22 SPICE Matbench QMOF Figure 2: Relative performance improvement across all tasks of all fine-tuning datasets, in percentages, of (a) Scratch Large (GN-OC-L) over Scratch Small (GN-OC-S), (b) Fine-tuned Large (JMP-L) over Fine-tuned Small (JMP-S), and (c) Fine-tuned Large (JMP-L) over Scratch Large (GN-OC-L). GN-OC shows poor scaling to large models, a clear sign of overfitting, whereasJMP reverses this, exhibiting much improved scaling dynamics. JMP also consistently outperforms GN-OC across all domains, datasets, and targets. The shaded rectangles indicate the average relative performance across all tasks for each dataset. The exact percentages can be found in Appendix C.1 pre-training and fine-tuning can be found in Section 3. Details on the pre-training and fine-tuning setup, such as the optimizers, learning rate schedules, and early stopping information, can be found in Appendix F. Exact hyperparameters can be found in Appendix J. 0 20 40 60 Relative Improvement (%) HOMO LUMO GNS-TAT-NN ET-NN ET-OREO JMP-S JMP-L Figure 3: Relative improvement, over training from scratch, of different pre-training methods on QM9\u2019s \u03f5LUMO and \u03f5HOMO. Common Observations: We begin by highlighting some common observations across all experiments. First, when training from scratch, GN-OC-L performs 8% worse on average than GN-OC-S, as shown in Figure 2 (a). This is a clear indication of overfitting and has been consistently observed in low-data regimes (Gasteiger et al., 2022). Second, this problem of overfitting is nearly eliminated by JMP, illustrated in Figure 2 (b). On average, JMP-L exhibits an impressive 21% relative performance gain over JMP-S. This indicates that the JMP training procedure is able to effectively leverage the additional capacity of the large model, even in low-data regimes. Third, we observe that pre-training with JMP elevates performance across all domains, datasets, and tasks (Figure 2 (c)), with an average relative improvement of 59% for JMP-L over GN-OC-L. Results on Small Molecules QM9 and rMD17: For each target of QM9 (Wu et al., 2018), we fine-tune a dedicated model using a simple prediction head with sum pooling for all targets. For R2, we use the same prediction head formulation as Th\u00f6lke and De Fabritiis (2022). Our results can be found in Table 2 compared against previous SOTA works (Liao and Smidt, 2022; Batatia et al., 2022; Musaelian et al., 2023; Feng et al., 2023a; Zaidi et al., 2022). With the sole exception of R2, our JMP-L model achieves SOTA results on all QM9 targets. For the R2 target, a similar phenomenon has been observed in previous pre-training works (Zaidi et al., 2022) where the benefits of using pre-trained models are not as pronounced. In addition to their impressive performance, our JMP-S and JMP-L models demonstrate a large improvement relative to their scratch-trained counterparts. Figure 3 compares this relative improvement \u2014 measured on the \u03f5LUMO and \u03f5HOMO targets \u2014 to other SOTA pre-training and transfer learning methods for QM9. As shown, JMP outperforms all previous methods by a significant margin. This is a strong signal that our pre-training approach is effective at learning generalizable representations for small molecules. We also report additional pretraining comparisons on all our finetuning benchmarks with a pre-trained model from (Zaidi et al., 2022) and demonstrate significant improvements on all tasks in Appendix A. Data overlap: Due to the limited complexity of small molecules, there is some data overlap between our pre-training datasets (ANI-1x and Transition-1x) and QM9. To check the impact of this overlap on our results, we evaluate the fine-tuning performance of our JMP-L on a QM9 dataset that excludes 6 \fPublished as a conference paper at ICLR 2024 the overlapping molecules. Using molecular compositions to identify overlaps, we observe that the exclusion of overlapping molecules has a negligible impact on our results (see Appendix I). Target (Units) TorchMDEquiMACE Allegro Pretrained Pretrained GN-OCGN-OCJMPJMPNet former ET-OREO GNS+TAT+NN S L S L \u00b5 (D) 0.011 0.011 0.015 0.016 0.020 0.023 0.010 0.008 \u03b1 (a3 0) 0.059 0.046 0.038 0.040 0.052 0.056 0.037 0.032 \u03b5HOMO (meV ) 20.3 15.0 22.0 16.8 14.9 21.8 22.7 11.1 8.8 \u03b5LUMO (meV ) 18.6 14.0 19.0 14.5 14.7 17.3 18.6 10.8 8.6 \u2206\u03b5 (meV ) 36.1 30.0 42.0 26.4 22.0 38.5 40.6 23.1 19.1 R2 (a2 0) 0.033 0.251 0.210 0.440 0.210 0.171 0.200 0.163 ZPVE (meV ) 1.8 1.3 1.2 1.0 1.2 1.2 1.0 0.9 U0 (meV ) 6.2 6.6 4.1 4.7 5.8 7.2 9.4 3.3 2.9 U (meV ) 6.4 6.7 4.1 4.4 5.8 6.9 9.7 3.3 2.8 H (meV ) 6.2 6.6 4.7 4.4 5.8 7.3 8.7 3.3 2.8 G (meV ) 8.3 7.6 5.5 5.7 6.9 8.1 9.2 4.5 4.3 C\u03bd (Cal/MolK) 0.026 0.023 0.021 0.020 0.024 0.024 0.018 0.017 Table 2: MAE test split results on all targets of the QM9 dataset. SOTA results are bolded. For rMD17, we compute forces by taking the negative gradient of the energy with respect to the atomic positions. Table 3 shows our force prediction results on the rMD17 dataset. Similarly to QM9, we observe that JMP consistently outperforms GN-OC across all rMD17 targets. Our JMP-L model achieves state of the art performance in 5 molecules and is very competitive on the rest. Appendix B.1 also shows that JMP achieves SOTA on 6/10 targets on the few-shot 50-sample subset of rMD17. Molecules MACE Allegro GN-OC-S GN-OC-L JMP-S JMP-L Aspirin 6.6 7.3 24.3 24.7 6.7 5.1 Benzene 0.3 0.2 1.0 1.0 0.7 0.3 Ethanol 2.1 2.1 13.0 13.3 2.8 2.0 Malonaldehyde 4.1 4.1 21.1 25.7 5.3 4.0 Naphthalene 1.6 0.9 5.6 5.7 2.2 1.4 Salicylic acid 3.1 2.9 14.7 15.1 4.6 3.4 Toluene 1.5 1.8 6.8 7.2 2.3 1.5 Uracil 2.1 1.8 12.0 12.9 4.0 2.5 Paracetamol 4.8 4.9 17.3 18.4 5.3 4.0 Azobenzene 3.0 2.6 11.1 11.4 4.5 3.3 Table 3: Force MAE results in meV/\u00c5 on the test split of the rMD17 dataset. SOTA is bolded. Results on Materials MatBench and QMOF: In the materials domain, we fine-tune on the MatBench (Dunn et al., 2020) and QMOF datasets (Rosen et al., 2021). For MatBench, we evaluated all regression tasks that utilize a 3D structure as an input and compared them with competitive models on the leaderboard (De Breuck et al., 2021; Ruff et al., 2023). For QMOF, we predict the band gap target on a 10k split, similarly to Kang et al. (2022); Cao et al. (2023). We use mean pooling for all experiments, except MatBench\u2019s phonons, which is the measure of frequency of the highest frequency optical phonon mode peak and thus uses max pooling. Our results can be found in Table 4. We observe that JMP-L achieves SOTA performance across QMOF and on all MatBench tasks. These two datasets contain diverse out-of-domain chemical structures (materials) and out-of-domain target labels (i.e., not energies and forces), relative to the pre-training datasets. JMP\u2019s impressive performance is yet another positive signal indicating that JMP is learning generalizable representations. Materials (Units) MODNet coGN GN-OC-S GN-OC-L JMP-S JMP-L (fold0 / mean) (fold0 / mean) (fold0) (fold0) (fold0 / mean) (fold0 / mean) JDFT2D (meV/atom) 25.55 / 33.20 22.25 / 37.17 26.19 25.34 20.72 / 30.16 23.12 / 29.94 Phonons (cm\u22121) 34.77 / 34.28 32.12 / 29.71 93.45 88.74 26.6 / 22.77 21.28 / 20.57 Dielectric (unitless) 0.169 / 0.271 0.178 / 0.309 0.225 0.211 0.133 / 0.252 0.119 / 0.249 Log GVRH (log10(GP A)) 0.073 / 0.073 0.068 / 0.069 0.082 0.082 0.06 / 0.062 0.057 / 0.059 Log KVRH (log10(GP A)) 0.054 / 0.055 0.052 / 0.054 0.061 0.063 0.044 / 0.046 0.045 / 0.045 Perovskites (eV/unitcell) 0.093 / 0.091 0.027 / 0.027 0.045 0.045 0.029 / 0.028 0.026 / 0.026 MP Gap (eV ) 0.215 / 0.220 0.153 / 0.156 0.228 0.235 0.119 / 0.121 0.089 / 0.091 MP E Form (meV/atom) 40.2 / 44.8 17.4 / 17 31.4 33.1 13.6 / 13.3 10.3 / 10.1 PT CGCNN PT MOFTransformer QMOF 0.28 0.27 0.25 0.24 0.18 0.16 Table 4: MAE test split results on different targets in the materials domain. SOTA is bolded. Results on Large Molecules MD22 and SPICE: To further investigate the impact of pre-training on unseen domains, we evaluate two large molecule datasets, MD22 (Chmiela et al., 2023) and SPICE (Eastman et al., 2023) and compare our results to the previous SOTA (Kovacs et al., 2023; Li et al., 2023). For SPICE, we only use the large molecule sub-tasks: solvated amino acids and dipeptides. 7 \fPublished as a conference paper at ICLR 2024 Molecule sGDML MACE Allegro GN-OC-S GN-OC-L JMP-S JMP-L Ac-Ala3-NHMe 34.55 3.80 4.63 5.07 6.27 2.64 1.92 DHA 32.41 2.80 3.17 2.87 3.95 2.01 1.37 Stachyose 29.24 3.80 4.21 2.22 3.85 2.69 1.73 AT-AT 29.97 4.30 4.13 5.38 5.96 3.02 1.98 AT-AT-CG-CG 30.48 5.00 5.55 5.80 5.62 3.28 2.11 Buckyball Catcher 29.57 3.70 10.35 8.20 3.08 2.26 Double Walled Nanotubes 22.68 12.00 11.20 9.61 8.36 6.17 Solvated Amino Acids 22.14 28.64 5.71 4.75 Dipeptides 8.78 10.68 4.71 3.64 Table 5: Force MAE results in meV/\u00c5 on test splits of large molecule datasets. SOTA is bolded. Similar to rMD17, we compute forces by taking the negative gradient of the energy with respect to the atomic positions. However, for MD22\u2019s Buckyball Catcher and Double-Walled Nanotubes, we were unable to fit these large structures in memory with using gradient-based force predictions; therefore, we used direct force prediction heads instead. Our results can be found in Table 5. Once again, our model demonstrates SOTA results across all molecules of MD22 and all tasks of SPICE. 5.1 ABLATION STUDIES Our ablations demonstrate the impact of various changes to JMP on the downstream fine-tuning performance. We performed pre-training experiments including dataset sampling strategies, loss formulation, and regularization strategies, and observed their impact on fine-tuning. Given the computational cost of training models on the full pre-training dataset, ablation experiments were conducted on a scaled-down version of the full pre-training dataset containing a randomly selected \u223c2.5M examples. All pre-training models are trained for 10 epochs. Similarly, fine-tuning for these experiments was run on only one task from each of the fine-tuning datasets (MD17: Aspirin, MD22: Stachyose, QM9: \u2206\u03f5, MatBench: MP E Form, QMOF: Band Gap, and SPICE: Solvated Amino Acids). Additional ablations, including using fully balanced (T = \u221e) sampling, threshold regression loss for energies and forces, and automatic task weighting strategies such as PCGrad (Yu et al., 2020) are explored in Appendix B. Table 6 shows the mean improvement, relative to the base, across all the fine-tuning tasks described above. A summarized insight of each ablation study follows: Ablations E[RI](%) Base (Temperature 1.0) [B] 0% B + Temperature 2.0 [T2] 2.2% B + Temperature \u221e[T\u221e] 2.6% T2 + SW Loss Averaging [SWL] 7.7% SWL + Weight Decay [WD] 11.4% SWL + Dropout [DO] 11.4% WD + Edge Dropout [ED] 13.2% WD + ED + EMA Weights [EMA] 12.4% EMA + OC20 Only [OC20] -9.9% Table 6: Ablation results demonstrating the mean relative improvements of each method relative to the base method (B), averaged over ablation subsplits. Base (B): Base refers to the naive implementation of a multi-task pre-training model without temperature sampling, structure-wise loss reduction, or additional regularization. This model serves as the baseline for comparison. Temperature Sampling: Temperature-based sampling with T = 2 provides a moderate improvement, while higher values (e.g., T = \u221e) show diminishing returns. This is consistent with Shaham et al. (2023), which shows that for large-enough models, T = 2 provides ideal performance across both low and high resource datasets. Structure-Wise Loss Reduction (SWL): The application of the structure-wise loss reduction strategy proved to be a substantial improvement on the model\u2019s performance, with T2 + SWL offering a 7.7% improvement over B. Weight Decay (WD): Elevating the weight decay regularization parameter to 0.1 (from the default 0.01) brings the collective improvement to 11.4% over B. Dropout (DO): Using dropout with p = 0.1 on atom update layers yielded similar uplift to WD. Edge Dropout (ED): For this ablation, we drop p = 0.1 of the edges at every step. We then scale the embeddings of remaining edges by a factor of 1 1\u2212p. This yielded a small improvement over WD, increasing the collective improvement to 13.2% over B. Exponential Moving Average (EMA): Fine-tuning on EMA weights did not improve performance. OC20 Only (OC20): To understand the impact of multi-task pre-training, we trained a model on the OC20 dataset only. We selected a 120M subset of OC20 to match the number of examples in the full JMP pre-training dataset. Note that this means that the dataset used in the OC20 ablation contains 48\u00d7 more data points than the rest of our ablations. Despite this, OC20 performed substantially worse than B, indicating that diverse multi-task pre-training is important for generalization. 8 \fPublished as a conference paper at ICLR 2024 Based on our ablation results the two most important changes from B were SWL and regularization methods like WD, DO, and ED. These results are consistent with Kurin et al. (2022), which demonstrates the effectiveness of regularization in multi-task learning. Our final model integrates temperature-based sampling (T2), structure-wise loss reduction strategy (SWL), an amplified weight decay regularization parameter of 0.1 (WD), edge dropout with p = 0.1 (ED), and EMA despite not showing a performance boost as it has been standard for training GemNet (Gasteiger et al., 2022). 5.2 COMPUTATIONAL COST ANALYSIS QM9 MD17 MD22 SPICE Matbench QMOF 0 50 100 150 Time to Milestone (GPU Hours) JMP GN-OC Figure 4: The number of GPU hours, averaged for each dataset, required to train GNOC-L to convergence and to fine-tune JMP-L to match GN-OC-L\u2019s performance. Overall, fine-tuning JMP-L was able to match GN-OCL\u2019s performance in 1 12 the time. Pre-training JMP-L required significant computational resources, which is typical for foundation model approaches. We pre-trained JMP-L on 128x V100 32GB GPUs for 2 epochs, which took around 34,400 GPU hours in total (see Appendix G for exact training times and CO2 impact). While this is a substantial upfront investment, it enables efficient fine-tuning across a diverse set of downstream tasks. We evaluated JMP-L fine-tuning performance versus training models from scratch (i.e., GN-OC-L). Training GN-OC-L on the downstream tasks until convergence based on our stopping criteria took around 3,300 GPU hours in total across all tasks. In contrast, fine-tuning JMP-L on the same tasks took only around 275 GPU hours total to match the performance of the models trained from scratch. This 12x reduction in compute demonstrates the significant benefits of pre-training. Figure 4 shows this difference in compute requirements, averaged for each fine-tuning dataset. 6"
+                }
+            ],
+            "Abhishek Das": [
+                {
+                    "url": "http://arxiv.org/abs/2108.05185v3",
+                    "title": "Evolution of Curvature in Riemannian Geometry",
+                    "abstract": "In this paper shall we endeavour to substantiate that the evolution of the\nRiemann- Christoffel tensor or curvature tensor can be expressed entirely by an\narbitrary timelike vector field and that the curvature tensor returns to its\ninitial value with respect to change in a particular index. This implies that\nPoincare's recurrence theorem is valid in this cosmological scenario. Also, it\nhas been shown that geodesics can diverge just as they can converge. As is\nostensible, this result indicates the existence the of a point of exclusivity -\nthe opposite of a singularity.",
+                    "authors": "Abhishek Das",
+                    "published": "2021-08-09",
+                    "updated": "2022-05-28",
+                    "primary_cat": "physics.gen-ph",
+                    "cats": [
+                        "physics.gen-ph"
+                    ],
+                    "main_content": "Introduction General theory of relativity, the geometric theory of gravitation is based on the plinth of Riemannian geometry that studies di\ufb00erentiable manifolds. There is a plethora of literature on these two intricately related topics. As is known, there are a multitude of implications resulting from Riemannian geometry when applied to the general relativity the Einstein \ufb01eld equations [1], solutions to such equations in the form of Schwarzchild metric [2], Friedmann-Lemaitre-Robertson-Walker metric [3, 4, 5, 6, 7, 8, 9, 10] and others [11, 12], singularities [13] and black holes [14, 15]. However, in essence, the current paper is not innately related to the general theory of relativity. It is essentially devoted to the geometric aspect of space-time, somewhat in contradistinction to a recent paper that was based on a Hamiltonian formulation [16]. The inception begins with the feasibility of existence of a point of exclusivity (the opposite of a singularity), that is corroborated considering the Raychaudhuri equation [13]. And then, we consider parallel transported vectors and thereby manipulating of their connection with the geometry of space-time we are led to an evolution equation for the Riemann-Christo\ufb00el tensor. As a consequence, several interesting implications are drawn out. Most importantly, it has been argued that with necessary conditions ful\ufb01lled one can indeed discern the existence of a point of exclusivity. 1 \f2 The positive and negative values of the expansion scalar In his seminal paper of 1955 [13], Raychaudhuri had obtained an equation that led to the focusing theorem which in turn substantiated the existence of singularities and black holes, eventually. We would \ufb01nd in the present section that the result derived by Raychaudhuri plays a signi\ufb01cant role in some novel aspects. Let us commence with the expansion scalar (\u03b8) of the Raychaudhuri equation. We know that \u03b8 = \u2202 \u2202t(ln G) where, G = \u221a\u2212g. Also, we know that Ai ;i = 1 \u221a\u2212g[Ai \u2202 \u2202xi \u221a\u2212g + \u221a\u2212g \u2202 \u2202xi Ai] So, considering the coordinate t (i = 0) we may write A0 ;0 = [A0\u03b8 + A0 ,0] Now, since the Kronecker delta function is independent of both the normal and covariant derivatives the last equation can be equivalently expressed as Ak;0 = Ak\u03b8 + Ak,0 (1) Again, we also know that Ak;i = Ak,i \u2212\u0393l kiAl Therefore, using equation (1) we obtain Ak\u03b8 + \u0393l k0Al = 0 (2) or, Ak\u03b8 + \u0393l k0\u03b4k l Ak = 0 Since, the vector \ufb01elds Ak\u2019s are arbitrary, we have the following equation for the expansion scalar \u03b8 + \u0393l k0\u03b4k l = 0 (3) 2 \fOne can make two immediate observations from equation (3). If k \u0338= l, we have \u03b8 = 0 (4) And, if k = l, we have \u03b8 + \u0393l l0 = 0 (5) Again, on account of the relation \u0393l l0 = 1 \u221a\u2212g \u2202 \u2202x0 \u221a\u2212g = \u2202 \u2202x0 (ln \u221a\u2212g) we derive from equation (5) \u03b8 + \u03b8 = 0 (6) which is, in essence, similar to equations (37). Either the expansion scalar is zero, or |\u03b8| = \u00b1\u03b8 (7) which seems erroneous from a mathematical perspective. But, suppose \u03b8 can actually have the same value with opposite signs; then the last two equations imply some physical meaning. This shall be further corroborated by the following methodology. Equation (2) can also be written as (with, Ak 0 = \u0393l k0Al) Ak\u03b8 + Ak 0 = 0 Di\ufb00erentiating with respect to x0 and using the above equation again we have Ak \u02d9 \u03b8 \u2212Ak,0AkAk 0 A2 + Ak 0,0 = 0 (8) where, A2 = AkAk. Again, Raychaudhuri\u2019s equation is as follows: \u02d9 \u03b8 = \u03b82 3 \u22122\u03c32 + 2\u03c92 \u2212Rs (9) where, the symbols have their usual meanings. Thus, from equations (8) and (9) we have A2\u03b82 3 + A2(2\u03c92 \u2212\u03c32 + Rs) + Ak,0Ak\u03b8 + AkAk 0,0 = 0 Writing \u01eb = 2\u03c92 \u22122\u03c32 \u2212Rs, and solving the quadratic equation we \ufb01nally derive \u03b8 = 3 2 \uf8ee \uf8f0\u2212Ak,0Ak A2 \u00b1 v u u t(Ak,0Ak)(Ak,0Ak) A4 \u22124 3 (Ak,0Ak A2 + \u01eb )\uf8f9 \uf8fb (10) 3 \fHere, we can derive some interesting conclusions. When the discriminant of the above solution is zero, we would have another quadratic equation of the form: p2 \u22124 3p \u22124 3\u01eb = 0 where, p = Ak,0Ak A2 . The solution is of the form p = 2 3 \u00b1 2 3 \u221a 1 \u22123\u01eb which implies that \u01eb \u22651 3. On the other hand, if we have (Ak,0Ak)(Ak,0Ak) A4 = 4 3 (Ak,0Ak A2 + \u01eb ) then, in such a scenario \u03b8 = \u22123 2 Ak,0Ak A2 This may be considered the initial value of \u03b8 during the birth of the universe when all geodesics were focused at a particular point and then after the big bang as the universe began to evolve the value of \u03b8 evolved accumulating the discriminant term. Again, if the Raychaudhri scalar is such that Rs > Ak,0Ak A2 + 2\u03c92 \u22122\u03c32 then v u u t(Ak,0Ak)(Ak,0Ak) A4 \u22124 3 (Ak,0Ak A2 + \u01eb ) > Ak,0Ak A2 So, \u03b8 can have positive values as well. Therefore, writing \u03b5 = \u2212Ak,0Ak A2 \u00b1 v u u t(Ak,0Ak)(Ak,0Ak) A4 \u22124 3 (Ak,0Ak A2 + \u01eb ) we would have \u03b8 = \u00b13 2\u03b5 (11) This is what insinuates from equation (7) too. Hence, essentially, \u02d9 \u03b8 can diverge to both positive and negative in\ufb01nity. The physical signi\ufb01cance of this conclusion is novel and 4 \fsigni\ufb01cant geodesics can diverge just as they can converge. As a consequence, there will exist a point of exclusivity akin to a point of singularity. This point of exclusivity can be looked upon as the origin of all matter and energy and the formation of spacetime as we know it. We understand immdiately how important the Raychaudhuri scalar and expansion scalar are. However, it should be borne in the mind that the notion and methodology innovated here is in stark contrast with the theory of Big Rip [17], particularly for the fact that no speculative, mysterious energy has been talked about here it is essentially the geometry trying to explain the evolution of the universe. 3 The evolution equation Now, we shall substantiate the novel result of the preceding section taking a di\ufb00erent route. At \ufb01rst, let us consider a covariant vector Bi whose transport between two di\ufb00erent points of a Riemannian manifold is independent of the path and that is not covariantly constant; then it is known that the derivative of this vector \ufb01eld would be given as [18] \u2202Bi \u2202xk = \u0393l ikBl (12) We assume that Bi is thrice di\ufb00erentiable. Thus we have the following from relation (12) \u22022Bi \u2202xn\u2202xk = \u2202\u0393l ik \u2202xn Bl + \u0393l ik \u2202Bl \u2202xn (13) and \u22023Bi \u2202xm\u2202xn\u2202xk = Bi,mnk = \u22022\u0393l ik \u2202xm\u2202xn Bl + \u0393l ik \u22022Bl \u2202xm\u2202xn + \u2202\u0393l ik \u2202xn \u2202Bl \u2202xm + \u2202\u0393l ik \u2202xm \u2202Bl \u2202xn (14) Now, computing Bi,mkn, subtracting the resultant equation from (14) and then changing the indices as l \u2192j, we have the equation Bi,mnk \u2212Bi,mkn = \u0393j ik,mnBj \u2212\u0393j in,mkBj + \u0393j ikBj,mn \u2212\u0393j inBj,mk + \u03b8 (15) where, \u03b8 = \u0393j ik,nBj,m + \u0393j ik,mBj,n \u2212\u0393j in,kBj,m \u2212\u0393j in,mBj,k and the \u2019comma\u2019 implies normal derivative. Now, we know the following relations regarding the Riemann-Christo\ufb00el (RC) tensor Bi;kn \u2212Bi,nk = \u2212Rm k niBm and Bi;nk \u2212Bi,kn = \u2212Rm n kiBm 5 \fwhere, the \u2019semi-colon\u2019 implies covariant derivative. Therefore, we can write the following from equation (15) \u2202m[Bi,nk \u2212Bi,kn] = \u2202m[2Rj i knBj + \u03b7Bi] = \u0393j ik,mnBj \u2212\u0393j in,mkBj\u0393j ikBj,mn \u2212\u0393j inBj,mk + \u03b8 where, \u03b7Bi = Bi;kn \u2212Bi;nk (\u03b7 being a covariant derivative operator). This yields the premature evolution equation for the RC tensor or the curvature tensor as 2\u2202m[Rj i kn]Bj+2Rj i knBj,m+[\u0393j in,mk\u2212\u0393j ik,mn]Bj+\u0393j inBj,mk\u2212\u0393j ikBj,mn+\u2202m(\u03b7Bi)\u2212\u03b8 = 0 (16) It is worth noting that the vector \ufb01elds Bi can be related to all the necessary features of the manifold and do not depend explicitly on the Christo\ufb00el symbols, in this regard. Now, with the di\ufb00erential equation \u2202Bi \u2202xn = \u0393m inBm we shall have equation (13) as Bi,kn = \u0393m in,kBm + \u0393m in\u0393p mkBp Di\ufb00erentiating this again and rearranging we get \u0393m in,jkBm = Bi,jkn \u2212\u0393m in,k\u0393r mjBr \u2212\u0393m in\u0393p mk,jBp \u2212\u0393m in,j\u0393p mkBp \u2212\u0393m in\u0393p mkBp,j Now, interchanging the indices m and j (m \u2194j) we have \u0393j in,mkBj = Bi,mkn \u2212\u0393j in,k\u0393r jmBr \u2212\u0393j in\u0393p jk,mBp \u2212\u0393j in,m\u0393p jkBp \u2212\u0393j in\u0393p jkBp,m (17) Similarly, we would have \u0393j ik,mnBj = Bi,mnk \u2212\u0393j ik,n\u0393r jmBr \u2212\u0393j in\u0393p jn,mBp \u2212\u0393j in,m\u0393p jnBp \u2212\u0393j ik\u0393p jnBp,m (18) Again, since \u0393j in,kBj = Bi,kn \u2212\u0393j in\u0393p jkBp, the equations (17) and (18) can be rewritten respectively as \u0393j in,mkBj = Bi,mkn \u2212\u0393j in\u0393p jk\u0393q pmBq \u2212\u03b4j r\u0393r jm{Bi,kn \u2212\u0393j in\u0393p jkBp} \u2212\u0393j in{Bj,mk \u2212\u0393p jk\u0393q pmBq} \u2212\u03b4j p\u0393p jk{Bi,mn \u2212\u0393j in\u0393p jmBp} (19) and \u0393j ik,mnBj = Bi,mnk \u2212\u0393j ik\u0393p jn\u0393q pmBq \u2212\u03b4j r\u0393r jm{Bi,nk \u2212\u0393j ik\u0393p jnBp} \u2212\u0393j ik{Bj,mn \u2212\u0393p jn\u0393q pmBq} \u2212\u03b4j p\u0393p jn{Bi,mk \u2212\u0393j ik\u0393p jmBp} (20) 6 \fNow, subtracting equation (20) from equation (19) and rearranging, we have Bj[\u0393j in,mk \u2212\u0393j ik,mn] = (Bi,mkn \u2212Bi,mnk) + \u0393j jm(Bi,nk \u2212Bi,kn) + \u0393j ikBj,mn \u2212\u0393j inBj,mk + \u03b4j p(\u0393p jnBi,mk \u2212\u0393p jkBi,mn) + 2\u03b4j pBj,m(\u0393j in\u0393p jk \u2212\u0393j ik\u0393p jn) (21) Let us consider the second term with parenthesis on the right hand side of the last equation. Multiplying by B2Bj = BjBjBj = BjB2 j we shall have B2Bj\u0393j jm(Bi,nk \u2212Bi,kn) = BjBjBj\u0393j jm(Bi,nk \u2212Bi,kn) = B2Bj,m(Bi,nk \u2212Bi,kn) Clearly, there is a breakdown of index notation, pertinent to the index \u2019j\u2019. We shall elaborate this scenario now and we shall \ufb01nd the above equation to be useful subsequently. We make an ansatz that in this special scenario, the \ufb01rst term in the parenthesis in equation (21), namely (Bi,mkn \u2212Bi,mnk), becomes explicitly independent of the index j present in the left hand side. The rationale can be attributed to when some structure preserving endomorphism that preserves the geometry of the manifold breaks down and thereby the RC tensor accrues a new upper index and the index notation breaks down. The feasibility of the rationale introduced above will be evident later while elucidating equation (30). So, essentially, the RC tensor arising from the breakdown will be independent of the index j. This causes the breakdown of the \u2019index notation\u2019 mentioned earlier. Therefore, considering this ansatz we may write Bi,mkn \u2212Bi,mnk = \u2212\u2202m[2Rp i knBp + \u03b7Bi] = \u22122\u2202m[Rp i kn]Bp \u22122[Rp i kn]Bp,m \u2212\u2202m(\u03b7Bi) where we have considered a new index p, such that j \u0338= p. Thus, using (21) and multiplying both sides of equation (16) by B2 = BjBj = \u03b4jjBjBj we derive 2BjB2 j [\u2202m{Rj i kn}Bj \u2212\u2202m{Rp i kn}Bp] + 2B2Bj[{Rj i kn}Bj,m \u2212{Rp i kn}Bp,m] + B2Bj,m(Bi,nk \u2212Bi,kn) + \u03b4j pBjB2 j (\u0393p jnBi,mk \u2212\u0393p jkBi,mn) + 2\u03b4j pBjB2 j Bj,m(\u0393j in\u0393p jk \u2212\u0393j ik\u0393p jn) \u2212BjB2 j \u03b8 = 0 (22) Now, we shall use the relation \u0393j in,kBj = Bi,kn \u2212\u0393j in\u0393p jkBp, and compute \u03b8 as follows: B2 j \u03b8 = BjBj,m(Bi,nk \u2212\u0393j ik\u0393p jnBp) + BjBj,n(Bi,km \u2212\u0393j im\u0393p jkBp) \u2212BjBj,m(Bi,kn \u2212\u0393j in\u0393p jkBp) \u2212BjBj,k(Bi,nm \u2212\u0393j im\u0393p jnBp) (23) from which we obtain B2 j \u03b8 = Bj,m(BjBi,nk \u2212Bi,kBj,n) + Bj,n(BjBi,km \u2212Bi,mBj,k) \u2212Bj,m(BjBi,kn \u2212Bi,nBj,k) \u2212Bj,k(BjBi,nm \u2212Bi,mBj,n) (24) Rearranging the terms we have B2 j \u03b8 = BjBj,m(Bi,nk \u2212Bi,kn) + Bj(Bj,nBi,km \u2212Bj,kBi,nm) + Bj,m(Bi,nBj,k \u2212Bi,kBj,n) 7 \fAlso, we have BjBj\u03b4j pBj(\u0393p jnBi,mk \u2212\u0393p jkBi,mn) = B2(Bj,nBi,mk \u2212Bj,kBi,mn) and Bj\u03b4j pB2 j Bj,m(\u0393j in\u0393p jk \u2212\u0393j ik\u0393p jn) = BjBj,m(Bi,nBj,k \u2212Bi,kBj,n) Therefore, the revised form of equation (22) is given as 2B2Bj[\u2202m{Rj i kn}Bj \u2212\u2202m{Rp i kn}Bp] + 2B2Bj[{Rj i kn}Bj,m \u2212{Rp i kn}Bp,m] + B2Bj(Bj,nBi,mk \u2212Bj,kBi,mn) + 2BjBj,m(Bi,nBj,k \u2212Bi,kBj,n) \u2212B2Bj(Bj,nBi,km \u2212Bj,kBi,nm) \u2212BjBj,m(Bi,nBj,k \u2212Bi,kBj,n) = 0 (25) Again, as we have seen before (Bi,nk \u2212Bi,kn) = (2Rj i kn + \u03b7Bj) where, \u03b7Bj = Bj;kn \u2212Bj;nk. And, we know the relation for the contravariant vector as Bn ;ik \u2212Bn ;ki = Rn i klBl Now since, the metric tensor is invariant with respect to the covariant derivative, lowering the index of the vector we obtain Bl;ik \u2212Bl;ki = Rn i klBn Thus (Bi,nk \u2212Bi,kn) = 2Rj i knBj + Rq k ijBq Using this relation we \ufb01nally derive the curvature evolution equation as follows: 2B2Bj[\u2202m{Rj i kn}Bj \u2212\u2202m{Rp i kn}Bp] + 2B2Bj[{Rj i kn}Bj,m \u2212{Rp i kn}Bp,m] + B2BjBj,n(2Rj i kmBj + Rq k ijBq) + B2BjBj,k(2Rj i mnBj + Rq m ijBq) + BjBj,m(Bi,nBj,k \u2212Bi,kBj,n) = 0 (26) The mathematical signi\ufb01cance is immediately apparent. The physical signi\ufb01cance will be manifest in the subsequent parts of the paper. Let us consider the special case where k = n. 8 \fIn such a scenario, the preceding equation reduces to 2B2Bj[\u2202m{Rj i}Bj\u2212\u2202m{Rp i }Bp]+2B2Bj[{Rj i}Bj,m\u2212{Rp i }Bp,m]+B2BjBj,nBq(Rq n ij+Rq m ij) = 0 (27) Again, Rp i = \u03b4p j Rj i and Bp = \u03b4j pBj. Also, taking into consideration another special case: n = q = m, we have 2B2B2 j \u2202mRj i[1 \u2212\u03b4p j \u03b4j p] + 2B2BjBj,mRj i[1 \u2212\u03b4p j \u03b4j p] + B2BjBmBj,mRij = 0 (28) which is another form of the evolution equation (26), with m = n = q = k. Now, if the ansatz breaks down and j = p then we have from (28) \u03b4jm\u03b4ijBj,mB2RijBiBj = 0 where, Rs = RijBiBj is the Raychaudhuri scalar. So Bj,mB2Rs = \u2202m(B2Rs) \u2212B2\u2202mRs = 0 \u21d2B2 = const. Assuming that this constant term doesn\u2019t change the structural form and properties of Rs we can write without loss of any generality B2Rs \u223cRs (29) which can be looked upon as an endomorphism of the set of Raychaudhuri scalar, in this particular cosmology (with respect to the index j), given as B2 : Rs 7\u2212 \u2192Rs (30) which implies B2 \u25e6Rs = Rs So, when the ansatz breaks down it corresponds to the consideration of an endomorphism. On the other hand, we assumed earlier that when an endomorphism breaks down the ansatz will hold. This substantiates the plausibility of why we had associated the breakdown of an endomorphism with the ansatz we introduced. Essentially, these two can be considered to be correlated and complementary. Another interpretation with regard to the endomorphism is that Bj is a timelike unit vector \ufb01eld with the respect to the index j, which insinuates that the Raychaudhuri scalar precludes all vector \ufb01elds the self scalar products of which are not constant. 9 \fNow, let us get back to equation (26). Using the Kronecker delta it can be rewritten as 2B2B2 j \u2202m{Rj i kn}[1 \u2212\u03b4p j \u03b4j p] + 2B2BjBj,m{Rj i kn}[1 \u2212\u03b4p j \u03b4j p] + B2BjBj,n(2Rj i kmBj + Rq k ijBq) + B2BjBj,k(2Rj i mnBj + Rq m ijBq) + BjBj,m(Bi,nBj,k \u2212Bi,kBj,n) = 0 (31) Again, we know that the Brouwer\u2019s \ufb01xed point theorem [19] states: For any continuous function f mapping a compact convex set to itself there is a \ufb01xed point. Therefore, considering a continuous mapping f in our Riemannian manifold and a compact, geodesically convex vector \ufb01eld (B) comprised of timelike vectors Bi, we shall have f : B 7\u2212 \u2192B and a \ufb01xed point under this automorphism. Incidentally, choosing the index j we can infer that there is a \ufb01xed point with respect to this index in the vector \ufb01eld (or the geodesic \ufb01eld), through which the family of vectors Bj are parallel transported. Consequently, the vectors Bj will be constant irrespective of the coordinates and the geometric structure of the manifold. Thus, equation (31) becomes 2B2B2 j \u2202m{Rj i kn}[1 \u2212\u03b4p j \u03b4j p] = 0 Rj i kn = const. (32) Hence, the curvature tensor returns to its initial state with respect to the index j. This corresponds to the statement of Poincare\u2019s recurrence theorem [20], if one considers the whole manifold to be a system. Now, let us consider equation (28). Writing, 1 \u2212\u03b4p j \u03b4j p = \u03b4, we have 2B2B2 j \u03b4\u03b4jk\u2202mRik + 2B2BjBj,m\u03b4\u03b4jkRik + B2BjBmBj,m\u03b4k j Rik = 0 or, 2B2B2 j \u03b4jk\u2202mRik + 2BjBj,m\u03b4jiRikBiBk + B2Bj,m\u03b4k j \u03b4ji\u03b4mkRikBiBk = 0 Since, we have considered the case where j \u0338= p, \u03b4 = 1. So, using the expression for the Raychaudhuri scalar (Rs) and after rearranging we have \ufb01nally Bj,mRs = \u2212\u03c1B2B2 j Rik,m (33) where, \u03c1 = 2\u03b4jk(2\u03b4jiB\u22121 j + \u03b4k j \u03b4ji\u03b4mk)\u22121 = 2\u03b4jk(2\u03b4jiB\u22121 j + \u03b4mi)\u22121. Thus, we have obtained the result that under certain conditions the Raychaudhuri scalar depends on the derivative of the Ricci tensor (Rik), the associated timelike vector \ufb01eld (Bj) and its derivatives. 10 \fSince, it is known that Rs is the trace of the tidal tensor epitomizing the relative acceleration due to gravity of two objects separated by an in\ufb01nitesimal distance and that Rik measures the change in geometry as an object moves along geodesics in the space, we can conclude: For some particular timelike vector \ufb01eld in a Riemannian or pseudo-Riemannian manifold, the relative acceleration due to gravity decreases with the increase in curvature and vice versa. 4 Sectional curvature In this section, we analyze and discuss the results of the preceding section. Firstly, let us consider the parameter \u03c1 of equation (33). Now, for j = k \u0338= i and m = i we have Bj,iRs = \u22122B2B2 j Rij,i (34) where, Rs = RijBiBj. On the other hand, for i = j = k, i \u0338= m we have with Rs = RjjBjBj. Hence, we shall obtain Bj,mRs = \u2212B2B3 j Rjj,m (35) Now, equation (34) can also be written as Bj,iRs = \u2212B2B2 j [Rij,i + Rij,i] (36) Now, if we take into consideration the automorphism introduced in the previous section, then Bj,i = 0 Consequently, we have Rij,i + Rij,i = 0 (37) The general implication of this equation is that the Ricci curvature, Rij, is constant with respect to the index i. However, there can be another implication that the \ufb01rst term gives a positive value and the second gives a negative value a notion that might seem erroneous, but can be used as an alternative explanation. To be precise |Rij,i| = \u00b1Rij,i (38) which essentially insinuates that |Rij| = \u00b1Rij (39) i.e. the Ricci curvature can have both positive and negative values. Again, since Rij is obtained by contracting the RC tensor which in turn is related to the sectional curvature of 11 \fa Riemannian manifold (with respect to the given manifold and two linearly independent tangent vectors at the same point), we can conclude that the sectional curvature also will have both positive and negative values. This bespeaks for both geodesic convergence and divergence on account of Rauch\u2019s comparison theorem [21, 22] that states: for positive sectional curvature, geodesics tend to converge and for negative sectional curvature, geodesics tend to diverge. Essentially, under special circumstances, geodesics can diverge. Geodesic convergence leads to a singularity; similarly geodesic divergence would lead to a point of exclusivity as shown previously by resorting to Raychaudhuri\u2019s equation. It is interesting to point out that during the in\ufb02ationary era, curvature played a signi\ufb01cant role pertinent to the dynamics, as researchers have found out [23]. 5 Discussions In the present article, we have established that an equation epitomizing the evolution of curvature, resorting to an ansatz originating from the breakdown of an endomorphism correlated to the Raychaudhuri scalar. It is also shown that the ansatz and the endomorphism are interrelated in the sense that one precludes the other. From the aforementioned considerations we have also found that the Riemann-Christo\ufb00el curvature tensor tends to follow Poincare\u2019s recurrence theorem and thereby the curvature returns to its initial value after certain period of time. Another interesting consequence is that the existence and feasibility of negative curvature which entails geodesic divergence. This is also validated by using the expansion scalar which is show to have both positive and negative values. This negative value and that of the Riemann-Christo\ufb00el tensor indicates that there might exist a point of exclusivity which is the opposite of the point of singularity a result that has been derived from the Raychaudhuri equation too. Ostensibly, the notion of such a point is in a sense a speculative extrapolation and demands ample amount of study and research. But, the prospect is something worth investigating, at the very least."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2101.01670v1",
+                    "title": "Rain Sensing Automatic Car Wiper Using AT89C51 Microcontroller",
+                    "abstract": "The turn of the century has seen a tremendous rise in technological advances\nin the field of automobiles. With 5G technology on its way and the development\nin the IoT sector, cars will start interacting with each other using V2V\ncommunications and become much more autonomous. In this project, an effort is\nmade to move in the same direction by proposing a model for an automatic car\nwiper system that operates on sensing rain and snow on the windshield of a car.\nWe develop a prototype for our idea by integrating a servo motor and raindrop\nsensor with an AT89C51 Microcontroller.",
+                    "authors": "Abhishek Das, Vivek Dhuri, Ranjushree Pal",
+                    "published": "2021-01-02",
+                    "updated": "2021-01-02",
+                    "primary_cat": "eess.SP",
+                    "cats": [
+                        "eess.SP"
+                    ],
+                    "main_content": "Introduction Today\u2019s car wipers are manual systems that work on the principle of manual switching. So here we propose an automatic wiper system that automatically switches ON on detecting rain and stops when the rain stops. Our project brings forward this system to automate the wiper system not need manual intervention. For this purpose, we use a rain sensor along with a microcontroller to drive the wiper motor. Our system uses a rain sensor to detect rain, this signal is then processed by a microcontroller to take the desired action. The rain sensor works on the principle of using water for completing its circuit, so when rain falls on it, the circuit gets completed and sends out a signal to the microcontroller. The microcontroller now processes this data and controls the motor. This system is equally useful for Aircraft and a smaller version of this can be used by motorbikers in their helmets so that they can drive easily in rains. Figure 1 shows the block diagram for our proposed idea. We use Assembly Language Coding using arm Keil \u00b5Vision 5 interface. For PCB designing, we use EAGLE software (Monk & Amos, 2017) and for circuit simulation, we have used Proteus Design Suite (Su & Wang, 2010). In what follows, we discuss the related prior work for such a problem in the next section (3), followed by de\ufb01ning the 1UG. Student, Department of Electronics and Telecommunication, DJSCE, Mumbai, India. Figure 1. Block Diagram of Proposed Model problem statement (4) and discussing our novel approaches in section (5). We then present our Experimental setup in section (6) followed with its results and discussion in section (7). Finally, we end the discussion with conclusion and future directions in the last section (8). The scripts and circuit designs are publicly available here. 3. Related Work In the current scenario, only high-end vehicles employ intelligent rain-sensing automatic wiper systems. Our system is modeled to demonstrate how useful is an automatic wiper system that adjusts speed itself based on rainfall intensity. Such a system improves the safety of a ride. There are many instances of accidents occurring during heavy rainfall due to lack of proper vision. In many cases, these accidents were due to manual errors (for example: not increasing the speed of the wiper) from the driver. An automatic, intelligent system like ours removes any manual errors. Our system adjusts wiper speed according to the intensity of rainfall and hence improves safety. Nowadays some models of Ford and Hyundai are also implementing an automatic wiper system in their vehicles[1]. Now, we discuss some prior work in this area. Since the rain sensor used in the automatic wiper system is expensive (Kulkarni & Holalad) designed a semi-automatic rain wiper arXiv:2101.01670v1  [eess.SP]  2 Jan 2021 \fRain Sensing Automatic Car Wiper Using AT89C51 Microcontroller Figure 2. 8051 Microcontroller pin con\ufb01guration that could be installed in economic vehicles. Their semiautomatic rain wiper had Cup sensor which was based on the principle of rate of water \ufb02ow and volume of water. The cup sensor was made up of a conical cup with probes at different levels of height. These levels of the probe were used to increase the wiper speed. Therefore, depending on the rain intensity the wiper system could change the speed. Their design was economical and had three different stages of rain intensity. Similarly, (Ashik & Basavaraju, 2014) designed automatic wipers with mist control that worked with three different rain intensities, which are drizzling, medium rain and heavy rain. The automatic wiper and internal wiper uses the combination of a sensor, microcontroller and the wiper motor. The external sensor and internal sensor is based on the principle of conductance. The microcontroller actuates the speed of the wiper motor by measuring the rain intensity as detected by the external motor. Similarly, the internal mist controllers are placed on the windshield which detects the mist signalling the controller to actuate the internal wiper motor. 4. Experimental Setup 4.1. AT89C51 Microcontroller The AT89C51 (Mazidi et al., 2005) is a low-power, highperformance CMOS 8-bit microcomputer with 4K bytes of Flash programmable and erasable read-only memory. The on-chip Flash allows the program memory to be reprogrammed in-system or by a conventional non-volatile Figure 3. Rain Sensor Module Figure 4. Servo Motor memory programmer. By combining a versatile 8-bit CPU with Flash on a monolithic chip, the Atmel AT89C51 is a powerful microcomputer that provides a highly-\ufb02exible and cost-effective solution to many embedded control applications. Figure 2 shows the pin con\ufb01guration of 8051 Microcontroller. 4.2. Rain Sensor Module A rain sensor module is an easy tool for rain detection (Gupta et al.). It can be used as a switch when a raindrop falls through the raining board and for measuring rainfall intensity. Figure 3 shows a depiction of a typical Rain Sensor Module. Due to its compact design and light weight, it can be easily attached into any system. The module features, a rain board, and the control board that is separate for more convenience, a power indicator LED, and sensitivity adjustable through a potentiometer. A raindrop sensor is a board coated with nickel in the form of lines. It works on the principle of ohms law. When there is no raindrop on board. Resistance is high so we get high voltage according to V=IR. When raindrop present it reduces the resistance because water is a conductor of electricity and the presence of water connects nickel lines in parallel so reduced resistance and the reduced voltage drop across it. 4.3. Servo Motor Servo motors (Sachin & Gaonkar, 2013) are self-contained mechanical devices that are used to control the machines with great precision. . Usually the servo motor is used to \fRain Sensing Automatic Car Wiper Using AT89C51 Microcontroller Figure 5. Operation of Servo based on Pulse Width Modulation control the angular motion from 0\u00b0 to 180\u00b0 and 0\u00b0 to 90\u00b0. The servo motor can be moved to a desired angular position by sending Pulse Width Modulated (Holtz, 1992) signals on the control wire. The servo understands the language of pulse position modulation. A pulse of width varying from 1 millisecond to 2 milliseconds in a repeated time frame is sent to the servo around 50 times in a second. The width of the pulse determines the angular position. For example, a pulse of 1 millisecond moves the servo towards 0\u00b0, while a 2 milliseconds wide pulse would take it to 180\u00b0. The pulse width for in-between angular positions can be interpolated accordingly. Thus a pulse of width 1.5 milliseconds will shift the servo to 90\u00b0. It must be noted that these values are only approximations. The actual behavior of the servos differs based on their manufacturer. A sequence of such pulses (50 in one second) is required to be passed to the servo to sustain a particular angular position. When the servo receives a pulse, it can retain the corresponding angular position for the next 20 milliseconds. So a pulse in every 20 millisecond time frame must be fed to the servo. Figure 4 shows an example of the servo motor we have used in our implementation, while Figure 5 shows the operation of servo motor based on Pulse Width Modulated signals. 4.4. Circuit Simulation and PCB Designing Proteus Design Suite by Labcenter Electronics provides a simple interface to design and simulate various circuits. It has a variety of electronic components and settings for each of them to choose from and is an ef\ufb01cient method to test the initial circuits for sanity checks before implementation. It has the option of adding various switches and connect and visualize the \ufb02ow in real-time, providing error logs and failure cases. Figure 7 demonstrates the simulation of our circuit design. After selecting all the components and verifying the simulation on the software, we then start implementing the actual PCB designing process which includes Figure 6. Flowchart steps like printing a layout from EAGLE PCB design software, etching the PCB, drilling, integrating and soldering all the components and \ufb01nally testing the prototype. Figure 8 shows the Layout of our printed circuit board on the EAGLE Software and Figure 9 shows the board after etching process. 5. Results and"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2101.00496v1",
+                    "title": "Smart Car Features using Embedded Systems and IoT",
+                    "abstract": "There has been a tremendous rise in technological advances in the field of\nautomobiles and autonomous vehicles. With the increase in the number of driven\nvehicles, the safety concerns with the same have also risen. The cases of\naccidents and life-threatening injuries have skyrocketed. It has become a\nnecessity to provide adequate safety measures in automobiles. This project aims\nto develop a prototype for a smart vehicle system that provides real-time\nlocation of the vehicle on detection of a crash and alert the police station\nand relatives of the user, it has a panic button feature for a passenger's\nsafety. We also demonstrate a mechanism for cabin monitoring and an interactive\ninterface between a user and a car, where the user can inquire about the\ntemperature, humidity, and other variables inside the car remotely by sending a\ntext message to the GSM module which is present in the car. The GSM module\nconnects to the Arduino, which fetches the readings from sensors attached to it\nand sends it back to the user through a text message. We show the integration\nof MQ3 Alcohol sensor with Arduino for drunk driving prevention.",
+                    "authors": "Abhishek Das, Vivek Dhuri, Aditya Desai, Suyash Ail, Ameya Kadam",
+                    "published": "2021-01-02",
+                    "updated": "2021-01-02",
+                    "primary_cat": "cs.HC",
+                    "cats": [
+                        "cs.HC",
+                        "cs.SY",
+                        "eess.SY"
+                    ],
+                    "main_content": "Introduction Driver safety has been an important feature in automobiles that have been made compulsion in various countries. An increasing number of amateur rash drivers, careless driving, and delayed access to \ufb01rst aid to victims has been a major cause of deaths. Cases of harassment, robbery in cabs are rising with more people using modern-day cab services. Driver fatigue monitoring, accident prevention measures, GPS-based location and nearest hospital alert, smart braking systems, smart airbags, etc. are some of the features currently implemented in a few of the high-end luxury-level vehicles. There hasn\u2019t been a cost-ef\ufb01cient model developed for the low-end budget cars. It is important to provide accessible safety measures in the vehicle to minimize the risk of loss of life. This project aims to develop a cost-ef\ufb01cient smart vehicle system that can help aid the cause. Figure 1 1UG. Student, Department of Electronics and Telecommunication, DJSCE, Mumbai, India 2Assistant Professor, Department of Electronics and Telecommunication, DJSCE, Mumbai, India. Figure 1. Block Diagram of Proposed Model shows the block diagram for our prototype. The primary objective of this project is to show how various sensors can be integrated to the Arduino or any microcontroller system, how to communicate with such a system remotely using technologies like GSM and GPS technology, send commands to inquire about the sensor readings and perform desired actions by using the actuators connected to the system. We have developed a low-cost prototype to demonstrate our ideas and create a baseline implementation for research purposes in this relatively new domain of the Internet of Things. With the recent developments in the capacity to process enormous amount of data from sensors, as well as communication technologies such as 5G, we believe our ideas can be scaled and deployed in real-time. The scripts have been made publicly available to the research community for further development here. In what follows, we discuss the related prior work for such a problem in the next section (3), followed by de\ufb01ning the Experimental Setup (4) and discussing our novel approaches in section (5), followed with its results and discussion in section (6). Finally, we end the discussion with conclusion and future directions in the last section (7). arXiv:2101.00496v1  [cs.HC]  2 Jan 2021 \fSmart Car Features using Embedded Systems and IoT Figure 2. GSM SIM 900A module 3. Related Work According to a Statistical Report (ind, 2016) published by the Department of Roads and Highways Transport on Vehicle Mishaps in the country in 2016, the country has recorded 4,60,852 accidents in the year resulting in 1,45,685 deaths. Approximately 423 people died in 1,227 vehicle accidents every day. The data also states that at least 16 deaths occurring in vehicle mishaps out of 55 accidents in every hour in a particular period were primarily because victims were unable to receive suitable treatment within time. Thus, if an alert system is made and an alarm is raised, it might become possible to save many lives. There has been prior work in the area of using GSM and GPS systems along with microcontrollers (Shinde et al., 2015) developed a similar tracking system using an Embedded Linux board namely Raspberry Pi and a GSM SIM900A module. The objective of their tracker was to raise an alert whenever the vehicle deviated from the prede\ufb01ned route which was set in the Raspberry Pi by the user. It also had features for sending noti\ufb01cations when the vehicle exceeded a set speed limit. (Saaid et al., 2014) implemented a vehicle location \ufb01nder using a similar combination of GSM and GPS systems particularly for the task of vehicle thefts. The use of panic buttons in vehicles is a idea which hasn\u2019t been deployed in real-life applications yet. According to the newspaper article (hin, 2016), the Parliament of India will make it compulsory from 1st of April 2018 for all public transport vehicles which include buses and cabs to have a location tracker device and one or more panic buttons to alert the authorities in case of an emergency. Although, the government has not made the installation of cameras in these vehicles mandatory, primarily citing privacy concerns and due to the factor that it will generate tremendous amounts of data every second. The technology to process such huge Figure 3. Basic AT commands used with GSM module data sets is currently unavailable. However, with the development in Internet of Things, in the future, this might be possible using Vehicle to Vehicle Communications. Nowadays, a large population of people chooses to travel by cabs and hence, keeping in mind the safety of the commuters, it is the need of the hour for developing such products. Another study by (deu) mentions that according to National Crime Records Bureau (NCRB) report, drunk driving was a major factor in road accidents. 99 per cent of the fatal accidents that occur on the Highways are due to drunk driving and there is no check on this. Majority of these accidents involved trucks since the truck drivers drive irresponsibly when they are fully drunk. Until and unless the nation starts a new system of checking drunk driving on the highways, these fatalities cannot be reduced, as mentioned by a Joint Commissioner of Police. The current system of Drunk driver checking requires traf\ufb01c police to make people blow into the breath-analyzers. However, it is not suf\ufb01cient to check every instance of drunk driving cases due to the presence of an enormous number of vehicles on roads and especially outside cities and highways. Thus, an automatic monitoring system is needed to tackle this problem. 4. Experimental Setup 4.1. GSM SMS Alert System SIM900A Modem is built with Dual Band GSM/ GPRS based SIM900A modem from SIMCOM. It works on frequencies 900/ 1800 MHz SIM900A can search these two bands automatically. The baud rate is con\ufb01gurable from 1200-115200 through AT command. This is a complete GSM module in an SMT type and designed with a very powerful single-chip processor integrating AMR926EJ-S core, allowing you to bene\ufb01t from small dimensions and cost-effective solutions. Figure 2 shows a GSM SIM900A \fSmart Car Features using Embedded Systems and IoT Figure 4. Neo 6m GPS module Figure 5. Triggering of Airbag circuitry and Accident Alert module. 4.2. GPS Tracking The NEO-6m module shown in Figure 4 is a stand-alone GPS receiver featuring the high-performance u-blox 6 positioning engines. It is a \ufb02exible and cost-effective receiver that offers numerous connectivity options in a mini 160 x 122 x 24 cm package. It has a compact architecture and power and memory options which makes NEO-6m modules optimal for space constraint, low-cost devices. It has an acquisition engine, and 2 million effective correlators, and can make enormous parallel frequency searches, thus it can \ufb01nd a satellite within a small time. This 50-channel u-blox 6 positioning engine gives a Time to First Fix of around 1-2 seconds. It has an anti-jamming technology, Eeprom for storing settings which gives these receivers fantastic navigation performance even when placed in extremely dif\ufb01cult environments. 4.3. Arduino Uno Development Board Arduino Uno is a development board based on a dual-inlinepackage ATmega328 AVR microcontroller (Mazidi et al., 2005). It has 20 digital input/ output pins, 6 of them can be used as Pulse Width Modulated (Holtz, 1992) outputs and 6 can be used as analog inputs. It has a 16 MHz crystal, a USB port, an ICSP header. Programs can be loaded onto it from Figure 6. SW420 Impact Sensor Figure 7. MQ3 Ethanol Sensor the Arduino computer program software which is an opensource IDE. The Arduino has a vast support community, which makes it a very easy way to get started working with it. 5. Proposed Methodology 5.1. Accident Detection When a car hits something with a strong force, it starts to decelerate very rapidly. An Impact Sensor detects the change of velocity/amount of vibration. If the impact is great enough, the impact sensor triggers the airbag circuit and at the same time, it signals the Arduino to send an alert. Thus, when the impact is severe, the Arduino extracts the location by signalling the GPS module which connects with the GPS satellites and retrieves the location of the car. This location co-ordinates along with a google map link are sent to the designated mobile number in an SMS form through the GSM module. The SW420 sensor module gives outputs as \u20181\u2019s or \u20180\u2019s depending on vibration, tilt, and external force applied to it. In absence of vibration, this module gives logic \u20180\u2019 as output and in presence of vibration, it gives logic \u20181\u2019 as output. It has sensitivity control on the board. Figure 6 shows the SW420 impact sensor which we have used in our prototype. \fSmart Car Features using Embedded Systems and IoT Figure 8. Rain Sensor Module 5.2. Passenger Safety A panic button is placed such that whenever a passenger feels terror and discomfort due to certain reasons, an alert message is raised by sending an SMS on pressing the button. There can be multiple panic buttons placed at different spots in the vehicle and connected to the Arduino. 5.3. Drunk Driver Prevention In this proposed system, an MQ-3 Ethanol Sensor as in Figure 7 is placed on the steering of the car or seat belt of the driver seat, such that it can monitor the percentage of alcohol in the breath of the driver. If it is found to be higher than set limits, then the Arduino signals the GSM to send an alert for the same to the driver\u2019s prede\ufb01ned safety number (such as a home number). Measures can also include not to start the car engine unless the alcohol percentage reduces. When the user exhales, any ethanol present in their breath is oxidized to acetic acid. At the cathode, oxygen from the atmosphere is reduced. The overall reaction is the oxidation of ethyl alcohol. The charge \ufb02ow produced by this reaction is measured and resistance is calculated, which results in the different levels of intoxication that the Arduino will determine. 5.4. Rain Sensing Automatic Wiper Car wipers in existing models are controlled manually by the driver. Some of the high-end cars have this feature, but due to cost factors, they have not yet made their way into normal vehicles. A cost-effective version of it is proposed in this project which includes a rain-drop sensor shown in Figure 8 connected to the microcontroller, which is Arduino in this case. The rain sensor detects rain and sends the corresponding signal to the Arduino. This signal is then processed by the Arduino to take the desired action. The rain sensor consists of nickel tracks which when gets connected by water droplets in between the two tracks, the circuit gets connected and it detects rain. The raindrop sensor module is low cost and precise for raindrop detection. Its sensitivity can be changed by rotating the screw on the board. It has a Figure 9. Demonstration of Prototype digital output pin to indicate whether water is present or not and an analog output pin to give a measure of the intensity of water. The module has a power indicator led and separate control board. A servo motor primarily contains a suitable motor, a gear reduction unit, a position measurement sensor, an control circuitry. Servo motor is a highly precise motor in terms of rotating angle. These are lightweight, low cost, compact motors that can be easily integrated into any circuits. The DC motor is connected to the gear unit which gives feedback to the position sensor. The potentiometer adjusts displacement according to the present position of the motor shaft. As the resistance changes, the differential voltage is generated. A PWM wave is given to the control wire which is transformed into voltage and is compared to the signal generated from the position sensor module. The control pin is connected to the Arduino\u2019s PWM enabled pins. 6. Results and Discussion Figure 9 shows the prototype we have developed for demonstration. Figure 10 shows the screenshot of an alert sent by our system on detection of an accident. The text message has information about the location coordinates of the car. The message could be sent to a police station or an designated relative by presetting the number in the system. Similarly, Figure 11 shows the screenshot of the message delivered when the panic button in the vehicle is pressed. It has the location coordinated and the link to open it on the Google Map. In addition to these alerts, various other information can be sent in case of an emergency by modifying the code in the system. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2101.00350v1",
+                    "title": "Multi-Image Steganography Using Deep Neural Networks",
+                    "abstract": "Steganography is the science of hiding a secret message within an ordinary\npublic message. Over the years, steganography has been used to encode a lower\nresolution image into a higher resolution image by simple methods like LSB\nmanipulation. We aim to utilize deep neural networks for the encoding and\ndecoding of multiple secret images inside a single cover image of the same\nresolution.",
+                    "authors": "Abhishek Das, Japsimar Singh Wahi, Mansi Anand, Yugant Rana",
+                    "published": "2021-01-02",
+                    "updated": "2021-01-02",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Steganography refers to the technique to hide secret messages within a non-secret message in order to avoid detection during message transmission. The secret data is then extracted from the encoded non-secret message at its destination. The use of steganography can be combined with encryption as an extra step for hiding or protecting data. Traditionally, steganography is performed to embed lowresolution images onto a high-resolution image using naive methods like LSB manipulation. Motivation for the project comes from the recent works, like that of (Baluja, 2017), (Hayes & Danezis, 2017), and (Zhu et al., 2018). These papers suggest the use of deep neural networks to model the data-hiding pipeline. These methods have signi\ufb01cantly improved the ef\ufb01ciency in terms of maintaining the secrecy and quality of the encoded messages. Recently, similar work in terms of audio signal steganography, like (Kreuk et al., 2019), has shown that deep neural networks can be used to encode multiple audio messages onto a single cover message. We aim to make an effort in a similar direction, by utilizing the ideas from the aforementioned papers to encode multiple images into the single cover image. Unlike traditional methods, we use the same resolution cover and secret images and we aim to keep the changes to the encoded cover image unnoticeable to human perception and statistical analysis, while at the same time keeping the decoded images highly intelligible. 1Electrical and Computer Engineering, Carnegie Mellon University 2Information Networking Institute, Carnegie Mellon University. The scripts have been made publicly available to the research community for further development here. In what follows, we discuss the related prior work for such a problem in the next section (3), followed by Baseline Implementations in section (4) Datasets in section (5), our Proposed Methodology in section (6) followed with its Results and Discussion in section (7). Finally, we end the discussion with Future Directions, Conclusion and Acknowledgement in the sections (8), (9), and (10) respectively. 3. Related Work Out of the several implementations, below two are most aligned and important to our goal. 3.1. Hiding Images in Plain Sight: Deep Steganography (Baluja, 2017) attempts to place a full-sized color image within another image of the same size. Deep neural networks are simultaneously trained to create the hiding and revealing processes and are designed to speci\ufb01cally work as a pair. The system is trained on images drawn randomly from the ImageNet database and works well on natural images from a wide variety of sources. Unlike many popular steganographic methods that encode the secret message within the least signi\ufb01cant bits of the cover image, their approach compresses and distributes the secret image\u2019s representation across all of the available bits. The three components involved in the system include1. Preparation Network prepares the secret image to be hidden. If the secret-image (size M\u00d7M) is smaller than the cover image (N\u00d7N), the preparation network progressively increases the size of the secret image to the size of the cover, thereby distributing the secret image\u2019s bits across the entire N \u00d7 N pixels. 2. Hiding Network takes as input the output of the preparation-network and the cover image, and creates the Container image. The input to this network is an N \u00d7 N pixel \ufb01eld, with depth concatenated RGB channels of the cover image and the transformed channels of the secret image. 3. Reveal Network used by the receiver of the image; arXiv:2101.00350v1  [cs.CV]  2 Jan 2021 \fMulti-Image Steganography Using Deep Neural Networks Figure 1. The three components of the full system. Left: Secret-Image preparation. Center: Hiding the image in the cover image. Right: Uncovering the hidden image with the reveal network; this is trained simultaneously, but is used by the receiver. it is the decoder. It receives only the Container image (neither the cover nor the secret image). The decoder network removes the cover image to reveal the secret image. The paper by (Baluja, 2017) talks about how a trained system must learn to compress the information from the secret image into the least noticeable portions of the cover image. However, no explicit attempt has been made to actively hide the existence of that information from machine detection. They trained the steganalysis networks as binary classi\ufb01ers, using the unperturbed ImageNet images as negative samples, and their containers as positive examples. The paper serves a baseline for single secret image encoding. However, it does not talk about multi-image steganography. Figure 2. Model overview: the encoder E gets as input the carrier c and the message m, it encodes c using the carrier encoder Ec and concatenates Ec(c) to c and m to generate h. Then, the decoder carrier Dc generates the new encoded carrier, from which the decoder message Dm decodes the message \u02c6 m . During training the reconstruction loss is applied between c and m to \u02c6 c and \u02c6 m, respectively 3.2. Hide and Speak: Deep Neural Networks for Speech Steganography (Kreuk et al., 2019) implements steganography for speech data using deep neural networks. It is based on an architecture that comprises 3 subparts, i.e, Encoder Network, Carrier Decoder Network, and a Message Decoder Network. They utilize ideas from (Zhu et al., 2018) to extend the encoder network to audio signals. The architecture of the model comprises of 3 sub-parts: 1. An Encoder Network (Ec) 2. A Carrier Decoder Network (Dc) 3. A Message Decoder Network (Dm) In the Carrier/cover encoder network, the encoded carrier (Ec(c)) is appended with the carrier (c) and the secret message (m), forming, [Ec(c);c;m]. This output is fed to the Carrier Decoder (Dc) which outputs the carrier embedded with a hidden message. Finally, this is fed to the Message Decoder (Dm) which reconstructs the hidden message. The \ufb01rst part learns to extract a map of potential redundancies from the carrier signal. The second part utilizes the map to best \u201cstuff\u201d a secret message into the carrier such that the carrier is minimally affected. The part third learns to extract the hidden message from the steganographically-modi\ufb01ed carrier. All the components in these networks are Gated Convs. Ec is composed of 3 blocks of Gated Convs, Dc 4 and Dm 6 blocks of Gated Convs. Each block contains 64 kernels of size 3*3. This paper demonstrates the capability to hide multiple secret messages in a single carrier, which aligns with our goals. In the paper, \ufb01ve independent speech messages have been hidden in a single speech recording. This is achieved by 2 different approaches. One approach utilizes multiple decoders, with each decoder trained to decode a different message. The other approach utilizes a conditional decoder that also takes in as input a code indicating the message index to be encoded. We borrowed the concept of multiple decoders from this paper and used it to fetch multiple secret images from the \fMulti-Image Steganography Using Deep Neural Networks Figure 3. The following results shows the cover and hidden images before and after running the model for the \ufb01rst 2 images after 20 epochs. cover image which looks like the cover image but consists of the secret images being hidden inside the cover image via passing through separate prep networks and then concatenated together. For this, we utilize the loss as de\ufb01ned by this paper for the multiple decoders. For our case, we extend the loss function de\ufb01ned in this paper for our use case. We take the reveal loss for each decoder as RevealLoss = \u03bbs \u2217P ||S \u2212S\u2032||2 and for the entire system as summation of reveal losses for each decoder added with the loss calculated for cover image as well. L(c, M) = \u03bbc \u2217||C \u2212Dc(E(c, M))||2 + \u03bbm \u2217P ||mi \u2212 Dm(Dc(E(c, M)))||2 , where M = mik We used this approach to introduce multi-image steganography using the above idea from this paper and extending it to images to add multiple images in one cover image and then retrieving it. 4. Baseline Implementations We aimed to implement a baseline single image steganography model over which we could preform out extensions. Since (Baluja, 2017) has several implementations, we implemented two of the most popular implementations and analyzed them to be suitable for the extensions for our model. The details of these implementations are as follows: 4.1. Ingham\u2019s Implementation (Ingham) is a PyTorch based implementation which follows the architecture shown in Figure 1. The architecture includes a Prep Network, a Hidden Network, and a Reveal Network and embeds a single secret image onto a single cover image. The model architecture is de\ufb01ned as follows: 1. Prep Network Two sets of three sequential layers consisting of (Conv2d Relu) combinations, concatenated and fed into the next set. 2. Hidden Network Similar to the prep network above but includes an extra Conv2D sequential layer for adding Gaussian noise to the cover. This allows the hidden information to be encoded in bits other than the LSB of the cover image. 3. Reveal Network Similar to the above networks, with an extra Conv2d at the end. 4.1.1. IMPLEMENTATION DETAILS For full code please refer here. The model is explained below: 1. Transformations: Scaling, random crop and normalization 2. Optimizer: Adam, with learning rate 0.001 3. Customized Loss, as suggested by (Baluja, 2017). coverloss + \u03b2\u00d7hiddenloss) \fMulti-Image Steganography Using Deep Neural Networks Figure 4. Baluja\u2019s implementation by Alexandre 4.1.2. RESULTS The author of the implementation had shown desirable results for higher resolution images. Since we used lowresolution images with an equal resolution for both secret and cover, we did not see similar results in our implementations. See Figure 3. The implementation generated lossy secret images, [left-side images in results] while retaining the cover image almost completely [right-side images in results]. Since other baseline models performed signi\ufb01cantly better, we decided not to proceed with this implementation. 4.2. Alexandre\u2019s Implementation Alexandre\u2019s is another implementation (Muzio) of Baluja\u2019s paper, based on Keras. There are new features introduced in this model that others have not implemented like noise addition, mean stabilization. We have tried various models to see which con\ufb01guration provides the best results and is fastest. The model architecture has three parts: 1. Preparation Network : Transforms secret image to be concatenated with cover. 2. Hiding Network: Converts the concatenated image into encoded cover. 3. Reveal Network: Extracts the secret image from the encoded cover. Hiding and reveal networks use aggregated Conv2D layers: 5 layers of 65 \ufb01lters [50 3x3 \ufb01lters, 10 4x4 \ufb01lters, and 5 5x5 \ufb01lters]. Prep network uses 2 layers of similar structure. All Conv 2D layers are followed by ReLU activation. 4.2.1. IMPLEMENTATION DETAILS For full code please refer here. Model is explained below: 1. Adam optimizer, with learning rate as 0.001 and a custom scheduler. 2. Model has been trained for 800 epochs with a batch size of 256 and an additional 100 epochs with batch size of 32. 3. To make sure weights are updated only once, reveal network weights are frozen before adding it to the full model. 4. Gaussian noise with 0.01 standard deviation is added to the encoder\u2019s output before passing it through the decoder. 5. Mean sum of squared error has been used for calculating the decoder\u2019s loss. 4.2.2. RESULTS As it can be seen in Figure 5, the model generates decent results in the Tiny ImageNet dataset. The generated images showed a minimal loss for both the cover and the secret images. Owing to the performance of the model with a single image steganography we decided to use this implementation for our work going ahead. 5. Datasets Since our model does not have speci\ufb01c requirements pertaining to the classes of the images, we used the Tiny ImageNet \fMulti-Image Steganography Using Deep Neural Networks Figure 5. The results showing the cover and hidden images before and after running the model for 900 epochs. Left to Right Columns are: Cover Image, Secret Image, Encoded Cover Image, Decoded Secret Image, Diff Cover Image, Diff Secret Image. We can notice than the differences between the original cover and the encoded cover is almost going null. Same with the original secret and the decoded secret image. (tin) dataset in order to obtain the secret and cover images. The dataset is the collection of 64\u00d764\u00d73 images, used by the Stanford CS231 class. Further extensions of the \ufb01nal model can also be applied to larger images from datasets like ImageNet (Deng et al., 2009). We have also used Tiny ImageNet for faster training. Our training set is made of a random subset of images from all 200 classes. 2000 images are randomly sampled. The image vectors are normalized across RGB values. We split the entire training data into four halves, one for the cover image and the other three halves for three secret images. 6. Proposed Methodology We aim to perform multi-image steganography, hiding three or more images in a single cover image. The embedded secret images must be retrievable with minimum loss. The encoded cover image must look like the original cover image. To perform this, we combine the idea of (Baluja, 2017) and (Kreuk et al., 2019). We take the network implementation idea of having a prep and hiding network as an encoder and a reveal network as a decoder from (Baluja, 2017). To extend this for multiple images, we pass multiple secret images via the prep network and then concatenating these resulting data with the carrier image and then \ufb01nally send this via the Hiding network. We then take the idea of having multiple decoders, one per secret image, from (Kreuk et al., 2019) to retrieve all the secret images from the container image.To improve the security of our image retrieval model, we extend the idea presented by (Baluja, 2017) of putting secret images with noise in the original cover image instead of putting the secret images at the LSBs of the original cover image. Using of Multiple Prep and Reveal Networks. In their implementation for multiple audio signal steganography, (Kreuk et al., 2019), suggested the use of multiple decoders to derive the decoded secret from a single encoded cover. This technique is an extension of the same idea in the image domain. It does not require scaling down the size of the image or sacri\ufb01cing on the color channels of the secret images. There was another technique of using conditional decoders instead of multiple decoders but in this implementation, we have only worked on implementing multiple prep/reveal networks. Hence, we decided to build our extension based on this technique. A brief description of the encoder/decoder framework as per this technique is as follows: \u2022 ENCODER: Consists of multiple prep networks, each corresponding to separate secret image input. Prep \fMulti-Image Steganography Using Deep Neural Networks Figure 6. The image shows the DeepSteg architecture with multiple CNN based sub-networks. Inside the encoder, the prep networks convert the input secret images into images that can be concatenated to the cover. The concatenation is then passed through the hiding network to generate the encode cover. In the decoder network, separate reveal networks are deployed to generate the decoded secrets out of the encoded cover. network outputs are concatenated together with the cover image and then fed through the Hiding network. \u2022 DECODER: The decoder network comprises of multiple reveal networks, each of which is trained separately to decode its corresponding message. 6.1. Model Architecture The model architecture is as described in the Figure 6. The underlying architecture of each of the sub-networks is as follows: 1. Prep Networks: Each of the prep networks consists of the aggregation of 2 layers. With each of the layers made up of 3 separate Conv2D layers. The number of channels for these 3 Conv2D layers is 50, 10, and 5 respectively with the kernel sizes like 3, 4, and 5 for each layer. The stride length constantly remains 1 along both the axis. Appropriate padding is added to each Conv2D layer so as to keep output image in the same dimensions. Each Conv2d layer is followed with a ReLU activation. 2. Hiding Network: The hiding network is an aggregation of 5 layers. With each of these layers made up of the 3 separate Conv2D layers. The underlying structure of the Conv2D layers in the hiding network is similar to the Conv2D layers in the Prep Network. 3. Reveal Network: Each of the reveal networks shares a similar underlying architecture with the hiding network, using 5 layers of similarly formed Conv2D layers. 6.2. Implementation Details For full code please refer here. The training details are explained below: 1. Adam optimizer has been used with a custom LR scheduler. 2. Learning rate remains constant with 0.001 till \ufb01rst 200 epochs, decreasing to 0.0003 from 200 epochs to 400 epochs and further decreasing it 0.00003 for remaining iterations. 3. Model has been trained for 750 epochs with a batch size of 256 and an additional 400 epochs with a batch size of 32. 4. Tiny Image Dataset has been used, where images are 64x64. Dataset is created by taking 10 images per class for train and 2000 images in total for train and test. 5. Train set is divided into 2 sections. First 1000 images are used for training as secret images and rest 1000 for cover images. \fMulti-Image Steganography Using Deep Neural Networks Figure 7. Result of hiding two secret images. Left to Right Columns are: Cover Image, Secret Image1, Secret Image2, Encoded Cover Image, Decoded Secret Image1, Decoded Secret Image2. 6. Preparation and Hiding networks share the same stacked Keras model and loss. Reveal network has its own stacked model and its own loss function. 7. Currently, the learning rate is 0.001. 8. To make sure weights are updated only once, reveal network weights are frozen before adding it to the full model. 9. Gaussian noise with 0.01 standard deviation is added to encoder\u2019s output before passing it through the decoder. 10. Mean sum of squared error has been used for calculating decoder\u2019s loss. 11. The loss used is for the full model is represented as: Loss = \u03bbc \u2217||C \u2212C\u2032||2 + \u03bbs \u2217||S1 \u2212S\u2032 1||2 + \u03bbs \u2217 ||S2 \u2212S\u2032 2||2 + \u03bbs \u2217||S3 \u2212S\u2032 3||2 12. While training the reveal network we only consider the secret image component of the loss. 13. During the full model training, the loss for both cover and secret image is taken into consideration. 14. Currently we are taking both the \u03bbs and \u03bbc as 1.0. 7. Results and Discussion Figure 7 depicts the results of hiding two secret images over a single cover image. The input images are depicted on the left side while the encoder/decoder outputs are presented on the left hand size. The encoded cover image looks similar to the original cover to a great extent, and it does not reveal information about the secret images. The results of hiding three secret images are shown in Figure 8. The encoded cover is more lossy as compared to the case when only two secret images are used. The secret images are retrieved successfully in both cases. The losses received for the below results after 750 epochs were as below 1. Loss of Entire Setup 182053.70 2. Loss secret1 51495.24 3. Loss secret2 39911.16 4. Loss secret3 39337.07 5. Loss Cover 51310.23 Currently, the above two results were taken for two and three secret images added to the cover image and then retrieved. As we increase the number of images the loss for all the \fMulti-Image Steganography Using Deep Neural Networks Figure 8. Result of hiding three secret images. Left to Right Columns are: Cover Image, Secret Image1, Secret Image2, Secret Image2, Secret Image3,Encoded Cover Image, Decoded Secret Image1, Decoded Secret Image2, Decoded Secret Image3. values is expected to increase as more image features are being hidden in one single image. So, we need to \ufb01nd some threshold with respect to how many images can be added to the cover image to get decent results. We also have not explored the \u03bbs value for the secret messages and the \u03bbc for the cover image. This parameter may help incorrectly de\ufb01ning the loss equation and help in getting clearer results for the secret and encoded image. Currently for both the experiments we have taken the \u03bbs and \u03bbc as 1. 8. Future Directions From the implementation perspective, we aim to, 1. Increase the number of secret Images with lower loss. 2. Exploring \u03bbs and \u03bbc to see how it affects our results. 3. Use conditional decoders instead of multiple decoders. 4. We have used visual inspection as our primary evaluation metric, we can improve this by passing the encoded cover image through security software for pixel details veri\ufb01cation. This project can enable exploration with steganography and, more generally, in placing supplementary information in images. Several previous methods have attempted to use neural networks to either augment or replace a small portion of an image-hiding system. We are trying to demonstrate a method to create a fully trainable system that provides visually excellent results in unobtrusively placing multiple full-size, color images into a carrier image. Extensions can be towards a complete steganographic system, hiding the existence of the message from statistical analyzers. This will likely necessitate a new objective in training and encoding smaller images within large cover images. 9."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2012.14891v1",
+                    "title": "Detecting Hate Speech in Multi-modal Memes",
+                    "abstract": "In the past few years, there has been a surge of interest in multi-modal\nproblems, from image captioning to visual question answering and beyond. In\nthis paper, we focus on hate speech detection in multi-modal memes wherein\nmemes pose an interesting multi-modal fusion problem. We aim to solve the\nFacebook Meme Challenge \\cite{kiela2020hateful} which aims to solve a binary\nclassification problem of predicting whether a meme is hateful or not. A\ncrucial characteristic of the challenge is that it includes \"benign\nconfounders\" to counter the possibility of models exploiting unimodal priors.\nThe challenge states that the state-of-the-art models perform poorly compared\nto humans. During the analysis of the dataset, we realized that majority of the\ndata points which are originally hateful are turned into benign just be\ndescribing the image of the meme. Also, majority of the multi-modal baselines\ngive more preference to the hate speech (language modality). To tackle these\nproblems, we explore the visual modality using object detection and image\ncaptioning models to fetch the \"actual caption\" and then combine it with the\nmulti-modal representation to perform binary classification. This approach\ntackles the benign text confounders present in the dataset to improve the\nperformance. Another approach we experiment with is to improve the prediction\nwith sentiment analysis. Instead of only using multi-modal representations\nobtained from pre-trained neural networks, we also include the unimodal\nsentiment to enrich the features. We perform a detailed analysis of the above\ntwo approaches, providing compelling reasons in favor of the methodologies\nused.",
+                    "authors": "Abhishek Das, Japsimar Singh Wahi, Siyao Li",
+                    "published": "2020-12-29",
+                    "updated": "2020-12-29",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction In today\u2019s world, social media platforms play a major role in in\ufb02uencing people\u2019s everyday life. Though having numerous bene\ufb01ts, it also has the capability of shaping public opinion and religious beliefs across the world. It can be used to attack people directly or indirectly based on race, caste, immigration status, religion, ethnicity, nationality, sex, gender identity, sexual orientation, and disability or disease. Hate Speech on online social media can trigger social polarization, hateful crimes. On large platforms such as Facebook and Twitter, it becomes practically impossible for a human to monitor the source and spreading of such maFigure 1. Multi-modal \u201cmean\u201d meme and Benign confounders. Mean meme (left), Benign text confounder (middle) and Benign image confounder (right) licious activities, thus it is the responsibility of the machine learning and arti\ufb01cial intelligence research community to address and solve this problem of detecting hate speech ef\ufb01ciently. In tasks such as VQA and multi-modal machine translation, it has been observed that baseline models using the language domain perform well without even exploiting the multimodal understanding and reasoning(Devlin et al., 2015). However, the Facebook Hateful Memes Challenge dataset is designed in such a manner that unimodal models exploiting just the language or vision modalities separately will fail, and only the models that can learn the true multi-modal reasoning and understanding will be able to perform well. They achieve this by the introduction of \u201cbenign confounders\u201d in the dataset, i.e. for every hateful meme, they \ufb01nd an alternative image or caption which when replaced, is enough to make the meme harmless or non-hateful, thus \ufb02ipping the label. Consider a sentence like \u201cdishwasher for sale, missing parts\u201d. Unimodally, this sentence is harmless, but when combined with an equally harmless image of a girl without a hand, suddenly it becomes mean. See Figure 1 for an illustration. Thus, this challenge set is an excellent stage that aims to facilitate the development of robust multi-modal models, and at the same time addresses an important realworld problem of detecting hateful speech on online social media platforms. Majority of the prior work baselines aim arXiv:2012.14891v1  [cs.CV]  29 Dec 2020 \fDetecting Hate Speech in multi-modal Memes at solving this problem by \ufb01nding an alignment between the two modalities, but it faces the hardship of not knowing the context behind the image and the text combination. In this paper, we introduce two major ideas wherein we try to explore the two modalities using pre-trained Image captioning models and sentiment analysis to understand the context and relationship between the two modalities. Many of the baselines tend to focus more on the text modality for hate speech. Also, during the data analysis, we realized that majority of the hateful memes are converted into benign just by describing the image, i.e., benign text confounders. In our \ufb01rst approach, we try to balance the representations of the two modalities and tackle the benign text confounders by fetching a deeper understanding of the image via object detection and captioning. We then use this representation and fuse it with the multi-modal representation from the stateof-the-art models to improve the performance. Through the error analysis, we also found that the \ufb01netuning a model with pretrained multi-modal representations does not always provide desirable results. It may because those embeddings are pretrained to predict the semantic correlation between image and text but semantic information are dif\ufb01cult to capture and may be insuf\ufb01cient for solving this challenge. Thus, we try to include some high-level features like text and image sentiments to aid the prediction because the sentiment analysis is a related and relatively simple task. On the Facebook Hateful Memes Challenge Dataset, we show both our approaches bene\ufb01t the prediction. In what follows, we discuss the related prior work for such a problem in the next section (3), followed by de\ufb01ning the problem statement (4) and discussing our novel approaches in section (5). We then present our Experimental setup in section (6) followed with its results and discussion in section (7). Finally, we end the discussion with conclusion and future directions in the last section (8). 3. Related Work Hate speech detection has gained more and more attentions in recent years. There have been several text-only hate speech datasets released, mostly based on Twitter [(Waseem, 2016),(Waseem & Hovy, 2016),(Davidson et al., 2017)], and various architectures have been proposed for classi\ufb01ers [(Kumar et al., 2018), (Malmasi & Zampieri, 2017)]. Also, in the past few years, there has been a surge in multi-modal tasks and problems, ranging from visual question answering[(Goyal et al., 2017)] to image captioning[(Sidorov et al., 2020),(Gurari et al., 2020)] and beyond. However, there has been surprisingly little work related to multi-modal hate speech, with only a few papers including both image and text modality. Some of the works related to multi-modal hate detection based on image and text modality are as follows. Figure 2. Mean memes and their benign text confounders Yang et al. [(Yang et al., 2019)] reported that augmenting text with image embedding information immediately boosts performance in hate speech detection. In this paper, the image embeddings are formed by using the second last layer of the pre-trained ResNet neural network on ImageNet and then hashing these values for ef\ufb01cient photo indexing, searching, and clustering. The most straightforward way of integrating text with photo features is to concatenate both image and text vectors. The concatenated vector is followed by dropout, MLP, and softmax operations for the \ufb01nal hate speech classi\ufb01cation. They also explore other fusion techniques like gated summation and bi-linear transformation. Gomez et al.(Gomez et al., 2020) highlighted the issue that most of the previous work on hate speech is done using textual data only and that hate-speech detection on multi-modal publications has not been addressed yet. So, they created MMHS150k, a manually annotated multi-modal hate speech dataset formed by 150,000 tweets, each one of them containing text and an image. The data points are labeled into one of the six categories: No attacks to any community, racist, sexist, homophobic, religion-based attacks, or attacks to other communities. They trained a LSTM model which considered just the tweets text as a baseline for the task of detecting hate speech in multi-modal publications. Their further objective was to exploit the information in the visual domain to outperform their baseline. They did this by proposing two models. The \ufb01rst one was the Feature Concatenation Model (FCM), which is an MLP that concatenates the image representation extracted by a CNN and the textual features of both the tweet text and the image text extracted by an LSTM. Their second model named Textual Kernels Model (TKM) was inspired by VQA tasks and was based on the intuition of looking for patterns in the image corresponding to the associated texts. This was done by learning kernels from textual representations and convolving them with CNN feature maps. Our \ufb01rst approach extends this idea of a deeper understanding of the visual domain. To our knowledge, this paper is the \ufb01rst to use pre-trained image captioning models to generate the \u201dactual caption\u201d from the image along with \fDetecting Hate Speech in multi-modal Memes Figure 3. Approach 1 Model architecture for Image captioning the image embeddings and add these through fusion techniques like concatenation and bilinear transformations with the multi-modal embedding of the state-of-the-art baselines. Now, we describe some relevant work in Image Captioning. (Xu et al., 2015) introduced an encoder-decoder architecture which uses attention mechanism to generate captions. It is trainable by standard back-propagation methods. Most conventional approaches use a top-down mechanism for captioning tasks. A recent method (Anderson et al., 2018) combines Bottom-Up and Top-Down Attention which utilizes a Faster R-CNN based object detection to extract k image features, V = {v1, ..., vk}, vi \u2208RD that enables the attention to be calculated at the level of objects. Each image feature here encodes a salient image region. The captioning model uses a soft-top down approach given the features and partial output sequences as context. It consists of a 2-Layer LSTM, the \ufb01rst layer is called as Top-Down Attention LSTM, the output of which is used to \ufb01nd the attention weights. These attended image features are used by the second LTSM layer which is characterized as a Language Model. They further use cross-entropy loss minimization. The quality of captions generated is vastly improved using this combined technique. Their method is highly modular and allows using various architectures in captioning stage for the features generated using object detection. One can also use different object detection mechanisms in place of Faster R-CNN, or even replace it with the spatial output of a CNN. Multi-modal sentiment analysis is a relatively new topic. However, extensive research (Soleymani et al., 2017) (Shenoy & Sardana, 2020) (Kumar & Vepa, 2020) (Ghosal et al., 2018) (Zadeh et al., 2018) (Majumder et al., 2018) has already been done in this \ufb01eld and yielded fruitful results. Some (Kumar & Vepa, 2020) (Ghosal et al., 2018) tends to improve the prediction accuracy by developing more sophisticated attention mechanism to better capture the interaction between two modalities, while some (Zadeh et al., 2018) (Majumder et al., 2018) introduce very innovative ways of fusion, which utilizes graph or hierarchical architecture. In addition, (Shenoy & Sardana, 2020) leverages the sentiments to improve multi-modal dialogue task. However, very little has been done to improve the hateful media detection with multi-modal sentiment information. We introduce a sentiment analysis approach as our second experiment wherein we carry out uni-modal sentiment analysis on both text and visual domains to \ufb01nd the orientation of both the modalities. 4. Proposed Approaches 4.1. Problem Statement The objective of this challenge is to classify memes as hateful or benign while considering their information from both text and visual modality. Denote the visual components of all memes by X1 = {I1, ..., Ii} where i is the index of the memes, and in our case, the visual component I is the meme itself. Let X2 = {T1, ..., Ti} denotes the text extracted from the memes. If phrases locate in multiple regions of a single meme, the corresponding T will include all the text information by concatenation. Let Y = {y1, ..., yi} be the corresponding labels of all memes, where each y \u2208{0, 1} \fDetecting Hate Speech in multi-modal Memes Figure 4. Approach 2 Model architecture using Sentiment analysis with 0 means benign and 1 indicates a hateful meme. Thus, our task can be formulated as a binary classi\ufb01cation problem with X1 and X2 as input. The goal of our paper is to model the P(Y |X1, X2), denoted by p\u03b8, which minimize the following cost function: J(\u03b8) = X i \u2212(Y log(p\u03b8) + (1 \u2212Y )log(1 \u2212p\u03b8)) (1) 4.2. Image Captioning As discussed above, this paper tackles the benign text confounders present in the dataset which converts an originally hateful meme into a benign one just by describing what is happening in the image. Figure 2 shows some of these adversarial samples. As shown in Figure 5, they account for 20% of the dataset and thus our hypothesis is that if we can provide our model with this extra knowledge, it will combat these adversarial examples and provide a boost in accuracy. Using object detection and image captioning helps in learning this aspect of the dataset and understanding the behavior of the benign text confounders and thus gives a better performance than the baseline models. Comparing the \u201dactual caption\u201d with the \u201dpre-extracted caption\u201d of the meme will help in understanding whether both are aligned or not. Also, most of the multi-modal baselines tend to focus more on the text modality for the hate speech. Our intuition behind this approach is to \ufb01nd a deeper relationship between the text and the image modalities. As we can see in the Figure 3, we \ufb01st pass the hateful dataset (both the modalities,i.e., Xpre-extracted captions and YImages of the meme) into the Visual Bert model pre-trained on the COCO dataset. This fetches us the multi-modal representation of the two modalities, i,e., a 786 tensor of the multimodal representation (m1,m2,m3,...). Parallelly, we also pass the image into an Image Captioning model (Show, Attend, and Tell, Bottom up top down), which fetches us a caption for the image present in the meme (X3 = {C1, ..., Ci} denotes the caption extracted from the images.). We then pass this text caption via a pre-trained Bert model to get a textual representation of another 768 dimensional tensor (h1,h2,h3,...). Then, we fuse the two tensors using fusion techniques like concatenation and bilinear transformations. Bilinear transformation is a \ufb01lter to integrate the information of two vectors into one vector. Mathematically we have bilinear (m\u2019,h\u2019, dim) = m\u2019T .M.h + b, where dim is a hyper-parameter indicating the expected dimension of the output vector (768), M is a weight matrix of dimension (dim,|m\u2019|,|h\u2019|), and b is a bias vector of dimension dim. Again we concatenate m, h, and bilinear(m\u2019,h\u2019,dim) for hate speech classi\ufb01cation. Finally, we then pass the output via an Multi-layer perceptron to get a binary classi\ufb01cation of hateful and non-hateful memes (0/1). We \ufb01ne tune the Visual Bert model and the Bert model from the Facebook hateful dataset and the captions generated on the images of the Facebook hateful dataset. This new approach of combining the image captioning and multimodal baselines helps in tackling the previous mentioned challenges and increases the performance signi\ufb01cantly. 4.3. Sentiment Analysis Another approach is to utilize the sentiment information of both modalities to generate richer representations for further prediction. We \ufb01rst obtain the multi-modal contextual representations em from input T and I by using a pre-trained model. In our experiment, we use VisualBERT (Li et al., 2019). However, similar to some other pre-trained models, the VisualBERT focuses more on the correlation between the input modalities, but the text and image in hateful memes \fDetecting Hate Speech in multi-modal Memes Figure 5. Types of memes in the Facebook Hateful Memes Challenge Dataset are usually connected indirectly. Thus, unimodal sentiments, which are closely related to hate detection, can bene\ufb01t the prediction. A RoBERTa (Liu et al., 2019) model is then used to obtain the text sentiment embeddings et from T, while a VGG (Simonyan & Zisserman, 2014) is applied for visual sentiments ev from I. However, due to the limitation of annotated data, we are unable to \ufb01ne-tune those two models on our dataset. Instead, the RoBERTa is trained on Stanford Sentiment Treebank (Socher et al., 2013) and the visual sentiment model parameters are learned from T4SA dataset (Vadicamo et al., 2017). Then, em et ev are fused through concatenation and passed to multi-layer perceptrons to make the \ufb01nal prediction yhat. The framework of the entire model is shown in Figure 4. 5. Experimental Setup 5.1. Dataset We have used the Facebook Memes Challenge Dataset (Kiela et al., 2020) which comprises 10k memes. These memes are carefully designed for this task by annotators who are specially trained to employ hate-speech as de\ufb01ned by Facebook. The features in this dataset are the meme images themselves and string representations of the text in the image. The dataset comprises \ufb01ve different types of memes as shown in Figure 5: multi-modal hate, where benign confounders were found for both modalities, unimodal hate where one or both modalities were already hateful on their own, benign image and benign text confounders and \ufb01nally random not-hateful examples. The Training, Validation and Test split is 85, 5 and 10 respectively, and the individual sets are fully balanced. Each meme in the training and validation set are annotated as either 1 or 0 which corresponds to the hateful and benign classes respectively. 5.2. Multi-modal Baselines For analysis, we select VisualBERT(Li et al., 2019), a baseline model pretrained on COCO dataset with a multimodal objective. We \ufb01ne-tune the model on our dataset following the same training guidelines as mentioned in the original challenge paper(Kiela et al., 2020) and then evaluate it on the validation set comprising of 500 memes. Figure 6 shows the Confusion matrix for the same, which gives an approximate of the error cases made by the baseline. 5.2.1. VISUALBERT In order to utilize the VisualBERT, multiple region features f1, f2, . . . , fn are \ufb01rst extracted from input image I using Faster RCNN (Ren et al., 2015). Each region feature f is then converted to visual embedding ev by following equation. ev = f + es (2) where es stands for segment embedding, which indicates whether the input is text or image. For the text input, the textual embedding et is obtained in a similar way: et = ft + es + ep (3) where ft is the token embedding for each token in the sentence, and ep is the positional embedding to indicate the relative position of each token. After concatenating ev and et, the embedding is sent into pre-trained VisualBERT model for further processing. VisualBERT (Li et al., 2019) is a pre-trained model for learning joint contextualized representations of vision and language. It contains multiple transformer blocks on top of the visual and text embedding. It is pre-trained on Microsoft COCO captions (Chen et al., 2015) with two objectives: masked language modelling and sentence-image prediction task. The masked language modelling is very similar to the approach in sentence BERT (Devlin et al., 2018), where some input text tokens are masked randomly, and the model needs to predict what are the original tokens. The sentenceimage prediction requires the model to decide whether the input text matches the image. The VisualBERT output of the \ufb01rst token is used as the multi-modal representation em. An MLP is then used to make the \ufb01nal prediction. The model is \ufb01ne-tuned for the current task by using the following loss function. l(\u03b8) = CrossEntropyLoss(W \u00b7 em, y) (4) where em is a vector of size h. h is the hidden size of VisualBERT. W, which has a shape of 2 by h, is the learnable matrix of the MLP. \u03b8 denotes the parameters of the entire model, including the W. \fDetecting Hate Speech in multi-modal Memes Figure 6. Confusion Matrix for baseline VisualBERT COCO model 5.3. Methodology For both approaches, we use mmf (Singh et al., 2020), a modular framework from Facebook AI Research, to build the main neural architectures. We use mmf\u2019s version of Visual BERT to generate multi-modal representations. The model is pre-trained on MS COCO dataset with a hidden dimension of 768. For the Image Captioning models in our \ufb01rst approach, we use two implementations. The \ufb01rst one is an implementation of Show, Attend, and Tell by Xu et al. (Xu et al., 2015) and second one using Bottom-Up Top-Down approach by Anderson et al. (Anderson et al., 2018). We take the top 10000 words from the vocabulary and process the images via Inception V3 model. The pre-trained Bert model used to encode generated caption has a dimension of 768. These two result are then fused together and then passed via an MLP classi\ufb01er. In the second approach, we directly use the \ufb01nal logits of sentiment analysis models and their sum as the sentiment embedding. The MLP classifer consists of 2 layers with 768 hidden units. 5.4. Evaluation Metrics We have evaluated the performance of our classi\ufb01er using the following two metrics as suggested in the challenge 5.4.1. AREA UNDER THE RECEIVER OPERATING CHARACTERISTICS (AUCROC) Receiver Operating Characteristics curve is a graph of True Positive Rate (TPR) v/s False Positive Rate (FPR). It measures how well the binary classi\ufb01er discriminates between the classes as its decision threshold is varied.(Bradley, 1997). Figure 7. AUCROC/Accuracy for Different Experiments A perfect classi\ufb01er will have an area under the curve of 1, where the top left corner in the plot is the ideal point with a TPR of 1 and a FPR of 0. Thus, a larger area under the curve is desirable for any classi\ufb01er to maximize TPR and minimize FPR. 5.4.2. CLASSIFICATION ACCURACY We \ufb01nd the accuracy of the predictions which is given by the ratio of correct predictions to the total number of predictions made, since it is easier to interpret. Thus, for each test sample, we output the label \u2208{0, 1} and the probabilities with which the classi\ufb01er predicts a sample to be hateful. This probability is used to plot the AUCROC curve. 6. Results and Discussions 6.1. Image Captioning We use two frameworks to test our experiments, \ufb01rst being the MMF framework designed by the Facebook research lab who conducted this challenge and the second being creating all the models locally by using simple baselines like Concat BERT. Initially, we tested the image captioning locally by fusing it with Concat BERT baseline model . The baseline accuracy for this model turned out to be 57%. Then, we built an Image captioning model based on Xu et al. (Xu et al., 2015) and passed the caption via a Bert model to get the textual representation. When we fused this textual representation with the Concat BERT results, the accuracy increased by 2% verifying the importance of the captioning and tackling the presence of the benign text confounders. Then, we shifted to the MMF framework to test it on better baseline models like Visual Bert. As can be seen in the Figure 7, the image captioning approach gives a signi\ufb01cant improvement \fDetecting Hate Speech in multi-modal Memes Figure 8. Mean meme (left), Benign Text Confounder and the testing meme (middle), Object Detection Visualization before captioning(right) in the AUCROC and the accuracy on the test set. There is increase of 3.6 % in the AUCROC score and an increase of 6.7 % in the accuracy of the model. This shows us that the image captioning model tackles these benign confounders and give a better representation to the image modality and thus improves the results. The Figure 8 comprises of three images. The \ufb01rst image is the original hateful meme, the second image is the one being tested which is created by adding benign text confounder by just describing the image and thus making it a non-hateful meme with a label of \u20190\u2019. i.e., non-hateful. The third image shows the visualization of object detection bounding boxes on the test image. For the test image as input, the baseline VisualBert predicts a label \u20191\u2019, thus, misclassifying it as a hateful meme because it is not able to understand the benign text confounder. However, using our approach, it is correctly labeled as benign. Our model captions the image similar to the benign text confounder and thus the model learns about its similarity and that its benign behavior. This helps the classi\ufb01er to classify this as a benign result. There are many such examples present in the dataset which are correctly classi\ufb01ed by our model, thus improving the accuracy and the AUCROC value. We also ran Bilinear Transformation as the fusion technique but it brought the performance down and also it ran very slow on the dataset, thus, we decided to move ahead with concatenation itself for the results. 6.2. Sentiment Analysis For the sentiment analysis approach, although the model doesn\u2019t improve the AUCROC value by a large margin, we still see a signi\ufb01cant gain of 4% in the accuracy. We directly compare our models\u2019 results against the Visual BERT baseline and observe two common cases when sentiment analysis bene\ufb01ts the prediction. The \ufb01rst case is when the text and image have opposite sentiments, as shown in the Figure 9. Sample images from dev set. The sentiment value under the image are ranged from 0 to 1 with 1 as positive. Green label denotes the ground truth. \ufb01rst image of Figure 9. The baseline considers this meme as benign, but our model can clearly indicate its irony and then guide the prediction. The other is when both modalities have a positive sentiment as the meme shown in the second image of Figure 9. Sentiment information can help to con\ufb01rm benign memes. However, since we do not have the annotated data to \ufb01ne-tune the sentiment analysis models or perform multi-task learning, the accuracy of sentiment prediction is undesirable. As shown in the third meme in Figure 9, the text doesn\u2019t seem very negative, and the image seems neutral, but our model predicts both as negative. Also in some complicated cases, the sentiments are not very helpful. For example, when sentiments of both modalities are negative as the last two visuals in the \ufb01gure, our model does not work well because the meme has a similar chance to be benign or hateful. 6.3. Combining both Image Captioning and Sentiment Analysis We also performed an experiment wherein we concatenated both the image captioning results as well as the sentiment analysis features along with the Visual Bert multimodal representation and \ufb01ne tuned it on the dataset. Again, we saw a signi\ufb01cant increase in the AUCROC and the accuracy of the model in comparison to the baseline model. We expected the results to give an even better performance than the captioning results, as it would have different features to learn from, but the value of the accuracy decreased in comparison to the Image captioning results. Some reasons for this behavior could be due to con\ufb02icts in the two representations being concatenated together which could lead to lower accuracy and AUCROC value. Another reason could be due to presence of redundant features in different representations and thus reducing the performance. We also performed an analysis on some data points related to this test. As can be seen in the \ufb01gure 10, the middle image of the benign \fDetecting Hate Speech in multi-modal Memes Figure 10. Mean meme (left), Benign Text Confounder and the testing meme with positive text sentiment and positive visual sentiment (middle), Object Detection Visualization before captioning(right) confounder is wrongly classi\ufb01ed by the baseline model as hateful but the combined approach learns the alignment of the caption and the pre-extracted caption along with the sentiment of both the modalities (i.e. positive in this case) and gives a correct prediction of non-hateful label. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2006.01016v1",
+                    "title": "Probing Emergent Semantics in Predictive Agents via Question Answering",
+                    "abstract": "Recent work has shown how predictive modeling can endow agents with rich\nknowledge of their surroundings, improving their ability to act in complex\nenvironments. We propose question-answering as a general paradigm to decode and\nunderstand the representations that such agents develop, applying our method to\ntwo recent approaches to predictive modeling -action-conditional CPC (Guo et\nal., 2018) and SimCore (Gregor et al., 2019). After training agents with these\npredictive objectives in a visually-rich, 3D environment with an assortment of\nobjects, colors, shapes, and spatial configurations, we probe their internal\nstate representations with synthetic (English) questions, without\nbackpropagating gradients from the question-answering decoder into the agent.\nThe performance of different agents when probed this way reveals that they\nlearn to encode factual, and seemingly compositional, information about\nobjects, properties and spatial relations from their physical environment. Our\napproach is intuitive, i.e. humans can easily interpret responses of the model\nas opposed to inspecting continuous vectors, and model-agnostic, i.e.\napplicable to any modeling approach. By revealing the implicit knowledge of\nobjects, quantities, properties and relations acquired by agents as they learn,\nquestion-conditional agent probing can stimulate the design and development of\nstronger predictive learning objectives.",
+                    "authors": "Abhishek Das, Federico Carnevale, Hamza Merzic, Laura Rimell, Rosalia Schneider, Josh Abramson, Alden Hung, Arun Ahuja, Stephen Clark, Gregory Wayne, Felix Hill",
+                    "published": "2020-06-01",
+                    "updated": "2020-06-01",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Since the time of Plato, philosophers have considered the apparent distinction between \u201cknowing how\u201d (procedural knowledge or skills) and \u201cknowing what\u201d (propositional knowledge or facts). It is uncontroversial that deep rein*Equal contribution 1Georgia Institute of Technology 2DeepMind. Correspondence to: <felixhill@google.com>. Proceedings of the 37 th International Conference on Machine Learning, Vienna, Austria, PMLR 108, 2020. Copyright 2020 by the author(s). forcement learning (RL) agents can effectively acquire procedural knowledge as they learn to play games or solve tasks. Such knowledge might manifest in an ability to \ufb01nd all of the green apples in a room, or to climb all of the ladders while avoiding snakes. However, the capacity of such agents to acquire factual knowledge about their surroundings \u2013 of the sort that can be readily hard-coded in symbolic form in classical AI \u2013 is far from established. Thus, even if an agent successfully climbs ladders and avoids snakes, we have no certainty that it \u2018knows\u2019 that ladders are brown, that there are \ufb01ve snakes nearby, or that the agent is currently in the middle of a three-level tower with one ladder left to climb. The acquisition of knowledge about objects, properties, relations and quantities by learning-based agents is desirable for several reasons. First, such knowledge should ultimately complement procedural knowledge when forming plans that enable execution of complex, multi-stage cognitive tasks. Second, there seems (to philosophers at least) to be something fundamentally human about having knowledge of facts or propositions (Stich, 1979). If one of the goals of AI is to build machines that can engage with, and exhibit convincing intelligence to, human users (e.g. justifying their behaviour so humans understand/trust them), then a need for uncovering and measuring such knowledge in learning-based agents will inevitably arise. Here, we propose the question-conditional probing of agent internal states as a means to study and quantify the knowledge about objects, properties, relations and quantities encoded in the internal representations of neural-networkbased agents. Couching an analysis of such knowledge in terms of question-answering has several pragmatic advantages. First, question-answering provides a general purpose method for agent-analysis and an intuitive investigative tool for humans \u2013 one can simply ask an agent what it knows about its environment and get an answer back, without having to inspect internal activations. Second, the space of questions is essentially open-ended \u2013 we can pose arbitrarily complex questions to an agent, enabling a comprehensive analysis of the current state of its propositional knowledge. Question-answering has previously been studied in textual (Rajpurkar et al., 2016; 2018), visual (Malinowski & Fritz, 2014; Antol et al., 2015; Das et al., 2017) and embodied (Gordon et al., 2018; Das et al., 2018a) setarXiv:2006.01016v1  [cs.AI]  1 Jun 2020 \fProbing Emergent Semantics in Predictive Agents via Question Answering Figure 1. We train predictive agents to explore a visually-rich 3D environment with an assortment of objects of different shapes, colors and sizes. As the agent navigates (trajectory shown in white on the top-down map), an auxiliary network learns to simulate representations of future observations (labeled \u2018Simulation Network\u2019) k steps into the future, self-supervised by a loss against the ground-truth egocentric observation at t+k. Simultaneously, another decoder network is trained to extract answers to a variety of questions about the environment, conditioned on the agent\u2019s internal state but without affecting it (notice \u2018stop gradient\u2019 \u2013 gradients from the QA decoder are not backpropagated into the agent). We use this question-answering paradigm to decode and understand the internal representations that such agents develop. Note that the top-down map is only shown for illustration and not available to the agent. tings. Crucially, however, these systems are trained end-toend for the goal of answering questions. Here, we utilize question-answering simply to probe an agent\u2019s internal representation, without backpropagating gradients from the question-answering decoder into the agent. That is, we view question-answering as a general purpose (conditional) decoder of environmental information designed to assist the development of agents by revealing the extent (and limits) of their knowledge. Many techniques have been proposed for endowing agents with general (i.e. task-agnostic) knowledge, based on both hard-coding and learning. Here, we speci\ufb01cally focus on the effect of self-supervised predictive modeling \u2013 a learningbased approach \u2013 on the acquisition of propositional knowledge. Inspired by learning in humans (Elman, 1990; Rao & Ballard, 1999; Clark, 2016; Hohwy, 2013), predictive modeling, i.e. predicting future sensory observations, has emerged as a powerful method to learn general-purpose neural network representations (Elias, 1955; Atal & Schroeder, 1970; Schmidhuber, 1991; Schaul & Ring, 2013; Schaul et al., 2015; Silver et al., 2017; Wayne et al., 2018; Guo et al., 2018; Gregor et al., 2019; Recanatesi et al., 2019). These representations can be learned while exploring in and interacting with an environment in a task-agnostic manner, and later exploited for goal-directed behavior. We evaluate predictive vs. non-predictive agents (both trained for exploration) on our question-answering testbed to investigate how much knowldge of object shapes, quantities, and spatial relations they acquire solely by egocentric prediction. The set includes a mix of questions that can plausibly be answered from a single observation or a few consecutive observations, and those that require the agent to integrate global knowledge of its entire surroundings. Concretely, we make the following contributions: \u2022 In a visually-rich 3D room environment developed in the Unity engine, we develop a set of questions designed to probe a diverse body of factual knowledge about the environment \u2013 from identifying shapes and colors (\u2018What shape is the red object?\u2019) to counting (\u2018How many blue objects are there?\u2019) to spatial relations (\u2018What is the color of the chair near the table?\u2019), exhaustive search (\u2018Is there a cushion?\u2019), and comparisons (\u2018Are there the same number of tables as chairs?\u2019). \u2022 We train RL agents augmented with predictive loss functions \u2013 1) action-conditional CPC (Guo et al., 2018) and 2) SimCore (Gregor et al., 2019) \u2013 for an exploration task and analyze the internal representations they develop by decoding answers to our suite of questions. Crucially, the QA decoder is trained independent of the predictive agent and we \ufb01nd that QA performance is indicative of the agent\u2019s ability to capture global environment structure and semantics solely through egocentric prediction. We compare these predictive agents to strong non-predictive LSTM baselines as well as to an agent that is explicitly optimized for the question-answering task. \u2022 We establish generality of the encoded knowledge by testing zero-shot generalization of a trained QA decoder to compositionally novel questions (unseen combinations of seen attributes), suggesting a degree of \fProbing Emergent Semantics in Predictive Agents via Question Answering compositionality in the internal representations captured by predictive agents. 2. Background and related work Our work builds on studies of predictive modeling and auxiliary objectives in reinforcement learning as well as grounded language learning and embodied question answering. Propositional knowledge is knowledge that a statement, expressed in natural or formal language, is true (Truncellito, 2007). Since at least Plato, epistemologist philosophers have contrasted propositional knowledge with procedural knowledge (knowledge of how to do something), and some (but not all) distinguish this from perceptual knowledge (knowledge obtained by the senses that cannot be translated into a proposition) (Dretske, 1995). An ability to exhibit this sort of knowledge in a convincing way is likely to be crucial for the long-term goal of having agents achieve satisfying interactions with humans, since an agent that cannot express its knowledge and beliefs in human-interpretable form may struggle to earn the trust of users. Predictive modeling and auxiliary loss functions in RL. The power of predictive modeling for representation learning has been known since at least the seminal work of (Elman, 1990) on emergent language structures. More recent examples include Word2Vec (Mikolov et al., 2013), Skip-Thought vectors (Kiros et al., 2015), and BERT (Devlin et al., 2019) in language, while in vision similar principles have been applied to context prediction (Doersch et al., 2015; Noroozi & Favaro, 2016), unsupervised tracking (Wang & Gupta, 2015), inpainting (Pathak et al., 2016) and colorization (Zhang et al., 2016). More related to us is the use of such techniques in designing auxiliary loss functions for training model-free RL agents, such as successor representations (Dayan, 1993; Zhu et al., 2017a), value and reward prediction (Jaderberg et al., 2016; Hermann et al., 2017; Wayne et al., 2018), contrastive predictive coding (CPC) (Oord et al., 2018; Guo et al., 2018), and SimCore (Gregor et al., 2019). Grounded language learning. Inspired by the work of (Winograd, 1972) on SHRDLU, several recent works have explored linguistic representation learning by grounding language into actions and pixels in physical environments \u2013 in 2D gridworlds (Andreas et al., 2017; Yu et al., 2018; Misra et al., 2017), 3D (Chaplot et al., 2018; Das et al., 2018a; Gordon et al., 2018; Cangea et al., 2019; Puig et al., 2018; Zhu et al., 2017a; Anderson et al., 2018; Gupta et al., 2017; Zhu et al., 2017b; Oh et al., 2017; Shu et al., 2018; Vogel & Jurafsky, 2010; Hill et al., 2020) and textual (Matuszek et al., 2013; Narasimhan et al., 2015) environments. Closest to our work is the task of Embodied Question Answering (Gordon et al., 2018; Das et al., 2018a;b; Yu et al., 2019; Wijmans et al., 2019) \u2013 where an embodied agent in an environment (e.g. a house) is asked to answer a question (e.g. \u201cWhat color is the piano?\u201d). Typical approaches to EmbodiedQA involve training agents to move for the goal of answering questions. In contrast, our focus is on learning a predictive model in a goal-agnostic exploration phase and using question-answering as a post-hoc testbed for evaluating the semantic knowledge that emerges in the agent\u2019s representations from predicting the future. Neural population decoding. Probing an agent with a QA decoder can be viewed as a variant of neural population decoding, used as an analysis tool in neuroscience (Georgopoulos et al., 1986; Bialek et al., 1991; Salinas & Abbott, 1994) and more recently in deep learning (Guo et al., 2018; Gregor et al., 2019; Azar et al., 2019; Alain & Bengio, 2016; Conneau et al., 2018; Tenney et al., 2019). The idea is to test whether speci\ufb01c information is encoded in a learned representation, by feeding the representation as input to a probe network, generally a classi\ufb01er trained to extract the desired information. In RL, this is done by training a probe to predict parts of the ground-truth state of the environment, such as an agent\u2019s position or orientation, without backpropagating through the agent\u2019s internal state. Prior work has required a separate network to be trained for each probe, even for closely related properties such as position vs. orientation (Guo et al., 2018) or grammatical features of different words in the same sentence (Conneau et al., 2018). Moreover, each probe is designed with property-speci\ufb01c inductive biases, such as convnets for topdown views vs. MLPs for position (Gregor et al., 2019). In contrast, we train a single, general-purpose probe network that covers a variety of question types, with an inductive bias for language processing. This generality is possible because of the external conditioning, in the form of the question, supplied to the probe. External conditioning moreover enables agent analysis using novel perturbations of the probe\u2019s training questions. Neuroscience. Predictive modeling is thought to be a fundamental component of human cognition (Elman, 1990; Hohwy, 2013; Seth, 2015). In particular, it has been proposed that perception, learning and decision-making rely on the minimization of prediction error (Rao & Ballard, 1999; Clark, 2016). A well-established strand of work has focused on decoding predictive representations in brain states (Nortmann et al., 2013; Huth et al., 2016). The question of how prediction of sensory experience relates to higherorder conceptual knowledge is complex and subject to debate (Williams, 2018; Roskies & Wood, 2017), though some have proposed that conceptual knowledge, planning, reasoning, and other higher-order functions emerge in deeper layers of a predictive network. We focus on the emergence of propositional knowledge in a predictive agent\u2019s internal representations. \fProbing Emergent Semantics in Predictive Agents via Question Answering Table 1. QA task templates. In every episode, objects and their con\ufb01gurations are randomly generated, and these templates get translated to QA pairs for all unambiguous <shape, color> combinations. There are 50 shapes and 10 colors in total. See A.4 for details. Question type Template Level codename # QA pairs Attribute What is the color of the <shape>? color 500 What shape is the <color> object? shape 500 Count How many <shape> are there? count shape 200 How many <color> objects are there? count color 40 Exist Is there a <shape>? existence shape 100 Compare + Count Are there the same number of <color1> objects as <color2> objects? compare n color 180 Are there the same number of <shape1> as <shape2>? compare n shape 4900 Relation + Attribute What is the color of the <shape1> near the <shape2>? near color 24500 What is the <color> object near the <shape>? near shape 25000 3. Environment & Tasks Environment. We use a Unity-based visually-rich 3D environment (see Figure 1). It is a single L-shaped room that can be programmatically populated with an assortment of objects of different colors at different spatial locations and orientations. In total, we use a library of 50 different objects, referred to as \u2018shapes\u2019 henceforth (e.g. chair, teddy, glass, etc.), in 10 different colors (e.g. red, blue, green, etc.). For a complete list of environment details, see Sec. A.4. At every step, the agent gets a 96 \u00d7 72 \ufb01rst-person RGB image as its observation, and the action space consists of movements (move-{forward,back,left,right}), turns (turn-{up,down,left,right}), and object pickup and manipulation (4 DoF: yaw, pitch, roll, and movement along the axis between the agent and object). See Table 5 in the Appendix for the full set of actions. Question-Answering Tasks. We develop a range of question-answering tasks of varying complexity that test the agent\u2019s local and global scene understanding, visual reasoning, and memory skills. Inspired by (Johnson et al., 2017; Das et al., 2018a; Gordon et al., 2018), we programmatically generate a dataset of questions (see Table 1). These questions ask about the presence or absence of objects (existence shape), their attributes (color, shape), counts (count color, count shape), quantitative comparisons (compare count color, compare count shape), and elementary spatial relations (near color, near shape). Unlike the fully-observable setting in CLEVR (Johnson et al., 2017), the agent does not get a global view of the environment, and must answer these questions from a sequence of partial egocentric observations. Moreover, unlike prior work on EmbodiedQA (Gordon et al., 2018; Das et al., 2018a), the agent is not being trained end-to-end to move to answer questions. It is being trained to explore, and answers are being decoded (without backpropagating gradients) from its internal representation. Thus, in order to answer these questions, the agent must learn to encode relevant aspects of the environment in a representation amenable to easy decoding into symbols (e.g. what does the word \u201cchair\u201d mean? or what representations does computing \u201chow many\u201d require?). 4. Approach Learning an exploration policy. Predictive modeling has proven to be effective for an agent to develop general knowledge of its environment as it explores and behaves towards its goal, typically maximising environment returns (Gregor et al., 2019; Guo et al., 2018). Since we wish to evaluate the effectiveness of predictive modeling independent of the agent\u2019s speci\ufb01c goal, we de\ufb01ne a simple task that stimulates the agent to visit all of the \u2018important\u2019 places in the environment (i.e. to acquire an exploratory but otherwise task-neutral policy). This is achieved by giving the agent a reward of +1.0 every time it visits an object in the room for the \ufb01rst time. After visiting all objects, rewards are refreshed and available to be consumed by the agent again (i.e. re-visiting an object the agent has already been to will now again lead to a +1.0 reward), and this process continues for the duration of each episode (30 seconds or 900 steps). During training on this exploration task, the agent receives a \ufb01rst-person RGB observation xt at every timestep t, and processes it using a convolutional neural network to produce zt. This is input to an LSTM policy whose hidden state is ht and output a discrete action at. The agent optimizes the discounted sum of future rewards using an importanceweighted actor-critic algorithm (Espeholt et al., 2018). Training the QA-decoder. The question-answering decoder is operationalized as an LSTM that is initialized with the agent\u2019s internal representation ht and receives the question as input at every timestep (see Fig. 2). The question is a string that we tokenise into words and then map to learned embeddings. The question decoder LSTM is then unrolled for a \ufb01xed number of computation steps after which it predicts a softmax distribution over the vocabulary of oneword answers to questions in Table 1, and is trained via a cross-entropy loss. Crucially, this QA decoder is trained \fProbing Emergent Semantics in Predictive Agents via Question Answering Figure 2. Approach: at every timestep t, the agent receives an RGB observation xt as input, processes it using a convolutional neural network to produce zt, which is then processed by an LSTM to select action at. The agent learns to explore \u2013 it receives a reward of 1.0 for navigating to each new object. As it explores the environment, it builds up an internal representation ht, which receives pressure from an auxiliary predictive module to capture environment semantics so as to accurately predict consequences of its actions multiple steps into the future. We experiment with a vanilla LSTM agent and two recent predictive approaches \u2013 CPC|A (Guo et al., 2018) and SimCore (Gregor et al., 2019). The internal representations are then probed via a question-answering decoder whose gradients are not backpropagated into the agent. The QA decoder is an LSTM initialized with ht and receiving the question at every timestep. independent of the agent policy; i.e. gradients from this decoder are not allowed to \ufb02ow back into the agent. We evaluate question-answering performance by measuring top1 accuracy at the end of the episode \u2013 we consider the agent\u2019s top predicted answer at the last time step of the episode and compare that with the ground-truth answer. The QA decoder can be seen as a general purpose decoder trained to extract object-speci\ufb01c knowledge from the agent\u2019s internal state without affecting the agent itself. If this knowledge is not retained in the agent\u2019s internal state, then this decoder will not be able to extract it. This is an important difference with respect to prior work (Gordon et al., 2018; Das et al., 2018a) \u2013 wherein agents were trained to move to answer questions, i.e. all parameters had access to linguistic information. Recall that the agent\u2019s navigation policy has been trained for exploration, and so the visual information required to answer a question need not be present in the observation at the end of the episode. Thus, through question-answering, we are evaluating the degree to which agents encode relevant aspects of the environment (object colors, shapes, counts, spatial relations) in their internal representations and maintain this information in memory beyond the point at which it was initially received. See A.1.3 for more details about the QA decoder. 4.1. Auxiliary Predictive Losses We augment the baseline architecture described above with an auxiliary predictive head consisting of a simulation network (operationalized as an LSTM) that is initialized with the agent\u2019s internal state ht and deterministically simulates future latent states s1 t, . . . , sk t , . . . in an open-loop manner, receiving the agent\u2019s action sequence as input. We evaluate two predictive losses \u2013 action-conditional CPC (Guo et al., 2018) and SimCore (Gregor et al., 2019). See Fig. 2 for overview, A.1.2 for details. Action-conditional CPC (CPC|A, (Guo et al., 2018)) makes use of a noise contrastive estimation model to discriminate between true observations processed by the convolutional neural network z+ t+k (k steps into the future) and negatives randomly sampled from the dataset z\u2212 t+k, in our case from other episodes in the minibatch. Speci\ufb01cally, at each timestep t + k (up to a maximum), the output of the simulation core sk t and z+ t+k are fed to an MLP to predict 1, and sk t and z\u2212 t+k are used to predict 0. SimCore (Gregor et al., 2019) uses the simulated state sk t to condition a generative model based on ConvDRAW (Gregor et al., 2016) and GECO (Rezende & Viola, 2018) that predicts the distribution of true observations p(xt+k|ht, at,...,(t+k)) in pixel space. Baselines. We evaluate and compare the above approaches with 1) a vanilla RL agent without any auxiliary predictive losses (referred to as \u2018LSTM\u2019), and 2) a question-only agent that receives zero-masked observations as input and is useful to measure biases in our question-answering testbed. Such a baseline is critical, particularly when working with simulated environments, as it can uncover biases in the environment\u2019s generation of tasks that can result in strong but uninteresting performance from agents capable of powerful function approximation (Thomason et al., 2019). \fProbing Emergent Semantics in Predictive Agents via Question Answering Figure 3. L \u2013 Reward in an episode. R \u2013 Top-1 QA accuracy. Averaged over 3 seeds. Shaded region is 1 SD. No stop gradient. We also compare against an agent without blocking the QA decoder gradients (labeled \u2018No SG\u2019). This model differs from the above in that it is trained endto-end \u2013 with supervision \u2013 to answer the set of questions in addition to the exploration task. Hence, it represents an agent receiving privileged information about how to answer and its performance provides an upper bound for how challenging these question-answering tasks are in this context. 5. Experiments & Results 5.1. Question-Answering Performance We begin by analyzing performance on a single question \u2013 shape \u2013 which are of the form \u201cwhat shape is the <color> object?\u201d. Figure 3 shows the average reward accumulated by the agent in one episode (left) and the QA accuracy at the last timestep of the episode (right) for all approaches over the course of training. We make the following observations: \u2022 All agents learn to explore. With the exception \u2018question-only\u2019, all agents achieve high reward on the exploration task. This means that they visited all objects in the room more than once each and therefore, in principle, have been exposed to suf\ufb01cient information to answer all questions. \u2022 Predictive models aid navigation. Agents equipped with auxiliary predictive losses \u2013 CPC|A and SimCore \u2013 collect the most rewards, suggesting that predictive modeling helps navigate the environment ef\ufb01ciently. This is consistent with \ufb01ndings in (Gregor et al., 2019). \u2022 QA decoding from LSTM and CPC|A representations is no better than chance. \u2022 SimCore\u2019s representations lead to best QA accuracy. SimCore gets to a QA accuracy of \u223c72% indicating that its representations best capture propositional knowledge and are best suited for decoding answers to questions. Figure 4 (Left) shows example predictions. \u2022 Wide gap between SimCore and No SG. There is a \u223c24% gap between SimCore and the No SG oracle, suggesting scope for better auxiliary predictive losses. It is worth emphasizing that answering this shape question from observations is not a challenging task in and of itself. The No SG agent, which is trained end-to-end to optimize both for exploration and QA, achieves almost-perfect accuracy (\u223c96%). The challenge arises from the fact that we are not training the agent end-to-end \u2013 from pixels to navigation to QA \u2013 but decoding the answer from the agent\u2019s internal state, which is learned agnostic to the question. The answer can only be decoded if the agent\u2019s internal state contains relevant information represented in an easily-decodable way. Decoder complexity. To explore the possibility that answer-relevant information is present in the agent\u2019s internal state but requires a more powerful decoder, we experiment with QA decoders of a range of depths. As detailed in Figure 7 in the appendix, we \ufb01nd that using a deeper QA decoder with SimCore does lead to higher QA accuracy (from 1 \u219212 layers), although greater decoder depths become detrimental after 12 layers. Crucially, however, in the nonpredictive LSTM agent, the correct answer cannot be decoded irrespective of QA decoder capacity. This highlights an important aspect of our question-answering evaluation paradigm \u2013 that while the absolute accuracy at answering questions may also depend on decoder capacity, relative differences provide an informative comparison between internal representations developed by different agents. Table 2 shows QA accuracy for all QA tasks (see Figure 8 in appendix for training curves). The results reveal large variability in dif\ufb01culty across question types. Questions about attributes (color and shape), which can be answered from a single well-chosen frame of visual experience, are the easiest, followed by spatial relationship questions (near color and near shape), and the hardest are counting questions (count color and count shape). We further note that: \u2022 All agents perform better than the question-only baseline, which captures any biases in the environment or question distributions (enabling strategies such as constant prediction of the most-common answer). \u2022 CPC|A representations are not better than LSTM \fProbing Emergent Semantics in Predictive Agents via Question Answering Table 2. Top-1 accuracy on question-answering tasks. Overall shape color exist count shape count color compare n color compare n shape near shape near color Baseline: Question-only 29 \u00b1 3 04 \u00b1 2 10 \u00b1 2 63 \u00b1 4 24 \u00b1 3 24 \u00b1 3 49 \u00b1 3 70 \u00b1 3 04 \u00b1 2 09 \u00b1 3 LSTM 31 \u00b1 4 04 \u00b1 1 10 \u00b1 2 54 \u00b1 6 34 \u00b1 3 38 \u00b1 3 53 \u00b1 3 70 \u00b1 3 04 \u00b1 2 09 \u00b1 3 CPC|A 32 \u00b1 3 06 \u00b1 2 08 \u00b1 2 64 \u00b1 3 39 \u00b1 3 39 \u00b1 3 50 \u00b1 4 70 \u00b1 3 06 \u00b1 2 10 \u00b1 3 SimCore 60 \u00b1 3 72 \u00b1 3 81 \u00b1 3 72 \u00b1 3 39 \u00b1 3 57 \u00b1 3 56 \u00b1 3 73 \u00b1 3 30 \u00b1 3 59 \u00b1 3 Oracle: No SG 63 \u00b1 3 96 \u00b1 2 81 \u00b1 2 60 \u00b1 3 45 \u00b1 3 57 \u00b1 3 51 \u00b1 3 76 \u00b1 3 41 \u00b1 3 72 \u00b1 3 Figure 4. (Left): Sample trajectory (1 \u21924) and QA decoding predictions (for top 5 most probable answers) for the \u2018What shape is the green object?\u2019 from SimCore. Note that top-down map is not available to the agent. (Right): QA accuracy on disjoint train and test splits. on most question types. \u2022 SimCore representations achieve higher QA accuracy than other approaches, substantially above the question-only baseline on count color (57% vs. 24%), near shape (30% vs. 4%) and near color (59% vs. 9%), demonstrating a strong tendency for encoding and retaining information about object identities, properties, and both spatial and temporal relations. Finally, as before, the No SG agent trained to answer questions without stopped gradients achieves highest accuracy for most questions, although not all \u2013 perhaps due to tradeoffs between simultaneously optimizing performance for different QA losses and the exploration task. 5.2. Compositional Generalization While there is a high degree of procedural randomization in our environment and QA tasks, overparameterized neuralnetwork-based models in limited environments are always prone to over\ufb01tting or rote memorization. We therefore constructed a test of the generality of the information encoded in the internal state of an agent. The test involves a variant of the shape question type (i.e. questions like \u201cwhat shape is the <color> object?\u201d), but in which the possible question-answer pairs are partitioned into mutually exclusive training and test splits. Speci\ufb01cally, the test questions are constrained such that they are compositionally novel \u2013 the <color, shape> combination involved in the question-answer pair is never observed during training, but both attributes are observed in other contexts. For instance, a test question-answer pair \u201cQ: what shape is the blue object?, A: table\u201d is excluded from the training set of the QA decoder, but \u201cQ: what shape is the blue object?, A: car\u201d and \u201cQ: What shape is the green object?, A: table\u201d are part of the training set (but not the test set). We evaluate the SimCore agent on this test of generalization (since other agents perform poorly on the original task). Figure 4 (right) shows that the QA decoder applied to SimCore\u2019s internal states performs at substantially above-chance (and all baselines) on the held-out test questions (although somewhat lower than training performance). This indicates that the QA decoder extracts and applies information in a comparatively factorized (or compositional) manner, and suggests (circumstantially) that the knowledge acquired by the SimCore agent may also be represented in this way. 5.3. Robustness of the results To check if our results are robust to the choice of environment, we developed a similar setup using the DeepMind Lab environment (Beattie et al., 2016) and ran the same experiments without any change in hyperparameters. The environment consists of a rectangular room that is populated with a random selection of objects of different shapes and colors in each episode. There are 6 distinct objects in each room, selected from a pool of 20 objects and 9 different colors. We use a similar exploration reward structure as in our earlier environment to train the agents to navigate and observe all objects. Finally, in each episode, we introduce a question of the form \u2018What is the color of the <shape>?\u2019 where <shape> is replaced by the name of an object present in the room. \fProbing Emergent Semantics in Predictive Agents via Question Answering Figure 5. (Left) DeepMind Lab environment (Beattie et al., 2016): Rectangular-shaped room with 6 randomly selected objects out of a pool of 20 different objects of different colors. (Right) QA accuracy for color questions (What is the color of the <shape>?) in DeepMind Lab. Consistent with results in the main paper, internal representations of the SimCore agent lead to the highest accuracy while CPC|A and LSTM perform worse and similar to each other. Figure 5 shows question-answering accuracies in the DeepMind Lab environment. Consistent with the results presented above, internal representations of the SimCore agent lead to the highest answering accuracy while CPC|A and the vanilla LSTM agent perform worse and similar to each other. Crucially, for running experiments in DeepMind Lab, we did not change any hyperparameters from the experimental setup described before. This demonstrates that our approach is not speci\ufb01c to a single environment and that it can be readily applied in a variety of settings. 6. Discussion Developing agents with world models of their environments is an important problem in AI. To do so, we need tools to evaluate and diagnose the internal representations forming these world models in addition to studying task performance. Here, we marry together population or glass-box decoding techniques with a question-answering paradigm to discover how much propositional (or declarative) knowledge agents acquire as they explore their environment. We started by developing a range of question-answering tasks in a visually-rich 3D environment, serving as a diagnostic test of an agent\u2019s scene understanding, visual reasoning, and memory skills. Next, we trained agents to optimize an exploration objective with and without auxiliary self-supervised predictive losses, and evaluated the representations they form as they explore an environment, via this question-answering testbed. We compared model-free RL agents alongside agents that make egocentric visual predictions and found that the latter (in particular SimCore (Gregor et al., 2019)) are able to reliably capture detailed propositional knowledge in their internal states, which can be decoded as answers to questions, while non-predictive agents do not, even if they optimize the exploration objective well. Interestingly, not all predictive agents are equally good at acquiring knowledge of objects, relations and quantities. We compared a model learning the probability distribution of future frames in pixel space via a generative model (SimCore (Gregor et al., 2019)) with a model based on discriminating frames through contrastive estimation (CPC|A (Guo et al., 2018)). We found that while both learned to navigate well, only the former developed representations that could be used for answering questions about the environment. (Gregor et al., 2019) previously showed that the choice of predictive model has a signi\ufb01cant impact on the ability to decode an agent\u2019s position and top-down map reconstructions of the environment from its internal representations. Our experiments extend this result to decoding factual knowledge, and demonstrate that the question-answering approach has utility for comparing agents. Finally, the fact that we can even decode answers to questions from an agent\u2019s internal representations learned solely from egocentric future predictions, without exposing the agent itself directly to knowledge in propositional form, is encouraging. It indicates that the agent is learning to form and maintain invariant object identities and properties (modulo limitations in decoder capacity) in its internal state without explicit supervision. It is \u223c30 years since (Elman, 1990) showed how syntactic structures and semantic organization can emerge in the units of a neural network as a consequence of the simple objective of predicting the next word in a sequence. This work corroborates Elman\u2019s \ufb01ndings, showing that language-relevant general knowledge can emerge in a situated neural-network agent that predicts future low-level visual observations via suf\ufb01ciently powerful generative mechanism. The result also aligns with perspectives that emphasize the importance of between sensory modalities in supporting the development of conceptual or linguistic knowledge (McClelland et al., 2019). Our study is a small example of how language can be used as a channel to probe and understand what exactly agents can learn from their environments. We hope it motivates future research in evaluating predictive agents using natural linguistic interactions. \fProbing Emergent Semantics in Predictive Agents via Question Answering"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1901.05531v1",
+                    "title": "Response to \"Visual Dialogue without Vision or Dialogue\" (Massiceti et al., 2018)",
+                    "abstract": "In a recent workshop paper, Massiceti et al. presented a baseline model and\nsubsequent critique of Visual Dialog (Das et al., CVPR 2017) that raises what\nwe believe to be unfounded concerns about the dataset and evaluation. This\narticle intends to rebut the critique and clarify potential confusions for\npractitioners and future participants in the Visual Dialog challenge.",
+                    "authors": "Abhishek Das, Devi Parikh, Dhruv Batra",
+                    "published": "2019-01-16",
+                    "updated": "2019-01-16",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction . Figure 1: Visual Dialog task: given an image, dialog history, and follow-up question, predict the answer. Task. The goal of Visual Dialog is to develop conversation agents that can talk about images. Towards this end, in previous work [3], we proposed a task \u2013 given an image, dialog history, and follow-up question, predict a free-form natural language answer to the question (Fig. 1) \u2013 and a large-scale dataset1, evaluation metrics and server2, and baseline models3 for this task. Key challenge. A fundamental challenge in dialog systems is automatic evaluation of long free-form answers since existing metrics such as BLEU, METEOR, and ROUGE are known to correlate poorly with human judgement [6]. Thus, as proposed in our initial paper [3], to evaluate Visual Dialog, models are provided a list of 100 candidate answers for each question \u2013 consisting of the ground-truth answer from the dataset mixed with nearest neighbors, popular, and random answers \u2013 and evaluated on how well they rank the ground-truth answer on retrieval metrics such as mean reciprocal rank (MRR), recall (R@1, 5, 10), and mean rank. 1visualdialog.org/data 2evalai.cloudcv.org/web/challenges/challenge-page/103/overview 3github.com/batra-mlp-lab/visdial arXiv:1901.05531v1  [cs.CV]  16 Jan 2019 \fAs we describe in our paper [3], these candidate answers for each question are programmatically curated from other answers in the dataset and not human-generated, and so, some candidate answers may be semantically identical (e.g. \u2018yeah\u2019 and \u2018yes\u2019). Thus, more recently, we conducted new human studies \u2013 asking four human subjects to annotate whether each of the 100 candidate answers is correct or not for all questions in the VisDial test split. For evaluation, we report the normalized discounted cumulative gain (NDCG) over the top K ranked options, where K is the number of answers marked as correct by at least one annotator. For this computation, we consider the relevance of an answer to be the fraction of annotators that marked it as correct. This was the primary evaluation criterion for the 1st Visual Dialog Challenge4. As described in [3], there are two broad families of dialog models (unfortunately with names that are overloaded in machine learning) \u2013 \u2018generative\u2019 models (that produce a response wordby-word given some context and are evaluated on the ranking of the likelihood scores they assign to candidate answers), and \u2018discriminative\u2019 models (that simply learn to rank a list of candidate answers and cannot produce a new response). This retrieval-based evaluation holds for both families. Compatibility of the evaluation metric with generative models is crucial, since they are more useful for real-world applications where answer options are not available. 2 Concern 1: Suitability of NDCG evaluation . Massiceti et al. [7] note that \u2018the VisDial dataset was recently updated to version 1.0, where the curators try to ameliorate some of the issues with the single-\u201cground-truth\" answer approach. They incorporate a human-agreement scores for candidate answers, and introduce a modi\ufb01ed evaluation which weighs the predicted rankings by these scores. However, in making this change, the primary evaluation for this data has now become an explicit classi\ufb01cation task on the candidate answers \u2013 requiring access, at train time, to all 100 candidates for every question-image pair. For the stated goals of Visual Dialog, this change can be construed as unsuitable as it falls into the category of rede\ufb01ning the problem to match a potentially unsuitable evaluation measure \u2013 how can one get better ranks in the candidate-answer-ranking task.\u2019 The claim that \u201cthe primary evaluation for this data has now become an explicit classi\ufb01cation task on the candidate answers\u201d is incorrect and thus the conclusion drawn from it is inaccurate and confusing. First, the task has not changed, only the evaluation metric (from MRR to NDCG). The task did not and does not \u201crequire access, at train time, to all 100 candidates\u201d. Discriminative models use 100 candidate answers at train time; generative models do not. This was discussed in our initial paper [3] and continues to be true. Perhaps what the authors [7] are trying to say and express concern for is \u2013 this metric (NDCG) will favor one kind of model family over another. This is possible and something we have given a lot of thought to. Empirical \ufb01ndings from the 1st Visual Dialog Challenge5 indicate that these generative models perform comparably (or even better sometimes) than discriminative models on the NDCG metric \u2013 for example, 53.67 vs. 49.58 on VisDial v1.0 test-std for Memory Network + Attention with generative vs. discriminative decoding respectively. Code and models available here: https://github.com/batra-mlp-lab/visdial#pretrained-models-1. While this is still a potentially weak surrogate for human-in-the-loop evaluation of Visual Dialog models, it is encouraging that there now seems to be an automatic evaluation criterion on which generative models, which do not have access to candidate answers during training, outperform discriminative models. As we describe on visualdialog.org, the reason why we chose a single track for the challenge was that in practice, the distinction between the two model families can get blurry (e.g., non-parametric models that internally maintain a large list of answer options), and the separation would be dif\ufb01cult to enforce. Note that our choice of ranking for evaluation isn\u2019t an endorsement of either approach (generative or discriminative). 4visualdialog.org/challenge/2018#evaluation 5visualdialog.org/challenge/2018#winners 2 \f3 Concern 2: Comparison to proposed CCA baseline [7] . Massiceti et al. [7] proposed a simple CCA baseline with two variants \u2013 1) question-only (ignoring image and dialog history), 2) question + image (ignoring dialog history), which they show outperforms state-of-the-art models on the mean rank metric. They further note that \u2018an important takeaway from our analyses is that it is highly effective to begin exploration with the simplest possible tools one has at one\u2019s disposal. This is particularly apposite in the era of deep neural networks, where the prevailing attitude appears to be that it is preferable to start exploration with complicated methods that aren\u2019t well understood, as opposed to older, perhaps even less fashionable methods that have the bene\ufb01t of being rigorously understood.\u2019 We agree that simple and strong baselines are important, and are pleasantly surprised to see that a CCA baseline performs so well on mean rank. However, there are a few problems with this analysis. First, the baseline proposed by Massiceti et al. [7] is not close to state-ofthe-art \u2013 the authors cherry-pick the mean rank metric and ignore trends on all other metrics (see Tab. 1). Second, it ignores that a similar \ufb01nding has already been presented in the original Visual Dialog paper [3], that question-only and question + image models perform close to but slightly worse than full Q+I+H models. We recreate Tab. 1 from [3]. Third, the authors [7] ignore that the CCA baselines perform worse than not just state-of-the-art models, but also these Q and Q+I ablations [3], and comparable to answer prior and nearest neighbor (NN) baselines [3] on MRR and R@k. Finally, the results presented in [7] are not directly comparable. The proposed CCA baselines use Resnet-34 [5] features and FastText [1] embeddings, while the baselines in [3] use VGG-16 [8] and learn word embeddings from scratch respectively. Model NDCG MRR R@1 R@5 R@10 Mean Rank v0.9 val \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Answer prior 0.3735 23.55 48.52 53.23 26.50 NN-Q 0.4570 35.93 54.07 60.26 18.93 NN-QI 0.4274 33.13 50.83 58.69 19.62 LF-Q-G 0.5048 39.78 60.58 66.33 17.89 LF-QI-G 0.5204 42.04 61.65 67.66 16.84 LF-QIH-G 0.5199 41.83 61.78 67.59 17.07 HRE-QIH-G 0.5237 42.29 62.18 67.92 17.07 HREA-QIH-G 0.5242 42.28 62.33 68.17 16.79 MN-QIH-G 0.5259 42.29 62.85 68.88 17.06 A-Q (Massiceti et al. [7]) 0.3031 16.77 44.86 58.06 16.21 A-QI (Massiceti et al. [7]) 0.2427 12.17 35.38 50.57 18.29 v1.0 test-std \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 LF-QIH-G 0.5121 0.4568 35.08 55.92 64.02 18.81 HRE-QIH-G 0.5245 0.4561 34.78 56.18 63.72 18.78 MN-QIH-G 0.5280 0.4580 35.05 56.35 63.92 19.31 A-Q (Massiceti et al. [7]) 0.2832 15.95 40.10 55.10 17.08 A-QI (Massiceti et al. [7]) 0.2393 12.73 33.05 48.68 19.24 Table 1: Performance of methods on VisDial v0.9 and v1.0, measured by normalized discounted cumulative gain (NDCG), mean reciprocal rank (MRR), recall@k and mean rank. Higher is better for NDCG, MRR, and recall@k, while lower is better for mean rank. 4"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1810.11187v2",
+                    "title": "TarMAC: Targeted Multi-Agent Communication",
+                    "abstract": "We propose a targeted communication architecture for multi-agent\nreinforcement learning, where agents learn both what messages to send and whom\nto address them to while performing cooperative tasks in partially-observable\nenvironments. This targeting behavior is learnt solely from downstream\ntask-specific reward without any communication supervision. We additionally\naugment this with a multi-round communication approach where agents coordinate\nvia multiple rounds of communication before taking actions in the environment.\nWe evaluate our approach on a diverse set of cooperative multi-agent tasks, of\nvarying difficulties, with varying number of agents, in a variety of\nenvironments ranging from 2D grid layouts of shapes and simulated traffic\njunctions to 3D indoor environments, and demonstrate the benefits of targeted\nand multi-round communication. Moreover, we show that the targeted\ncommunication strategies learned by agents are interpretable and intuitive.\nFinally, we show that our architecture can be easily extended to mixed and\ncompetitive environments, leading to improved performance and sample complexity\nover recent state-of-the-art approaches.",
+                    "authors": "Abhishek Das, Th\u00e9ophile Gervet, Joshua Romoff, Dhruv Batra, Devi Parikh, Michael Rabbat, Joelle Pineau",
+                    "published": "2018-10-26",
+                    "updated": "2020-02-22",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI",
+                        "cs.MA",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Effective communication is a key ability for collaboration. Indeed, intelligent agents (humans or arti\ufb01cial) in realworld scenarios can signi\ufb01cantly bene\ufb01t from exchanging information that enables them to coordinate, strategize, and utilize their combined sensory experiences to act in the physical world. The ability to communicate has wide-ranging applications for arti\ufb01cial agents \u2013 from multi-player gameplay in simulated (e.g. DoTA, StarCraft) or physical worlds (e.g. robot soccer), to self-driving car networks communicating with each other to achieve safe 1Georgia Tech 2McGill University 3Facebook AI Research. \u2039Work done during an internship at Facebook AI Research. Correspondence to: Abhishek Das <abhshkdz@gatech.edu>. Proceedings of the 36 th International Conference on Machine Learning, Long Beach, California, PMLR 97, 2019. Copyright 2019 by the author(s). and swift transport, to teams of robots on search-and-rescue missions deployed in hostile, fast-evolving environments. A salient property of human communication is the ability to hold targeted interactions. Rather than the \u2018one-size\ufb01ts-all\u2019 approach of broadcasting messages to all participating agents, as has been previously explored (Sukhbaatar et al., 2016; Foerster et al., 2016; Singh et al., 2019), it can be useful to direct certain messages to speci\ufb01c recipients. This enables a more \ufb02exible collaboration strategy in complex environments. For example, within a team of searchand-rescue robots with a diverse set of roles and goals, a message for a \ufb01re-\ufb01ghter (e.g. \u201csmoke is coming from the kitchen\u201d) is largely meaningless for a bomb-defuser. We develop TarMAC, a Targeted Multi-Agent Communication architecture for collaborative multi-agent deep reinforcement learning. Our key insight in TarMAC is to allow each individual agent to actively select which other agents to address messages to. This targeted communication behavior is operationalized via a simple signature-based soft attention mechanism: along with the message, the sender broadcasts a key which encodes properties of agents the message is intended for, and is used by receivers to gauge the relevance of the message. This communication mechanism is learned implicitly, without any attention supervision, as a result of end-to-end training using task reward. The inductive bias provided by soft attention in the communication architecture is suf\ufb01cient to enable agents to 1) communicate agent-goal-speci\ufb01c messages (e.g. guide \ufb01re\ufb01ghter towards \ufb01re, bomb-defuser towards bomb, etc.), 2) be adaptive to variable team sizes (e.g. the size of the local neighborhood a self-driving car can communicate with changes as it moves), and 3) be interpretable through predicted attention probabilities that allow for inspection of which agent is communicating what message and to whom. Our results however show that just using targeted communication is not enough. Complex real-world tasks might require large populations of agents to go through multiple rounds of collaborative communication and reasoning, involving large amounts of information to be persistent in memory and exchanged via high-bandwidth communication channels. To this end, our actor-critic framework combines centralized training with decentralized exarXiv:1810.11187v2  [cs.LG]  22 Feb 2020 \fTarMAC: Targeted Multi-Agent Communication Decentralized Targeted Multi-Round Reinforcement Execution Communication Decisions Learning DIAL (Foerster et al., 2016) Yes No No Yes (Q-Learning) CommNet (Sukhbaatar et al., 2016) Yes No Yes Yes (REINFORCE) VAIN (Hoshen, 2017) No Yes Yes No (Supervised) ATOC (Jiang & Lu, 2018) Yes No No Yes (Actor-Critic) IC3Net (Singh et al., 2019) Yes No Yes Yes (REINFORCE) TarMAC (this paper) Yes Yes Yes Yes (Actor-Critic) Table 1: Comparison with previous work on collaborative multi-agent communication with continuous vectors. ecution (Lowe et al., 2017), thus enabling scaling to large team sizes. In this context, our inter-agent communication architecture also supports multiple rounds of targeted interactions at every time-step, wherein the agents\u2019 recurrent policies persist relevant information in internal states. While natural language, i.e. a \ufb01nite set of discrete tokens with pre-speci\ufb01ed human-conventionalized meanings, may seem like an intuitive protocol for inter-agent communication \u2013 one that enables human-interpretability of interactions \u2013 forcing machines to communicate among themselves in discrete tokens presents additional training challenges. Since our work focuses on machine-only multiagent teams, we allow agents to communicate via continuous vectors (rather than discrete symbols), as has been explored in (Sukhbaatar et al., 2016; Singh et al., 2019), and agents have the \ufb02exibility to discover and optimize their communication protocol as per task requirements. We provide extensive empirical evaluation of our approach across a range of tasks, environments, and team sizes. \u2022 We begin by benchmarking TarMAC and its ablation without attention on a cooperative navigation task derived from the SHAPES environment (Andreas et al., 2016) in Section 5.1. We show that agents learn intuitive attention behavior across task dif\ufb01culties. \u2022 Next, we evaluate TarMAC on the traf\ufb01c junction environment (Sukhbaatar et al., 2016) in Section 5.2, and show that agents are able to adaptively focus on \u2018active\u2019 agents in the case of varying team sizes. \u2022 We then demonstrate its ef\ufb01cacy in 3D environments with a cooperative \ufb01rst-person point-goal navigation task in House3D (Wu et al., 2018) (Section 5.3). \u2022 Finally, in Section 5.4, we show that TarMAC can be easily combined with IC3Net (Singh et al., 2019), thus extending its applicability to mixed and competitive environments, and leading to signi\ufb01cant improvements in performance and sample complexity. 2. Related Work Multi-agent systems fall at the intersection of game theory, distributed systems, and Arti\ufb01cial Intelligence in general (Shoham & Leyton-Brown, 2008), and thus have a rich and diverse literature. Our work builds on and is related to prior work in deep multi-agent reinforcement learning, the centralized training and decentralized execution paradigm, and emergent communication protocols. Multi-Agent Reinforcement Learning (MARL). Within MARL (see Busoniu et al. (2008) for a survey), our work is related to efforts on using recurrent neural networks to approximate agent policies (Hausknecht & Stone, 2015), stabilizing algorithms for multi-agent training (Lowe et al., 2017; Foerster et al., 2018), and tasks in novel domains e.g. coordination and navigation in 3D environments (Peng et al., 2017; OpenAI, 2018; Jaderberg et al., 2018). Centralized Training & Decentralized Execution. Both Sukhbaatar et al. (2016) and Hoshen (2017) adopt a centralized framework at both training and test time \u2013 a central controller processes local observations from all agents and outputs a probability distribution over joint actions. In this setting, the controller (e.g. a fully-connected network) can be viewed as implicitly encoding communication. Sukhbaatar et al. (2016) propose an ef\ufb01cient controller architecture that is invariant to agent permutations by virtue of weight-sharing and averaging (as in Zaheer et al. (2017)), and can, in principle, also be used in a decentralized manner at test time since each agent just needs its local state vector and the average of incoming messages to take an action. Meanwhile, Hoshen (2017) proposes to replace averaging by an attentional mechanism to allow targeted interactions between agents. While closely related to our communication architecture, this work only considers fully-supervised one-next-step prediction tasks, while we study the full reinforcement learning problem with tasks requiring planning over long time horizons. Moreover, a centralized controller quickly becomes intractable in real-world tasks with many agents and high-dimensional observation spaces e.g. navigation in House3D (Wu et al., 2018). To address these weaknesses, we adopt the framework of centralized learning but decentralized execution (following Foerster et al. (2016); Lowe et al. (2017)) and further relax it by allowing agents to communicate. While agents can use extra information during training, at test time, they pick actions solely based on local observations and communication messages. \fTarMAC: Targeted Multi-Agent Communication Emergent Communication Protocols. Our work is also related to recent work on learning communication protocols in a completely end-to-end manner with reinforcement learning \u2013 from perceptual input (e.g. pixels) to communication symbols (discrete or continuous) to actions (e.g. navigating in an environment). While (Foerster et al., 2016; Jorge et al., 2016; Das et al., 2017; Kottur et al., 2017; Mordatch & Abbeel, 2017; Lazaridou et al., 2017) constrain agents to communicate with discrete symbols with the explicit goal to study emergence of language, our work operates in the paradigm of learning a continuous communication protocol in order to solve a downstream task (Sukhbaatar et al., 2016; Hoshen, 2017; Jiang & Lu, 2018; Singh et al., 2019). Jiang & Lu (2018); Singh et al. (2019) also operate in a decentralized execution setting and use an attentional communication mechanism, but in contrast to our work, they use attention to decide when to communicate, not who to communicate with. In Section 5.4, we discuss how to potentially combine the two approaches. Table 1 summarizes the main axes of comparison between our work and previous efforts in this exciting space. 3. Technical Background Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs). A Dec-POMDP is a multiagent extension of a partially observable Markov decision process (Oliehoek, 2012). For N agents, it is de\ufb01ned by a set of states S describing possible con\ufb01gurations of all agents, a global reward function R, a transition probability function T, and for each agent i P 1, ..., N a set of allowed actions Ai, a set of possible observations \u2126i and an observation function Oi. At each time step every agent picks an action ai based on its local observation \u03c9i following its own stochastic policy \u03c0\u03b8ipai|\u03c9iq. The system randomly transitions to the next state s1 given the current state and joint action Tps1|s, a1, ..., aNq. The agent team receives a global reward r \u201c Rps, a1, ..., aNq while each agent receives a local observation of the new state Oip\u03c9i|s1q. Agents aim to maximize the total expected return J \u201c \u0159T t\u201c0 \u03b3trt where \u03b3 is a discount factor and T is the episode time horizon. Actor-Critic Algorithms. Policy gradient methods directly adjust the parameters \u03b8 of the policy in order to maximize the objective Jp\u03b8q \u201c Es\u201ep\u03c0,a\u201e\u03c0\u03b8psq rRps, aqs by taking steps in the direction of \u2207Jp\u03b8q. We can write the gradient with respect to the policy parameters as the following: \u2207\u03b8Jp\u03b8q \u201c Es\u201ep\u03c0,a\u201e\u03c0\u03b8psq r\u2207\u03b8 log \u03c0\u03b8pa|sqQ\u03c0ps, aqs , where Q\u03c0ps, aq is the action-value. It is the expected remaining discounted reward if we take action a in state s and follow policy \u03c0 thereafter. Actor-Critic algorithms learn an approximation \u02c6 Qps, aq of the unknown true action-value function by e.g. temporal-difference learning (Sutton & Barto, 1998). This \u02c6 Qps, aq is the Critic and \u03c0\u03b8 is the Actor. Multi-Agent Actor-Critic. Lowe et al. (2017) propose a multi-agent Actor-Critic algorithm adapted to centralized learning and decentralized execution wherein each agent learns its own policy \u03c0\u03b8ipai|\u03c9iq conditioned on local observation \u03c9i using a central Critic that estimates the joint action-value \u02c6 Qps, a1, ..., aNq conditioned on all actions. 4. TarMAC: Targeted Multi-Agent Communication We now describe our multi-agent communication architecture in detail. Recall that we have N agents with policies t\u03c01, ..., \u03c0Nu, respectively parameterized by t\u03b81, ..., \u03b8Nu, jointly performing a cooperative task. At every timestep t, the ith agent for all i P t1, ..., Nu sees a local observation \u03c9t i, and must select a discrete environment action at i \u201e \u03c0\u03b8i and send a continuous communication message mt i, received by other agents at the next timestep, in order to maximize global reward rt \u201e R. Since no agent has access to the underlying complete state of the environment st, there is incentive in communicating with each other and being mutually helpful to do better as a team. Policies and Decentralized Execution. Each agent is essentially modeled as a Dec-POMDP augmented with communication. Each agent\u2019s policy \u03c0\u03b8i is implemented as a 1-layer Gated Recurrent Unit (Cho et al., 2014). At every timestep, the local observation \u03c9t i and a vector ct i aggregating messages sent by all agents at the previous timestep (described in more detail below) are used to update the hidden state ht i of the GRU, which encodes the entire messageaction-observation history up to time t. From this internal state representation, the agent\u2019s policy \u03c0\u03b8i pat i | ht iq predicts a categorical distribution over the space of actions, and another output head produces an outgoing message vector mt i. Note that for our experiments, agents are symmetric and policy parameters are shared across agents, i.e. \u03b81 \u201c ... \u201c \u03b8N. This considerably speeds up learning. Centralized Critic. Following prior work (Lowe et al., 2017; Foerster et al., 2018), we operate under the centralized learning and decentralized execution paradigm wherein during training, a centralized Critic guides the optimization of individual agent policies. The Critic takes as input predicted actions tat 1, ..., at Nu and internal state representations tht 1, ..., ht Nu from all agents to estimate the joint action-value \u02c6 Qt at every timestep. The centralized Critic is learned by temporal difference (Sutton & Barto, 1998) and the gradient of the expected return Jp\u03b8iq \u201c ErRs with respect to policy parameters is approximated by: \u2207\u03b8iJp\u03b8iq \u201c E \u201d \u2207\u03b8i log \u03c0\u03b8ipat i|ht iq \u02c6 Qtpht 1, ..., ht N, a1 t, ..., aN t q \u0131 . Note that compared to an individual Critic \u02c6 Qipht i, at iq per agent, having a centralized Critic leads to considerably \fTarMAC: Targeted Multi-Agent Communication Figure 1: Overview of our multi-agent architecture with targeted communication. Left: At every timestep, each agent policy gets a local observation \u03c9t i and aggregated message ct i as input, and predicts an environment action at i and a targeted communication message mt i. Right: Targeted communication between agents is implemented as a signature-based soft attention mechanism. Each agent broadcasts a message mt i consisting of a signature kt i, which can be used to encode agent-speci\ufb01c information and a value vt i, which contains the actual message. At the next timestep, each receiving agent gets as input a convex combination of message values, where the attention weights are obtained by a dot product between sender\u2019s signature kt i and a query vector qt`1 j predicted from the receiver\u2019s hidden state. lower variance in policy gradient estimates since it takes into account actions from all agents. At test time, the Critic is not needed and policy execution is fully decentralized. Targeted, Multi-Round Communication. Establishing complex collaboration strategies requires targeted communication i.e. the ability to address speci\ufb01c messages to speci\ufb01c agents, as well as multi-round communication i.e. multiple rounds of back-and-forth interactions between agents. We use a signature-based soft-attention mechanism in our communication structure to enable targeting. Each message mt i consists of 2 parts: a signature kt i P Rdk to encode properties of intended recipients and a value vt i P Rdv: mt i \u201c r signature kt i vt i value s . (1) At the receiving end, each agent (indexed by j) predicts a query vector qt`1 j P Rdk from its hidden state ht`1 j , which is used to compute a dot product with signatures of all N messages. This is scaled by 1{?dk followed by a softmax to obtain attention weights \u03b1ji for each incoming message: \u03b1j \u201c softmax \u00bb \u2014 \u2014 \u2013 qt`1 j T kt 1 ?dk ... qt`1 j T kt i ?dk \u03b1ji ... qt`1 j T kt N ?dk \ufb01 \ufb03 \ufb03 \ufb02 (2) used to compute ct`1 j , the input message for agent j at t`1: ct`1 j \u201c N \u00ff i\u201c1 \u03b1jivt i. (3) Intuitively, attention weights are high when both sender and receiver predict similar signature and query vectors respectively. Note that Equation 2 also includes \u03b1ii corresponding to the ability to self-attend (Vaswani et al., 2017), which we empirically found to improve performance, especially in situations when an agent has found the goal in a coordinated navigation task and all it is required to do is stay at the goal, so others bene\ufb01t from attending to this agent\u2019s message but return communication is not necessary. Note that the targeting mechanism in our formulation is implicit i.e. agents implicitly encode properties without addressing speci\ufb01c recipients. For example, in a self-driving car network, a particular message may be for \u201ccars travelling on the west to east road\" (implicitly encoding properties) as opposed to speci\ufb01cally for \u201ccar 2\u201d (explicit addressing). For multi-round communication, aggregated message vector ct`1 j and internal state ht j are \ufb01rst used to predict the next internal state h1t j taking into account the \ufb01rst round: h1t j \u201c tanh ` Wh\u00d1h1r ct`1 j } ht j s \u02d8 . (4) Next, the updated hidden state h1t j is used to predict signature, query, value followed by repeating Equations 1-4 \fTarMAC: Targeted Multi-Agent Communication Figure 2: Visualizations of learned targeted communication in SHAPES. Figure best viewed in color. 4 agents have to \ufb01nd rred, red, green, blues respectively. t \u201c 1: inital spawn locations; t \u201c 2: 4 was on red at t \u201c 1 so 1 and 2 attend to messages from 4 since they have to \ufb01nd red. 3 has found its goal (green) and is self-attending; t \u201c 6: 4 attends to messages from 2 as 2 is on 4\u2019s target \u2013 blue; t \u201c 8: 1 \ufb01nds red, so 1 and 2 shift attention to 1; t \u201c 21: all agents are at their respective goal locations and primarily self-attending. for multiple rounds until we get a \ufb01nal aggregated message vector ct`1 j to be used as input at the next timestep. Number of rounds of communication is treated as a hyperparameter. Our entire communication architecture is differentiable, and message vectors are learnt through backpropagation. 5. Experiments We evaluate TarMAC on a variety of tasks and environments. All our models were trained with a batched synchronous version of the multi-agent Actor-Critic described above, using RMSProp with a learning rate of 7 \u02c6 10\u00b44 and \u03b1 \u201c 0.99, batch size 16, discount factor \u03b3 \u201c 0.99 and entropy regularization coef\ufb01cient 0.01 for agent policies. All our agent policies are instantiated from the same set of shared parameters; i.e. \u03b81 \u201c ... \u201c \u03b8N. Each agent\u2019s GRU hidden state is 128-d, message signature/query is 16-d, and message value is 32-d (unless speci\ufb01ed otherwise). All results are averaged over 5 independent seeds (unless noted otherwise), and error bars show standard error of means. 5.1. SHAPES The SHAPES dataset was introduced by Andreas et al. (2016)1, and originally created for testing compositional visual reasoning for the task of visual question answering. It consists of synthetic images of 2D colored shapes arranged in a grid (3 \u02c6 3 cells in the original dataset) along with corresponding question-answer pairs. There are 3 shapes (circle, square, triangle), 3 colors (red, green, blue), and 2 sizes (small, big) in total (see Figure 2). 1github.com/jacobandreas/nmn2/tree/shapes We convert each image from SHAPES into an active environment where agents can now be spawned at different regions of the image, observe a 5\u02c65 local patch around them and their coordinates, and take actions to move around \u2013 tup, down, left, right, stayu. Each agent is tasked with navigating to a speci\ufb01ed goal state in the environment within a max no. of steps \u2013 t\u2018red\u2019, \u2018blue square\u2019, \u2018small green circle\u2019, etc. u \u2013 and the reward for each agent at every timestep is based on team performance i.e. rt \u201c # agents on goal # agents . Having a symmetric, team-based reward incentivizes agents to cooperate in \ufb01nding each agent\u2019s goal. How does targeting work? Recall that each agent predicts a signature and value vector as part of the message it sends, and a query vector to attend to incoming messages. The communication is targeted because the attention probabilities are a function of both the sender\u2019s signature and receiver\u2019s query vectors. So it is not just the receiver deciding how much of each message to listen to. The sender also sends out signatures that affects how much of each message is sent to each receiver. The sender\u2019s signature could encode parts of its observation most relevant to other agents\u2019 goals (e.g. it would be futile to convey coordinates in the signature), and the message value could contain the agent\u2019s own location. For example, in Figure 2, at t \u201c 6, we see that when agent 2 passes by blue, agent 4 starts attending to agent 2. Here, agent 2\u2019s signature likely encodes the color it observes (which is blue), and agent 4\u2019s query encodes its goal (which is also blue) leading to high attention probability. Agent 2\u2019s message value encodes coordinates agent 4 has to navigate to, which it ends up reaching by t \u201c 21. SHAPES serves as a \ufb02exible testbed for carefully controlling and analyzing the effect of changing the size of the en\fTarMAC: Targeted Multi-Agent Communication 30 \u02c6 30, 4 agents, \ufb01ndrreds 50 \u02c6 50, 4 agents, \ufb01ndrreds 50 \u02c6 50, 4 agents, \ufb01ndrred,red,green,blues No communication 95.3\u02d82.8% 83.6\u02d83.3% 69.1\u02d84.6% No attention 99.7\u02d80.8% 89.5\u02d81.4% 82.4\u02d82.1% TarMAC 99.8\u02d80.9% 89.5\u02d81.7% 85.8\u02d82.5% Table 2: Success rates on 3 different settings of cooperative navigation in the SHAPES environment. vironment, no. of agents, goal con\ufb01gurations, etc. Figure 2 visualizes learned protocols, and Table 2 reports quantitative evaluation for three different con\ufb01gurations \u2013 1) 4 agents, all tasked with \ufb01nding red in 30 \u02c6 30 images, 2) 4 agents, all tasked with \ufb01nding red in 50 \u02c6 50 images, 3) 4 agents, tasked with \ufb01nding rred,red,green,blues respectively in 50\u02c650 images. We compare TarMAC against two baselines \u2013 1) without communication, and 2) with communication but where broadcasted messages are averaged instead of attention-weighted, so all agents receive the same message vector, similar to Sukhbaatar et al. (2016). Bene\ufb01ts of communication and attention increase with task complexity (30 \u02c6 30 \u00d1 50 \u02c6 50 & \ufb01ndrreds \u00d1 \ufb01ndrred,red,green,blues). 5.2. Traf\ufb01c Junction Environment and Task. The simulated traf\ufb01c junction environments from (Sukhbaatar et al., 2016) consist of cars moving along pre-assigned, potentially intersecting routes on one or more road junctions. The total number of cars is \ufb01xed at Nmax, and at every timestep new cars get added to the environment with probability parrive. Once a car completes its route, it becomes available to be sampled and added back to the environment with a different route assignment. Each car has a limited visibility of a 3 \u02c6 3 region around it, but is free to communicate with all other cars. The action space for each car at every timestep is gas and brake, and the reward consists of a linear time penalty \u00b40.01\u03c4, where \u03c4 is the number of timesteps since car has been active, and a collision penalty rcollision \u201c \u00b410. Easy Hard No communication 84.9\u02d84.3% 74.1\u02d83.9% CommNet (Sukhbaatar et al., 2016) 99.7\u02d80.1% 78.9\u02d83.4% TarMAC 1-round 99.9\u02d80.1% 84.6\u02d83.2% TarMAC 2-round 99.9\u02d80.1% 97.1\u02d81.6% Table 3: Success rates on traf\ufb01c junction. Our targeted 2-round communication architecture gets a success rate of 97.1\u02d81.6% on the \u2018hard\u2019 variant, signi\ufb01cantly outperforming Sukhbaatar et al. (2016). Note that 1and 2-round refer to the number of rounds of communication between actions (Equation 4). Quantitative Results. We compare our approach with CommNet (Sukhbaatar et al., 2016) on the easy and hard dif\ufb01culties of the traf\ufb01c junction environment. The easy task has one junction of two one-way roads on a 7 \u02c6 7 grid with Nmax \u201c 5 and parrive \u201c 0.30, while the hard task has four connected junctions of two-way roads on a 18\u02c618 grid with Nmax \u201c 20 and parrive \u201c 0.05. See Figure 4a, 4b for an example of the four two-way junctions in the hard task. As shown in Table 3, a no communication baseline has success rates of 84.9\u02d84.3% and 74.1\u02d83.9% on easy and hard respectively. On easy, both CommNet and TarMAC get close to 100%. On hard, TarMAC with 1-round signi\ufb01cantly outperforms CommNet with a success rate of 84.6\u02d83.2%, while 2-round further improves on this at 97.1\u02d81.6%, which is an \u201e18% absolute improvement over CommNet. We did not see gains going beyond 2 rounds in this environment. Message size vs. multi-round communication. We study performance of TarMAC with varying message value size and number of rounds of communication on the hard variant of the traf\ufb01c junction task. As can be seen in Figure 3, multiple rounds of communication leads to signi\ufb01cantly higher performance than simply increasing message size, demonstrating the advantage of multi-round communication. In fact, decreasing message size to a single scalar performs almost as well as 64-d, perhaps because even a single real number can be suf\ufb01ciently partitioned to cover the space of meanings/messages that need to be conveyed. Figure 3: Success rates for 1 vs. 2-round vs. message size on hard. Performance does not decrease signi\ufb01cantly even when the message vector is a single scalar, and 2-round communication before taking an action leads to signi\ufb01cant improvements over 1-round. Model Interpretation. Interpreting the learned policies of TarMAC, Figure 4a shows braking probabilities at different locations \u2013 cars tend to brake close to or right before entering traf\ufb01c junctions, which is reasonable since junctions have the highest chances for collisions. Turning our attention to attention probabilities (Figure 4b), we can see that cars are most-attended to when in the \u2018internal grid\u2019 \u2013 right after crossing the 1st junction and before hitting the 2nd junction. These attention probabilities are intuitive \u2013 cars \fTarMAC: Targeted Multi-Agent Communication (a) Brake probabilities at different locations on the hard traf\ufb01c junction environment. Cars tend to brake close to or right before entering junctions. (b) Attention probabilities at different locations. Cars are most attended to in the \u2018internal grid\u2019 \u2013 right after the 1st junction and before the 2nd. (c) No. of cars being attended to 1) is positively correlated with total cars, indicating that TarMAC is adaptive to dynamic team sizes, and 2) is slightly right-shifted, since it takes few steps of communication to adapt. Figure 4: Interpretation of model predictions from TarMAC in the traf\ufb01c junction environment. Figure 5: Agents navigating to the fireplace in House3D (marked in yellow). Note in particular that agent 4 is spawned facing away from it. It communicates with others, turns to face the fireplace, and moves towards it. learn to attend to speci\ufb01c sensitive locations with the most relevant local observations to avoid collisions. Finally, Figure 4c compares total number of cars in the environment vs. number of cars being attended to with probability \u0105 0.1 at any time. Interestingly, these are (loosely) positively correlated, with Spearman\u2019s \u03c3 \u201c 0.49, which shows that TarMAC is able to adapt to variable number of agents. Crucially, agents learn this dynamic targeting behavior purely from task rewards with no hand-coding! Note that the right shift between the two curves is expected, as it takes a few timesteps of communication for team size changes to propagate. At a relative time shift of 3, the Spearman\u2019s rank correlation between the two curves goes up to 0.53. 5.3. House3D Next, we benchmark TarMAC on a cooperative point-goal navigation task in House3D (Wu et al., 2018). House3D provides a rich and diverse set of publicly-available2 3D indoor environments, wherein agents do not have access to the top-down map and must navigate purely from \ufb01rstperson vision. Similar to SHAPES, the agents are tasked with \ufb01nding a speci\ufb01ed goal (such as \u2018\ufb01replace\u2019) within a max no. of steps, spawned at random locations in the environment and allowed to communicate and move around. Each agent gets a shaped reward based on progress towards the speci\ufb01ed target. An episode is successful if all agents end within 0.5m of the target object in 500 navigation steps. 2github.com/facebookresearch/house3d Table 4 shows success rates on a find[fireplace] task in House3D. A no-communication navigation policy trained with the same reward structure gets a success rate of 62.1\u02d85.3%. Mean-pooled communication (no attention) performs slightly better with a success rate of 64.3\u02d82.3%, and TarMAC achieves the best success rate at 68.9\u02d81.1%. TarMAC agents take 82.5 steps to reach the target on average vs. 101.3 for no attention vs. 186.5 for no communication. Figure 5 visualizes a predicted navigation trajectory of 4 agents. Note that the communication vectors are signi\ufb01cantly more compact (32-d) than the high-dimensional observation space (224\u02c6224 image), making our approach particularly attractive for scaling to large agent teams. Success rate Avg. # steps No communication 62.1\u02d85.3% 186.5 No attention 64.3\u02d82.3% 101.3 TarMAC 68.9\u02d81.1% 82.5 Table 4: 4-agent find[fireplace] navigation task in House3D. Note that House3D is a challenging testbed for multi-agent reinforcement learning. To get to \u201e100% accuracy, agents have to deal with high-dimensional visual observations, be able to navigate long action sequences (up to \u201e500 steps), and avoid getting stuck against objects, doors, and walls. \fTarMAC: Targeted Multi-Agent Communication (a) 3 agents, 5 \u02c6 5 grid, vision=0, max steps=20 (b) 10 agents, 20 \u02c6 20 grid, vision=1, max steps=80 Figure 6: Average no. of steps to complete an episode (lower is better) during training in the Predator-Prey mixed environment. IC3Net + TarMAC converges much faster than IC3Net, demonstrating that attentional communication helps. Shaded region shows 95% CI. 3 agents, 5 \u02c6 5, 5 agents, 10 \u02c6 10, 10 agents, 20 \u02c6 20, vision=0, max steps=20 vision=1, max steps=40 vision=1, max steps=80 CommNet (Sukhbaatar et al., 2016) 9.1\u02d80.1 13.1\u02d80.01 76.5\u02d81.3 IC3Net (Singh et al., 2019) 8.9\u02d80.02 13.0\u02d80.02 52.4\u02d83.4 IC3Net ` TarMAC 8.31\u02d80.06 12.74\u02d80.08 41.67\u02d85.82 IC3Net ` TarMAC (2-round) 7.24\u02d80.08 \u2013 35.57\u02d83.96 Table 5: Average number of steps taken to complete an episode (lower is better) at convergence in the Predator-Prey mixed environment. 5.4. Mixed and Competitive Environments Finally, we look at how to extend TarMAC to mixed and competitive scenarios. Communication via sender-receiver soft attention in TarMAC is poorly suited for competitive scenarios, since there is always \u201cleakage\u201d of the agent\u2019s state as a message to other agents via a low but non-zero attention probability, thus compromising its strategy and chances of success. Instead, an agent should \ufb01rst be able to independently decide if it wants to communicate at all, and then direct its message to speci\ufb01c recipients if it does. The recently proposed IC3Net architecture by Singh et al. (2019) addresses the former \u2013 learning when to communicate. At every timestep, each agent in IC3Net predicts a hard gating action to decide if it wants to communicate. At the receiving end, messages from agents who decide to communicate are averaged to be the next input message. Replacing this message averaging with our sender-receiver soft attention, while keeping the rest of the architecture and training details the same as IC3Net, should provide an inductive bias for more \ufb02exible communication strategies, since this model (IC3Net + TarMAC) can learn both when to communicate and whom to address messages to. We evaluate IC3Net + TarMAC on the Predator-Prey environment from Singh et al. (2019), consisting of n predators, with limited vision, moving around (with a penalty of rexplore \u201c \u00b40.05 per timestep) in search of a stationary prey. Once a predator reaches a prey, it keeps getting positive reward rprey \u201c 0.05 till end of episode i.e. till other agents reach prey or maximum no. of steps. The prey gets 0.05 per timestep only till the \ufb01rst predator reaches it, so it has incentive to not communicate its location. We compare average no. of steps for agents to reach the prey during training (Figure 6) and at convergence (Table 5). Figure 6 shows that using TarMAC with IC3Net leads to signi\ufb01cantly faster convergence than IC3Net alone, and Table 5 shows that TarMAC agents reach the prey faster. Results are averaged over 3 independent runs with different seeds. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1711.11543v2",
+                    "title": "Embodied Question Answering",
+                    "abstract": "We present a new AI task -- Embodied Question Answering (EmbodiedQA) -- where\nan agent is spawned at a random location in a 3D environment and asked a\nquestion (\"What color is the car?\"). In order to answer, the agent must first\nintelligently navigate to explore the environment, gather information through\nfirst-person (egocentric) vision, and then answer the question (\"orange\").\n  This challenging task requires a range of AI skills -- active perception,\nlanguage understanding, goal-driven navigation, commonsense reasoning, and\ngrounding of language into actions. In this work, we develop the environments,\nend-to-end-trained reinforcement learning agents, and evaluation protocols for\nEmbodiedQA.",
+                    "authors": "Abhishek Das, Samyak Datta, Georgia Gkioxari, Stefan Lee, Devi Parikh, Dhruv Batra",
+                    "published": "2017-11-30",
+                    "updated": "2017-12-01",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI",
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction The embodiment hypothesis is the idea that intelligence emerges in the interaction of an agent with an environment and as a result of sensorimotor activity. Smith and Gasser [1] Our long-term goal is to build intelligent agents that can perceive their environment (through vision, audition, or other sensors), communicate (i.e., hold a natural language dialog grounded in the environment), and act (e.g. aid humans by executing API calls or commands in a virtual or embodied environment). In addition to being a fundamental scienti\ufb01c goal in arti\ufb01cial intelligence (AI), even a small advance towards such intelligent systems can fundamentally change our lives \u2013 from assistive dialog agents for the visually impaired, to natural-language interaction with selfdriving cars, in-home robots, and personal assistants. As a step towards goal-driven agents that can perceive, communicate, and execute actions, we present a new AI task \u2013 Embodied Question Answering (EmbodiedQA), along \u2039Work partially done during an internship at Facebook AI Research. Figure 1: Embodied Question Answering \u2013 EmbodiedQA\u2013 tasks agents with navigating rich 3D environments in order to answer questions. These embodied agents must jointly learn language understanding, visual reasoning, and navigation to succeed. with virtual environments, evaluation metrics, and a novel deep reinforcement learning (RL) model for this task. Concretely, the EmbodiedQA task is illustrated in Fig. 1 \u2013 an agent is spawned at a random location in an environment (a house or building) and asked a question (e.g. \u2018What color is the car?\u2019). The agent perceives its environment through \ufb01rst-person vision (a single RGB camera) and can perform a few atomic actions: move-tforward, backward, right, leftu and turn-tright, leftu. The goal of the agent is to intelligently navigate the environment and gather the visual information necessary to answer the question. EmbodiedQA is a challenging task that subsumes several fundamental AI problems as sub-tasks. Clearly, the agent must understand language (what is the question asking?) and vision (what does a car look like?), but a successful agent must also learn to perform: Active Perception: The agent may be spawned anywhere in the environment and may not immediately \u2018see\u2019 the pixels containing the answer to the visual question (i.e. the car may not be visible). Thus, the agent must move to succeed \u2013 controlling the pixels that it will perceive. The agent must learn to map its visual input to the correct action based on its perception of the world, the underlying physical constraints, and its understanding of the question. arXiv:1711.11543v2  [cs.CV]  1 Dec 2017 \fCommon Sense Reasoning: The agent is not provided a \ufb02oor-plan or map of the environment, and must navigate from egocentric views alone. Thus, it must learn common sense (where am I? where are cars typically found in a housing compound? and where is the garage with respect to me?) similar to how humans may navigate in a house they have never visited (the car is probably in the garage outside, so I should \ufb01nd a door that leads out). Language Grounding: One commonly noted shortcoming of modern vision-and-language models is their lack of grounding \u2013 these models often fail to associate entities in text with corresponding image pixels, relying instead on dataset biases to respond seemingly intelligently even when attending to irrelevant regions [2, 3]. In EmbodiedQA, we take a goal-driven view of grounding \u2013 our agent grounds a visual question not into pixels but into a sequence of actions (\u2018garage\u2019 means to navigate towards the house exterior where the \u2018car\u2019 is usually parked). Credit Assignment: From a reinforcement learning perspective, EmbodiedQA presents a particularly challenging learning problem. Consider the question \u2018How many rooms contain chairs?\u2019. How does an agent discover that this question involves exploring the environment to visit \u2018rooms\u2019, detecting \u2018chairs\u2019, incrementing a count every time a \u2018chair\u2019 is in the view (except while the agent is in the same \u2018room\u2019), and stopping when no more \u2018rooms\u2019 can be found? All without knowing what a \u2018room\u2019 is or how to \ufb01nd it, what a \u2018chair\u2019 looks like, or what counting is. To succeed, the agent must execute a somewhat precise sequence of hundreds of inter-dependent actions (forward, forward, turn-right, forward, forward, ..., turn-left, \u20185\u2019) \u2013 all to be learned from a reward signal that says \u20184\u2019 is the right answer and anything else is incorrect. The task is complex enough that most random action sequences result in negative reward, and when things do go wrong, it\u2019s dif\ufb01cult for the agent to know why \u2013 was the question misunderstood? Can the agent not detect chairs? Did the agent navigate incorrectly? Was the counting incorrect? As the \ufb01rst step in this challenging space, we judiciously scope out a problem space \u2013 environments, question types, learning paradigm \u2013 that allow us to augment the sparse RL rewards with imitation learning (showing the agent example trajectories) and reward shaping [4] (giving intermediate \u2018getting closer or farther\u2019 navigation rewards). Speci\ufb01cally, our approach follows the recent paradigm from robotics and deep RL [5,6] \u2013 that the training environments are assumed to be suf\ufb01ciently instrumented \u2013 i.e., provide access to the agent location, depth and semantic annotations of the environment, and allow for computing obstacle-avoiding shortest paths from the agent to any target location. Crucially, at test time, our agents operate entirely from egocentric RGB vision alone \u2013 no structured representation of the environments, no access to a map, no explicit localization of the agent or mapping of the environment, no A* or any other heuristic planning, and no pre-processing or handcoded knowledge about the environment or the task of any kind. The agent in its entirety \u2013 vision, language, navigation, answering modules \u2013 is trained completely end-to-end \u2013 from raw sensory input (pixels and words) to goal-driven multi-room indoor navigation to visual question answering! Contributions. We make the following contributions: \u2022 We propose a new AI task: EmbodiedQA, where an agent spawned in an environment must intelligently navigate from an egocentric view to gather the necessary information to answer visual questions about its environment. \u2022 We introduce a novel Adaptive Computation Time [7] navigator \u2013 that decomposes navigation into a \u2018planner\u2019 that selects actions, and a \u2018controller\u2019 that executes these primitive actions a variable number of times before returning control to the planner. When the agent decides it has seen the required visual information to answer the question, it stops navigating and outputs an answer. \u2022 We initialize our agents via imitation learning and show that agents can answer questions more accurately after \ufb01ne-tuning with reinforcement learning \u2013 that is, when allowed to control their own navigation for the express purpose of answering questions accurately. Unlike some prior work, we explicitly test and demonstrate generalization of our agents to unseen environments. \u2022 We evaluate our agents in House3D [8], a rich, interactive environment based on human-designed 3D indoor scenes from the SUNCG dataset [9]. These diverse virtual environments enable us to test generalization of our agent across \ufb02oor-plans, objects, and room con\ufb01gurations \u2013 without the concerns of safety, privacy, and expense inherent to real robotic platforms. \u2022 We introduce the EQA dataset of visual questions and answers grounded in House3D. The different question types test a range of agent abilities \u2013 scene recognition (location), spatial reasoning (preposition), color recognition (color). While the EmbodiedQA task de\ufb01nition supports free-fom natural language questions, we represent each question in EQA as as a functional program that can be programmatically generated and executed on the environment to determine the answer. This gives us the ability to control the distribution of question-types and answers in the dataset, deter algorithms from exploiting dataset bias [3, 10], and provide \ufb01ne-grained breakdown of performance by skill. \u2022 We integrated House3D renderer with Amazon Mechanical Turk (AMT), allowing subjects to remotely operate the agent, and collected expert demonstrations of questionbased navigation that serve as a benchmark to compare our proposed and future algorithms. All our code and data will be made publicly available. \f2. Related Work We place our work in context by arranging prior work along the axes of vision (from a single-frame to video), language (from single-shot question answering to dialog), and action (from passive observers to active agents). When viewed from this perspective, EmbodiedQA Language Single Frame Video Single-Shot QA Dialog Vision VideoQA VQA Embodied QA Visual  Dialog presents a novel problem con\ufb01guration \u2013 single-shot QA about videos captured by goal-driven active agents. Next, we contrast this against various 2D slices in this space. VQA: Vision + Language. Like EmbodiedQA, image and video question answering tasks [11\u201315] require reasoning about natural language questions posed about visual content. The crucial difference is the lack of control \u2013 these tasks present answering agents with a \ufb01xed view of the environment (i.e. one or more images from some \ufb01xed trajectory through the world) from which the agent must answer the question, never allowing the agents to actively perceive. In contrast, EmbodiedQA agents control their trajectory and fate, for good or ill. The task is signi\ufb01cantly harder than VQA (i.e. most random paths are useless) but the agent has the \ufb02exibility to avoid confusing viewpoints and seek visual input that will maximize answer con\ufb01dence. Visual Navigation: Vision + Action. The problem of navigating in an environment based on visual perception has long been studied in vision and robotics (see [16] for an extensive survey). Classical techniques divide navigation into two distinct phases \u2013 mapping (where visual observations are used to construct a 3D model of the environment), and planning (which selects paths based on this map). Recent developments in deep RL have proposed fused architectures that go directly from egocentric visual observations to navigational actions [17\u201323]. We model our agents as similar pixel-to-action navigators. The key distinction in EmbodiedQA is how the goals are speci\ufb01ed. Visual navigation typically speci\ufb01es agent goals either implicitly via the reward function [19, 20] (thus training a separate policy for each goal/reward), or explicitly by conditioning on goal state representations [24] including images of target objects [18]. In contrast, EmbodiedQA speci\ufb01es agent goals via language, which is inherently compositional and renders training a separate policy for every task (question) infeasible. Situated Language Learning: Language + Action. Inspired by the classical work of Winograd [25], a number of recent works have revisited grounded language learning by situating agents in simple globally-perceived environments and tasking them with goals speci\ufb01ed in natural language. The form and structure of these goal speci\ufb01cations range from declarative programs [26], to simple templated commands [27,28], to free-form natural language instructions [29,30]. One key distinction in EmbodiedQA, of course, is visual sensing \u2013 the environment is only partially observable, i.e. the agent does not have access to the \ufb02oor plan, object labels, attributes, etc., and must extract this information purely from \ufb01rst-person visual sensing. Embodiment: Vision + Language + Action. Closest to EmbodiedQA are recent works that extend the situated language learning paradigm to settings where agents\u2019 perceptions are local, purely visual, and change based on their actions \u2013 a setting we refer to as embodied language learning. In concurrent and unpublished work, Hermann et al. [21] and Chaplot et al. [17] both develop embodied agents in simple game-like environments consisting of 1-2 rooms and a handful of objects with variable color and shape. In both settings, agents were able to learn to understand simple \u2018go to X\u2019/\u2018pick up X\u2019 style commands where X would specify an object (and possibly some of its attributes). Similarly, Oh et al. [23] present embodied agents in a simple mazeworld and task them to complete a series of instructions. In contrast to these approaches, our EmbodiedQA environments consist of multi-room homes (\u201e8 per home) that are densely populated by a variety of objects (\u201e54 unique objects per home). Furthermore, the instructions and commands in these works are low-level and more closely relate to actions than the questions presented in EmbodiedQA. Interactive Environments. There are a number of interactive environments commonly used in the community, ranging from simple 2D grid-worlds (e.g. XWORLD [27]), to 3D game-like environments with limited realism (e.g. DeepMind Lab [31] or Doom [17]), to more complex, realistic environments (e.g. AI2-THOR [19] or Stanford 2D-3DS [32]). While realistic environments provide rich representations of the world, most consist of only a handful of environments due to the high dif\ufb01culty of their creation. On the other hand, large sets of synthetic environments can be programmatically generated; however, they typically lack realism (either in appearance or arrangement). In this work, we use the House3D [8] environment as it strikes a useful middle-ground between simple synthetic and realistic environments. See Sec. 3.1 for more details. Hierarchical Agents. We model our EmbodiedQA agents as deep hierarchical agents that decompose the overall control problem such that a higher-level planner invokes lowerlevel controls to issue primitive actions. Such hierarchical modeling has recently shown promise in the deep reinforcement learning setting [23, 28, 33]. Our model also draws inspiration from the work on Adaptive Computation Time models of Graves [7]. \f(a) Sample Environments garage kitchen elevator office balcony patio lobby gym bathroom living room bedroom dining room (b) Queryable Rooms ironing board food processor sink rug cup desk pan bed sofa toilet piano xbox vase table chessboard towel rack television whiteboard range oven dishwasher fireplace fish tank stereo set shoe rack fruit bowl knife rack wardrobe cabinet cutting board vacuum cleaner utensil holder water dispenser coffee machine loudspeaker playstation dressing table refrigerator bookshelf microwave bathtub ottoman dresser computer washer tv stand mirror heater dryer kettle plates shower (c) Queryable Objects Figure 2: The EQA dataset is built on a subset of the environments and objects from the SUNCG [9] dataset. We show (a) sample environments and the (b) rooms and (c) objects that are asked about in the EmbodiedQA task. 3. EQA Dataset: Questions In Environments Having placed EmbodiedQA in context, we now dive deeper by outlining the environments in which our agents are embodied and the questions they must answer. We will publicly release the environments, our curated EQA dataset, and our code to aid research in this nascent area. 3.1. House3D: Interactive 3D Environments We instantiate EmbodiedQA in House3D [8], a recently introduced rich, interactive environment based on 3D indoor scenes from the SUNCG dataset [9]. Concretely, SUNCG consists of synthetic 3D scenes with realistic room and furniture layouts, manually designed using an online interior design interface (Planner5D [34]). Scenes were also further \u2018veri\ufb01ed\u2019 as realistic by majority vote of three human annotators. In total, SUNCG contains over 45k environments with 49k valid \ufb02oors, 404k rooms containing 5 million object instances of 2644 unique objects from 80 different categories. House3D converts SUNCG from a static 3D dataset to a set of virtual environments, where an agent (approximated as a cylinder 1 meter high) may navigate under simple physical constraints (not being able to pass through walls or objects). Fig. 2a shows top-down views of sample environments. Full details may be found in [8]. We build the EQA dataset on a pruned subset of environments from House3D. First, we only consider environments for which all three SUNCG annotators consider the scene layout realistic. Next, we \ufb01lter out atypical environments such as those lacking ground or those that are too small or large (only keeping houses with an internal area of 300800m2 covering at least 1{3 the total ground area). Finally, we exclude non-home environments by requiring at least one kitchen, living room, dining room, and bedroom. 3.2. Question-Answer Generation We would like to pose questions to agents that test their abilities to ground language, use common sense, reason visually, and navigate the environments. For example, answering the question \u2018What color is the car?\u2019 ostensibly requires grounding the symbol \u2018car\u2019, reasoning that cars are typically outside, navigating outside and exploring until the car is found, and visually inspecting its color. We draw inspiration from the CLEVR [35] dataset, and programmatically generate a dataset (EQA) of grounded questions and answers. This gives us the ability to control the distribution of question-types and answers in the dataset, and deter algorithms from exploiting dataset bias. Queryable Rooms and Objects. Figs. 2c, 2b show the queryable rooms (12) and objects (50) in EQA. We exclude objects and rooms from SUNCG that are obscure (e.g. loggia rooms) or dif\ufb01cult to resolve visually (e.g. very small objects like light switches). We merge some semantically similar object categories (e.g. teapot, coffee kettle) and singular vs plural forms of the same object type (e.g. (books, book)) to reduce ambiguity. Questions as Functional Programs. Each question in EQA is represented as a functional program that can be executed on the environment yielding an answer1. These functional programs are composed of a small set of elementary operations (selectp\u00a8q, uniquep\u00a8q, queryp\u00a8q, etc.) that operate on sets of room or object annotations. The number and the order of evaluation of these elementary operations de\ufb01nes a question type or template. For instance, one question type in EQA is the location template: location: \u2018What room is the <OBJ> located in?\u2019 where <OBJ> refers to one of the queryable objects. The sequence of elementary operations for this question type is: selectpobjectsq \u00d1 uniquepobjectsq \u00d1 queryplocationq. The \ufb01rst function, selectpobjectsq, gets all the object names from the environment. The second, uniquepobjectsq, retains only the objects that have a single instance in the entire house. The third, queryplocationq, generates a question (by \ufb01lling in the appropriate template) for each such object. The 2nd operation, uniquepobjectsq, is particularly important to generate unambiguous questions. For instance, if there are two air conditioners in the house, the question \u2018What room is the air conditioner lo1or a response that the question is inapplicable (e.g. referring to objects not in the environment) or ambiguous (having multiple valid answers). \fcated in?\u2019 is ambiguous, with potentially two different answers depending on which instance is being referred to. Question Types. Associated with each question type is a template for generating a question about the rooms and objects, their attributes and relationships. We de\ufb01ne nine question types and associated templates in EQA: EQA v1 $ \u2019 \u2019 \u2019 \u2019 \u2019 \u2019 & \u2019 \u2019 \u2019 \u2019 \u2019 \u2019 % location: \u2018What room is the <OBJ> located in?\u2019 color: \u2018What color is the <OBJ>?\u2019 color_room: \u2018What color is the <OBJ> in the <ROOM>?\u2019 preposition: \u2018What is <on/above/below/next-to> the <OBJ> in the <ROOM>?\u2019 existence: \u2018Is there a <OBJ> in the <ROOM>?\u2019 logical: \u2018Is there a(n) <OBJ1> and a(n) <OBJ2> in the <ROOM>?\u2019 count: \u2018How many <OBJs> in the <ROOM>?\u2019 room_count: \u2018How many <ROOMs> in the house?\u2019 distance: \u2018Is the <OBJ1> closer to the <OBJ2> than to the <OBJ3> in the <ROOM>?\u2019 The <ROOM> and <OBJ> tags above can be \ufb01lled by any valid room or object listed in Fig. 2b and Fig. 2c respectively. Given these question templates, the possible answers are room names (location), object names (preposition), yes/no (existence, logical and distance), color names (color) or numbers (count). These questions test a range of agent abilities including object detection (existence), scene recognition (location), counting (count), spatial reasoning (preposition), color recognition (color), and logical operators (logic). Moreover, many of these questions require multiple capabilities: e.g., answering a distance question requires recognizing the room and objects as well as reasoning about their spatial relation. Furthermore, the agent must do this by navigating the environment to \ufb01nd the room, looking around the room to \ufb01nd the objects, and possibly remembering their positions through time (if all three objects are not simultaneously visible). Different question types also require different degrees of navigation and memory. For instance, \u2018How many bedrooms in the house?\u2019 requires signi\ufb01cant navigation (potentially exploring the entire environment) and long-term memory (keeping track of the count), while a question like \u2018What color is the chair in the living room?\u2019 requires \ufb01nding a single room, the living room, and looking for a chair. EQA is easily extensible to include new elementary operations, question types, and templates as needed to increase the dif\ufb01culty of the task to match the development of new models. As a \ufb01rst step in this challenging space, our experiments focus on EQA v1, which consists of 4 question types \u2013 location, color, color_room, preposition. One virtue of these questions is that there is a single target queried object (<OBJ>), which enables the use of shortest paths from the agent\u2019s spawn location to the target as expert demonstrations for imitation learning (details in Section 4.1). We stress that EQA is not a static dataset, rather a curriculum of capabilities that we would like to achieve in embodied communicating agents. Question-Answer Generation and Dataset Bias. In principle, we now have the ability to automatically generate all valid questions and their associated answers for each environment by executing the functional programs on the environment\u2019s annotations provided by SUNCG. However, careful consideration is needed to make sure the developed dataset is balanced over question types and answers. For each \ufb01lled question template (e.g. \u2018What room is the refrigerator located in?\u2019), we execute its functional form on all associated environments in the dataset (i.e. those containing refrigerators) to compute the answer distribution for this question. We exclude questions for which the normalized entropy of the answer distribution is below 0.5 \u2013 e.g., an agent can simply memorize that refrigerators are almost always in kitchens, so this question would be discarded. We also exclude questions occurring in fewer than four environments as the normalized entropy estimates are unreliable. Finally, in order to benchmark performance of agents vs human performance on EQA, it is important for the questions to not be tedious or frustrating for humans to answer. We do not ask count questions for objects with high counts (>=5) or distance questions between object triplets without clear differences in distance. We set these thresholds and room / object blacklists manually based on our experience performing these tasks. Complete discussion of the question templates, functional programs, elementary operations, and various checks-andbalances can be found in the supplement. Environments Unique Questions Total Questions train 643 147 4246 val 67 104 506 test 57 105 529 25.9% 32.5% 38.7% 2.9% color color_room preposition location Figure 3: Overview of the EQA v1 dataset including dataset split statistics (left) and question type breakdown (right). EQA v1 Statistics. The EQA v1 dataset consists of over 5000 question across over 750 environments, referring to a total of 45 unique objects in 7 unique room types. The dataset is split into train, val, test such that there is no overlap in environments across splits. Fig. 3 shows the dataset splits and question type distribution. Approximately 6 questions are asked per environment on average, 22 at most, and 1 at fewest. There are relatively few \fPLNR CNN STOP \ud835\udc3c\"#$ % \ud835\udc4e\" #$ PLNR CNN TURN LEFT \u210e\"#( \ud835\udc3c\"#( % \ud835\udc4e\"#( \ud835\udc4e\" #( CNN CTRL 1 \ud835\udc4e\" #( TURN LEFT \ud835\udc3c\"#( ) CNN CTRL 0 RETURN \ud835\udc3c\"#( * PLNR CNN FORWARD \u210e\"#* \ud835\udc3c\"#* % \ud835\udc4e\"#* \ud835\udc4e\" #* CNN CTRL 1 \ud835\udc4e\" #* FORWARD \ud835\udc3c\"#* ) CNN CTRL 1 \ud835\udc4e\" #* FORWARD \ud835\udc3c\"#* * CNN CTRL 1 \ud835\udc4e\" #* FORWARD \ud835\udc3c\"#* ( CNN CTRL 0 RETURN \ud835\udc3c\"#* $ PLNR CNN TURN RIGHT \u210e\"#) \ud835\udc3c\"#) % \ud835\udc4e\"#) \ud835\udc4e\" #) CNN CTRL 0 RETURN \ud835\udc3c\"#) ) \u210e\" \ud835\udc4e\" \ud835\udc44 \ud835\udc44 \ud835\udc44 \ud835\udc44 Figure 4: Our Adaptive Computation Time (ACT) navigator splits the navigation task between a planner and a controller module. The planner selects actions and the controller decides to continue performing that action for a variable number of time steps \u2013 resulting in a decoupling of direction (\u2018turn left\u2019) and velocity (\u20185 times\u2019) and strengthening the long-term gradient \ufb02ows of the planner module. preposition questions as many frequently occurring spatial relations are too easy to resolve without exploration and fail the entropy thresholding. We will make EQA v1 and the entire generation engine publicly available. 4. A Hierarchical Model for EmbodiedQA We now introduce our proposed neural architecture for an EmbodiedQA agent. Recall that the agent is spawned at a random location in the environment, receives a question, and perceives only through a single egocentric RGB camera. Importantly, unlike some prior work [26\u201330], in EmbodiedQA, the agent does not receive any global or structured representation of the environment (map, location, objects, rooms), or of the task (the functional program that generated the question). Overview of the Agent. The agent has 4 natural modules \u2013 vision, language, navigation, answering \u2013 and is trained from raw sensory input (pixels and words) to goal-driven multi-room indoor navigation to visual question answering. The modules themselves are built up largely from conventional neural building blocks \u2013 Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs). One key technical novelty in our model is the use of Adaptive Computation Time (ACT) RNNs by Graves [7], which is an elegant approach for allowing RNNs to learn how many computational steps to take between receiving an input and emitting an output by back-propagating through a \u2018halting\u2019 layer. We make use of this idea in our navigation module to cleanly separate the decision between \u2013 direction (where to move, decided by a \u2018planner\u2019) and velocity (how far to move, decided by a \u2018controller\u2019). Fig. 4 illustrates the different modules in our agent, which we describe next. Vision. Our agent takes egocentric 224\u02c6224 RGB images from the House3D renderer as input, which we process with a CNN consisting of 4 t5\u02c65 Conv, ReLU, BatchNorm, 2\u02c62 MaxPoolu blocks, producing a \ufb01xed-size representation. A strong visual system for EmbodiedQA should encode information about object attributes (i.e. colors and textures), semantics (i.e. object categories), and environmental geometry (i.e. depth). As such, we pretrain the CNN under a multi-task pixel-to-pixel prediction framework \u2013 treating the above CNN as an encoder network, we train multiple network heads to decode the 1) original RGB values, 2) semantic class, and 3) depth of each pixel (which can be obtained from the House3D renderer). See the supplementary material for full model and training details. Language. Our agents also receive questions which we encode with 2-layer LSTMs with 128-dim hidden states. Note that we learn separate question encoders for the navigation and answering modules \u2013 as each may need to focus on different parts of the question. For instance, in the question \u2018What color is the chair in the kitchen?\u2019, \u2018color\u2019 is irrelevant for navigation and \u2018kitchen\u2019 matters little for question answering (once in the kitchen). Navigation. We introduce a novel Adaptive Computation Time (ACT) navigator that decomposes navigation into a \u2018planner\u2019, that selects actions (forward, left, right), and a \u2018controller\u2019, that executes these primitive actions a variable number of times (1,2, ...) before returning control back to the planner. Intuitively, this structure separates the intention of the agent (i.e. get to the other end of the room) from the series of primitive actions required to achieve this directive (i.e. \u2018forward, forward, forward, ...\u2019), and is reminiscent of hierarchical RL approaches [23, 28, 33]. This division also allows the planner to have variable time steps between decisions, strengthening long-term gradient \ufb02ows. Formally, let t \u201c 1, 2, . . . , T denote planner timestamps, and n \u201c 0, 1, 2, . . . Nptq denote the variable number of controller steps. Let In t denote the encoding of the image observed at t-th planner-time and n-th controller-step. We instantiate the planner as an LSTM. Thus, the planner maintains a hidden state ht (that is updated only at planner timesteps), and samples an action at P tforward, turn-left, turn-right, stop-navigationu: \fat, ht \u00d0 PLNR ` ht\u00b41, I0 t , Q, at\u00b41 \u02d8 , (1) where Q is the question encoding. After taking this action, the planner passes control to the controller, which considers the planner\u2019s state and the current frame to decide to continue performing at or to return control to the planner, i.e. t0, 1u Q cn t \u00d0 CTRL pht, at, In t q (2) If cn t \u201c 1 then the action at repeats and CTRL is applied to the next frame. Else if cn t \u201c 0 or a max of 5 controllersteps has been reached, control is returned to the planner. We instantiate the controller as a feed-forward multi-layer perceptron with 1 hidden layer. Intuitively, the planner encodes \u2018intent\u2019 into the state encoding ht and the chosen action at, and the controller keeps going until the visual input In t aligns with the intent of the planner. Question Answering. After the agent decides to stop (or a max number of actions have been taken), the question answering module is executed to provide an answer based on the sequence of frames I1 1, . . . , In T the agent has observed throughout its trajectory. The answering module attends to each of the last \ufb01ve frame, computes an attention pooled visual encoding based on image-question similarity, combines these with an LSTM encoding of the question, and outputs a softmax over the space of 172 possible answers. 4.1. Imitation Learning and Reward Shaping We employ a two-stage training process. First, the navigation and answering modules are independently trained using imitation/supervised learning on automatically generated expert demonstrations of navigation. Second, the entire architecture is jointly \ufb01ne-tuned using policy gradients. Independent Pretraining via Imitation Learning. Most questions that could be asked in EmbodiedQA do not have a natural \u2018correct\u2019 navigation required to answer them. As mentioned in Section 3.2, one virtue of EQA v1 questions is that they contain a single target queried object (<OBJ>). This allows us to use the shortest path from the agent\u2019s spawn location to the target as an expert demonstration. The navigation module is trained to mimic the shortest path actions in a teacher forcing setting i.e., given the history encoding, question encoding, and the current frame, the model is trained to predict the action that would keep it on the shortest path. We use a cross-entropy loss and train the model for 15 epochs. We \ufb01nd that even in this imitation learning case, it is essential to train the navigator under a distance-based curriculum. In the \ufb01rst epoch, we backtrack 10 steps from the target along the shortest path and initialize the agent at this point with the full history of the trajectory from the spawned location. We step back an additional 10 steps at each successive training epoch. We train for 15 epochs total with batch size ranging from 5 to 20 questions (depending on path length due to memory limitations). The question answering module is trained into predict the correct answer based on the question and the frames seen on the shortest path. We apply standard cross-entropy training over 50 epochs with a batch size of 20. Target-aware Navigational Fine-tuning. While the navigation and answering modules that result from imitation learning perform well on their independent tasks, they are poorly suited to dealing with each other. Speci\ufb01cally, both modules are used to following the provided shortest path, but when in control the navigator may generalize poorly and provide the question answerer with unhelpful views of target (if it \ufb01nds it at all). Rather than try to force the answering agent to provide correct answers from noisy or absent views, we freeze it and \ufb01ne-tune the navigator. We provide two types of reward signals to the navigator: the question answering accuracy achieved at the end of the navigation and a reward shaping [4] term that gives intermediate rewards for getting closer to the target. Speci\ufb01cally, the answering reward is 5 if the agent chooses to stop and answers correctly and 0 otherwise. The navigational reward for forward actions is 0.005 times the change in distance to target object (there is no reward or penalty for turning). We train the agent with REINFORCE [36] policy gradients with a running average baseline for the answer reward. As in the imitation learning setting, we follow a curriculum of increasing distance between spawn and target locations. Training details. All LSTMs are 2-layered with a 128-d hidden state. We use Adam [37] with a learning rate of 10\u00b43, and clamp gradients to r\u00b45, 5s. We incrementally load environments in memory and use a batch size of 10 both during the imitation learning and REINFORCE \ufb01netuning stages. One forward step corresponds to at most 0.25 metres, and it takes 40 turns to turn 360\u02dd, i.e. one right or left turn action leads to 9\u02dd change in viewing angle. Backward and strafe motions are not allowed. We snap the continuous renderer space to a 1000\u02c61000 grid to check for obstacles. Our entire codebase will be publicly available. 5. Experiments and Results The ultimate goal of an EmbodiedQA agent is to answer questions accurately. However, it is important to disentangle success/failure at the intermediate task of navigation from the ultimate downstream task of question answering. Question Answering Accuracy. Our agent (and all baselines) produce a probability distribution over 172 possible answers (colors, rooms, objects). We report the mean rank (MR) of the ground-truth answer in the answer list sorted by the agent\u2019s beliefs, where the mean is computed over all test questions and environments. Navigation Accuracy. We evaluate navigation performance on EQA v1 by reporting the distance to the target object at navigation termination pdTq, change in distance \fNavigation QA dT d\u2206 dmin %rT %r\u00ea %stop MR T\u00b410 T\u00b430 T\u00b450 T\u00b410 T\u00b430 T\u00b450 T\u00b410 T\u00b430 T\u00b450 T\u00b410 T\u00b430 T\u00b450 T\u00b410 T\u00b430 T\u00b450 T\u00b410 T\u00b430 T\u00b450 T\u00b410 T\u00b430 T\u00b450 Baselines $ \u2019 \u2019 \u2019 & \u2019 \u2019 \u2019 % Reactive 2.09 2.72 3.14 -1.44 -1.09 -0.31 0.29 1.01 1.82 50% 49% 47% 52% 53% 48% 3.18 3.56 3.31 LSTM 1.75 2.37 2.90 -1.10 -0.74 -0.07 0.34 1.06 2.05 55% 53% 44% 59% 57% 50% 80% 75% 80% 3.35 3.07 3.55 Reactive+Q 1.58 2.27 2.89 -0.94 -0.63 -0.06 0.31 1.09 1.96 52% 51% 45% 55% 57% 54% 3.17 3.54 3.37 LSTM+Q 1.13 2.23 2.89 -0.48 -0.59 -0.06 0.28 0.97 1.91 63% 53% 45% 64% 59% 54% 80% 71% 68% 3.11 3.39 3.31 Us \" ACT+Q 0.46 1.50 2.74 0.16 0.15 0.12 0.42 1.42 2.63 58% 54% 45% 60% 56% 46% 100% 100% 100% 3.09 3.13 3.25 ACT+Q-RL 1.67 2.19 2.86 -1.05 -0.52 0.01 0.24 0.93 1.94 57% 56% 45% 65% 62% 52% 32% 32% 24% 3.13 2.99 3.22 Oracle \" HumanNav\u02da 0.81 0.81 0.81 0.44 1.62 2.85 0.33 0.33 0.33 86% 86% 86% 87% 89% 89% ShortestPath+VQA 0.85 2.78 4.86 3.21 3.21 3.21 Table 1: Quantitative evaluation of EmbodiedQA agents on navigation and answering metrics for the EQA v1 test set. Ill-de\ufb01ned cells are marked with \u2018-\u2019 because 1) reactive models don\u2019t have a stopping action, 2) humans pick a single answer from a drop-down list, so mean rank is not de\ufb01ned, 3) most distance metrics are trivially de\ufb01ned for shortest paths since they always end at the target object by design. what color is the fish tank/bowl in the living room? what room is the vase located in? Figure 5: Sample trajectories from ACT+Q-RL agent projected on a \ufb02oor plan (white areas are unoccupiable) and on-path egocentric views. The agent moves closer to already visible objects \u2013 potentially improving its perception of the objects. Note that the \ufb02oor plan is shown only for illustration and not available to the agents. to target from initial to \ufb01nal position pd\u2206q, and the smallest distance to the target at any point in the episode pdminq. All distances are measured in meters along the shortest path to the target. We also record the percentage of questions for which an agent either terminates in p%rTq or ever enters p%r\u00eaq the room containing the target object(s). Finally, we also report the percent of episodes in which agents choose to terminate navigation and answer before reaching the maximum episode length p%stopq. To sweep the dif\ufb01culty of the task at test time, we spawn the agent 10, 30, or 50 actions away from the target and report each metric for T\u00b410, T\u00b430, T\u00b450 settings. Navigation Baselines. We compare our ACT navigator with a number of sophisticated baselines and ablations. Reactive CNN. This is a feedforward network that uses the last-n frames to predict the next action. We tried n \u201c t1, 3, 5, 10u and report n \u201c 5, which worked best. Note that this is a target-agnostic baseline (i.e., is not aware of the question). The purpose of this baseline is to check whether simply memorizing frames from training environments generalizes to test (it does not). Reactive CNN+Question. This combines the frame representation (as above) with an LSTM encoding of the question to predict the next action. This is similar to the approach of [18], with the difference that the goal is speci\ufb01ed via a question encoding instead of a target image. Note that the action space for both the reactive baselines is tforward, turn-left, turn-rightu, there is no stop token. At test time, the model is run for max no. of actions (\u201c 100). LSTM+Question. The above two are memoryless navigators. This LSTM navigator takes as input the encodings of the question, current frame, and previous action, and predicts the next action. Note that these are identical inputs/outputs as our ACT navigator. The purpose of comparing to this ablation of our approach is to establish the bene\ufb01t of our proposed planner-controller architecture. We also compare against an ablated version of this baseline without the question encoding as input (LSTM). Navigation Oracles. We compare against two oracles: HumanNav* denotes goal-driven navigations by AMT workers remotely operating the agent (\u02da denotes that data human studies were conducted on a subset set of test). ShortestPaths+VQA denotes the question answering performance achieved by our answering module when fed in shortest path at test time. Table 1 shows the results of all baselines compared with our approach trained with just imitation learning (ACT+Q) and our approach \ufb01ne-tuned with RL (ACT+Q-RL)2. We make a few key observations: \u2022 All baselines are poor navigators. All baselines methods have negative d\u2206, i.e. they end up farther from the target than where they start. This con\ufb01rms our intuition that EmbodiedQA is indeed a dif\ufb01cult problem. \u2022 Memory helps. All models start equally far away from the target. Baselines augmented with memory (LSTM vs Reactive and LSTM-Q vs Reactive-Q) end closer to the target, i.e. achieve smaller dT, than those without. 2youtube.com/watch?v=gVj-TeIJfrk shows example navigation and answer predictions by our agent. \f\u2022 ACT Navigators performs best. Our proposed navigator (ACT+Q) achieves the smallest distance to target at the end of navigation (dT), and the RL-\ufb01netuned navigator (ACT+Q-RL) achieves the highest answering accuracy. \u2022 RL agent overshoots. Interestingly, we observe that while our RL-\ufb01netuned agent (ACT+Q-RL) gets closest to the target in its trajectory (i.e., achieves least dmin) and enters the target room most often (i.e., achieves highest %r\u00ea), it does not end closest to the target (i.e., does not achieve highest d\u2206). These statistics and our qualitative analysis suggests that this is because RL-\ufb01netuned agents learn to explore, with a lower stopping rate (%stop), and often overshoot the target. This is consistent with observations in literature [6]. In EmbodiedQA, this overshooting behavior does not hurt the question answering accuracy because the answering module can attend to frames along the trajectory. This behavior can be corrected by including a small negative reward for each action. \u2022 Shortest paths are not optimal for VQA. A number of methods outperform ShortestPath+VQA in terms of answering accuracy. This is because while the shortest path clearly takes an agent to the target object, it may not provide the best vantage to answer the question. In future work, these may be improved by ray tracing methods to appropriately frame of the target object at termination. 6."
+                }
+            ],
+            "Aditya Grover": [
+                {
+                    "url": "http://arxiv.org/abs/1906.09531v2",
+                    "title": "Bias Correction of Learned Generative Models using Likelihood-Free Importance Weighting",
+                    "abstract": "A learned generative model often produces biased statistics relative to the\nunderlying data distribution. A standard technique to correct this bias is\nimportance sampling, where samples from the model are weighted by the\nlikelihood ratio under model and true distributions. When the likelihood ratio\nis unknown, it can be estimated by training a probabilistic classifier to\ndistinguish samples from the two distributions. We employ this likelihood-free\nimportance weighting method to correct for the bias in generative models. We\nfind that this technique consistently improves standard goodness-of-fit metrics\nfor evaluating the sample quality of state-of-the-art deep generative models,\nsuggesting reduced bias. Finally, we demonstrate its utility on representative\napplications in a) data augmentation for classification using generative\nadversarial networks, and b) model-based policy evaluation using off-policy\ndata.",
+                    "authors": "Aditya Grover, Jiaming Song, Alekh Agarwal, Kenneth Tran, Ashish Kapoor, Eric Horvitz, Stefano Ermon",
+                    "published": "2019-06-23",
+                    "updated": "2019-11-03",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG",
+                        "cs.NE"
+                    ],
+                    "main_content": "Introduction Learning generative models of complex environments from high-dimensional observations is a longstanding challenge in machine learning. Once learned, these models are used to draw inferences and to plan future actions. For example, in data augmentation, samples from a learned model are used to enrich a dataset for supervised learning [1]. In model-based off-policy policy evaluation (henceforth MBOPE), a learned dynamics model is used to simulate and evaluate a target policy without real-world deployment [2], which is especially valuable for risk-sensitive applications [3]. In spite of the recent successes of deep generative models, existing theoretical results show that learning distributions in an unbiased manner is either impossible or has prohibitive sample complexity [4, 5]. Consequently, the models used in practice are inherently biased,1 and can lead to misleading downstream inferences. In order to address this issue, we start from the observation that many typical uses of generative models involve computing expectations under the model. For instance, in MBOPE, we seek to \ufb01nd the expected return of a policy under a trajectory distribution de\ufb01ned by this policy and a learned dynamics model. A classical recipe for correcting the bias in expectations, when samples from a different distribution than the ground truth are available, is to importance weight the samples according to the likelihood ratio [6]. If the importance weights were exact, the resulting estimates are unbiased. But in practice, the likelihood ratio is unknown and needs to be estimated since the true data distribution is unknown and even the model likelihood is intractable or ill-de\ufb01ned for many deep generative models, e.g., variational autoencoders [7] and generative adversarial networks [8]. Our proposed solution to estimate the importance weights is to train a calibrated, probabilistic classi\ufb01er to distinguish samples from the data distribution and the generative model. As shown in prior work, the output of such classi\ufb01ers can be used to extract density ratios [9]. Appealingly, this estimation procedure is likelihood-free since it only requires samples from the two distributions. 1We call a generative model biased if it produces biased statistics relative to the true data distribution. 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada. arXiv:1906.09531v2  [stat.ML]  3 Nov 2019 \fTogether, the generative model and the importance weighting function (speci\ufb01ed via a binary classi\ufb01er) induce a new unnormalized distribution. While exact density estimation and sampling from this induced distribution is intractable, we can derive a particle based approximation which permits ef\ufb01cient sampling via resampling based methods. We derive conditions on the quality of the weighting function such that the induced distribution provably improves the \ufb01t to the the data distribution. Empirically, we evaluate our bias reduction framework on three main sets of experiments. First, we consider goodness-of-\ufb01t metrics for evaluating sample quality metrics of a likelihood-based and a likelihood-free state-of-the-art (SOTA) model on the CIFAR-10 dataset. All these metrics are de\ufb01ned as Monte Carlo estimates from the generated samples. By importance weighting samples, we observe a bias reduction of 23.35% and 13.48% averaged across commonly used sample quality metrics on PixelCNN++ [10] and SNGAN [11] models respectively. Next, we demonstrate the utility of our approach on the task of data augmentation for multi-class classi\ufb01cation on the Omniglot dataset [12]. We show that, while naively extending the model with samples from a data augmentation, a generative adversarial network [1] is not very effective for multiclass classi\ufb01cation, we can improve classi\ufb01cation accuracy from 66.03% to 68.18% by importance weighting the contributions of each augmented data point. Finally, we demonstrate bias reduction for MBOPE [13]. A typical MBOPE approach is to \ufb01rst estimate a generative model of the dynamics using off-policy data and then evaluate the policy via Monte Carlo [2, 14]. Again, we observe that correcting the bias of the estimated dynamics model via importance weighting reduces RMSE for MBOPE by 50.25% on 3 MuJoCo environments [15]. 2 Preliminaries Notation. Unless explicitly stated otherwise, we assume that probability distributions admit absolutely continuous densities on a suitable reference measure. We use uppercase notation X, Y, Z to denote random variables and lowercase notation x, y, z to denote speci\ufb01c values in the corresponding sample spaces X, Y, Z. We use boldface for multivariate random variables and their vector values. Background. Consider a \ufb01nite dataset Dtrain of instances x drawn i.i.d. from a \ufb01xed (unknown) distribution pdata. Given Dtrain, the goal of generative modeling is to learn a distribution p\u03b8 to approximate pdata. Here, \u03b8 denotes the model parameters, e.g. weights in a neural network for deep generative models. The parameters can be learned via maximum likelihood estimation (MLE) as in the case of autoregressive models [16], normalizing \ufb02ows [17], and variational autoencoders [7, 18], or via adversarial training e.g., using generative adversarial networks [8, 19] and variants. Monte Carlo Evaluation We are interested in use cases where the goal is to evaluate or optimize expectations of functions under some distribution p (either equal or close to the data distribution pdata). Assuming access to samples from p as well some generative model p\u03b8, one extreme is to evaluate the sample average using the samples from p alone. However, this ignores the availability of p\u03b8, through which we have a virtually unlimited access of generated samples ignoring computational constraints and hence, could improve the accuracy of our estimates when p\u03b8 is close to p. We begin by presenting a direct motivating use case of data augmentation using generative models for training classi\ufb01ers which generalize better. Example Use Case: Suf\ufb01cient labeled training data for learning classi\ufb01cation and regression system is often expensive to obtain or susceptible to noise. Data augmentation seeks to overcome this shortcoming by arti\ufb01cially injecting new datapoints into the training set. These new datapoints are derived from an existing labeled dataset, either by manual transformations (e.g., rotations, \ufb02ips for images), or alternatively, learned via a generative model [1, 20]. Consider a supervised learning task over a labeled dataset Dcl. The dataset consists of feature and label pairs (x, y), each of which is assumed to be sampled independently from a data distribution pdata(x, y) de\ufb01ned over X \u00d7 Y. Further, let Y \u2286Rk. In order to learn a classi\ufb01er f\u03c8 : X \u2192Rk with parameters \u03c8, we minimize the expectation of a loss \u2113: Y \u00d7 Rk \u2192R over the dataset Dcl: Epdata(x,y)[\u2113(y, f\u03c8(x))] \u2248 1 |Dcl| X (x,y)\u223cDcl \u2113(y, f\u03c8(x)). (1) 2 \fE.g., \u2113could be the cross-entropy loss. A generative model for the task of data augmentation learns a joint distribution p\u03b8(x, y). Several algorithmic variants exist for learning the model\u2019s joint distribution and we defer the speci\ufb01cs to the experiments section. Once the generative model is learned, it can be used to optimize the expected classi\ufb01cation loss in Eq. (1) under a mixture distribution of empirical data distributions and generative model distributions given as: pmix(x, y) = mpdata(x, y) + (1 \u2212m)p\u03b8(x, y) (2) for a suitable choice of the mixture weights m \u2208[0, 1]. Notice that, while the eventual task here is optimization, reliably evaluating the expected loss of a candidate parameter \u03c8 is an important ingredient. We focus on this basic question \ufb01rst in advance of leveraging the solution for data augmentation. Further, even if evaluating the expectation once is easy, optimization requires us to do repeated evaluation (for different values of \u03c8) which is signi\ufb01cantly more challenging. Also observe that the distribution p under which we seek expectations is same as pdata here, and we rely on the generalization of p\u03b8 to generate transformations of an instance in the dataset which are not explicitly present, but plausibly observed in other, similar instances [21]. 3 Likelihood-Free Importance Weighting Whenever the distribution p, under which we seek expectations, differs from p\u03b8, model-based estimates exhibit bias. In this section, we start out by formalizing bias for Monte Carlo expectations and subsequently propose a bias reduction strategy based on likelihood-free importance weighting (LFIW). We are interested in evaluating expectations of a class of functions of interest f \u2208F w.r.t. the distribution p. For any given f : X \u2192R, we have Ex\u223cp[f(x)] = R p(x)f(x)dx. Given access to samples from a generative model p\u03b8, if we knew the densities for both p and p\u03b8, then a classical scheme to evaluate expectations under p using samples from p\u03b8 is to use importance sampling [6]. We reweight each sample from p\u03b8 according to its likelihood ratio under p and p\u03b8 and compute a weighted average of the function f over these samples. Ex\u223cp[f(x)] = Ex\u223cp\u03b8 \u0014 p(x) p\u03b8(x)f(x) \u0015 \u22481 T T X i=1 w(xi)f(xi) (3) where w(xi) := p(xi)/p\u03b8(xi) is the importance weight for xi \u223cp\u03b8. The validity of this procedure is subject to the use of a proposal p\u03b8 such that for all x \u2208X where p\u03b8(x) = 0, we also have f(x)p(x) = 0.2 To apply this technique to reduce the bias of a generative sampler p\u03b8 w.r.t. p, we require knowledge of the importance weights w(x) for any x \u223cp\u03b8. However, we typically only have a sampling access to p via \ufb01nite datasets. For instance, in the data augmentation example above, where p = pdata, the unknown distribution used to learn p\u03b8. Hence we need a scheme to learn the weights w(x), using samples from p and p\u03b8, which is the problem we tackle next.In order to do this, we consider a binary classi\ufb01cation problem over X \u00d7 Y where Y = {0, 1} and the joint distribution is denoted as q(x, y). Let \u03b3 = q(y=0) q(y=1) > 0 denote any \ufb01xed odds ratio. To specify the joint q(x, y), we additionally need the conditional q(x|y) which we de\ufb01ne as follows: q(x|y) = \u001ap\u03b8(x) if y = 0 p(x) otherwise. (4) Since we only assume sample access to p and p\u03b8(x), our strategy would be to estimate the conditional above via learning a probabilistic binary classi\ufb01er. To train the classi\ufb01er, we only require datasets of samples from p\u03b8(x) and p(x) and estimate \u03b3 to be the ratio of the size of two datasets. Let c\u03c6 : X \u2192[0, 1] denote the probability assigned by the classi\ufb01er with parameters \u03c6 to a sample x belonging to the positive class y = 1. As shown in prior work [9, 22], if c\u03c6 is Bayes optimal, then the importance weights can be obtained via this classi\ufb01er as: w\u03c6(x) = p(x) p\u03b8(x) = \u03b3 c\u03c6(x) 1 \u2212c\u03c6(x). (5) 2A stronger suf\ufb01cient, but not necessary condition that is independent of f, states that the proposal p\u03b8 is valid if it has a support larger than p, i.e., for all x \u2208X, p\u03b8(x) = 0 implies p(x) = 0. 3 \f(a) Setup (b) n = 50 (c) n = 100 (d) n = 1000 Figure 1: Importance Weight Estimation using Probabilistic Classi\ufb01ers. (a) A univariate Gaussian (blue) is \ufb01t to samples from a mixture of two Gaussians (red). (b-d) Estimated class probabilities (with 95% con\ufb01dence intervals based on 1000 bootstraps) for varying number of points n, where n is the number of points used for training the generative model and multilayer perceptron. In practice, we do not have access to a Bayes optimal classi\ufb01er and hence, the estimated importance weights will not be exact. Consequently, we can hope to reduce the bias as opposed to eliminating it entirely. Hence, our default LFIW estimator is given as: Ex\u223cp[f(x)] \u22481 T T X i=1 \u02c6 w\u03c6(xi)f(xi) (6) where \u02c6 w\u03c6(xi) = \u03b3 c\u03c6(xi) 1\u2212c\u03c6(xi) is the importance weight for xi \u223cp\u03b8 estimated via c\u03c6(x). Practical Considerations. Besides imperfections in the classi\ufb01er, the quality of a generative model also dictates the ef\ufb01cacy of importance weighting. For example, images generated by deep generative models often possess distinct artifacts which can be exploited by the classi\ufb01er to give highly-con\ufb01dent predictions [23, 24]. This could lead to very small importance weights for some generated images, and consequently greater relative variance in the importance weights across the Monte Carlo batch. Below, we present some practical variants of LFIW estimator to offset this challenge. 1. Self-normalization: The self-normalized LFIW estimator for Monte Carlo evaluation normalizes the importance weights across a sampled batch: Ex\u223cp[f(x)] \u2248 T X i=1 \u02c6 w\u03c6(xi) PT j=1 \u02c6 w\u03c6(xj) f(xi) where xi \u223cp\u03b8. (7) 2. Flattening: The \ufb02attened LFIW estimator interpolates between the uniform importance weights and the default LFIW weights via a power scaling parameter \u03b1 \u22650: Ex\u223cp[f(x)] \u22481 T T X i=1 \u02c6 w\u03c6(xi)\u03b1f(xi) where xi \u223cp\u03b8. (8) For \u03b1 = 0, there is no bias correction, and \u03b1 = 1 returns the default estimator in Eq. (6). For intermediate values of \u03b1, we can trade-off bias reduction with any undesirable variance introduced. 3. Clipping: The clipped LFIW estimator speci\ufb01es a lower bound \u03b2 \u22650 on the importance weights: Ex\u223cp[f(x)] \u22481 T T X i=1 max( \u02c6 w\u03c6(xi), \u03b2)f(xi) where xi \u223cp\u03b8. (9) When \u03b2 = 0, we recover the default LFIW estimator in Eq. (6). Finally, we note that these estimators are not exclusive and can be combined e.g., \ufb02attened or clipped weights can be normalized. Con\ufb01dence intervals. Since we have real and generated data coming from a \ufb01nite dataset and parametric model respectively, we propose a combination of empirical and parametric bootstraps to derive con\ufb01dence intervals around the estimated importance weights. See Appendix A for details. Synthetic experiment. We visually illustrate our importance weighting approach in a toy experiment (Figure 1a). We are given a \ufb01nite set of samples drawn from a mixture of two Gaussians (red). The model family is a unimodal Gaussian, illustrating mismatch due to a parametric model. The mean 4 \fAlgorithm 1 SIR for the Importance Resampled Generative Model p\u03b8,\u03c6 Input: Generative Model p\u03b8, Importance Weight Estimator \u02c6 w\u03c6, budget T 1: Sample x1, x2, . . . , xT independently from p\u03b8 2: Estimate importance weights \u02c6 w(x1), \u02c6 w(x2), . . . , \u02c6 w(xT ) 3: Compute \u02c6 Z \u2190PT t=1 \u02c6 w(xt) 4: Sample j \u223cCategorical \u0010 \u02c6 w(x1) \u02c6 Z , \u02c6 w(x2) \u02c6 Z , . . . , \u02c6 w(xT ) \u02c6 Z \u0011 5: return xj and variance of the model are estimated by the empirical means and variances of the observed data. Using estimated model parameters, we then draw samples from the model (blue). In Figure 1b, we show the probability assigned by a binary classi\ufb01er to a point to be from true data distribution. Here, the classi\ufb01er is a single hidden-layer multi-layer perceptron. The classi\ufb01er is not Bayes optimal, which can be seen by the gaps between the optimal probabilities curve (black) and the estimated class probability curve (green). However, as we increase the number of real and generated examples n in Figures 1c-d, the classi\ufb01er approaches optimality. Furthermore, even its uncertainty shrinks with increasing data, as expected. In summary, this experiment demonstrates how a binary classi\ufb01er can mitigate this bias due to a mismatched generative model. 4 Importance Resampled Generative Modeling In the previous section, we described a procedure to augment any base generative model p\u03b8 with an importance weighting estimator \u02c6 w\u03c6 for debiased Monte Carlo evaluation. Here, we will use this augmentation to induce an importance resampled generative model with density p\u03b8,\u03c6 given as: p\u03b8,\u03c6(x) \u221dp\u03b8(x) \u02c6 w\u03c6(x) (10) where the partition function is expressed as Z\u03b8,\u03c6 = R p\u03b8(x) \u02c6 w\u03c6(x)dx = Ep\u03b8[ \u02c6 w\u03c6(x)]. Density Estimation. Exact density estimation requires a handle on the density of the base model p\u03b8 (typically intractable for models such as VAEs and GANs) and estimates of the partition function. Exactly computing the partition function is intractable. If p\u03b8 permits fast sampling and importance weights are estimated via LFIW (requiring only a forward pass through the classi\ufb01er network), we can obtain unbiased estimates via a Monte Carlo average, i.e., Z\u03b8,\u03c6 \u22481 T PT i=1 \u02c6 w\u03c6(xi) where xi \u223cp\u03b8. To reduce the variance, a potentially large number of samples are required. Since samples are obtained independently, the terms in the Monte Carlo average can be evaluated in parallel. Sampling-Importance-Resampling. While exact sampling from p\u03b8,\u03c6 is intractable, we can instead perform sample from a particle-based approximation to p\u03b8,\u03c6 via sampling-importance-resampling [25, 26] (SIR). We de\ufb01ne the SIR approximation to p\u03b8,\u03c6 via the following density: pSIR \u03b8,\u03c6 (x; T) := Ex2,x3,...,xT \u223cp\u03b8 \" \u02c6 w\u03c6(x) \u02c6 w\u03c6(x) + PT i=2 \u02c6 w\u03c6(xi) p\u03b8(x) # (11) where T > 0 denotes the number of independent samples (or \u201cparticles\"). For any \ufb01nite T, sampling from pSIR \u03b8,\u03c6 is tractable, as summarized in Algorithm 1. Moreover, any expectation w.r.t. the SIR approximation to the induced distribution can be evaluated in closed-form using the self-normalized LFIW estimator (Eq. 7). In the limit of T \u2192\u221e, we recover the induced distribution p\u03b8,\u03c6: lim T \u2192\u221epSIR \u03b8,\u03c6 (x; T) = p\u03b8,\u03c6(x) \u2200x (12) Next, we analyze conditions under which the resampled density p\u03b8,\u03c6 provably improves the model \ufb01t to pdata. In order to do so, we further assume that pdata is absolutely continuous w.r.t. p\u03b8 and p\u03b8,\u03c6. We de\ufb01ne the change in KL via the importance resampled density as: \u2206(pdata, p\u03b8, p\u03b8,\u03c6) := DKL(pdata, p\u03b8,\u03c6) \u2212DKL(pdata, p\u03b8). (13) Substituting Eq. 10 in Eq. 13, we can simplify the above quantity as: \u2206(pdata, p\u03b8, p\u03b8,\u03c6) = Ex\u223cpdata[\u2212log(p\u03b8(x) \u02c6 w\u03c6(x)) + log Z\u03b8,\u03c6 + log p\u03b8(x)] (14) = Ex\u223cpdata[log \u02c6 w\u03c6(x)] \u2212log Ex\u223cp\u03b8[ \u02c6 w\u03c6(x)]. (15) 5 \fTable 1: Goodness-of-\ufb01t evaluation on CIFAR-10 dataset for PixelCNN++ and SNGAN. Standard errors computed over 10 runs. Higher IS is better. Lower FID and KID scores are better. Model Evaluation IS (\u2191) FID (\u2193) KID (\u2193) Reference 11.09 \u00b1 0.1263 5.20 \u00b1 0.0533 0.008 \u00b1 0.0004 PixelCNN++ Default (no debiasing) 5.16 \u00b1 0.0117 58.70 \u00b1 0.0506 0.196 \u00b1 0.0001 LFIW 6.68 \u00b1 0.0773 55.83 \u00b1 0.9695 0.126 \u00b1 0.0009 SNGAN Default (no debiasing) 8.33\u00b1 0.0280 20.40 \u00b1 0.0747 0.094 \u00b1 0.0002 LFIW 8.57 \u00b1 0.0325 17.29 \u00b1 0.0698 0.073 \u00b10.0004 The above expression provides a necessary and suf\ufb01cient condition for any positive real valued function (such as the LFIW classi\ufb01er in Section 3) to improve the KL divergence \ufb01t to the underlying data distribution. In practice, an unbiased estimate of the LHS can be obtained via Monte Carlo averaging of logimportance weights based on Dtrain. The empirical estimate for the RHS is however biased.3 To remedy this shortcoming, we consider the following necessary but insuf\ufb01cient condition. Proposition 1. If \u2206(pdata, p\u03b8, p\u03b8,\u03c6) \u22650, then the following conditions hold: Ex\u223cpdata[ \u02c6 w\u03c6(x)] \u2265Ex\u223cp\u03b8[ \u02c6 w\u03c6(x)], (16) Ex\u223cpdata[log \u02c6 w\u03c6(x)] \u2265Ex\u223cp\u03b8[log \u02c6 w\u03c6(x)]. (17) The conditions in Eq. 16 and Eq. 17 follow directly via Jensen\u2019s inequality applied to the LHS and RHS of Eq. 15 respectively. Here, we note that estimates for the expectations in Eqs. 16-17 based on Monte Carlo averaging of (log-) importance weights are unbiased. 5 Application Use Cases In all our experiments, the binary classi\ufb01er for estimating the importance weights was a calibrated deep neural network trained to minimize the cross-entropy loss. The self-normalized LFIW in Eq. (7) worked best. Additional analysis on the estimators and experiment details are in Appendices B and C. 5.1 Goodness-of-\ufb01t testing In the \ufb01rst set of experiments, we highlight the bene\ufb01ts of importance weighting for a debiased evaluation of three popularly used sample quality metrics viz. Inception Scores (IS) [27], Frechet Inception Distance (FID) [28], and Kernel Inception Distance (KID) [29]. All these scores can be formally expressed as empirical expectations with respect to the model. For all these metrics, we can simulate the population level unbiased case as a \u201creference score\" wherein we arti\ufb01cially set both the real and generated sets of samples used for evaluation as \ufb01nite, disjoint sets derived from pdata. We evaluate the three metrics for two state-of-the-art models trained on the CIFAR-10 dataset viz. an autoregressive model PixelCNN++ [10] learned via maximum likelihood estimation and a latent variable model SNGAN [11] learned via adversarial training. For evaluating each metric, we draw 10,000 samples from the model. In Table 1, we report the metrics with and without the LFIW bias correction. The consistent debiased evaluation of these metrics via self-normalized LFIW suggest that the SIR approximation to the importance resampled distribution (Eq. 11) is a better \ufb01t to pdata. 5.2 Data Augmentation for Multi-Class Classi\ufb01cation We consider data augmentation via Data Augmentation Generative Adversarial Networks (DAGAN) [1]. While DAGAN was motivated by and evaluated for the task of meta-learning, it can also be applied for multi-class classi\ufb01cation scenarios, which is the setting we consider here. We trained a DAGAN on the Omniglot dataset of handwritten characters [12]. The DAGAN training procedure is described in the Appendix. The dataset is particularly relevant because it contains 1600+ classes but only 20 examples from each class and hence, could potentially bene\ufb01t from augmented data. 3If \u02c6 Z is an unbiased estimator for Z, then log \u02c6 Z is a biased estimator for log Z via Jensen\u2019s inequality. 6 \f(a) (b) (c) (d) (e) (f) Figure 2: Qualitative evaluation of importance weighting for data augmentation. (a-f) Top row shows held-out data samples from a speci\ufb01c class in Omniglot. Bottom row shows generated samples from the same class ranked in decreasing order of importance weights. Table 2: Classi\ufb01cation accuracy on the Omniglot dataset. Standard errors computed over 5 runs. Dataset Dcl Dg Dg w/ LFIW Dcl + Dg Dcl + Dg w/ LFIW Accuracy 0.6603 \u00b1 0.0012 0.4431 \u00b1 0.0054 0.4481 \u00b1 0.0056 0.6600 \u00b1 0.0040 0.6818 \u00b1 0.0022 Once the model has been trained, it can be used for data augmentation in many ways. In particular, we consider ablation baselines that use various combinations of the real training data Dcl and generated data Dg for training a downstream classi\ufb01er. When the generated data Dg is used, we can either use the data directly with uniform weighting for all training points, or choose to importance weight (LFIW) the contributions of the individual training points to the overall loss. The results are shown in Table 2. While generated data (Dg) alone cannot be used to obtain competitive performance relative to the real data (Dcl) on this task as expected, the bias it introduces for evaluation and subsequent optimization overshadows even the naive data augmentation (Dcl + Dg). In contrast, we can obtain signi\ufb01cant improvements by importance weighting the generated points (Dcl + Dg w/ LFIW). Qualitatively, we can observe the effect of importance weighting in Figure 2. Here, we show true and generated samples for 6 randomly choosen classes (a-f) in the Omniglot dataset. The generated samples are ranked in decreasing order of the importance weights. There is no way to formally test the validity of such rankings and this criteria can also prefer points which have high density under pdata but are unlikely under p\u03b8 since we are looking at ratios. Visual inspection suggests that the classi\ufb01er is able to appropriately downweight poorer samples, as shown in Figure 2 (a, b, c, d bottom right). There are also failure modes, such as the lowest ranked generated images in Figure 2 (e, f bottom right) where the classi\ufb01er weights reasonable generated samples poorly relative to others. This could be due to particular artifacts such as a tiny disconnected blurry speck in Figure 2 (e bottom right) which could be more revealing to a classi\ufb01er distinguishing real and generated data. 5.3 Model-based Off-policy Policy Evaluation So far, we have seen use cases where the generative model was trained on data from the same distribution we wish to use for Monte Carlo evaluation. We can extend our debiasing framework to more involved settings when the generative model is a building block for specifying the full data generation process, e.g., trajectory data generated via a dynamics model along with an agent policy. In particular, we consider the setting of off-policy policy evaluation (OPE), where the goal is to evaluate policies using experiences collected from a different policy. Formally, let (S, A, r, P, \u03b7, T) denote an (undiscounted) Markov decision process with state space S, action space A, reward function r, transition P, initial state distribution \u03b7 and horizon T. Assume \u03c0e : S \u00d7 A \u2192[0, 1] is a known policy that we wish to evaluate. The probability of generating a certain trajectory \u03c4 = {s0, a0, s1, a1, ..., sT , aT } of length T with policy \u03c0e and transition P is given as: p\u22c6(\u03c4) = \u03b7(s0) T \u22121 Y t=0 \u03c0e(at|st)P(st+1|st, at). (18) The return on a trajectory R(\u03c4) is the sum of the rewards across the state, action pairs in \u03c4: R(\u03c4) = PT t=1 r(st, at), where we assume a known reward function r. 7 \fTable 3: Off-policy policy evaluation on MuJoCo tasks. Standard error is over 10 Monte Carlo estimates where each estimate contains 100 randomly sampled trajectories. Environment v(\u03c0e) (Ground truth) \u02dc v(\u03c0e) \u02c6 v(\u03c0e) (w/ LFIW) \u02c6 v80(\u03c0e) (w/ LFIW) Swimmer 36.7 \u00b1 0.1 100.4 \u00b1 3.2 25.7 \u00b1 3.1 47.6 \u00b1 4.8 HalfCheetah 241.7 \u00b1 3.56 204.0 \u00b1 0.8 217.8 \u00b1 4.0 219.1 \u00b1 1.6 HumanoidStandup 14170 \u00b1 53 8417 \u00b1 28 9372 \u00b1 375 9221 \u00b1 381 0 20 40 60 80 100 H 20 40 60 | (v)| Swimmer 0 20 40 60 80 100 H 20 30 | (v)| HalfCheetah 0 20 40 60 80 100 H 4500 5000 5500 | (v)| HumanoidStandup Figure 3: Estimation error \u03b4(v) = v(\u03c0e) \u2212\u02c6 vH(\u03c0e) for different values of H (minimum 0, maximum 100). Shaded area denotes standard error over different random seeds. We are interested in the value of a policy de\ufb01ned as v(\u03c0e) = E\u03c4\u223cp\u2217(\u03c4) [R(\u03c4)]. Evaluating \u03c0e requires the (unknown) transition dynamics P. The dynamics model is a conditional generative model of the next states st+1 conditioned on the previous state-action pair (st, at). If we have access to historical logged data D\u03c4 of trajectories \u03c4 = {s0, a0, s1, a1, . . . , } from some behavioral policy \u03c0b : S \u00d7 A \u2192[0, 1], then we can use this off-policy data to train a dynamics model P\u03b8(st+1|st, at). The policy \u03c0e can then be evaluated under this learned dynamics model as \u02dc v(\u03c0e) = E\u03c4\u223c\u02dc p(\u03c4)[R(\u03c4)], where \u02dc p uses P\u03b8 instead of the true dynamics in Eq. (18). However, the trajectories sampled with P\u03b8 could signi\ufb01cantly deviate from samples from P due to compounding errors [30]. In order to correct for this bias, we can use likelihood-free importance weighting on entire trajectories of data. The binary classi\ufb01er c(st, at, st+1) for estimating the importance weights in this case distinguishes between triples of true and generated transitions. For any true triple (st, at, st+1) extracted from the off-policy data, the corresponding generated triple (st, at,\u02c6 st+1) only differs in the \ufb01nal transition state, i.e., \u02c6 st+1 \u223cP\u03b8(\u02c6 st+1|st, at). Such a classi\ufb01er allows us to obtain the importance weights \u02c6 w(st, at,\u02c6 st+1) for every predicted state transition (st, at,\u02c6 st+1). The importance weights for the trajectory \u03c4 can be derived from the importance weights of these individual transitions as: p\u22c6(\u03c4) \u02dc p(\u03c4) = QT \u22121 t=0 P(st+1|st, at) QT \u22121 t=0 P\u03b8(st+1|st, at) = T \u22121 Y t=0 P(st+1|st, at) P\u03b8(st+1|st, at) \u2248 T \u22121 Y t=0 \u02c6 w(st, at,\u02c6 st+1). (19) Our \ufb01nal LFIW estimator is given as: \u02c6 v(\u03c0e) = E\u03c4\u223c\u02dc p(\u03c4) \"T \u22121 Y t=0 \u02c6 w(st, at,\u02c6 st+1) \u00b7 R(\u03c4) # . (20) We consider three continuous control tasks in the MuJoCo simulator [15] from OpenAI gym [31] (in increasing number of state dimensions): Swimmer, HalfCheetah and HumanoidStandup. High dimensional state spaces makes it challenging to learning a reliable dynamics model in these environments. We train behavioral and evaluation policies using Proximal Policy Optimization [32] with different hyperparameters for the two policies. The dataset collected via trajectories from the behavior policy are used train a ensemble neural network dynamics model. We the use the trained dynamics model to evaluate \u02dc v(\u03c0e) and its IW version \u02c6 v(\u03c0e), and compare them with the ground truth returns v(\u03c0e). Each estimation is averaged over a set of 100 trajectories with horizon T = 100. Speci\ufb01cally, for \u02c6 v(\u03c0e), we also average the estimation over 10 classi\ufb01er instances trained with different random seeds on different trajectories. We further consider performing IW over only the \ufb01rst H steps, and use uniform weights for the remainder, which we denote as \u02c6 vH(\u03c0e). This allow us to interpolate between \u02dc v(\u03c0e) \u2261\u02c6 v0(\u03c0e) and \u02c6 v(\u03c0e) \u2261\u02c6 vT (\u03c0e). Finally, as in the other experiments, we used the self-normalized variant (Eq. (7)) of the importance weighted estimator in Eq. (20). We compare the policy evaluations under different environments in Table 3. These results show that the rewards estimated with the trained dynamics model differ from the ground truth by a large margin. 8 \fBy importance weighting the trajectories, we obtain much more accurate policy evaluations. As expected, we also see that while LFIW leads to higher returns on average, the imbalance in trajectory importance weights due to the multiplicative weights of the state-action pairs can lead to higher variance in the importance weighted returns. In Figure 3, we demonstrate that policy evaluation becomes more accurate as more timesteps are used for LFIW evaluations, until around 80 \u2212100 timesteps and thus empirically validates the bene\ufb01ts of importance weighting using a classi\ufb01er. Given that our estimates have a large variance, it would be worthwhile to compose our approach with other variance reduction techniques such as (weighted) doubly robust estimation in future work [33], as well as incorporate these estimates within a framework such as MAGIC to further blend with model-free OPE [14]. In Appendix C.5.1, we also consider a stepwise LFIW estimator for MBOPE which applies importance weighting at the level of every decision as opposed to entire trajectories. Overall. Across all our experiments, we observe that importance weighting the generated samples leads to uniformly better results, whether in terms of evaluating the quality of samples, or their utility in downstream tasks. Since the technique is a black-box wrapper around any generative model, we expect this to bene\ufb01t a diverse set of tasks in follow-up works. However, there is also some caution to be exercised with these techniques as evident from the results of Table 1. Note that in this table, the con\ufb01dence intervals (computed using the reported standard errors) around the model scores after importance weighting still do not contain the reference scores obtained from the true model. This would not have been the case if our debiased estimator was completely unbiased and this observation reiterates our earlier claim that LFIW is reducing bias, as opposed to completely eliminating it. Indeed, when such a mismatch is observed, it is a good diagnostic to either learn more powerful classi\ufb01ers to better approximate the Bayes optimum, or \ufb01nd additional data from pdata in case the generative model fails the full support assumption. 6 Related Work & Discussion Density ratios enjoy widespread use across machine learning e.g., for handling covariate shifts, class imbalance etc. [9, 34]. In generative modeling, estimating these ratios via binary classi\ufb01ers is frequently used for de\ufb01ning learning objectives and two sample tests [19, 35, 35\u201341]. In particular, such classi\ufb01ers have been used to de\ufb01ne learning frameworks such as generative adversarial networks [8, 42], likelihood-free Approximate Bayesian Computation (ABC) [43] and earlier work in unsupervised-as-supervised learning [44] and noise contrastive estimation [43] among others. Recently, [45] used importance weighting to reweigh datapoints based on differences in training and test data distributions i.e., dataset bias. The key difference is that these works are explicitly interested in learning the parameters of a generative model. In contrast, we use the binary classi\ufb01er for estimating importance weights to correct for the model bias of any \ufb01xed generative model. Recent concurrent works [46\u201348] use MCMC and rejection sampling to explicitly transform or reject the generated samples. These methods require extra computation beyond training a classi\ufb01er, in rejecting the samples or running Markov chains to convergence, unlike the proposed importance weighting strategy. For many model-based Monte Carlo evaluation usecases (e.g., data augmentation, MBOPE), this extra computation is unnecessary. If samples or density estimates are explicitly needed from the induced resampled distribution, we presented a particle-based approximation to the induced density where the number of particles is a tunable knob allowing for trading statistical accuracy with computational ef\ufb01ciency. Finally, we note resampling based techniques have been extensively studied in the context of improving variational approximations for latent variable generative models [49\u201352]. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1905.12892v2",
+                    "title": "AlignFlow: Cycle Consistent Learning from Multiple Domains via Normalizing Flows",
+                    "abstract": "Given datasets from multiple domains, a key challenge is to efficiently\nexploit these data sources for modeling a target domain. Variants of this\nproblem have been studied in many contexts, such as cross-domain translation\nand domain adaptation. We propose AlignFlow, a generative modeling framework\nthat models each domain via a normalizing flow. The use of normalizing flows\nallows for a) flexibility in specifying learning objectives via adversarial\ntraining, maximum likelihood estimation, or a hybrid of the two methods; and b)\nlearning and exact inference of a shared representation in the latent space of\nthe generative model. We derive a uniform set of conditions under which\nAlignFlow is marginally-consistent for the different learning objectives.\nFurthermore, we show that AlignFlow guarantees exact cycle consistency in\nmapping datapoints from a source domain to target and back to the source\ndomain. Empirically, AlignFlow outperforms relevant baselines on image-to-image\ntranslation and unsupervised domain adaptation and can be used to\nsimultaneously interpolate across the various domains using the learned\nrepresentation.",
+                    "authors": "Aditya Grover, Christopher Chute, Rui Shu, Zhangjie Cao, Stefano Ermon",
+                    "published": "2019-05-30",
+                    "updated": "2019-12-21",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.NE",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction In recent years, there has been an increase in the availability of both labeled and unlabeled datasets from multiple sources. For example, many variants of face datasets scraped from sources such as Wikipedia and IMDB are publicly available. Given data from two or more domains, we expect sample-ef\ufb01cient learning algorithms to be able to learn and align the shared structure across these domains for accurate downstream tasks. This perspective has a broad range of applications across machine learning, including relational learning (Kim et al. 2017), domain adaptation (Taigman, Polyak, and Wolf 2016; Hoffman et al. 2017; Bousmalis et al. 2017), image and video translation for computer vision (Isola et al. 2017; Wang et al. 2018), and machine translation for low resource languages (Gu et al. 2018). Many variants of the domain alignment problem have been studied in prior work. For instance, unpaired cross-domain translation refers to the task of learning a mapping from one domain to another given datasets from the two domains (Zhu \u2217Equal Contribution Copyright c \u20dd2020, Association for the Advancement of Arti\ufb01cial Intelligence (www.aaai.org). All rights reserved. et al. 2017b). This task can be used as a subproblem in the domain adaptation setting, where the goal is to learn a classi\ufb01er for the unlabeled domain given labeled data from a related source domain (Saenko et al. 2010). Many of these problems are underconstrained due to the limitations on the labelled supervision available for target domain. An amalgam of inductive biases need to be explicitly enforced to learn meaningful solutions, e.g., cycle-consistency (Zhu et al. 2017b), entropic regularization (Courty et al. 2017) etc. These inductive biases can be speci\ufb01ed via additional loss terms or by specifying constraints on the model family. We present AlignFlow, a latent variable generative framework that seeks to discover the shared structure across multiple data domains using normalizing \ufb02ows (Rezende and Mohamed 2015; Dinh, Krueger, and Bengio 2014; Dinh, Sohl-Dickstein, and Bengio 2017). Latent variable generative models are highly effective for inferring hidden structure within observed data from a single domain. In AlignFlow, we model the data from each domain via an invertible generative model with a single latent space shared across all the domains. If we let the two domains to be A and B with a shared latent space, say Z, then the latent variable generative model for A may additionally share some or all parameters with the model of domain B. Akin to a single invertible model, the collection of invertible models in AlignFlow provide great \ufb02exibility in specifying learning objectives and can be trained via maximum likelihood estimation, adversarial training or a hybrid variant accounting for both objectives. By virtue of an invertible design, AlignFlow naturally extends as a cross-domain translation model. To translate data across two domains, say A to B, we can invert a data point from A \u2192Z \ufb01rst followed by a second inversion from Z \u2192B. Appealingly, we show that this composition of invertible mappings is exactly cycle-consistent, i.e., translating a datapoint from A to B using the forward mapping and backwards using the reverse mapping gives back the original datapoint and vice versa from B to A. Cycle-consistency was \ufb01rst introduced in CycleGAN (Zhu et al. 2017a) and has been shown to be an excellent inductive bias for underconstrained problems, such as unpaired domain alignment. While models such as CycleGAN only provide approximate cycle-consistency by incorporating additional loss arXiv:1905.12892v2  [cs.LG]  21 Dec 2019 \fterms, AlignFlow can omit these terms and guarantee exact cycle-consistency by design. We analyze the AlignFlow framework extensively. Theoretically, we derive conditions under which the AlignFlow objective is consistent in the sense of recovering the true marginal distributions. For objectives that use adversarial loss terms, we derive optimal critics in this setting. Empirically, we consider two sets of tasks: image-to-image translation and unsupervised domain adaptation. On both these tasks, we observe consistent improvements over other approximately cycle-consistent generative frameworks on three benchmark pairs of high-dimensional image datasets. 2 Preliminaries In this section, we discuss the necessary background and notation on generative adversarial networks and normalizing \ufb02ows. We overload uppercase notation X, Y, Z to denote random variables and their sample spaces and use lowercase notation x, y, z to denote values assumed by these variables. 2.1 Generative Adversarial Networks Generative adversarial networks (GAN) are a class of latent variable generative models that specify the generator as a deterministic mapping h : Z \u2192X between a set of latent variables Z and a set of observed variables X (Goodfellow et al. 2014). In order to sample from a GAN, we need a prior density over Z that permits ef\ufb01cient sampling. The generator of a GAN can also be conditional, where the conditioning is on another set of observed variables and optionally the latent variables Z as before (Mirza and Osindero 2014). A GAN is trained via adversarial training, wherein the generator h plays a minimax game with an auxiliary critic C. The goal of the critic C : X \u2192R is to distinguish real samples from the observed dataset with samples generated via h. The generator, on the other hand, tries to generate samples that can maximally confuse the critic. Many learning objectives have been proposed for adversarial training, such as those based on f-divergences (Nowozin, Cseke, and Tomioka 2016) and Wasserstein Distance (Arjovsky, Chintala, and Bottou 2017). For the standard cross-entropy GAN, the critic outputs a probability of a datapoint being real and optimizes the following objective w.r.t. a data distribution p\u2217 X : X \u2192R\u22650: LGAN(C, h) = Ex\u223cp\u2217 X[log C(x)] + Ez\u223cpZ[log(1 \u2212C(h(z)))]. (1) for a suitable choice of prior density pZ. The generator and the critic are both parameterized by deep neural networks and learned via alternating gradient updates. Because adversarial training only requires samples from the generative model, it can be used to train generative models with intractable or ill-de\ufb01ned likelihoods (Mohamed and Lakshminarayanan 2016). Hence, adversarial training is likelihood-free and in practice, it gives excellent performance for tasks that require data generation. However, these models are hard to train due to the alternating minimax optimization and suffer from issues such as mode collapse (Goodfellow 2016). 2.2 Normalizing Flows Normalizing \ufb02ows are a class of latent variable generative models that specify the generator as an invertible mapping h : Z \u2192X between a set of latent variables Z and a set of observed variables X. Let pX and pZ denote the marginal densities de\ufb01ned by the model over X and Z respectively. Using the change-of-variables formula, these marginal densities can be related as: pX(x) = pZ(z) \f \f \f \fdet\u2202h\u22121 \u2202X \f \f \f \f X=x (2) where z = h\u22121(x) due to the invertibility constraints. Here, the second term on the RHS corresponds to the absolute value of the determinant of the Jacobian of the inverse transformation and signi\ufb01es the change in volume when translating across the two sample spaces. For evaluating likelihoods via the change-of-variables formula, we require ef\ufb01cient and tractable evaluation of the prior density, the inverse transformation h\u22121, and the determinant of the Jacobian of h\u22121. To draw a sample from this model, we perform ancestral sampling, i.e., we \ufb01rst sample a latent vector z \u223cpZ(z) and obtain the sampled vector as given by x = h(z). This requires the ability to ef\ufb01ciently: (1) sample from the prior density and (2) evaluate the forward transformation h. Many transformations parameterized by deep neural networks that satisfy one or more of these criteria have been proposed in the recent literature on normalizing \ufb02ows, e.g., NICE (Dinh, Krueger, and Bengio 2014) and Autoregressive Flows (Kingma et al. 2016; Papamakarios, Murray, and Pavlakou 2017). By suitable design of transformations, both likelihood evaluation and sampling can be performed ef\ufb01ciently, as in Real-NVP (Dinh, Sohl-Dickstein, and Bengio 2017). Consequently, a \ufb02ow model can be trained ef\ufb01ciently to maximize the likelihood of the observed dataset (a.k.a. maximum likelihood estimation or MLE) as well as likelihood-free adversarial training (Grover, Dhar, and Ermon 2018). 3 The AlignFlow Framework In this section, we present the AlignFlow framework for learning generative models in the presence of unpaired data from multiple domains. For ease of presentation, we consider the case of two domains. Unless mentioned otherwise, our results naturally extend to more than two domains as well. 3.1 Problem Setup The learning setting we consider is as follows. We are given unpaired datasets DA and DB from two domains A and B respectively. We assume that the datapoints are sampled i.i.d. from some true but unknown marginal densities denoted as p\u2217 A and p\u2217 B respectively. We are interested in learning models for the following distributions: (a) the marginal likelihoods pA and pB that approximate p\u2217 A and p\u2217 B and (b) conditional distributions pA|B and pB|A. The unconditional models can be used for density estimation and sampling from A and B whereas the conditional models can be used for translating (i.e., conditional sampling) from B \u2192A and A \u2192B. \fBefore presenting the AlignFlow framework, we note two observations. For task (a), we need datasets from the domains A and B respectively for learning. For task (b), we note that the problem is underconstrained since we are only given data from the marginal distributions and hence, it is unclear how to learn the conditional distribution that relates the datapoints from the two domains. Hence, we need additional inductive biases on our learning algorithm that can learn useful conditional distributions, In practice, many such forms of inductive biases have been designed and shown to be useful across relavant tasks such as crossdomain translation and domain adaptation (Zhu et al. 2017b; Liu, Breuel, and Kautz 2017). 3.2 Representation We will use a graphical model to represent the relationships between the domains. Consider a Bayesian network A \u2190 Z \u2192B with two sets of observed random variables (domains) A \u2286Rn and B \u2286Rn and a parent set of latent random variables Z \u2286Z. The latent variables Z indicate a shared feature space between the observed variables A and B, which will be exploited later for ef\ufb01cient learning and inference. While Z is unobserved, we assume a prior density pZ over these variables, such as an isotropic Gaussian. Finally, to compactly specify the joint distribution over all sets of variables, we constrain the relationship between A and Z, and B and Z to be invertible. That is, we specify mappings GZ\u2192A and GZ\u2192B such that the respective inverses GA\u2192Z = G\u22121 Z\u2192A and GB\u2192Z = G\u22121 Z\u2192B exist. Notice that such a representation naturally provides a mechanism to translate from one domain to another as the composition of two invertible mappings: GA\u2192B = GZ\u2192B \u25e6GA\u2192Z (3) GB\u2192A = GZ\u2192A \u25e6GB\u2192Z. (4) Since composition of invertible mappings is invertible, both GA\u2192B and GB\u2192A are invertible. In fact, it is straightforward to observe that GA\u2192B and GB\u2192A are inverses of each other: G\u22121 A\u2192B = (GZ\u2192B \u25e6GA\u2192Z)\u22121 = G\u22121 A\u2192Z \u25e6G\u22121 Z\u2192B = GZ\u2192A \u25e6GB\u2192Z = GB\u2192A. (5) 3.3 Learning Algorithms & Objectives As discussed in the preliminaries, each of the individual \ufb02ows GZ\u2192A and GZ\u2192B express a model with density pA and pB respectively and can be trained independently via maximum likelihood estimation, adversarial learning, or a hybrid objective. However, our goal is to perform sampleef\ufb01cient learning by exploiting data from other domains as well as learn a conditional mapping across the two domains. For both these goals, we require learning algorithms which use data from both domains for parameter estimation. Unless mentioned otherwise, all our results that hold for a particular domain A will have a natural counterpart for the domain B. Adversarial Training Instead of adversarial training of GZ\u2192A and GZ\u2192B independently, we can directly perform adversarial training of the mapping GB\u2192A. That is, we \ufb01rst generate data from GB\u2192A using the prior density given as p\u2217 B. We also introduce a critic CA which distinguishes real samples a \u223cp\u2217 A with the generated samples GB\u2192A(b) for b \u223cp\u2217 B. For example, the cross-entropy GAN loss in this case is given as: LGAN(CA, GB\u2192A) = Ea\u223cp\u2217 A[log CA(a)] + Eb\u223cp\u2217 B[log(1 \u2212CA(GB\u2192A(b)))]. (6) The expectations above are approximated empirically via datasets DA and DB respectively. Maximum Likelihood Estimation Unlike adversarial training, \ufb02ow models trained with maximum likelihood estimation (MLE) explicitly require a prior pZ with a tractable density (e.g., isotropic Gaussian) to evaluate model likelihoods using the change-of-variables formula in Eq. 2. Due to this tractability requirement, we cannot substitute pZ with the unknown p\u2217 B for MLE. Instead, we can share parameters between the two mappings. The extent of parameter sharing depends on the similarity across the two domains; for highly similar domains, entire architectures could potentially be shared in which case GZ\u2192A = GZ\u2192B. Hybrid Training Both MLE and adversarial training objectives can be combined into a single training objective. The most general AlignFlow objective is given as: LAlignFlow(GB\u2192A, CA, CB; \u03bbA, \u03bbB) = LGAN(CA, GB\u2192A) + LGAN(CB, GA\u2192B) \u2212\u03bbALMLE(GZ\u2192A) \u2212\u03bbBLMLE(GZ\u2192B) (7) where \u03bbA \u22650 and \u03bbB \u22650 are hyperparameters that control the strength of the MLE terms for domains A and B respectively. The AlignFlow objective is minimized w.r.t. the parameters of the generator GA\u2192B and maximized w.r.t. parameters of the critics CA and CB. Notice that LAlignFlow is expressed as a function of the critics CA, CB and only GB\u2192A since the latter also encompasses the other parametric functions appearing in the objective (GA\u2192B, GZ\u2192A, GZ\u2192B) via the invertibility constraints in Eqs. 3-5. When \u03bbA = \u03bbB = 0, we perform pure adverarial training and the prior over Z plays no role in learning. On the other hand, when \u03bbA = \u03bbB \u2192\u221e, we can perform pure MLE training to learn the invertible generator. Here, the critics CA, CB play no role since the adversarial training terms are ignored. 3.4 Inference AlignFlow can be used for both conditional and unconditional sampling at test time. For conditional sampling as in the case of domain translation, we are given a datapoint b \u2208B and we can draw the corresponding cross-domain translation in domain A via the mapping GB\u2192A. For unconditional sampling, we require \u03bbA \u0338= 0 since doing so will activate the use of the prior pZ via the MLE terms in the learning objective. Thereafter, we can obtain samples by \ufb01rst drawing z \u223cpZ and then applying the mapping GZ\u2192A to z. Furthermore, the same z can be mapped to domain B via GZ\u2192B. Hence, we can sample paired data (GZ\u2192A(z), GZ\u2192B(z)) given z \u223cpZ. \f4 Theoretical Analysis The AlignFlow objective consists of three parametric models: one generator GB\u2192A \u2208G, and two critics CA \u2208CA, CB \u2208 CB. Here, G, CA, CB denote model families speci\ufb01ed e.g., via deep neural network based architectures. In this section, we analyze the optimal solutions to these parameterized models within well-speci\ufb01ed model families. 4.1 Optimal Generators Our \ufb01rst result characterizes the conditions under which the optimal generators exhibit marginal-consistency for the data distributions de\ufb01ned over the domains A and B. De\ufb01nition 1. (Marginal-consistency) Let pX,Y denote the joint distribution between two domains X and Y. An invertible mapping GY\u2192X : Y \u2192X is marginally-consistent w.r.t. two arbitrary distributions (pX, pY) iff for all x \u2208X, y \u2208Y: pX(x) = \u001a pY(y) \f \f \fdet \u2202GY\u2192X \u22121 \u2202X \f \f \f X=x , if x = GY\u2192X(y) 0, otherwise. (8) Next, we show that AlignFlow is marginally-consistent for well-speci\ufb01ed model families. Lemma 1. Let GA and GB denote the class of invertible mappings represented by the AlignFlow architecture for mapping Z \u2192A and Z \u2192B. For a given choice of prior distribution pZ, if there exist mappings G\u2217 Z\u2192A \u2208GA, G\u2217 Z\u2192B \u2208GB that are marginally-consistent w.r.t. (p\u2217 A, pZ) and (p\u2217 B, pZ) respectively, then the mapping G\u2217 B\u2192A = G\u2217 Z\u2192A \u25e6G\u2217\u22121 Z\u2192B is marginallyconsistent w.r.t. (p\u2217 A, p\u2217 B). The result follows directly from De\ufb01nition 1 and changeof-variables applied to the mapping G\u2217 B\u2192A. Theorem 1. Assume that the model families for the critics CA : A \u2192[0, 1] and CB : B \u2192[0, 1] are the set of all measurable functions for the cross-entropy GAN objective. Then, G\u2217 B\u2192A (as de\ufb01ned in Lemma 1) globally minimizes the AlignFlow objective in Eq. 7 for any \u03bbA \u22650, \u03bbB \u22650. Proof. See Appendix A.1. Theorem 1 suggests that optimizing the AlignFlow objective will recover the marginal data distributions p\u2217 A and p\u2217 B under suitable conditions. For the other goal of learning cross-domain mappings, we note that marginally-consistent mappings w.r.t. a target data distribution (such as p\u2217 A) and a target prior density (such as p\u2217 B) need not be unique. While a cycle-consistent, invertible model family mitigates the underconstrained nature of the crossdomain translation problem, it does not provably eliminate it. We provide some non-identi\ufb01able constructions in Appendix A.3 and leave open the exploration of additional constraints that guarantee identi\ufb01ability for future exploration. 4.2 Optimal Critics Unlike standard adversarial training of an unconditional normalizing \ufb02ow model (Grover, Dhar, and Ermon 2018; Danihelka et al. 2017), the AlignFlow model involves two critics. Here, we are interested in characterizing the dependence of the optimal critics for a given invertible mapping GA\u2192B. Consider the AlignFlow framework where the GAN A B YA YB GA\u2192B GB\u2192A CA CB (a) CycleGAN A B Z YA YB GA\u2192Z = G\u22121 Z\u2192A GB\u2192Z = G\u22121 Z\u2192B CA CB (b) AlignFlow Figure 1: CycleGAN v.s. AlignFlow for unpaired crossdomain translation. Unlike CycleGAN, AlignFlow speci\ufb01es a single invertible mapping GA\u2192Z \u25e6G\u22121 B\u2192Z that is exactly cycle-consistent, represents a shared latent space Z between the two domains, and can be trained via both adversarial training and exact maximum likelihood estimation. Doubleheaded arrows denote invertible mappings. YA and YB are random variables denoting the output of the critics used for adversarial training. loss terms in Eq. 7 are speci\ufb01ed via the cross-entropy objective in Eq. 1. For this model, we can relate the optimal critics using the following result. Theorem 2. Let p\u2217 A and p\u2217 B denote the true data densities for domains A and B respectively. Let C\u2217 A and C\u2217 B denote the optimal critics for the AlignFlow objective with the crossentropy GAN loss for any \ufb01xed choice of the invertible mapping GA\u2192B. Letting b = GA\u2192B(a) for any a \u2208A, we have: C\u2217 A(a) = C\u2217 B(b)p\u2217 A(a) p\u2217 A(a) + p\u2217 B(b)(1 \u2212C\u2217 B(b)) \f \f \fdet \u2202G\u22121 B\u2192A \u2202A \f \f \f A=a . (9) Proof. See Appendix A.2. In essence, the above result shows that the optimal critic for one domain, w.l.o.g. say A, can be directly obtained via the optimal critic of another domain B for any choice of the invertible mapping GA\u2192B, assuming access to the data marginals p\u2217 A and p\u2217 B. 4.3 Exact Cycle Consistency So far, we have only discussed objectives that are marginallyconsistent with respect to data distributions p\u2217 A and p\u2217 B. However, many domain alignment tasks such as cross-domain \ftranslation require can be cast as learning a joint distribution p\u2217 A,B. As discussed previously, this problem is underconstrained given unpaired datasets DA and DB and the learned marginal densities alone do not guarantee learning a mapping that is useful for downstream tasks. Cycle consistency, as proposed in CycleGAN (Zhu et al. 2017a), is a highly effective learning objective that encourages learning of meaningful cross-domain mappings such that the data translated from domain A to B via GA\u2192B to be mapped back to the original datapoints in A via GB\u2192A. That is, GB\u2192A(GA\u2192B(a)) \u2248a for all a \u2208A. Formally, the cycle-consistency loss for translation from A to B and back is de\ufb01ned as: LCycle(GB\u2192A, GA\u2192B) = Ea\u223cp\u2217 A[\u2225GB\u2192A(GA\u2192B(a)) \u2212a\u22251]. (10) Symmetrically, we have a cycle consistency term LCycle(GA\u2192B, GB\u2192A) in the reverse direction that encourages GA\u2192B(GB\u2192A(b)) \u2248b for all b \u2208B. Next, we show that AlignFlow is exactly cycle consistent. Proposition 1. Let G denote the class of invertible mappings represented by an arbitrary AlignFlow architecture. For any GB\u2192A \u2208G, we have: LCycle(GB\u2192A, GA\u2192B) = 0 (11) LCycle(GA\u2192B, GB\u2192A) = 0 (12) where GA\u2192B = G\u22121 B\u2192A by design. The proposition follows directly from the invertible design of the AlignFlow framework (Eq. 5) and Eq. 10. Comparison with CycleGAN We illustrate and compare AlignFlow and CycleGAN in Figure 1. CycleGAN parameterizes two independent cross-domain mappings GA\u2192B and GB\u2192A, whereas AlignFlow only speci\ufb01es a single, invertible mapping. Learning in a CycleGAN is restricted to an adversarial training objective along with additional cycleconsistent loss terms. In contrast, AlignFlow is exactly consistent and can be trained via adversarial learning, MLE, or a hybrid (Eq. 7) without the need for additional loss terms to enforce cycle consistency. Finally, inference in CycleGAN is restricted to conditional sampling since it does not involve any latent variables Z with easy-to-sample prior densities. As described previously, AlignFlow permits both conditional and unconditional sampling. Comparison with UNIT and CoGAN Models such as CoGAN (Liu and Tuzel 2016) and its extension UNIT (Liu, Breuel, and Kautz 2017) also consider adding a shared-space constraint between two different domain decoders. These models again can only enforce approximate cycle consistency and introduce additional encoder networks. Moreover, they only approximate lower bounds to the log-likelihood unlike AlignFlow which permits exact MLE training. 5 Experimental Evaluation To achieve our two goals of data-ef\ufb01cient modeling of individual domains and effective cross-domain mappings, we evaluate AlignFlow on two tasks: (a) unsupervised image-toimage translation, and (b) unsupervised domain adaptation. For additional experimental details, results, and analysis beyond those stated below, we refer the reader to Appendix B. Table 1: Mean Squared Error (MSE) comparing CycleGAN and variants of AlignFlow(AF) on paired test sets. MSE is computed pixelwise after normalizing images to (\u22121, 1). Dataset Model MSE (A \u2192B) MSE (B \u2192A) Facades CycleGAN 0.7129 0.3286 AF (ADV only) 0.6727 0.2679 AF (Hybrid) 0.5801 0.2512 AF (MLE only) 0.9014 0.5960 Maps CycleGAN 0.0245 0.0953 AF (ADV only) 0.0385 0.1123 AF (Hybrid) 0.0209 0.0897 AF (MLE only) 0.0452 0.1746 CityScapes CycleGAN 0.1252 0.1200 AF (ADV only) 0.2569 0.2196 AF (Hybrid) 0.1130 0.1462 AF (MLE only) 0.2526 0.2272 5.1 Image-To-Image Translation We evaluate AlignFlow on three image-to-image translation datasets used by Zhu et al. (2017a): Facades, Maps, and CityScapes (Cordts et al. 2016). These datasets are chosen because they provide one-to-one aligned image pairs, so one can quantitatively evaluate unpaired image-to-image translation models via a distance metric such as mean squared error (MSE) between generated examples and the corresponding ground truth. While MSE can be substituted for perceptual losses in other scenarios, it is a suitable metric for evaluating datasets with one-to-one ground pairings. Note that the task at hand is unpaired translation and hence, the pairing information is omitted during training and only used for evaluation. We report the MSE for translations on the test sets after cross-validation of hyperparameters in Table 1. For hybrid models, we set \u03bbA = \u03bbB and report results for the best values of these hyperparameters. We observe that while learning AlignFlow via adversarial training or MLE alone is not as competitive as CycleGAN, hybrid training of AlignFlow signi\ufb01cantly outperforms CycleGAN in almost all cases. Specifically, we observe that MLE alone typically performs worse than adversarial training, but together both these objectives seem to have a regularizing effect on each other. Qualitative interpolations on the Facades dataset are shown in Figure 2. 5.2 Unsupervised Domain Adaptation In unsupervised domain adaptation (Saenko et al. 2010), we are given data from two related domains: a source and a target domain. For the source, we have access to both the input datapoints and their labels. For the target, we are only provided with input datapoints without any labels. Using the available data, the goal is to learn a classi\ufb01er for the target domain. We extend Hoffman et al. (2017) to use an AlignFlow architecture and objective (adversarially trained Real-NVPs (Dinh, Sohl-Dickstein, and Bengio 2017) here) in place of CycleGAN for this task. A variety of algorithms have been proposed for the above task which seek to match pixel-level or feature-level distri\fFigure 2: Latent space interpolation on Facades. Top: Leftand right-most images are sampled from DA (red boxes). Interpolation is then performed in latent space and then decoded using GZ\u2192A. We see semantically meaningful changes across the row, e.g., in the shadow and the style of entrance to the building. Bottom: For each image in the top row, its latent representation is decoded into the target domain using GZ\u2192B. Inspection of the orange regions indicates a change from 3 \ufb02oors (left) to 4 \ufb02oors (right). Table 2: Test classi\ufb01cation accuracies for domain adaptation from source \u2192target. The source only and target only models directly use classi\ufb01ers trained on the source and target datasets respectively. Baseline numbers taken from the cited works. Model MNIST \u2192USPS USPS \u2192MNIST SVHN \u2192MNIST source only 82.2 \u00b1 0.8 69.6 \u00b1 3.8 67.1 \u00b1 0.6 ADDA (Tzeng et al. 2017) 89.4 \u00b1 0.2 90.1 \u00b1 0.8 76.0 \u00b1 1.8 CyCADA (Hoffman et al. 2017) 95.6 \u00b1 0.2 96.5 \u00b1 0.1 90.4 \u00b1 0.4 UNIT (Liu, Breuel, and Kautz 2017) 95.97 93.58 90.53 AlignFlow 96.2 \u00b1 0.2 96.7 \u00b1 0.1 91.0 \u00b1 0.3 target only 96.3 \u00b1 0.1 99.2 \u00b1 0.1 99.2 \u00b1 0.1 (a) (b) (c) (d) Figure 3: Examples of failure modes for CycleGAN reconstructions in SVHN \u2194MNIST cross-domain translation. In each group of three images in (a-d), a real example from the source domain (SVHN) is shown on the left, the translated image in the target domain (MNIST) is shown at center, and the reconstructed image in the source domain based on the translation is shown on the right. butions across the two domains. See Appendix B for more details. For fair and relevant comparison, we compare against baselines Cycada (Hoffman et al. 2017) and UNIT (Liu, Breuel, and Kautz 2017) which involve pixel-level translations and are closest to the current work. We evaluate across all pairs of source and target datasets as in Hoffman et al. (2017) and Liu, Breuel, and Kautz (2017): MNIST, USPS, SVHN, which are all image datasets of handwritten digits with 10 classes. In Table 2, we see that AlignFlow outperforms both Cycada (Hoffman et al. 2017) (based on CycleGAN) and UNIT (Liu, Breuel, and Kautz 2017) in all cases. Combining AlignFlow with other state-of-the-art adaptation approaches e.g., Shu et al. (2018), Long et al. (2018), Kumar et al. (2018), Liu et al. (2018a), Sankaranarayanan et al. (2018), Liu et al. (2018b) is an interesting direction for future work. In Figure 3, we show some failure modes of using approximately cycle-consistent objectives for the Cycada model. Notice that the image label and style changes or becomes unrecognizable in translating and reconstructing the input. In contrast, AlignFlow is exactly cycle consistent and hence, the source reconstructions based on the translated images will be exactly the source image by design. 5.3 Multi-Domain Concurrent Interpolations The use of a shared latent space in AlignFlow allows us to perform paired interpolations in two domains simultaneously. While pure MLE without any parameter sharing does not give good alignment, pure adversarial training cannot be used for unconditional sampling since the prior pZ is inactive. Hence, \f(a) MNIST\u2192USPS (b) USPS\u2192MNIST Figure 4: Multi-domain latent space interpolations. Top: Left-most and right-most images are sampled from DA (in red boxes). Interpolation is then performed in latent space and then decoded using GZ\u2192A. Bottom: For each corresponding image in the top row, its latent representation is decoded into the target domain using GZ\u2192B. Note how both class identity and style are preserved in the interpolated pairs of digits in the two domains. Also, notice that the USPS images (even the true ones in red boxes) are slightly blurred due to the upscaling applied as standard preprocessing. we use AlignFlow models trained via a hybrid objective for latent space interpolations. In particular, we sample two datapoints a\u2032, a\u2032\u2032 \u2208DA and obtain their latent representations z\u2032, z\u2032\u2032 \u2208Z via GZ\u2192A. Following Dinh, Sohl-Dickstein, and Bengio (2017), we compute interpolations in the polar space as \u02dc z = z\u2032 sin \u03c6 + z\u2032\u2032 cos \u03c6 for several values of \u03c6 \u2208(0, 2\u03c0). Finally, we map \u02dc z to either back to domain A via GZ\u2192A and B via GZ\u2192B. We show this empirically on the MNIST/USPS datasets in Figure 4. We see that many aspects of style and content are preserved in the interpolated samples. 6 Related Work A key assumption in unsupervised domain alignment is the existence of a deterministic or stochastic mapping GA\u2192B such that the distribution of B matches that of GA\u2192B(A), and vice versa. This assumption can be incorporated as a marginal distribution-matching constraint into the objective using an adversarially-trained GAN critic (Goodfellow et al. 2014). However, this objective is under-constrained. To partially mitigate this issue, CycleGAN (Zhu et al. 2017a), DiscoGAN (Kim et al. 2017), and DualGAN (Yi et al. 2017) added an approximate cycle-consistency constraint that encourages GB\u2192A \u25e6GA\u2192B and GA\u2192B \u25e6GB\u2192A to behave like identity functions on domains A and B respectively. While cycle-consistency is empirically very effective, alternatives based on variational autoencoders that do not require either cycles or adversarial training have also been proposed recently (Hoshen 2018; Hoshen and Wolf 2018). Models such as CoGAN (Liu and Tuzel 2016), UNIT (Liu, Breuel, and Kautz 2017), and CycleGAN (Zhu et al. 2017a) have since been extended to enable one-to-many mappings (Huang et al. 2018b; Zhu et al. 2017b) as well as multi-domain alignment (Choi et al. 2018). Our work focuses on the one-to-one unsupervised domain alignment setting. In contrast to previous models, AlignFlow leverages both a shared latent space and exact cycle-consistency. To our knowledge, AlignFlow provides the \ufb01rst demonstration that invertible models can be used successfully in lieu of the cycleconsistency objective. Furthermore, AlignFlow allows the incorporation of exact maximum likelihood training, which we demonstrated to induce a meaningful shared latent space that is amenable to interpolation. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1903.08850v2",
+                    "title": "Stochastic Optimization of Sorting Networks via Continuous Relaxations",
+                    "abstract": "Sorting input objects is an important step in many machine learning\npipelines. However, the sorting operator is non-differentiable with respect to\nits inputs, which prohibits end-to-end gradient-based optimization. In this\nwork, we propose NeuralSort, a general-purpose continuous relaxation of the\noutput of the sorting operator from permutation matrices to the set of unimodal\nrow-stochastic matrices, where every row sums to one and has a distinct arg\nmax. This relaxation permits straight-through optimization of any computational\ngraph involve a sorting operation. Further, we use this relaxation to enable\ngradient-based stochastic optimization over the combinatorially large space of\npermutations by deriving a reparameterized gradient estimator for the\nPlackett-Luce family of distributions over permutations. We demonstrate the\nusefulness of our framework on three tasks that require learning semantic\norderings of high-dimensional objects, including a fully differentiable,\nparameterized extension of the k-nearest neighbors algorithm.",
+                    "authors": "Aditya Grover, Eric Wang, Aaron Zweig, Stefano Ermon",
+                    "published": "2019-03-21",
+                    "updated": "2019-04-29",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG",
+                        "cs.NE"
+                    ],
+                    "main_content": "INTRODUCTION Learning to automatically sort objects is useful in many machine learning applications, such as topk multi-class classi\ufb01cation (Berrada et al., 2018), ranking documents for information retrieval (Liu et al., 2009), and multi-object target tracking in computer vision (Bar-Shalom & Li, 1995). Such algorithms typically require learning informative representations of complex, high-dimensional data, such as images, before sorting and subsequent downstream processing. For instance, the k-nearest neighbors image classi\ufb01cation algorithm, which orders the neighbors based on distances in the canonical pixel basis, can be highly suboptimal for classi\ufb01cation (Weinberger et al., 2006). Deep neural networks can instead be used to learn representations, but these representations cannot be optimized end-to-end for a downstream sorting-based objective, since the sorting operator is not differentiable with respect to its input. In this work, we seek to remedy this shortcoming by proposing NeuralSort, a continuous relaxation to the sorting operator that is differentiable almost everywhere with respect to the inputs. The output of any sorting algorithm can be viewed as a permutation matrix, which is a square matrix with entries in {0, 1} such that every row and every column sums to 1. Instead of a permutation matrix, NeuralSort returns a unimodal row-stochastic matrix. A unimodal row-stochastic matrix is de\ufb01ned as a square matrix with positive real entries, where each row sums to 1 and has a distinct arg max. All permutation matrices are unimodal row-stochastic matrices. NeuralSort has a temperature knob that controls the degree of approximation, such that in the limit of zero temperature, we recover a permutation matrix that sorts the inputs. Even for a non-zero temperature, we can ef\ufb01ciently project any unimodal matrix to the desired permutation matrix via a simple row-wise arg max operation. Hence, NeuralSort is also suitable for ef\ufb01cient straight-through gradient optimization (Bengio et al., 2013), which requires \u201cexact\u201d permutation matrices to evaluate learning objectives. As the second primary contribution, we consider the use of NeuralSort for stochastic optimization over permutations. In many cases, such as latent variable models, the permutations may be latent but directly in\ufb02uence observed behavior, e.g., utility and choice models are often expressed as distributions over permutations which govern the observed decisions of agents (Regenwetter et al., \u2217Equal contribution 1 arXiv:1903.08850v2  [stat.ML]  29 Apr 2019 \fPublished as a conference paper at ICLR 2019 2006; Chierichetti et al., 2018). By learning distributions over unobserved permutations, we can account for the uncertainty in these permutations in a principled manner. However, the challenge with stochastic optimization over discrete distributions lies in gradient estimation with respect to the distribution parameters. Vanilla REINFORCE estimators are impractical for most cases, or necessitate custom control variates for low-variance gradient estimation (Glasserman, 2013). In this regard, we consider the Plackett-Luce (PL) family of distributions over permutations (Plackett, 1975; Luce, 1959). A common modeling choice for ranking models, the PL distribution is parameterized by n scores, with its support de\ufb01ned over the symmetric group consisting of n! permutations. We derive a reparameterizable sampler for stochastic optimization with respect to this distribution, based on Gumbel perturbations to the n (log-)scores. However, the reparameterized sampler requires sorting these perturbed scores, and hence the gradients of a downstream learning objective with respect to the scores are not de\ufb01ned. By using NeuralSort instead, we can approximate the objective and obtain well-de\ufb01ned reparameterized gradient estimates for stochastic optimization. Finally, we apply NeuralSort to tasks that require us to learn semantic orderings of complex, highdimensional input data. First, we consider sorting images of handwritten digits, where the goal is to learn to sort images by their unobserved labels. Our second task extends the \ufb01rst one to quantile regression, where we want to estimate the median (50-th percentile) of a set of handwritten numbers. In addition to identifying the index of the median image in the sequence, we need to learn to map the inferred median digit to its scalar representation. In the third task, we propose an algorithm that learns a basis representation for the k-nearest neighbors (kNN) classi\ufb01er in an end-to-end procedure. Because the choice of the k nearest neighbors requires a non-differentiable sorting, we use NeuralSort to obtain an approximate, differentiable surrogate. On all tasks, we observe signi\ufb01cant empirical improvements due to NeuralSort over the relevant baselines and competing relaxations to permutation matrices. 2 PRELIMINARIES An n-dimensional permutation z = [z1, z2, . . . , zn]T is a list of unique indices {1, 2, . . . , n}. Every permutation z is associated with a permutation matrix Pz \u2208{0, 1}n\u00d7n with entries given as: Pz[i, j] = \u001a1 if j = zi 0 otherwise. Let Zn denote the set of all n! possible permutations in the symmetric group. We de\ufb01ne the sort : Rn \u2192Zn operator as a mapping of n real-valued inputs to a permutation corresponding to a descending ordering of these inputs. E.g., if the input vector s = [9, 1, 5, 2]T , then sort(s) = [1, 3, 4, 2]T since the largest element is at the \ufb01rst index, second largest element is at the third index and so on. In case of ties, elements are assigned indices in the order they appear. We can obtain the sorted vector simply via Psort(s)s. 2.1 PLACKETT-LUCE DISTRIBUTIONS The family of Plackett-Luce distributions over permutations is best described via a generative process: Consider a sequence of n items, each associated with a canonical index i = 1, 2, . . . , n. A common assumption in ranking models is that the underlying generating process for any observed permutation of n items satis\ufb01es Luce\u2019s choice axiom (Luce, 1959). Mathematically, this axiom de\ufb01nes the \u2018choice\u2019 probability of an item with index i as: q(i) \u221dsi where si > 0 is interpreted as the score of item with index i. The normalization constant is given by Z = P i\u2208{1,2,...,n} si. If we choose the n items one at a time (without replacement) based on these choice probabilities, we obtain a discrete distribution over all possible permutations. This distribution is referred to as the Plackett-Luce (PL) distribution, and its probability mass function for any z \u2208Zn is given by: q(z|s) = sz1 Z sz2 Z \u2212sz1 \u00b7 \u00b7 \u00b7 szn Z \u2212Pn\u22121 i=1 szi (1) where s = {s1, s2, . . . , sn} is the vector of scores parameterizing this distribution (Plackett, 1975). 2 \fPublished as a conference paper at ICLR 2019 s z \u03b8 f sort Figure 1: Stochastic computation graphs with a deterministic node z corresponding to the output of a sort operator applied to the scores s. 2.2 STOCHASTIC COMPUTATION GRAPHS The abstraction of stochastic computation graphs (SCG) compactly speci\ufb01es the forward value and the backward gradient computation for computational circuits. An SCG is a directed acyclic graph that consists of three kinds of nodes: input nodes which specify external inputs (including parameters), deterministic nodes which are deterministic functions of their parents, and stochastic nodes which are distributed conditionally on their parents. See Schulman et al. (2015) for a review. To de\ufb01ne gradients of an objective function with respect to any node in the graph, the chain rule necessitates that the gradients with respect to the intermediate nodes are well-de\ufb01ned. This is not the case for the sort operator. In Section 3, we propose to extend stochastic computation graphs with nodes corresponding to a relaxation of the deterministic sort operator. In Section 4, we further use this relaxation to extend computation graphs to include stochastic nodes corresponding to distributions over permutations. The proofs of all theoretical results in this work are deferred to Appendix B. 3 NEURALSORT: THE RELAXED SORTING OPERATOR Our goal is to optimize training objectives involving a sort operator with gradient-based methods. Consider the optimization of objectives written in the following form: L(\u03b8, s) = f(Pz; \u03b8) (2) where z = sort(s). Here, s \u2208Rn denotes a vector of n real-valued scores, z is the permutation that (deterministically) sorts the scores s, and f(\u00b7) is an arbitrary function of interest assumed to be differentiable w.r.t a set of parameters \u03b8 and z. For example, in a ranking application, these scores could correspond to the inferred relevances of n webpages and f(\u00b7) could be a ranking loss. Figure 1 shows the stochastic computation graph corresponding to the objective in Eq. 2. We note that this could represent part of a more complex computation graph, which we skip for ease of presentation while maintaining the generality of the scope of this work. While the gradient of the above objective w.r.t. \u03b8 is well-de\ufb01ned and can be computed via standard backpropogation, the gradient w.r.t. the scores s is not de\ufb01ned since the sort operator is not differentiable w.r.t. s. Our solution is to derive a relaxation to the sort operator that leads to a surrogate objective with well-de\ufb01ned gradients. In particular, we seek to use such a relaxation to replace the permutation matrix Pz in Eq. 2 with an approximation b Pz such that the surrogate objective f( b Pz; \u03b8) is differentiable w.r.t. the scores s. The general recipe to relax non-differentiable operators with discrete codomains N is to consider differentiable alternatives that map the input to a larger continuous codomain M with desirable properties. For gradient-based optimization, we are interested in two key properties: 1. The relaxation is continuous everywhere and differentiable (almost-)everywhere with respect to elements in the input domain. 2. There exists a computationally ef\ufb01cient projection from M back to N. Relaxations satisfying the \ufb01rst requirement are amenable to automatic differentiation for optimizing stochastic computational graphs. The second requirement is useful for evaluating metrics and losses that necessarily require a discrete output akin to the one obtained from the original, non-relaxed operator. E.g., in straight-through gradient estimation (Bengio et al., 2013; Jang et al., 2017), the 3 \fPublished as a conference paper at ICLR 2019 0 1/2 1/2 7/16 3/16 3/8 9/16 5/16 1/8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 P D R U 3/8 1/8 1/2 3/4 1/4 0 1/4 1/2 1/4 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 Figure 2: Center: Venn Diagram relationships between permutation matrices (P), doubly-stochastic matrices (D), unimodal row stochastic matrices (U), and row stochastic matrices (R). Left: A doubly-stochastic matrix that is not unimodal. Right: A unimodal matrix that is not doublystochastic. non-relaxed operator is used for evaluating the learning objective in the forward pass and the relaxed operator is used in the backward pass for gradient estimation. The canonical example is the 0/1 loss used for binary classi\ufb01cation. While the 0/1 loss is discontinuous w.r.t. its inputs (real-valued predictions from a model), surrogates such as the logistic and hinge losses are continuous everywhere and differentiable almost-everywhere (property 1), and can give hard binary predictions via thresholding (property 2). Note: For brevity, we assume that the arg max operator is applied over a set of elements with a unique maximizer and hence, the operator has well-de\ufb01ned semantics. With some additional bookkeeping for resolving ties, the results in this section hold even if the elements to be sorted are not unique. See Appendix C. Unimodal Row Stochastic Matrices. The sort operator maps the input vector to a permutation, or equivalently a permutation matrix. Our relaxation to sort is motivated by the geometric structure of permutation matrices. The set of permutation matrices is a subset of doubly-stochastic matrices, i.e., a non-negative matrix such that the every row and column sums to one. If we remove the requirement that every column should sum to one, we obtain a larger set of row stochastic matrices. In this work, we propose a relaxation to sort that maps inputs to an alternate subset of row stochastic matrices, which we refer to as the unimodal row stochastic matrices. De\ufb01nition 1 (Unimodal Row Stochastic Matrices). An n \u00d7 n matrix is Unimodal Row Stochastic if it satis\ufb01es the following conditions: 1. Non-negativity: U[i, j] \u22650 \u2200i, j \u2208{1, 2, . . . , n}. 2. Row Af\ufb01nity: Pn j=1 U[i, j] = 1 \u2200i \u2208{1, 2, . . . , n}. 3. Argmax Permutation: Let u denote an n-dimensional vector with entries such that ui = arg maxj U[i, j] \u2200i \u2208{1, 2, . . . , n}. Then, u \u2208Zn, i.e., it is a valid permuation. We denote Un as the set of n \u00d7 n unimodal row stochastic matrices. All row stochastic matrices satisfy the \ufb01rst two conditions. The third condition is useful for gradient based optimization involving sorting-based losses. The condition provides a straightforward mechanism for extracting a permutation from a unimodal row stochastic matrix via a row-wise arg max operation. Figure 2 shows the relationships between the different subsets of square matrices. NeuralSort. Our relaxation to the sort operator is based on a standard identity for evaluating the sum of the k largest elements in any input vector. Lemma 2. [Lemma 1 in Ogryczak & Tamir (2003)] For an input vector s = [s1, s2, . . . , sn]T that is sorted as s[1] \u2265s[2] \u2265. . . \u2265s[n], we have the sum of the k-largest elements given as: k X i=1 s[i] = min \u03bb\u2208{s1,s2,...,sn} \u03bbk + n X i=1 max(si \u2212\u03bb, 0). (3) The identity in Lemma 2 outputs the sum of the top-k elements. The k-th largest element itself can be recovered by taking the difference of the sum of top-k elements and the top-(k \u22121) elements. 4 \fPublished as a conference paper at ICLR 2019 s z \u03b8 f (a) Stochastic log s + g z \u03b8 f sort (b) Reparameterized Stochastic Figure 3: Stochastic computation graphs with stochastic nodes corresponding to permutations. Squares denote deterministic nodes and circles denote stochastic nodes. Corollary 3. Let s = [s1, s2, . . . , sn]T be a real-valued vector of length n. Let As denote the matrix of absolute pairwise differences of the elements of s such that As[i, j] = |si \u2212sj|. The permutation matrix Psort(s) corresponding to sort(s) is given by: Psort(s)[i, j] = \u001a1 if j = arg max[(n + 1 \u22122i)s \u2212As1] 0 otherwise (4) where 1 denotes the column vector of all ones. E.g., if we set i = \u230a(n + 1)/2\u230bthen the non-zero entry in the i-th row Psort(s)[i, :] corresponds to the element with the minimum sum of (absolute) distance to the other elements. As desired, this corresponds to the median element. The relaxation requires O(n2) operations to compute As, as opposed to the O(n log n) overall complexity for the best known sorting algorithms. In practice however, it is highly parallelizable and can be implemented ef\ufb01ciently on GPU hardware. The arg max operator is non-differentiable which prohibits the direct use of Corollary 3 for gradient computation. Instead, we propose to replace the arg max operator with soft max to obtain a continuous relaxation b Psort(s)(\u03c4). In particular, the i-th row of b Psort(s)(\u03c4) is given by: b Psort(s)[i, :](\u03c4) = soft max [((n + 1 \u22122i)s \u2212As1)/\u03c4] (5) where \u03c4 > 0 is a temperature parameter. Our relaxation is continuous everywhere and differentiable almost everywhere with respect to the elements of s. Furthermore, we have the following result. Theorem 4. Let b Psort(s) denote the continuous relaxation to the permutation matrix Psort(s) for an arbitrary input vector s and temperature \u03c4 de\ufb01ned in Eq. 5. Then, we have: 1. Unimodality: \u2200\u03c4 > 0, b Psort(s) is a unimodal row stochastic matrix. Further, let u denote the permutation obtained by applying arg max row-wise to b Psort(s). Then, u = sort(s). 2. Limiting behavior: If we assume that the entries of s are drawn independently from a distribution that is absolutely continuous w.r.t. the Lebesgue measure in R, then the following convergence holds almost surely: lim \u03c4\u21920+ b Psort(s)[i, :](\u03c4) = Psort(s)[i, :] \u2200i \u2208{1, 2, . . . , n}. (6) Unimodality allows for ef\ufb01cient projection of the relaxed permutation matrix b Psort(s) to the hard matrix Psort(s) via a row-wise arg max, e.g., for straight-through gradients. For analyzing limiting behavior, independent draws ensure that the elements of s are distinct almost surely. The temperature \u03c4 controls the degree of smoothness of our approximation. At one extreme, the approximation becomes tighter as the temperature is reduced. In practice however, the trade-off is in the variance of these estimates, which is typically lower for larger temperatures. 4 STOCHASTIC OPTIMIZATION OVER PERMUTATIONS In many scenarios, we would like the ability to express our uncertainty in inferring a permutation e.g., latent variable models with latent nodes corresponding to permutations. Random variables that assume values corresponding to permutations can be represented via stochastic nodes in the 5 \fPublished as a conference paper at ICLR 2019 stochastic computation graph. For optimizing the parameters of such a graph, consider the following class of objectives: L(\u03b8, s) = Eq(z|s) [f(Pz; \u03b8)] (7) where \u03b8 and s denote sets of parameters, Pz is the permutation matrix corresponding to the permutation z, q(\u00b7) is a parameterized distribution over the elements of the symmetric group Zn, and f(\u00b7) is an arbitrary function of interest assumed to be differentiable in \u03b8 and z. The SCG is shown in Figure 3a. In contrast to the SCG considered in the previous section (Figure 1), here we are dealing with a distribution over permutations as opposed to a single (deterministically computed) one. While such objectives are typically intractable to evaluate exactly since they require summing over a combinatorially large set, we can obtain unbiased estimates ef\ufb01ciently via Monte Carlo. Monte Carlo estimates of gradients w.r.t. \u03b8 can be derived simply via linearity of expectation. However, the gradient estimates w.r.t. s cannot be obtained directly since the sampling distribution depends on s. The REINFORCE gradient estimator (Glynn, 1990; Williams, 1992; Fu, 2006) uses the fact that \u2207sq(z|s) = q(z|s)\u2207s log q(z|s) to derive the following Monte Carlo gradient estimates: \u2207sL(\u03b8, s) = Eq(z|s) [f(Pz; \u03b8)\u2207s log q(z|s)] + Eq(z|s) [\u2207sf(Pz; \u03b8)] . (8) 4.1 REPARAMETERIZED GRADIENT ESTIMATORS FOR PL DISTRIBUTIONS REINFORCE gradient estimators typically suffer from high variance (Schulman et al., 2015; Glasserman, 2013). Reparameterized samplers provide an alternate gradient estimator by expressing samples from a distribution as a deterministic function of its parameters and a \ufb01xed source of randomness (Kingma & Welling, 2014; Rezende et al., 2014; Titsias & L\u00b4 azaro-Gredilla, 2014). Since the randomness is from a \ufb01xed distribution, Monte Carlo gradient estimates can be derived by pushing the gradient operator inside the expectation (via linearity). In this section, we will derive a reparameterized sampler and gradient estimator for the Plackett-Luce (PL) family of distributions. Let the score si for an item i \u2208{1, 2, . . . , n} be an unobserved random variable drawn from some underlying score distribution (Thurstone, 1927). Now for each item, we draw a score from its corresponding score distribution. Next, we generate a permutation by applying the deterministic sort operator to these n randomly sampled scores. Interestingly, prior work has shown that the resulting distribution over permutations corresponds to a PL distribution if and only if the scores are sampled independently from Gumbel distributions with identical scales. Proposition 5. [adapted from Yellott Jr (1977)] Let s be a vector of scores for the n items. For each item i, sample gi \u223cGumbel(0, \u03b2) independently with zero mean and a \ufb01xed scale \u03b2. Let \u02dc s denote the vector of Gumbel perturbed log-scores with entries such that \u02dc si = \u03b2 log si + gi. Then: q(\u02dc sz1 \u2265\u00b7 \u00b7 \u00b7 \u2265\u02dc szn) = sz1 Z sz2 Z \u2212sz1 \u00b7 \u00b7 \u00b7 szn Z \u2212Pn\u22121 i=1 szi . (9) For ease of presentation, we assume \u03b2 = 1 in the rest of this work. Proposition 5 provides a method for sampling from PL distributions with parameters s by adding Gumbel perturbations to the logscores and applying the sort operator to the perturbed log-scores. This procedure can be seen as a reparameterization trick that expresses a sample from the PL distribution as a deterministic function of the scores and a \ufb01xed source of randomness (Figure 3b). Letting g denote the vector of i.i.d. Gumbel perturbations, we can express the objective in Eq. 7 as: L(\u03b8, s) = Eg \u0002 f(Psort(log s+g); \u03b8) \u0003 . (10) While the reparameterized sampler removes the dependence of the expectation on the parameters s, it introduces a sort operator in the computation graph such that the overall objective is nondifferentiable in s. In order to obtain a differentiable surrogate, we approximate the objective based on the NeuralSort relaxation to the sort operator: Eg \u0002 f(Psort(log s+g); \u03b8) \u0003 \u2248Eg h f( b Psort(log s+g); \u03b8) i := b L(\u03b8, s). (11) Accordingly, we get the following reparameterized gradient estimates for the approximation: \u2207sb L(\u03b8, s) = Eg h \u2207sf( b Psort(log s+g); \u03b8) i (12) which can be estimated ef\ufb01ciently via Monte Carlo because the expectation is with respect to a distribution that does not depend on s. 6 \fPublished as a conference paper at ICLR 2019 5 DISCUSSION AND RELATED WORK The problem of learning to rank documents based on relevance has been studied extensively in the context of information retrieval. In particular, listwise approaches learn functions that map objects to scores. Much of this work concerns the PL distribution: the RankNet algorithm (Burges et al., 2005) can be interpreted as maximizing the PL likelihood of pairwise comparisons between items, while the ListMLE ranking algorithm in Xia et al. (2008) extends this with a loss that maximizes the PL likelihood of ground-truth permutations directly. The differentiable pairwise approaches to ranking, such as Rigutini et al. (2011), learn to approximate the comparator between pairs of objects. Our work considers a generalized setting where sorting based operators can be inserted anywhere in computation graphs to extend traditional pipelines e.g., kNN. Prior works have proposed relaxations of permutation matrices to the Birkhoff polytope, which is de\ufb01ned as the convex hull of the set of permutation matrices a.k.a. the set of doubly-stochastic matrices. A doubly-stochastic matrix is a permutation matrix iff it is orthogonal and continuous relaxations based on these matrices have been used previously for solving NP-complete problems such as seriation and graph matching (Fogel et al., 2013; Fiori et al., 2013; Lim & Wright, 2014). Adams & Zemel (2011) proposed the use of the Sinkhorn operator to map any square matrix to the Birkhoff polytope. They interpret the resulting doubly-stochastic matrix as the marginals of a distribution over permutations. Mena et al. (2018) propose an alternate method where the square matrix de\ufb01nes a latent distribution over the doubly-stochastic matrices themselves. These distributions can be sampled from by adding elementwise Gumbel perturbations. Linderman et al. (2018) propose a rounding procedure that uses the Sinkhorn operator to directly sample matrices near the Birkhoff polytope. Unlike Mena et al. (2018), the resulting distribution over matrices has a tractable density. In practice, however, the approach of Mena et al. (2018) performs better and will be the main baseline we will be comparing against in our experiments in Section 6. As discussed in Section 3, NeuralSort maps permutation matrices to the set of unimodal rowstochastic matrices. For the stochastic setting, the PL distribution permits ef\ufb01cient sampling, exact and tractable density estimation, making it an attractive choice for several applications, e.g., variational inference over latent permutations. Our reparameterizable sampler, while also making use of the Gumbel distribution, is based on a result unique to the PL distribution (Proposition 5). The use of the Gumbel distribution for de\ufb01ning continuous relaxations to discrete distributions was \ufb01rst proposed concurrently by Jang et al. (2017) and Maddison et al. (2017) for categorical variables, referred to as Gumbel-Softmax. The number of possible permutations grow factorially with the dimension, and thus any distribution over n-dimensional permutations can be equivalently seen as a distribution over n! categories. Gumbel-softmax does not scale to a combinatorially large number of categories (Kim et al., 2016; Mussmann et al., 2017), necessitating the use of alternate relaxations, such as the one considered in this work. 6 EXPERIMENTS We refer to the two approaches proposed in Sections 3, 4 as Deterministic NeuralSort and Stochastic NeuralSort, respectively. For additional hyperparameter details and analysis, see Appendix D. 6.1 SORTING HANDWRITTEN NUMBERS Dataset. We \ufb01rst create the large-MNIST dataset, which extends the MNIST dataset of handwritten digits. The dataset consists of multi-digit images, each a concatenation of 4 randomly selected individual images from MNIST, e.g., is one such image in this dataset. Each image is associated with a real-valued label, which corresponds to its concatenated MNIST labels, e.g., the label of is 1810. Using the large-MNIST dataset, we \ufb01nally create a dataset of sequences. Every sequence is this dataset consists of n randomly sampled large-MNIST images. Setup. Given a dataset of sequences of large-MNIST images, our goal is to learn to predict the permutation that sorts the labels of the sequence of images, given a training set of ground-truth permutations. Figure 4 (Task 1) illustrates this task on an example sequence of n = 5 large-MNIST images. This task is a challenging extension of the one considered by Mena et al. (2018) in sorting scalars, since it involves learning the semantics of high-dimensional objects prior to sorting. A 7 \fPublished as a conference paper at ICLR 2019 x1 x2 x3 x4 x5 si = CNN(xi) s3 s2 s1 s4 s5 z = NeuralSort(s) Task 1: Sorting Loss([3, 5, 1, 4, 2]T , z) \u02c6 y = CNN(\u02c6 x) Task 2: Median Regression Loss(2960, \u02c6 y) \u02c6 x = xz[3] Figure 4: Sorting and quantile regression. The model is trained to sort sequences of n = 5 largeMNIST images x1, x2, . . . , x5 (Task 1) and regress the median value (Task 2). In the above example, the ground-truth permutation that sorts the input sequence from largest to smallest is [3, 5, 1, 4, 2]T , 9803 being the largest and 1270 the smallest. Blue illustrates the true median image x1 with ground-truth sorted index 3 and value 2960. Table 1: Average sorting accuracy on the test set. First value is proportion of permutations correctly identi\ufb01ed; value in parentheses is the proportion of individual element ranks correctly identi\ufb01ed. Algorithm n = 3 n = 5 n = 7 n = 9 n = 15 Vanilla RS 0.467 (0.801) 0.093 (0.603) 0.009 (0.492) 0. (0.113) 0. (0.067) Sinkhorn 0.462 (0.561) 0.038 (0.293) 0.001 (0.197) 0. (0.143) 0. (0.078) Gumbel-Sinkhorn 0.484 (0.575) 0.033 (0.295) 0.001 (0.189) 0. (0.146) 0. (0.078) Deterministic NeuralSort 0.930 (0.951) 0.837 (0.927) 0.738 (0.909) 0.649 (0.896) 0.386 (0.857) Stochastic NeuralSort 0.927 (0.950) 0.835 (0.926) 0.741 (0.909) 0.646 (0.895) 0.418 (0.862) good model needs to learn to dissect the individual digits in an image, rank these digits, and \ufb01nally, compose such rankings based on the digit positions within an image. The available supervision, in the form of the ground-truth permutation, is very weak compared to a classi\ufb01cation setting that gives direct access to the image labels. Baselines. All baselines use a CNN that is shared across all images in a sequence to map each large-MNIST image to a feature space. The vanilla row-stochastic (RS) baseline concatenates the CNN representations for n images into a single vector that is fed into a multilayer perceptron that outputs n multiclass predictions of the image probabilities for each rank. The Sinkhorn and GumbelSinkhorn baselines, as discussed in Section 5, use the Sinkhorn operator to map the stacked CNN representations for the n objects into a doubly-stochastic matrix. For all methods, we minimized the cross-entropy loss between the predicted matrix and the ground-truth permutation matrix. Results. Following Mena et al. (2018), our evaluation metric is the the proportion of correctly predicted permutations on a test set of sequences. Additionally, we evaluate the proportion of individual elements ranked correctly. Table 1 demonstrates that the approaches based on the proposed sorting relaxation signi\ufb01cantly outperform the baseline approaches for all n considered. The performance of the deterministic and stochastic variants are comparable. The vanilla RS baseline performs well in ranking individual elements, but is not good at recovering the overall square matrix. We believe the poor performance of the Sinkhorn baselines is partly because these methods were designed and evaluated for matchings. Like the output of sort, matchings can also be represented as permutation matrices. However, distributions over matchings need not satisfy Luce\u2019s choice axiom or imply a total ordering, which could explain the poor performance on the tasks considered. 6.2 QUANTILE REGRESSION Setup. In this experiment, we extend the sorting task to regression. Again, each sequence contains n large-MNIST images, and the regression target for each sequence is the 50-th quantile (i.e., the median) of the n labels of the images in the sequence. Figure 4 (Task 2) illustrates this task on an example sequence of n = 5 large-MNIST images, where the goal is to output the third largest label. The design of this task highlights two key challenges since it explicitly requires learning both a suitable representation for sorting high-dimensional inputs and a secondary function that approximates the label itself (regression). Again, the supervision available in the form of the label of only a single image at an arbitrary and unknown location in the sequence is weak. 8 \fPublished as a conference paper at ICLR 2019 Table 2: Test mean squared error (\u00d710\u22124) and R2 values (in parenthesis) for quantile regression. Algorithm n = 5 n = 9 n = 15 Constant (Simulated) 356.79 (0.00) 227.31 (0.00) 146.94 ( 0.00) Vanilla NN 1004.70 (0.85) 699.15 (0.82) 562.97 (0.79) Sinkhorn 343.60 (0.25) 231.87 (0.19) 156.27 (0.04) Gumbel-Sinkhorn 344.28 (0.25) 232.56 (0.23) 157.34 (0.06) Deterministic NeuralSort 45.50 (0.95) 34.98 (0.94) 34.78 (0.92) Stochastic NeuralSort 33.80 (0.94) 31.43 (0.93) 29.34 (0.90) x0 x1 x2 \u00b7 \u00b7 \u00b7 xn ei = h\u03c6(xi) e2 e1 e0 \u00b7 \u00b7 \u00b7 en s1 = \u2212\u2225e1 \u2212e0\u22252 s2 = \u2212\u2225e2 \u2212e0\u22252 \u00b7 \u00b7 \u00b7 sn = \u2212\u2225en\u2212e0\u22252 z = NeuralSort(s) \u2113kNN(Pz, y0, y1, y2, . . . , yn) y0 top-k Figure 5: Differentiable kNN. The model is trained such that the representations ei for the training points {x1, . . . , xn} that have the same label y0 as x0 are closer to e0 (included in top-k) than others. Baselines. In addition to Sinkhorn and Gumbel-Sinkhorn, we design two more baselines. The Constant baseline always returns the median of the full range of possible outputs, ignoring the input sequence. This corresponds to 4999.5 since we are sampling large-MNIST images uniformly in the range of four-digit numbers. The vanilla neural net (NN) baseline directly maps the input sequence of images to a real-valued prediction for the median. Results. Our evaluation metric is the mean squared error (MSE) and R2 on a test set of sequences. Results for n = {5, 9, 15} images are shown in Table 2. The Vanilla NN baseline while incurring a large MSE, is competitive on the R2 metric. The other baselines give comparable performance on the MSE metric. The proposed NeuralSort approaches outperform the competing methods on both the metrics considered. The stochastic NeuralSort approach is the consistent best performer on MSE, while the deterministic NeuralSort is slightly better on the R2 metric. 6.3 END-TO-END, DIFFERENTIABLE k-NEAREST NEIGHBORS Setup. In this experiment, we design a fully differentiable, end-to-end k-nearest neighbors (kNN) classi\ufb01er. Unlike a standard kNN classi\ufb01er which computes distances between points in a prede\ufb01ned space, we learn a representation of the data points before evaluating the k-nearest neighbors. We are given access to a dataset D of (x, y) pairs of standard input data and their class labels respectively. The differentiable kNN algorithm consists of two hyperparameters: the number of training neighbors n, the number of top candidates k, and the sorting temperature \u03c4. Every sequence of items here consists of a query point x and a randomly sampled subset of n candidate nearest neighbors from the training set, say {x1, x2, . . . , xn}. In principle, we could use the entire training set (excluding the query point) as candidate points, but this can hurt the learning both computationally and statistically. The query points are randomly sampled from the train/validation/test sets as appropriate but the nearest neighbors are always sampled from the training set. The loss function optimizes for a representation space h\u03c6(\u00b7) (e.g., CNN) such that the top-k candidate points with the minimum Euclidean distance to the query point in the representation space have the same label as the query point. Note that at test time, once the representation space h\u03c6 is learned, we can use the entire training set as the set of candidate points, akin to a standard kNN classi\ufb01er. Figure 5 illustrates the proposed algorithm. Formally, for any datapoint x, let z denote a permutation of the n candidate points. The uniformlyweighted kNN loss, denoted as \u2113kNN(\u00b7), can be written as follows: \u2113kNN( b Pz, y, y1, y2, . . . , yn) = \u22121 k k X j=1 n X i=1 1(yi = y) b Pz[i, j] (13) 9 \fPublished as a conference paper at ICLR 2019 Table 3: Average test kNN classi\ufb01cation accuracies from n neighbors for best value of k. Algorithm MNIST Fashion-MNIST CIFAR-10 kNN 97.2% 85.8% 35.4% kNN+PCA 97.6% 85.9% 40.9% kNN+AE 97.6% 87.5% 44.2% kNN + Deterministic NeuralSort 99.5% 93.5% 90.7% kNN + Stochastic NeuralSort 99.4% 93.4% 89.5% CNN (w/o kNN) 99.4% 93.4% 95.1% where {y1, y2, . . . , yn} are the labels for the candidate points. Note that when b Pz is an exact permutation matrix (i.e., temperature \u03c4 \u21920), this expression is exactly the negative of the fraction of k nearest neighbors that have the same label as x. Using Eq. 13, the training objectives for Deterministic and Stochastic NeuralSort are given as: Deterministic: min \u03c6 1 |D| X (x,y)\u2208D \u2113kNN( b Psort(s), y, y1, . . . , yn) (14) Stochastic: min \u03c6 1 |D| X (x,y)\u2208D Ez\u223cq(z|s) h \u2113kNN( b Pz, y, y1, y2, . . . , yn) i (15) where each entry of s is given as sj = \u2212\u2225h\u03c6(x) \u2212h\u03c6(xj)\u22252 2. Datasets. We consider three benchmark datasetes: MNIST dataset of handwritten digits, FashionMNIST dataset of fashion apparel, and the CIFAR-10 dataset of natural images (no data augmentation) with the canonical splits for training and testing. Baselines. We consider kNN baselines that operate in three standard representation spaces: the canonical pixel basis, the basis speci\ufb01ed by the top 50 principal components (PCA), an autonencoder (AE). Additionally, we experimented with k = 1, 3, 5, 9 nearest neighbors and across two distance metrics: uniform weighting of all k-nearest neighbors and weighting nearest neighbors by the inverse of their distance. For completeness, we trained a CNN with the same architecture as the one used for NeuralSort (except the \ufb01nal layer) using the cross-entropy loss. Results. We report the classi\ufb01cation accuracies on the standard test sets in Table 3. On both datasets, the differentiable kNN classi\ufb01er outperforms all the baseline kNN variants including the convolutional autoencoder approach. The performance is much closer to the accuracy of a standard CNN. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1812.10539v3",
+                    "title": "Uncertainty Autoencoders: Learning Compressed Representations via Variational Information Maximization",
+                    "abstract": "Compressed sensing techniques enable efficient acquisition and recovery of\nsparse, high-dimensional data signals via low-dimensional projections. In this\nwork, we propose Uncertainty Autoencoders, a learning framework for\nunsupervised representation learning inspired by compressed sensing. We treat\nthe low-dimensional projections as noisy latent representations of an\nautoencoder and directly learn both the acquisition (i.e., encoding) and\namortized recovery (i.e., decoding) procedures. Our learning objective\noptimizes for a tractable variational lower bound to the mutual information\nbetween the datapoints and the latent representations. We show how our\nframework provides a unified treatment to several lines of research in\ndimensionality reduction, compressed sensing, and generative modeling.\nEmpirically, we demonstrate a 32% improvement on average over competing\napproaches for the task of statistical compressed sensing of high-dimensional\ndatasets.",
+                    "authors": "Aditya Grover, Stefano Ermon",
+                    "published": "2018-12-26",
+                    "updated": "2019-04-11",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG",
+                        "cs.NE"
+                    ],
+                    "main_content": "INTRODUCTION The goal of unsupervised representation learning is to learn transformations of the input data which succinctly capture the statistics of an underlying data distribution [1]. In this work, we propose a learning framework for unsupervised representation learning inspired by compressed sensing. Compressed sensing is a class of techniques used to e\ufb03ciently acquire and A preliminary version titled \u201dVariational Compressive Sensing using Uncertainty Autoencoders\u201d appeared at the Uncertainty in Deep Learning Workshop at UAI, 2018. Proceedings of the 22nd International Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2019, Naha, Okinawa, Japan. PMLR: Volume 89. Copyright 2019 by the author(s). recover high-dimensional data using compressed measurements much fewer than the data dimensionality. The celebrated results in compressed sensing posit that sparse, high-dimensional datapoints can be acquired using much fewer measurements (roughly logarithmic) than the data dimensionality [2, 3, 4]. The acquisition is done using certain classes of random matrices and the recovery procedure is based on LASSO [5, 6]. The assumptions of sparsity are fairly general and can be applied \u201cout-of-the-box\u201d for many data modalities, e.g., images and audio are typically sparse in the wavelet and Fourier basis respectively. However, such assumptions ignore the statistical nature of many realworld problems. For representation learning in particular, we have access to a training dataset from an underlying domain. In this work, we use this data to learn the acquisition and recovery procedures, thereby sidestepping generic sparsity assumptions. In particular, we view the compressed measurements as the latent representations of an uncertainty autoencoder. An uncertainty autoencoder (UAE) parameterizes both the acquisition and recovery procedures for compressed sensing. The learning objective for a UAE is based on the InfoMax principle, which seeks to learn encodings that maximize the mutual information between the observed datapoints and noisy representations [7]. Since the mutual information is typically intractable in high-dimensions, we instead maximize tractable variational lower bounds [8, 9]. In doing so, we introduce a parameteric decoder that is trained to recover the original datapoint via its noisy representation. Unlike LASSO-based recovery, a parametric decoder amortizes the recovery process, which requires only a forward pass through the decoder at test time and thus enables scalability to large datasets [10, 11]. Notably, the framework of uncertainty autoencoders uni\ufb01es and extends several lines of prior research in unsupervised representation learning. First, we show theoretically under suitable assumptions that an uncertainty autoencoder is an implicit generative model of the underlying data distribution [12], i.e., a UAE permits sampling from the learned data distribution even arXiv:1812.10539v3  [stat.ML]  11 Apr 2019 \fUncertainty Autoencoders though it does not specify an explicit likelihood function. Hence, it directly contrasts with variational autoencoders (VAE) which specify a likelihood function (which is intractable and approximated by a tractable evidence lower bound) [13]. Unlike a VAE, a UAE does not require specifying a prior over the latent representations and hence o\ufb00sets pathologically observed scenarios that cause the latent representations to be uninformative when used with expressive decoders [14]. Next, we show that an uncertainty autoencoder, under suitable assumptions, is a generalization of principal component analysis (PCA). While earlier results connecting standard autoencoders with PCA assume linear encodings and decodings [15, 16, 17], our result surprisingly holds even for non-linear decodings. In practice, linear encodings learned jointly with non-linear decodings based on the UAE objective vastly outperform the linear encodings obtained via PCA. For dimensionality reduction on the MNIST dataset, we observed an average improvement of 5.33% over PCA when the low-dimensional representations are used for classi\ufb01cation under a wide range of settings. We evaluate UAEs for statistical compressed sensing of high-dimensional datasets. On the MNIST, Omniglot, and CelebA datasets, we observe average improvements of 38%, 31%, and 28% in recovery over the closest benchmark across all measurements considered. Finally, we demonstrate that uncertainty autoencoders demonstrate good generalization performance across domains in experiments where the encoder/decoder trained on a source dataset are transferred over for compressed sensing of another target dataset. 2 PRELIMINARIES We use upper case to denote probability distributions and assume they admit absolutely continuous densities on a suitable reference measure, denoted by lower case notation. We also use upper and lower case for random variables and their realizations respectively. Compressed sensing (CS). Let the datapoint and measurements be denoted with multivariate random variables X \u2208Rn and Y \u2208Rm respectively. The goal is to recover X given the measurements Y . For the purpose of compressed sensing, we assume m < n and relate these variables through a measurement matrix W \u2208Rm\u00d7l and a parameterized acquisition function f\u03c8 : Rn \u2192Rl (for any integer l > 0) such that: y = Wf\u03c8(x) + \u03f5 (1) where \u03f5 is the measurement noise. If we let f\u03c8(\u00b7) be the identity function (i.e., f\u03c8(x) = x for all x), then we recover a standard system of underdetermined linear equations where measurements are linear combinations of the datapoint corrupted by noise. In all other cases, the acquisition function transforms x such that f\u03c8(x) is potentially more amenable for compressed sensing. For instance, f\u03c8(\u00b7) could specify a change of basis that encourages sparsity, e.g., a Fourier basis for audio. Note that we allow the codomain of the mapping f\u03c8(\u00b7) to be de\ufb01ned on a higher or lower dimensional space (i.e., l \u0338= n in general). Sparse CS. To obtain nontrivial solutions to an underdetermined system, X is assumed to be sparse in some basis B. We are not given any additional information about X. The measurement matrix W is a random Gaussian matrix and the recovery is done via LASSO [2, 3, 4]. LASSO solves for a convex \u21131minimization problem such that the reconstruction b x for any datapoint x is given as: \u02c6 x = arg minx \u2225Bx\u22251 + \u03bb\u2225y\u2212Wx\u22252 2 where \u03bb > 0 is a tunable hyperparameter. Statistical CS. In statistical compressed sensing [18], we are additionally given access to a set of signals D, such that each x \u2208D is assumed to be sampled i.i.d. from a data distribution Qdata. Using this dataset, we learn the the measurement matrix W and the acquisition function f\u03c8(\u00b7) in Eq. (1). At test time, we directly observe the measurements ytest that are assumed to satisfy Eq. (1) for a target datapoint xtest \u223cQdata(X) and the task is to provide an accurate reconstruction \u02c6 xtest. Evaluation is based on the reconstruction error between xtest and \u02c6 xtest. Particularly relevant to this work, we can optionally learn a recovery function g\u03b8 : Rm \u2192Rn to reconstruct X given the measurements Y . This amortized approach [11] is in contrast to standard LASSO-based decoding which solves an optimization problem for every new datapoint at test time. If we learned the recovery function g\u03b8(\u00b7) during training, then \u02c6 xtest = g\u03b8(ytest) and the \u21132 error is given by \u2225xtest \u2212g\u03b8(ytest)\u22252. Such a recovery process requires only a function evaluation at test time and permits scaling to large datasets [10, 11]. Autoencoders. An autoencoder is a pair of parameterized functions (e, d) designed to encode and decode datapoints. For a standard autoencoder, let e : Rn \u2192Rm and d : Rm \u2192Rn denote the encoding and decoding functions respectively for an ndimensional datapoint and an m-dimensional latent space. The learning objective minimizes the l2 reconstruction error over a dataset D: min e,d X x\u2208D \u2225x \u2212d(e(x))\u22252 2 (2) \fAditya Grover, Stefano Ermon where the encoding and decoding functions are typically parameterized using neural networks. 3 UNCERTAINTY AUTOENCODER Consider a joint distribution between the signals X and the measurements Y , which factorizes as Q\u03c6(X, Y ) = Qdata(X)Q\u03c6(Y |X). Here, Qdata(X) is a \ufb01xed data distribution and Q\u03c6(Y |X) is a parameterized observation model that depends on the measurement noise \u03f5, as given by Eq. (1). In particular, \u03c6 corresponds to collectively the set of measurement matrix parameters W and the acquisition function parameters \u03c8. For instance, for isotropic Gaussian noise \u03f5 with a \ufb01xed variance \u03c32, we have Q\u03c6(Y |X) = N(Wf\u03c8(X), \u03c32Im). In an uncertainty autoencoder, we wish to learn the parameters \u03c6 that permit e\ufb03cient and accurate recovery of a signal X using the measurements Y . In order to do so, we propose to maximize the mutual information between X and Y : max \u03c6 I\u03c6(X, Y ) = Z q\u03c6(x, y) log q\u03c6(x, y) qdata(x)q\u03c6(y)dxdy = H(X) \u2212H\u03c6(X|Y ) (3) where H denotes di\ufb00erential entropy. The intuition is simple: if the measurements preserve maximum information about the signal, we can hope that recovery will have low reconstruction error. We formalize this intuition by noting that this objective is equivalent to maximizing the average log-posterior probability of X given Y . In fact, in Eq. (3), we can omit the term corresponding to the data entropy (since it is independent of \u03c6) to get the following equivalent objective: max \u03c6 \u2212H\u03c6(X|Y ) = EQ\u03c6(X,Y )[log q\u03c6(x|y)]. (4) Even though the mutual information is maximized and equals the data entropy when Y = X, the dimensionality constraints on m \u226an, the parametric assumptions on f\u03c8(\u00b7), and the noise model prohibit learning an identity mapping. Note that the properties of noise \u03f5 such as the distributional family and su\ufb03cient statistics are externally speci\ufb01ed. For example, these could be speci\ufb01ed based on properties of the measurement device for compressed sensing. More generally for unsupervised representation learning, we treat these properties as hyperparameters tuned based on the reconstruction loss on a held-out set, or any other form of available supervision. It is not suggested to optimize for these statistics during learning as the UAE would tend to shrink this noise to zero to maximize mutual information, thus ignoring measurement uncertainty in the context of compressed sensing and preventing generalization to out-of-distribution examples for representation learning. The theoretical results in Section 4 analyze the e\ufb00ect of noise more formally. Estimating mutual information between arbitrary high dimensional random variables can be challenging. However, we can lower bound the mutual information by introducing a variational approximation to the model posterior Q\u03c6(X|Y ) [8]. Denoting this approximation as P\u03b8(X|Y ), we get the following lower bound: I\u03c6(X, Y ) \u2265H(X) + EQ\u03c6(X,Y ) [log p\u03b8(x|y)] . (5) Comparing Eqs. (3, 4, 5), we can see that the second term in Eq. (5) approximates the intractable negative conditional entropy, \u2212H\u03c6(X|Y ) with a variational lower bound. Optimizing this bound leads to a decoding distribution given by P\u03b8(X|Y ) with variational parameters \u03b8. The bound is tight when there is no distortion during recovery, or equivalently when the decoding distribution P\u03b8(X|Y ) matches the true posterior Q\u03c6(X|Y ) (i.e., the Bayes optimal decoder). Stochastic optimization. Formally, the uncertainty autoencoder (UAE) objective is given by: max \u03b8,\u03c6 EQ\u03c6(X,Y ) [log p\u03b8(x|y)] . (6) In practice, the data distribution Qdata(X) is unknown and accessible only via a \ufb01nite dataset D. Hence, expectations with respect to Qdata(X) and its gradients can be estimated using Monte Carlo methods. This allows us to express the UAE objective as: max \u03b8,\u03c6 X x\u2208D EQ\u03c6(Y |x) [log p\u03b8(x|y)] := L(\u03c6, \u03b8; D). (7) Tractable evaluation of the above objective is closely tied to the distributional assumptions on the noise model. This could be speci\ufb01ed externally based on, e.g., properties of the sensing device in compressed sensing. For the typical case of an isotropic Gaussian noise model, we know that Q\u03c6(Y |X) = N(Wf\u03c8(X), \u03c32Im), which is easy-to-sample. While Monte Carlo gradient estimates with respect to \u03b8 can be e\ufb03ciently obtained via linearity of expectation, gradient estimation with respect to \u03c6 is challenging since these parameters specify the sampling distribution Q\u03c6(Y |X). One solution is to evaluate score function gradient estimates along with control variates [19, 20, 21]. Alternatively, many continuous distributions (e.g., the isotropic Gaussian and Laplace distributions) can be reparameterized such that it is possible to obtain samples by applying a deterministic transformation to samples from a \ufb01xed distribution and typically leads to low-variance gradient estimates [13, 22, 23, 24]. \fUncertainty Autoencoders 4 THEORETICAL ANALYSIS In this section, we derive connections of uncertainty autoencoders with generative modeling and Principal Component Analysis (PCA). The proofs of all theoretical results in this section are in Appendix A. 4.1 Implicit generative modeling Starting from an arbitrary point x(0) \u2208Rn, de\ufb01ne a Markov chain over X, Y with the following transitions: y(t) \u223cQ\u03c6(Y |x(t)) (8) x(t+1) \u223cP\u03b8(X|y(t)) (9) Theorem 1. Let \u03b8\u2217, \u03c6\u2217denote an optimal solution to the UAE objective in Eq. (6). If there exists a \u03c6 such that q\u03c6(x|y) = p\u03b8\u2217(x|y) and the Markov chain de\ufb01ned in Eqs. (8, 9) is ergodic, then the stationary distribution of the chain for the parameters \u03c6\u2217and \u03b8\u2217 is given by Q\u03c6\u2217(X, Y ). The above theorem suggests an interesting insight into the behavior of UAEs. Under idealized conditions, the learned model speci\ufb01es an implicit generative model for Q\u03c6\u2217(X, Y ). Further, ergodicity can be shown to hold for the isotropic Gaussian noise model. Corollary 1. Let \u03b8\u2217, \u03c6\u2217denote an optimal solution to the UAE objective in Eq. (6). If there exists a \u03c6 such that q\u03c6(x|y) = p\u03b8\u2217(x|y) and the noise model is Gaussian, then the stationary distribution of the chain for the parameters \u03c6\u2217and \u03b8\u2217is given by Q\u03c6\u2217(X, Y ). The marginal of the joint distribution Q\u03c6(X, Y ) with respect to X corresponds to the data distribution. A UAE hence seeks to learn an implicit generative model of the data distribution [25, 12], i.e., even though we do not have a tractable estimate for the likelihood of the model, we can generate samples using the Markov chain transitions de\ufb01ned in Eqs. (8, 9). 4.2 Optimal encodings A UAE can also be viewed as a dimensionality reduction technique for the dataset D. While in general the encoding performing this reduction can be nonlinear, the case of a linear encoding is one where the projection vectors are given as the rows of the measurement matrix W. The result below characterizes the optimal encoding of the dataset D with respect to the UAE objective for an isotropic Gaussian noise model. Theorem 2. Assume a uniform data distribution over a \ufb01nite dataset D. Further, we assume that expectations in the UAE objective exist, and the signals and measurement matrices are bounded in \u21132/Frobenius norms, i.e., \u2225x\u22252 \u2264k1 for all x \u2208D, \u2225W\u2225F \u2264k2 10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 6 4 2 0 2 4 6 0.002 0.004 0.006 0.008 0.010 0.010 0.012 0.012 0.014 0.014 0.016 0.016 0.018 0.018 0.020 0.020 PCA projection UAE projection data PCA projected data UAE projected data Figure 1: Dimensionality reduction using PCA vs. UAE. Projections of the data (black points) on the UAE direction (green line) maximize the likelihood of decoding unlike the PCA projection axis (magenta line) which collapses many points in a narrow region. for some positive constants k1, k2 \u2208R+. For a linear encoder and isotropic Gaussian noise \u03f5 \u223cN(0, \u03c32I), the optimal measurement matrix W \u2217that maximizes the mutual information for an optimal decoder in the limit \u03c3 \u2192\u221eis given as: W \u2217= eigm \uf8eb \uf8edX xi,xj\u2208D \u0002 (xi \u2212xj)(xi \u2212xj)T \u0003 \uf8f6 \uf8f8 where eigm(M) denotes the top-m eigenvectors of the matrix M with the largest eigenvalues (speci\ufb01ed up to a positive scaling constant). Under the stated assumptions, the above result suggests an interesting connection between UAE and PCA. PCA seeks to \ufb01nd the directions that explain the most variance in the data. Theorem 2 suggests that when the noise in the projected signal is very high, the optimal projection directions (i.e., the rows of W \u2217) correspond to the principal components of the data signals. We note that this observation comes with a caveat; when the noise variance is high, it will dominate the contribution to the measurements Y in Eq. (1) as one would expect. Hence, the measurements and the signal will have low mutual information even under the optimal measurement matrix W \u2217. Our assumptions are notably di\ufb00erent from prior results in autoencoding drawing connections with PCA. Prior results show that linear encoding and decoding in a standard autoencoder recovers the principal components of the data (Eq. (3) in [15], Eq. (1) in [16]). In contrast, Theorem 2 is derived from variational principles and does not assume linear decoding. In general, the behaviors of UAE and PCA can be vastly di\ufb00erent. As noted in prior work [8, 26], the principal components may not be the the most informative low-dimensional projections for recovering the \fAditya Grover, Stefano Ermon 0 20 40 60 80 100 num measurements 2 3 4 5 6 7 8 9 l2 reconstruction error LASSO CS-VAE RP-UAE UAE (a) MNIST 0 20 40 60 80 100 num measurements 3 4 5 6 7 l2 reconstruction error (b) Omniglot Figure 2: Test \u21132 reconstruction error (per image) for compressed sensing. Original LASSO CS-VAE UAE (a) MNIST Original LASSO CS-VAE UAE (b) Omniglot Figure 3: Reconstructions for m = 25. Top: Original. Second: LASSO. Third: CS-VAE. Last: UAE. 25 projections of the data are su\ufb03cient for UAE to reconstruct the original image with high accuracy. original high-dimensional data back from its projections. A UAE, on the other hand, is explicitly designed to preserve as much information as possible (see Eq. (4)). We illustrate the di\ufb00erences in a synthetic experiment in Figure 1. The true data distribution is an equiweighted mixture of two Gaussians stretched along orthogonal directions. We sample 100 points (black) from this mixture and consider two dimensionality reductions. In the \ufb01rst case, we project the data on the \ufb01rst principal component (blue points on magenta line). This axis captures a large fraction of the variance in the data but collapses data sampled from the bottom right Gaussian in a narrow region. The projections of the data on the UAE axis (red points on green line) are more spread out. This suggest that recovery is easier, even if doing so increases the total variance in the projected space compared to PCA. 5 EXPERIMENTS 5.1 Statistical compressed sensing We perform compressed sensing on three datasets: MNIST [27], Omniglot [28], and CelebA dataset [29], with extremely low number of measurements m \u2208 {2, 5, 10, 25, 50, 100}. We discuss the MNIST and Omniglot datasets here since they have a similar setup. To save space, results on the CelebA dataset are deferred to Appendix B.3. Every image in MNIST and Omniglot has a dimensionality of 28 \u00d7 28. In all our experiments, we assume a Gaussian noise model with \u03c3 = 0.1. We evaluated UAE against: \u2022 LASSO decoding with random Gaussian matrices. The MNIST and Omniglot datasets are reasonably sparse in the canonical pixel basis, and hence, we did not observe any gains after applying Discrete Cosine Transform and Daubechies-1 Wavelet Transform. \u2022 CS-VAE. This approach to compressed sensing was proposed by [30] and learns a latent variable generative model over the observed variables X and the latent variables Z. Such a model de\ufb01nes a mapping G : Rk \u2192Rn from Z to X, which is given by either the mean function of the observation model for a VAE or the forward deterministic mapping to generate samples for a GAN. We use VAEs in our experiments. Thereafter, using a classic acquisition matrix satisfying a generalized Restricted Eigenvalue Condition (say W) (e.g., random Gaussian matrices), the reconstruction \u02c6 x for any datapoint is given as: \u02c6 x = G(arg minz \u2225y \u2212WG(z)\u22252). Intuitively, this procedure seeks the latent vector z such that the corresponding point on the range of G can best approximate the measurements y under the mapping W. We used the default parameter settings and architectures proposed in [30]. \fUncertainty Autoencoders 0 20 40 60 80 100 num measurements 2 3 4 5 6 7 8 9 l2 reconstruction error LASSO CS-VAE UAE-SD UAE-SE (a) Source: MNIST, Target: Omniglot 0 20 40 60 80 100 num measurements 2 3 4 5 6 7 8 9 l2 reconstruction error (b) Source: Omniglot, Target: MNIST Figure 4: Test \u21132 reconstruction error (per image) for transfer compressed sensing. Original CS-VAE UAE-SD UAE-SE (a) Source: MNIST, Target: Omniglot Original CS-VAE UAE-SD UAE-SE (b) Source: Omniglot, Target: MNIST Figure 5: Reconstructions for m = 25. Top: Target. Second: CS-VAE. Third: UAE-SD. Last: UAE-SE. \u2022 RP-UAE. To independently evaluate the e\ufb00ect of variational decoding, this ablation baseline encodes the data using Gaussian random projections (RP) and trains the decoder based on the UAE objective. Since LASSO and CS-VAE both use an RP encoding, the di\ufb00erences in performance would arise only due to the decoding procedures. The UAE decoder and the CS-VAE encoder/decoder are multi-layer perceptrons consisting of two hidden layers with 500 units each. For a fair comparison with random Gaussian matrices, the UAE encoder is linear. Further, we perform \u21132 regularization on the norm of W. This helps in generalization to test signals outside the train set and is equivalent to solving the Lagrangian of a constrained UAE objective: max \u03b8,\u03c6 EQ\u03c6(X,Y ) [log p\u03b8(x|y)] subject to \u2225W\u2225F \u2264k. The Lagrangian parameter is chosen by line search on the above objective. The constraint ensures that UAE does not learn encodings W that trivially scale the measurement matrix to overcome noise. For each m, we choose k to be the expected norm of a random Gaussian matrix of dimensions n \u00d7 m for fair comparisons with other baselines. In practice, the norm of the learned W for a UAE is much smaller than those of random Gaussian matrices suggesting that the observed performance improvements are non-trivial. Results. The \u21132 reconstruction errors on the standard test sets are shown in Figure 2. For both datasets, we observe that UAE drastically outperforms both LASSO and CS-VAE for all values of m considered. LASSO (blue curves) is unable to reconstruct with such few measurements. The CS-VAE (red) error decays much more slowly compared to UAE as m grows. Even the RP-UAE baseline (yellow), which trains the decoder keeping the encoding \ufb01xed to a random projection, outperforms CS-VAE. Jointly training the encoder and the decoder using the UAE objective (green) exhibits the best performance. These results are also re\ufb02ected qualitatively for the reconstructed test signals shown in Figure 3 for m = 25 measurements. 5.2 Transfer compressed sensing To test the generalization of the learned models to similar, unseen datasets, we consider the task of transfer compressed sensing task introduced in [31]. Experimental setup. We train the models on a source domain that is related to a target domain. Since the dimensions of MNIST and Omniglot images match, transferring from one domain to another requires no additional processing. For UAE, we consider two variants. In UAE-SE, we used the encodings from the source domain and retrain the decoder on the target domain. For UAE-SD, we use source decoder and retrain the encoder on the target domain. \fAditya Grover, Stefano Ermon Table 1: PCA vs. UAE. Average test classi\ufb01cation accuracy for the MNIST dataset. Dimensions Method kNN DT RF MLP AdaB NB QDA SVM 2 PCA 0.4078 0.4283 0.4484 0.4695 0.4002 0.4455 0.4576 0.4503 UAE 0.4644 0.5085 0.5341 0.5437 0.4248 0.5226 0.5316 0.5256 5 PCA 0.7291 0.5640 0.6257 0.7475 0.5570 0.6587 0.7321 0.7102 UAE 0.8115 0.6331 0.7094 0.8262 0.6164 0.7286 0.7961 0.7873 10 PCA 0.9257 0.6354 0.6956 0.9006 0.7025 0.7789 0.8918 0.8440 UAE 0.9323 0.5583 0.7362 0.9258 0.7165 0.7895 0.9098 0.8753 25 PCA 0.9734 0.6382 0.6889 0.9521 0.7234 0.8635 0.9572 0.9194 UAE 0.9730 0.5407 0.7022 0.9614 0.7398 0.8306 0.9580 0.9218 50 PCA 0.9751 0.6381 0.6059 0.9580 0.7390 0.8786 0.9632 0.9376 UAE 0.9754 0.5424 0.6765 0.9597 0.7330 0.8579 0.9638 0.9384 100 PCA 0.9734 0.6380 0.4040 0.9584 0.7136 0.8763 0.9570 0.9428 UAE 0.9731 0.6446 0.6241 0.9597 0.7170 0.8809 0.9595 0.9431 Results. The \u21132 reconstruction errors are shown in Figure 4. LASSO (blue curves) does not involve any learning, and hence its performance is same as Figure 2. The CS-VAE (red) performance degrades signi\ufb01cantly in comparison, even performing worse than LASSO in some cases. The UAE based methods outperform these approaches and UAE-SE (green) fares better than UAE-SD (yellow). Qualitative di\ufb00erences are highlighted in Figure 5 for m = 25 measurements. 5.3 Dimensionality reduction Dimensionality reduction is a common preprocessing technique for specifying features for classi\ufb01cation. We compare PCA and UAE on this task. While Theorem 2 posits that the two techniques are equivalent in the regime of high noise given optimal UAE decodings, we set the noise as a hyperparameter based on a validation set to enable out-of-sample generalization. Setup. We learn the principal components and UAE projections on the MNIST training set for varying number of dimensions. We then learn classi\ufb01ers based on the these projections. Again, we use a linear encoder for the UAE for a fair evaluation. Since the inductive biases vary across di\ufb00erent classi\ufb01ers, we considered 8 commonly used classi\ufb01ers: k-Nearest Neighbors (kNN), Decision Trees (DT), Random Forests (RF), Multilayer Perceptron (MLP), AdaBoost (AdaB), Gaussian Naive Bayes (NB), Quadratic Discriminant Analysis (QDA), and Support Vector Machines (SVM) with a linear kernel. Results. The performance of the PCA and UAE feature representations for di\ufb00erent number of dimensions is shown in Table 1. We \ufb01nd that UAE outperforms PCA in a majority of the cases. Further, this trend is largely consistent across classi\ufb01ers. The improvements are especially high when the number of dimensions is low, suggesting the bene\ufb01ts of UAE as a dimensionality reduction technique for classi\ufb01cation. 6 RELATED WORK In this section, we contrast uncertainty autoencoders with related works in autoencoding, compressed sensing, and mutual information maximization. Autoencoders. To contrast uncertainty autoencoders with other commonly used autoencoding schemes, consider a UAE with a Gaussian observation model with \ufb01xed isotropic covariance for the decoder of all the autoencoding objectives we discuss subsequently. The UAE objective can be simpli\ufb01ed as: min \u03b8,\u03c6 Ex,y\u223cQ\u03c6(X,Y ) \u0002 \u2225x \u2212g\u03b8(y)\u22252 2 \u0003 Standard Autoencoder. If we assume no measurement noise (i.e., \u03f5 = 0) and assume the observation model P\u03b8(X|Y ) to be a Gaussian with mean g\u03b8(Y ) and a \ufb01xed isotropic \u03a3, then the UAE objective reduces to minimizing the mean squared error between the true and recovered datapoint: min \u03b8,W,\u03c8 Ex\u223cQdata(X) \u0002 \u2225x \u2212g\u03b8(Wf\u03c8(x))\u22252 2 \u0003 This special case of a UAE corresponds to a standard autoencoder [32] where the measurements Y signify a hidden representation for X. However, this case lacks the interpretation of an implicit generative model since the assumptions of Theorem 1 do not hold. Denoising Autoencoders. A DAE [33] adds noise at the level of the input datapoint X to learn robust representations. For a UAE, the noise model is de\ufb01ned at the level of the compressed measurements. Again, with the assumptions of a Gaussian decoder, the DAE objective can be expressed as: min \u03b8,W,\u03c8 Ex\u223cQdata(X),\u02dc x\u223cC( \u02dc X|x) \u0002 \u2225x \u2212g(Wf\u03c8(\u02dc x))\u22252 2 \u0003 where C(\u00b7|X) is some prede\ufb01ned noise corruption model. Similar to Theorem 1, a DAE also learns an implicit model of the data distribution [34, 35]. \fUncertainty Autoencoders Variational Autoencoders. A VAE [13, 22] explicitly learns a latent variable model P\u03b8(X, Y ) for the dataset. The learning objective is a variational lower bound to the marginal log-likelihood assigned by the model to the data X, which notationally corresponds to EQdata(X)[log P\u03b8(x)]. The variational objective that maximizes this quantity can be simpli\ufb01ed as: min \u03b8,\u03c6 Ex,y\u223cQ\u03c6(X,Y ) \u0002 \u2225x \u2212g\u03b8(y)\u22252 2 \u0003 + Ex\u223cQdata [KL(Q\u03c6(Y |x), P(Y ))] The learning objective includes a reconstruction error term, akin to the UAE objective. Crucially, it also includes a regularization term to minimize the KL divergence of the variational posterior over Y with a prior distribution over Y . A key di\ufb00erence is that a UAE does not explicitly need to model the prior distribution over Y . On the downside, a VAE can perform e\ufb03cient ancestral sampling while a UAE requires running relatively expensive Markov Chains to obtain samples. Recent works have attempted to unify the variants of variational autoencoders through the lens of mutual information [36, 37, 14]. These works also highlight scenarios where the VAE can learn to ignore the latent code in the presence of a strong decoder thereby a\ufb00ecting the reconstructions to attain a lower KL loss. One particular variant, the \u03b2-VAE, weighs the additional KL regularization term with a positive factor \u03b2 and can e\ufb00ectively learn disentangled representations [38, 39]. Although [38] does not consider this case, the UAE can be seen as a \u03b2-VAE with \u03b2 = 0. To summarize, our uncertainty autoencoding formulation provides a combination of unique desirable properties for representation learning that are absent in prior autoencoders. As discussed, a UAE de\ufb01nes an implicit generative model without specifying a prior (Theorem 1) even under realistic conditions (Corollary 1; unlike DAEs) and has rich connections with PCA even for non-linear decoders (Theorem 2; unlike any kind of existing autoencoder). Generative modeling and compressed sensing. The closely related works of [30, 31] also use generative models for compressed sensing. As highlighted in Section 5, their approach is radically di\ufb00erent from UAE. Similar to [30], a UAE learns a data distribution. However, in doing so, it additionally learns an acquisition/encoding function and a recovery/decoding function, unlike [30, 31] which rely on generic random matrices and \u21132 decoding. The cost of implicit learning in a UAE is that some of its inference capabilities, such as likelihood evaluation and sampling, are intractable or require running Markov chains. However, these inference queries are orthogonal to compressed sensing. Finally, our decoding is amortized and scales to large datasets, unlike [30, 31] which solve an independent optimization problem for each test datapoint. Mutual information maximization. The principle of mutual information maximization, often referred to as InfoMax in prior work, was \ufb01rst proposed for learning encodings for communication over a noisy channel [7]. The InfoMax objective has also been applied for statistical compressed sensing for learning both linear and non-linear encodings [26, 40, 41]. Our work di\ufb00ers from these existing frameworks in two fundamental ways. First, we optimize for a tractable variational lower bound to the MI that which allows our method to scale to high-dimensional data. Second, we learn an amortized [10, 11] decoder in addition to the encoder that sidesteps expensive, per-example optimization for the test datapoints. Further, we improve upon the IM algorithm proposed originally for variational information maximization [8]. While the IM algorithm proposes to optimize the lower bound on the mutual information in alternating \u201cwake-sleep\u201d phases for optimizing the encoder (\u201cwake\u201d) and decoder (\u201csleep\u201d) analogous to the expectation-maximization procedure used in [26], we optimize the encoder and decoder jointly using a single consistent objective leveraging recent advancements in gradient based variational stochastic optimization. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1811.09813v1",
+                    "title": "Streamlining Variational Inference for Constraint Satisfaction Problems",
+                    "abstract": "Several algorithms for solving constraint satisfaction problems are based on\nsurvey propagation, a variational inference scheme used to obtain approximate\nmarginal probability estimates for variable assignments. These marginals\ncorrespond to how frequently each variable is set to true among satisfying\nassignments, and are used to inform branching decisions during search; however,\nmarginal estimates obtained via survey propagation are approximate and can be\nself-contradictory. We introduce a more general branching strategy based on\nstreamlining constraints, which sidestep hard assignments to variables. We show\nthat streamlined solvers consistently outperform decimation-based solvers on\nrandom k-SAT instances for several problem sizes, shrinking the gap between\nempirical performance and theoretical limits of satisfiability by 16.3% on\naverage for k=3,4,5,6.",
+                    "authors": "Aditya Grover, Tudor Achim, Stefano Ermon",
+                    "published": "2018-11-24",
+                    "updated": "2018-11-24",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.LG",
+                        "cs.LO",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Constraint satisfaction problems (CSP), such as boolean satis\ufb01ability (SAT), are useful modeling abstractions for many arti\ufb01cial intelligence and machine learning problems, including planning [13], scheduling [27], and logic-based probabilistic modeling frameworks such as Markov Logic Networks [30]. More broadly, the ability to combine constraints capturing domain knowledge with statistical reasoning has been successful across diverse areas such as ontology matching, information extraction, entity resolution, and computer vision [15, 4, 32, 29, 33]. Solving a CSP involves \ufb01nding an assignment to the variables that renders all of the problem\u2019s constraints satis\ufb01ed, if one exists. Solvers that explore the search space exhaustively do not scale since the state space is exponential in the number of variables; thus, the selection of branching criteria for variable assignments is the central design decision for improving the performance of these solvers [5]. Any CSP can be represented as a factor graph, with variables as nodes and the constraints between these variables (known as clauses in the SAT case) as factors. With such a representation, we can design branching strategies by inferring the marginal probabilities of each variable assignment. Intuitively, the variables with more extreme marginal probability for a particular value are more likely to assume that value across the satisfying assignments to the CSP. In fact, if we had access to an oracle that could perform exact inference, one could trivially branch on variable assignments with non-zero marginal probability and ef\ufb01ciently \ufb01nd solutions (if one exists) to hard CSPs such as SAT in time linear in the number of variables. In practice however, exact inference is intractable for even moderately sized CSPs and approximate inference techniques are essential for obtaining estimates of marginal probabilities. Variational inference is at the heart of many such approximate inference techniques. The key idea is to cast inference over an intractable joint distribution as an optimization problem over a family of tractable approximations to the true distribution [6, 34, 38]. Several such approximations exist, e.g., mean \ufb01eld, belief propagation etc. In this work, we focus on survey propagation. Inspired from 32nd Conference on Neural Information Processing Systems (NIPS 2018), Montr\u00e9al, Canada. arXiv:1811.09813v1  [cs.AI]  24 Nov 2018 \fa b c i j k l m Figure 1: Factor graph for a 3-SAT instance with 5 variables (circles) and 3 clauses (squares). A solid (dashed) edge between a clause and a variable indicates that the clause contains the variable as a positive (negative) literal. This instance corresponds to (\u00acxi \u2228xk \u2228\u00acxl) \u2227(xi \u2228xj \u2228\u00acxk) \u2227 (xk \u2228xl \u2228\u00acxm), with the clauses a, b, c listed in order. statistical physics, survey propagation is a message-passing algorithm that corresponds to belief propagation in a \u201clifted\u201d version of the original CSP and underlines many state-of-the-art solvers for random CSPs [24, 22, 21]. Existing branching rules for survey propagation iteratively pick variables with the most con\ufb01dent marginals and \ufb01x their values (by adding unary constraints on these variables) in a process known as decimation. This heuristic works well in practice, but struggles with a high variance in the success of branching, as the unary constraints leave the survey inspired decimation algorithm unable to recover in the event that a contradictory assignment (i.e., one that cannot be completed to form a satisfying assignment) is made. Longer branching predicates, de\ufb01ned over multiple variables, have lower variance and are more effective both in theory and practice [14, 1, 2, 36, 19, 18]. In this work, we introduce improved branching heuristics for survey propagation by extending this idea to CSPs; namely, we show that branching on more complex predicates than single-variable constraints greatly improves survey propagation\u2019s ability to \ufb01nd solutions to CSPs. Appealingly, the more complex, multi-variable predicates which we refer to as streamlining constraints, can be easily implemented as additional factors (not necessarily unary anymore) in message-passing algorithms such as survey propagation. For this reason, branching on more complex predicates is a natural extension to survey propagation. Using these new branching heuristics, we develop an algorithm and empirically benchmark it on families of random CSPs. Random CSPs exhibit sharp phase transitions between satis\ufb01able and unsatis\ufb01able instances and are an important model to analyze the average hardness of CSPs, both in theory and practice [25, 26]. In particular, we consider two such CSPs: k-SAT where constraints are restricted to disjunctions involving exactly k (possibly negated) variables [3] and XORSAT which substitutes disjunctions in k-SAT for XOR constraints of \ufb01xed length. On both these problems, our proposed algorithm outperforms the competing survey inspired decimation algorithm that branches based on just single variables, increasing solver success rates. 2 Preliminaries Every CSP can be encoded as a boolean SAT problem expressed in Conjunctive Normal Form (CNF), and we will use this representation for the remainder of this work. Let V and C denote index sets for n Boolean variables and m clauses respectively. A literal is a variable or its negation; a clause is a disjunction of literals. A CNF formula F is a conjunction of clauses, and is written as (l11 \u2228. . . \u2228l1k1) \u2227. . . \u2227(lm1 \u2228. . . \u2228lmkm). Each (lj1 \u2228. . . \u2228ljkj) is a clause with kj literals. For notational convenience, the variables will be indexed with letters i, j, k, . . . and the clauses will be indexed with letters a, b, c, . . .. Each variable i is Boolean, taking values xi \u2208{0, 1}. A formula is satis\ufb01able is there exists an assignment to the variables such that all the clauses are satis\ufb01ed, where a clause is satis\ufb01ed if at least one literal evaluates to true. Any SAT instance can be represented as an undirected graphical model where each clause corresponds to a factor, and is connected to the variables in its scope. Given an assignment to the variables in its scope, a factor evaluates to 1 if the corresponding clause evaluates to True, and 0 otherwise. The corresponding joint probability distribution is uniform over the set of satisfying assignments. An example factor graph illustrating the use of our notation is given in Figure 1. 2 \fk-SAT formulas are ones where all clauses (lj1 \u2228. . . \u2228ljkj) have exactly k literals, i.e., kj = k for j = 1, \u00b7 \u00b7 \u00b7 , m. Random k-SAT instances are generated by choosing each literal\u2019s variable and negation independently and uniformly at random in each of the m clauses. It has been shown that these instances have a very distinctive behavior where the probability of an instance having a solution has a phase transition explained as a function of the constraint density, \u03b1 = m/n, for a problem with m clauses and n variables for large enough k. These instances exhibit a sharp crossover at a threshold density \u03b1s(k): they are almost always satis\ufb01able below this threshold, and they become unsatis\ufb01able for larger constraint densities [12, 10]. Empirically, random instances with constraint density close to the satis\ufb01ability threshold are dif\ufb01cult to solve [23]. 2.1 Survey propagation The base algorithm used in many state-of-the-art solvers for constraint satisfaction problems such as random k-SAT is survey inspired decimation [7, 24, 16, 23]. The algorithm employs survey propagation, a message passing procedure that computes approximate single-variable marginal probabilities for use in a decimation procedure. Our approach uses the same message passing procedure, and we review it here for completeness. Survey propagation is an iterative procedure for estimating variable marginals in a factor graph. In the context of a factor graph corresponding to a Boolean formula, these marginals represent approximately the probability of a variable taking on a particular assignment when sampling uniformly from the set of satisfying assignments of the formula. Survey propagation considers three kinds of assignments for a variable: 0, 1, or unconstrained (denoted by \u2217). A high value for marginals corresponding to either of the \ufb01rst two assignments indicates that the variables assuming the particular assignment make it likely for the overall formula to be satis\ufb01able, whereas a high value for the unconstrained marginal indicates that satis\ufb01ablility is likely regardless of the variable assignment. In order to estimate these marginals from a factor graph, we follow a message passing protocol where we \ufb01rst compute survey messages for each edge in the graph. There are two kinds of survey messages: messages {\u03b7i\u2192a}i\u2208V,a\u2208C(i) from variable nodes i to clauses a, and messages {\u03b7a\u2192i}a\u2208C,i\u2208V (a) from clauses to variables. These messages can be interpreted as warnings of unsatis\ufb01ability. 1. If we let V (a) to be the set of variables appearing in clause a, then the message sent from a clause a to variable i, \u03b7a\u2192i, is intuitively the probability that all variables in V (a)\\{i} are in the state that violates clause a. Hence, clause a is issuing a warning to variable i. 2. The reverse message from variable i to clause a for some value xi, \u03b7i\u2192a, is interpreted as the probability of variable i assuming the value xi that violates clause a. As shown in Algorithm 1, the messages from factors (clauses) to variables \u03b7a\u2192i are initialized randomly [Line 2] and updated until a prede\ufb01ned convergence criteria [Lines 5-7]. Once the messages converge to \u03b7\u2217 a\u2192i, we can estimate the approximate marginals \u00b5i(0), \u00b5i(1), \u00b5i(\u2217) for each variable i. In case survey propagation does not converge even after repeated runs, or a contradiction is found, the algorithm output is UNSAT. The message passing updates SP-Update [Line 6] and the marginalization procedure Marginalize [Line 9] are deferred to Appendix A for ease of presentation. We refer the reader to [24] and [7] for a detailed analysis of the algorithm and connections to statistical physics. 2.2 Decimation and Simpli\ufb01cation The magnetization of a variable i, de\ufb01ned as M(i) := |\u00b5i(0) \u2212\u00b5i(1)|, is used as a heuristic bias to determine how constrained the variable is to take a particular value. The magnetization can be a maximum of one which occurs when either of the marginals is one and a minimum of zero when the estimated marginals are equal.1 The decimation procedure involves setting the variable(s) with the highest magnetization(s) to their most likely values based on the relative magnitude of \u00b5i(0) vs. \u00b5i(1) [Lines 12-13]. The algorithm then branches on these variable assignments and simpli\ufb01es the formula by unit propagation [Line 15]. In unit propagation, we recursively iterate over all the clauses that the decimated variable appears in. If the polarity of the variable in a literal matches its assignment, the clause is satis\ufb01ed and hence, the corresponding clause node and all its incident variable edges are 1Other heuristic biases are also possible. For instance, [23] use the bias 1 \u2212min(\u00b5i(1), \u00b5i(0)). 3 \fAlgorithm 1 SurveyInspiredDecimation(V, C) 1: Initialize V \u2190V and C \u2190C 2: Initialize messages {\u03b7a\u2192i}a\u2208C,i\u2208V (a) at random 3: while (P i |\u00b5i(0) \u2212\u00b5i(1)| > \u03f5) do 4: \u25b7Message passing inference 5: repeat 6: {\u03b7a\u2192i} \u2190SP-Update(V, C, {\u03b7a\u2192i}) 7: until Convergence to {\u03b7\u2217 a\u2192i} 8: for i = 1, . . . , |V| do 9: \u00b5i(0), \u00b5i(1), \u00b5i(\u2217) \u2190Marginalize(V, C, {\u03b7a\u2192i) 10: end for 11: \u25b7Branching (Decimation) 12: Choose i\u2217\u2190arg maxi\u2208V |\u00b5i(0) \u2212\u00b5i(1)| 13: Set y\u2217\u2190arg maxy\u2208{0,1} \u00b5i\u2217(y) 14: \u25b7Simpli\ufb01cation 15: Update V, C \u2190UnitPropagate(V, C \u222a{xi\u2217= y\u2217}) 16: end while 17: return LocalSearch(V, C) removed from the factor graph. If the polarity in the literal does not match the assignment, only the edge originating from this particular variable node incident to the clause node is removed from the graph. For example, setting variable k to 0 in Figure 1 leads to removal of edges incident to k from a and c, as well as all outgoing edges from b (because b is satis\ufb01ed). 2.3 Survey Inspired Decimation The full iterative process of survey propagation (on the simpli\ufb01ed graph from the previous iteration) followed by decimation is continued until a satisfying assignment is found, or a stopping condition is reached beyond which the instance is assumed to be suf\ufb01ciently easy for local search using a standard algorithm such as WalkSAT [31]. Note that when the factor graph is a tree and survey propagation converges to the exact warning message probabilities, Algorithm 1 is guaranteed to select good variables to branch on and to \ufb01nd a solution (assuming one exists). However, the factor graphs for CSPs are far from tree-like in practice and thus, the main factor affecting the success of survey inspired decimation is the quality of the estimated marginals. If these estimates are inaccurate, it is possible that the decimation procedure chooses to \ufb01x variables in contradictory con\ufb01gurations. To address this issue, we propose to use streamlining constraints. 3 Streamlining survey propagation Combinatorial optimization algorithms critically depend on good heuristics for deciding where to branch during search [5]. Survey propagation provides a strong source of information for the decimation heuristic. As discussed above, the approximate nature of message-passing implies that the \u201csignal\" might be misleading. We now describe a more effective way to use the information from survey propagation. Whenever we have a combinatorial optimization problem over X = {0, 1}n and wish to \ufb01nd a solution s \u2208S \u2286X, we may augment the original feasibility problem with constraints that partition the statespace X into disjoint statespaces and recursively search the resulting subproblems. Such partitioning constraints can signi\ufb01cantly simplify search by exploiting the structure of the solution set S and are known as streamlining constraints [17]. Good streamlining constraints will provide a balance between yielding signi\ufb01cant shrinkage of the search space and safely avoiding reductions in the solution density of the resulting subproblems. Partitioning the space based on the value of a single variable (like in decimation) performs well on the former at the cost of the latter. We therefore introduce a different constraining strategy that strives to achieve a more balanced trade-off. 4 \f3.1 Streamlining constraints for constraint satisfaction problems The success of survey inspired decimation relies on the fact that marginals carry some signal about the likely assignments of variables. However, the factor graph becomes more dense as the constraint density approaches the phase transition threshold, making it harder for survey propagation to converge in practice. This suggests that the marginals might provide a weaker signal to the decimation procedure in early iterations. Instead of selecting a variable to freeze in some con\ufb01guration as in decimation, e.g., xi = 1, we propose a strictly more general streamlining approach where we use disjunction constraints between subsets of highly magnetized variables, e.g., (xi \u2228xj) = 1. The streamlined constraints can cut out smaller regions of the search space while still making use of the magnetization signal. For instance, introducing a disjunction constraint between any pair of variables reduces the state-space by a factor of 4/3 (since three out of four possible variable assignments satisfy the clause), in contrast to the decimation procedure in Algorithm 1 which reduces the state space by a factor of 2. Intuitively, when branching with a length-2 clause such as (xi \u2228xj) we make an (irreversible) mistake only if we guess the value of both variables wrong. Decimation can also be seen as a special case of streamlining for the same choice of literal. To see why, we note that in the above example the acceptable variable assignments for decimation (xi, xj) = {(1, 0), (1, 1)} are a subset of the valid assignments for streamlining (xi, xj) = {(1, 0), (1, 1), (0, 1)}. The success of the streamlining constraints is strongly governed by the literals selected for participating in these added disjunctions. Disjunctions could in principle involve any number of literals, and longer disjunctions result in more conservative branching rules. But there are diminishing returns with increasing length, and so we restrict ourselves to disjunctions of length at most two in this paper. Longer clauses can in principle be handled by the inference procedure used by message-passing algorithms, and we leave an exploration of this extension to future work. 3.2 Survey Inspired Streamlining The pseudocode for survey inspired streamlining is given in Algorithm 2. The algorithm replaces the decimation step of survey inspired decimation with a streamlining procedure that adds disjunction constraints to the original formula [Line 16], thereby making the problem increasingly constrained until the search space can be ef\ufb01ciently explored by local search. For designing disjunctions, we consider candidate variables with the highest magnetizations, similar to decimation. If a variable i is selected, the polarity of the literal containing the variable is positive if \u00b5i(1) > \u00b5i(0) and negative otherwise [Lines 12-15]. Disjunctions use the signal from the survey propagation messages without overcommitting to a particular variable assignment too early (as in decimation). Speci\ufb01cally, without loss of generality, if we are given marginals \u00b5i(1) > \u00b5i(0) and \u00b5j(1) > \u00b5j(0) for variables i and j, the new update adds the streamlining constraint xi \u2228xj to the problem instead of overcommitting by constraining i or j to its most likely state. This approach leverages the signal from survey propagation, namely that it is unlikely for \u00acxi \u2227\u00acxj to be true, while also allowing for the possibility that one of the two marginals may have been estimated incorrectly. As long as streamlined constraints and decimation use the same bias signal (such as magnetization) for ranking candidate variables, adding streamlined constraints through the above procedure is guaranteed to not degrade performance compared with the decimation strategy in the following sense. Proposition 1. Let F be a formula under consideration for satis\ufb01ability, Fd be the formula obtained after one round of survey inspired decimation, and Fs be the formula obtained after one round of survey inspired streamlining. If Fd is satis\ufb01able, then so is Fs. Proof. Because unit-propagation is sound, the formula obtained after one round of survey inspired decimation is satis\ufb01able if and only if (F \u2227\u2113i\u2217) is satis\ufb01able, where the literal \u2113i\u2217denotes either xi\u2217 or \u00acxi\u2217. By construction, the formula obtained after one round of streamlining is F \u2227(\u2113i\u2217\u2228\u2113j\u2217). It is clear that if (F \u2227\u2113i\u2217) is satis\ufb01able, so is F \u2227(\u2113i\u2217\u2228\u2113j\u2217). Clearly, the converse need not be true. 3.3 Algorithmic design choices A practical implementation of survey inspired streamlining requires setting some design hyperparameters. These hyperparameters have natural interpretations as discussed below. 5 \fAlgorithm 2 SurveyInspiredStreamlining(V, C, T) 1: Initialize V \u2190V and C \u2190C 2: Initialize messages {\u03b7a\u2192i}a\u2208C,i\u2208V (a) at random 3: while P i |\u00b5i(0) \u2212\u00b5i(1)| \u2265\u03f5 do 4: repeat 5: {\u03b7a\u2192i} \u2190SP-Update(V, C, {\u03b7a\u2192i}) 6: until Convergence to {\u03b7\u2217 a\u2192i} 7: for i = 1, . . . , |V| do 8: \u00b5i(0), \u00b5i(1), \u00b5i(\u2217) \u2190Marginalize(V, C, {\u03b7a\u2192i) 9: end for 10: if t < T then 11: \u25b7Add Streamlining Constraints 12: Choose i\u2217\u2190arg maxi\u2208V |\u00b5i(0) \u2212\u00b5i(1)| 13: Choose j\u2217\u2190arg maxi\u2208V,i\u0338=i\u2217|\u00b5i(0) \u2212\u00b5i(1)| 14: Set y\u2217\u2190arg maxy\u2208{0,1} \u00b5i\u2217(y) 15: Set w\u2217\u2190arg maxy\u2208{0,1} \u00b5j\u2217(y) 16: C \u2190C \u222a{xi\u2217= y\u2217\u2228xj\u2217= w\u2217} 17: else 18: Choose i\u2217\u2190arg maxi\u2208V |\u00b5i(0) \u2212\u00b5i(1)| 19: Set y\u2217\u2190arg maxy\u2208{0,1} \u00b5i\u2217(y) 20: V, C \u2190UnitPropagate(V, C \u222a{xi\u2217= y\u2217}) 21: end if 22: end while 23: return LocalSearch(V, C) Disjunction pairing. Survey inspired decimation scales to large instances by taking the top R variables as decimation candidates at every iteration instead of a single candidate (Line 13 in Algorithm 1). The parameter R is usually set as a certain fraction of the total number of variables n in the formula, e.g., 1%. For the streamlining constraints, we take the top 2 \u00b7 R variables, and pair the variables with the highest and lowest magnetizations as a disjunction constraint. We remove these variables from the candidate list, repeating until we have added R disjunctions to the original set of constraints. For instance, if v1, \u00b7 \u00b7 \u00b7 , v2R are our top decimation candidates (with signs) in a particular round, we add the constraints (v1 \u2228v2R) \u2227(v2 \u2228v2R\u22121) \u2227\u00b7 \u00b7 \u00b7 \u2227(vR \u2228vR+1). Our procedure for scaling to top R decimation candidates ensures that Proposition 1 holds, because survey inspired decimation would have added (v1) \u2227(v2) \u2227\u00b7 \u00b7 \u00b7 \u2227(vR) instead. Other pairing mechanisms are possible, such as for example (v1 \u2228vR+1) \u2227(v2 \u2228vR+2) \u2227\u00b7 \u00b7 \u00b7 \u2227(vR \u2228 vR+R). Our choice is motivated by the observation that v2R is the variable we are least con\ufb01dent about we therefore choose to pair it with the one we are most con\ufb01dent about (v1). We have found our pairing scheme to perform slightly better in practice. Constraint threshold. We maintain a streamlining constraint counter for every variable which is incremented each time the variable participates in a streamlining constraint. When the counter reaches the constraint threshold, we no longer consider it as a candidate in any of the subsequent rounds. This is done to ensure that no single variable dominates the constrained search space. Iteration threshold. The iteration threshold T determines how many rounds of streamlining constraints are performed. While streamlining constraints smoothly guide search to a solution cluster, the trade-off being made is in the complexity of the graph. With every round of addition of streamlining constraints, the number of edges in the graph increases which leads to a higher chance of survey propagation failing to converge. To sidestep the failure mode, we perform T rounds of streamlining before switching to decimation. 4 Empirical evaluation We streamlining constraints for random k-SAT instances for k = {3, 4, 5, 6} with n = {5 \u00d7 104, 4 \u00d7 104, 3 \u00d7 104, 104} variables respectively and constraint densities close to the theoretical predictions of the phase transitions for satis\ufb01ability. 6 \f4.23 4.24 4.25 4.26 4.27 \u03b1 0.0 0.2 0.4 0.6 0.8 1.0 Solver Success Rate k=3 9.70 9.75 9.80 9.85 9.90 9.95 \u03b1 0.0 0.2 0.4 0.6 0.8 1.0 Solver Success Rate k=4 Phase Transition SID SIS 20.0 20.2 20.4 20.6 20.8 21.0 21.2 \u03b1 0.0 0.2 0.4 0.6 0.8 1.0 Solver Success Rate k=5 38 39 40 41 42 43 44 \u03b1 0.0 0.2 0.4 0.6 0.8 1.0 Solver Success Rate k=6 Figure 2: Random k-SAT solver rates (with 95% con\ufb01dence intervals) for k \u2208{3, 4, 5, 6}, for varying constraint densities \u03b1. The red line denotes the theoretical prediction for the phase transition of satis\ufb01ability. Survey inspired streamlining (SIS) drastically outperforms survey inspired decimation (SID) for all values of k. 4.1 Solver success rates In the \ufb01rst set of experiments, we compare survey inspired streamlining (SIS) with survey inspired decimation (SID). In line with [7], we \ufb01x R = 0.01n and each success rate is the fraction of 100 instances solved for every combination of \u03b1 and k considered. The constraint threshold is \ufb01xed to 2. The iteration threshold T is a hyperparameter set as follows. We generate a set of 20 random k-SAT instances for every \u03b1 and k. For these 20 \u201ctraining\" instances, we compute the empirical solver success rates varying T over {10, 20, . . . , 100}. The best performing value of T on these train instances is chosen for testing on 100 fresh instances. All results are reported on the test instances. Results. As shown in Figure 2, the streamlining constraints have a major impact on the solver success rates. Besides the solver success rates, we compare the algorithmic thresholds which we de\ufb01ne to be the largest constraint density for which the algorithm achieves a success rate greater than 0.05. The algorithmic thresholds are pushed from 4.25 to 4.255 for k = 3, 9.775 to 9.8 for k = 4, 20.1 to 20.3 for k = 5, and 39 to 39.5 for k = 6, shrinking the gap between the algorithmic thresholds and theoretical limits of satis\ufb01ability by an average of 16.3%. This is signi\ufb01cant as there is virtually no performance overhead in adding streamlining constraints. Distribution of failure modes. Given a satis\ufb01able instance, solvers based on survey propagation could fail for two reasons. First, the solver could fail to converge during message passing. Second, the local search procedure invoked after simpli\ufb01cation of the original formula could timeout which is likely to be caused due to a pathological simpli\ufb01cation that prunes away most (or even all) of the solutions. In our experiments, we \ufb01nd that the percentage of failures due to local search timeouts in SID and SIS are 36% and 24% respectively (remaining due to non-convergence of message passing). These observations can be explained by observing the effect of decimation and streamlining on the corresponding factor graph representation of the random k-SAT instances. Decimation simpli\ufb01es the factor graph as it leads to the deletion of variable and factor nodes, as well as the edges induced by the deleted nodes. This typically reduces the likelihood of non-convergence of survey propagation since the graph becomes less \u201cloopy\u201d, but could lead to overcon\ufb01dent (incorrect) branching decisions especially in the early iterations of survey propagation. On the other hand, streamlining takes smaller steps in reducing the search space (as opposed to decimation) and hence are less likely to make inconsistent variable assignments. However, a potential pitfall is that these constraints add factor nodes that make the graph more dense, which could affect the convergence of survey propagation. 7 \f0 20 40 60 80 100 Iteration 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Marginal Prediction Calibration 0 200 400 600 800 1000 Average Solution Distances Figure 3: Marginal prediction calibration (blue) and sampled solution distances (green) during solver run on 3-SAT with 5000 variables, \u03b1 = 4.15, T = 90. \u22121.5 \u22121.0 \u22120.5 0.0 0.5 1.0 1.5 Survey Propagation Magnetization \u22121.5 \u22121.0 \u22120.5 0.0 0.5 1.0 1.5 Variable Magnetization Iteration 0 \u22121.0 \u22120.5 0.0 0.5 1.0 Survey Propagation Magnetization 0 100 200 300 400 500 600 700 Number of Variables \u22121.5 \u22121.0 \u22120.5 0.0 0.5 1.0 1.5 Survey Propagation Magnetization \u22121.5 \u22121.0 \u22120.5 0.0 0.5 1.0 1.5 Variable Magnetization Iteration 50 \u22121.0 \u22120.5 0.0 0.5 1.0 Survey Propagation Magnetization 0 100 200 300 400 500 600 700 Number of Variables \u22121.5 \u22121.0 \u22120.5 0.0 0.5 1.0 1.5 Survey Propagation Magnetization \u22121.5 \u22121.0 \u22120.5 0.0 0.5 1.0 1.5 Variable Magnetization Iteration 95 \u22121.0 \u22120.5 0.0 0.5 1.0 Survey Propagation Magnetization 0 200 400 600 800 1000 1200 Number of Variables Figure 4: Top: Correlation between magnetization and estimated marginal probabilities for the same problem instance as we add streamlining constraints. Bottom: Histogram of variables magnetizations. As streamlining constraints are added, the average con\ufb01dence of assignments increases. 4.2 Solution cluster analysis Figures 3 and 4 reveal the salient features of survey inspired streamlining as it runs on an instance of 3-SAT with a constraint density of \u03b1 = 4.15, which is below the best achievable density but is known to be above the clustering threshold \u03b1d(3) \u22483.86. The iteration threshold, T was \ufb01xed to 90. At each iteration of the algorithm we use SampleSAT [35] to sample 100 solutions of the streamlined formula. Using these samples we estimate the marginal probabilities of all variables i.e., the fraction of solutions where a given variable is set to true. We use these marginal probabilities to estimate the marginal prediction calibration i.e., the frequency that a variable which survey propagation predicts has magnetization at least 0.9 has an estimated marginal at least as high as the prediction. The increase in marginal prediction calibrations during the course of the algorithm (Figure 3, blue curve) suggests that the streamlining constraints are selecting branches that preserve most of the solutions. This might be explained by the decrease in the average Hamming distance between pairs of sampled solutions over the course of the run (green curve). This decrease indicates that the streamlining constraints are guiding survey propagation to a subset of the full set of solution clusters. Over time, the algorithm is also \ufb01nding more extreme magnetizations, as shown in the bottom three histograms of Figure 4 at iterations 0, 50, and 95. Because magnetization is used as a proxy for how reliably one can branch on a given variable, this indicates that the algorithm is getting more and more con\ufb01dent on which variables it is \u201csafe\u201d to branch on. The top plots of Figure 4 show the empirical marginal of each variable versus the survey propagation magnetization. These demonstrate that overall the survey propagation estimates are becoming more and more risk-averse: by picking variables with high magnetization to branch on, it will only select variables with (estimated) marginals close to one. 8 \f20.0 20.2 20.4 20.6 20.8 21.0 21.2 \u03b1 0.0 0.2 0.4 0.6 0.8 1.0 Solver Success Rate k=5 Phase Transition Dimetheus Streamlining + Dimetheus (a) 38 39 40 41 42 43 44 \u03b1 0.0 0.2 0.4 0.6 0.8 1.0 Solver Success Rate k=6 (b) 0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56 0.63 \u03b1 0.0 0.2 0.4 0.6 0.8 1.0 Solver Success Rate k=2 Phase Transition SID SIS (c) Figure 5: (a, b) Random k-SAT solver rates (with 95% con\ufb01dence intervals) for k \u2208{5, 6} testing integration with Dimetheus. (c) XORSAT solver rates (with 95% con\ufb01dence intervals). 4.3 Integration with downstream solvers The survey inspired streamlining algorithm provides an easy \u201cblack-box\" integration mechanism with other solvers. By adding streamlining constraints in the \ufb01rst few iterations as a preprocessing routine, the algorithm carefully prunes the search space and modi\ufb01es the original formula that can be subsequently fed to any external downstream solver. We tested this procedure with Dimetheus [16] \u2013 a competitive ensemble solver that won two recent iterations of the SAT competitions in the random k-SAT category. We \ufb01xed the hyperparameters to the ones used previously. We did not \ufb01nd any statistically signi\ufb01cant change in performance for k = 3, 4; however, we observe signi\ufb01cant improvements in solver rates for higher k (Figure 5a,5b). 4.4 Extension to other constraint satisfaction problems The survey inspired streamlining algorithm can be applied to any CSP in principle. Another class of CSPs commonly studied is XORSAT. An XORSAT formula is expressed as a conjunction of XOR constraints of a \ufb01xed length. Here, we consider constraints of length 2. An XOR operation \u2295between any two variables can be converted to a conjunction of disjunctions by noting that xi \u2295xj = (\u00acxi \u2228\u00acxj) \u2227(xi \u2228xj), and hence, any XORSAT formula can be expressed in CNF form. Figure 5c shows the improvements in performance due to streamlining. While we note that the phase transition is not as sharp as the ones observed for random k-SAT (in both theory and practice [11, 28]), including streamlining constraints can improve the solver performance. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1806.06464v2",
+                    "title": "Learning Policy Representations in Multiagent Systems",
+                    "abstract": "Modeling agent behavior is central to understanding the emergence of complex\nphenomena in multiagent systems. Prior work in agent modeling has largely been\ntask-specific and driven by hand-engineering domain-specific prior knowledge.\nWe propose a general learning framework for modeling agent behavior in any\nmultiagent system using only a handful of interaction data. Our framework casts\nagent modeling as a representation learning problem. Consequently, we construct\na novel objective inspired by imitation learning and agent identification and\ndesign an algorithm for unsupervised learning of representations of agent\npolicies. We demonstrate empirically the utility of the proposed framework in\n(i) a challenging high-dimensional competitive environment for continuous\ncontrol and (ii) a cooperative environment for communication, on supervised\npredictive tasks, unsupervised clustering, and policy optimization using deep\nreinforcement learning.",
+                    "authors": "Aditya Grover, Maruan Al-Shedivat, Jayesh K. Gupta, Yura Burda, Harrison Edwards",
+                    "published": "2018-06-17",
+                    "updated": "2018-07-31",
+                    "primary_cat": "cs.MA",
+                    "cats": [
+                        "cs.MA",
+                        "cs.AI",
+                        "cs.LG",
+                        "cs.NE",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Intelligent agents rarely act in isolation in the real world and often seek to achieve their goals through interaction with other agents. Such interactions give rise to rich, complex behaviors formalized as per-agent policies in a multiagent system (Ferber, 1999; Wooldridge, 2009). Depending on the underlying motivations of the agents, interactions could be directed towards achieving a shared goal in a collaborative setting, opposing another agent in a competitive setting, or be a mixture of these in a setting where agents collaborate in teams to compete against other teams. Learning useful representations of the policies of agents based on their interactions is an important step towards characterization of the agent behavior and more generally inference and reasoning in multiagent systems. 1Stanford University 2Carnegie Mellon University 3OpenAI. Correspondence to: Aditya Grover <adityag@cs.stanford.edu>. Proceedings of the 35 th International Conference on Machine Learning, Stockholm, Sweden, PMLR 80, 2018. Copyright 2018 by the author(s). In this work, we propose an unsupervised encoder-decoder framework for learning continuous representations of agent policies given access to only a few episodes of interaction. For any given agent, the representation function is an encoder that learns a mapping from an interaction (i.e., one or more episodes of observation and action pairs involving the agent) to a continuous embedding vector. Using such embeddings, we condition a policy network (decoder) and train it simultaneously with the encoder to imitate other interactions involving the same (or a coupled) agent. Additionally, we can explicitly discriminate between the embeddings corresponding to different agents using triplet losses. For the embeddings to be useful, the representation function should generalize to both unseen interactions and unseen agents for novel downstream tasks. Generalization is wellunderstood in the context of supervised learning where a good model is expected to attain similar train and test performance. For multiagent systems, we consider a notion of generalization based on agent-interaction graphs. An agentinteraction graph provides an abstraction for distinguishing the agents (nodes) and interactions (edges) observed during training, validation, and testing. Our framework is agnostic to the nature of interactions in multiagent systems, and hence broadly applicable to competitive and cooperative environments. In particular, we consider two multiagent environments: (i) a competitive continuous control environment, RoboSumo (Al-Shedivat et al., 2018), and (ii) a ParticleWorld environment of cooperative communication where agents collaborate to achieve a common goal (Mordatch & Abbeel, 2018). For evaluation, we show how representations learned by our framework are effective for downstream tasks that include clustering of agent policies (unsupervised), classi\ufb01cation such as win or loss outcomes in competitive systems (supervised), and policy optimization (reinforcement). In the case of policy optimization, we show how these representations can serve as privileged information for better training of agent policies. In RoboSumo, we train agent policies that can condition on the opponent\u2019s representation and achieve superior win rates much more quickly as compared to an equally expressive baseline policy with the same number of parameters. In ParticleWorld, we train speakers that can communicate more effectively with a much wider range of listeners given knowledge of their representations. arXiv:1806.06464v2  [cs.MA]  31 Jul 2018 \fLearning Policy Representations in Multiagent Systems 2. Preliminaries In this section, we present the necessary background and notation relevant to the problem setting of this work. Markov games. We use the classical framework of Markov games (Littman, 1994) to represent multiagent systems. A Markov game extends the general formulation of partially observable Markov decision processes (POMDP) to the multiagent setting. In a Markov game, we are given a set of n agents on a state-space S with action spaces A1, A2, \u00b7 \u00b7 \u00b7 , An and observation spaces O1, O2, \u00b7 \u00b7 \u00b7 , On respectively. At every time step t, an agent i receives an observation o(t) i \u2208Oi and executes an action a(t) i \u2208Ai based on a stochastic policy \u03c0(i) : Oi \u00d7 Ai \u2192[0, 1]. Based on the executed action, the agent receives a reward r(t) i : S \u00d7 Ai \u2192R and the next observation o(t+1) i . The state dynamics are determined by a transition function T : S \u00d7A1 \u00d7\u00b7 \u00b7 \u00b7\u00d7An \u2192S. The agent policies are trained to maximize their own expected reward \u00af ri = PH t=1 r(t) i over a time horizon H. Extended Markov games. In this work, we are interested in interactions that involve not all but only a subset of agents. For this purpose, we generalize Markov games as follows. First, we augment the action space of each agent with a NO-OP (i.e., no action). Then, we introduce a problem parameter, 2 \u2264k \u2264n, with the following semantics. During every rollout of the Markov game, all but k agents deterministically execute the NO-OP operator while the k agents execute actions as per the policies de\ufb01ned on the original observation and action spaces. Accordingly, we assume that each agent receives rewards only in the interaction episode it participates in. Informally, the extension allows for multiagent systems where all agents do not necessarily have to participate simultaneously in an interaction. For instance, this allows to consider one-vs-one multiagent tournaments where only two players participate in any given match. To further introduce the notation, consider a multiagent system as a generalized Markov game. We denote the set of agent policies with P = {\u03c0(i)}n i=1, interaction episodes with E = {EMj}m j=1 where Mj \u2286{1, 2, \u00b7 \u00b7 \u00b7 , n}, |Mj| = k is the set of k agents participating in episode EMj. To simplify presentation for the rest of the paper, we assume k = 2 and, consequently, denote the set of interaction episodes between agents i and j as Eij. A single episode, eij \u2208Eij, consists of a sequence of observations and actions for the speci\ufb01ed time horizon, H. Imitation learning. Our approach to learning policy representations relies on behavioral cloning (Pomerleau, 1991)\u2014 a type of imitation learning where we train a mapping from observations to actions in a supervised manner. Although there exist other imitation learning algorithms (e.g., inverse reinforcement learning, Abbeel & Ng, 2004), our framework is largely agnostic to the choice of the algorithm, and we restrict our presentation to behavioral cloning, leaving other imitation learning paradigms to future work. 3. Learning framework The dominant paradigm for unsupervised representation learning is to optimize the parameters of a representation function that can best explain or generate the observed data. For instance, the skip-gram objective used for language and graph data learns representations of words and nodes predictive of representations of surrounding context (Mikolov et al., 2013; Grover & Leskovec, 2016). Similarly, autoencoding objectives, often used for image data, learn representations that can reconstruct the input (Bengio et al., 2009). In this work, we wish to learn a representation function that maps episode(s) from an agent policy, \u03c0(i) \u2208\u03a0 to a real-valued vector embedding where \u03a0 is a class of representable policies. That is, we optimize for the parameters \u03b8 for a function f\u03b8 : E \u2192Rd where E denotes the space of episodes corresponding to a policy and d is the dimension of the embedding. Here, we have assumed the agent policies are black-boxes, i.e., we can only access them based on interaction episodes with other agents in a Markov game. Hence, for every agent i, we wish to learn policies using Ei = \u222ajE(i) ij . Here, E(i) ij refers the episode data for interactions between agent i and j, but consisting of only the observation and action pairs of agent i. For a multiagent system, we propose the following auxiliary tasks for learning a good representation of an agent\u2019s policy: 1. Generative representations. The representation should be useful for simulating the agent\u2019s policy. 2. Discriminative representations. The representation should be able to distinguish the agent\u2019s policy with the policies of other agents. Accordingly, we now propose generative and discriminative objectives for representation learning in multiagent systems. 3.1. Generative representations via imitation learning Imitation learning does not require direct access to the reward signal, making it an attractive task for unsupervised representation learning. Formally, we are interested in learning a policy \u03c0(i) \u03c6 : S\u00d7A \u2192[0, 1] for an agent i given access to observation and action pairs from interaction episode(s) involving the agent. For behavioral cloning, we maximize the following (negative) cross-entropy objective: Ee\u223cEi \uf8ee \uf8f0X \u27e8o,a\u27e9\u223ce h log \u03c0(i) \u03c6 (a|o) i \uf8f9 \uf8fb where the expectation is over interaction episodes of agent i and the optimization is over the parameters \u03c6. \fLearning Policy Representations in Multiagent Systems Algorithm 1 Learn Policy Embedding Function (f\u03b8) input {Ei}n i=1 \u2013 interaction episodes, \u03bb \u2013 hyperparameter. 1: Initialize \u03b8 and \u03c6 2: for i = 1, 2, . . . , n do 3: Sample a positive episode pe \u2190e+ \u223cEi 4: Sample a reference episode re \u2190e\u2217\u223cEi\\e+ 5: Compute Im loss \u2190\u2212 P \u27e8o,a\u27e9\u223ce+ log \u03c0\u03c6,\u03b8(a|o, e\u2217) 6: for j = 1, 2, . . . , n do 7: if j \u0338= i then 8: Sample a negative episode ne \u2190e\u2212\u223cEj 9: Compute Id loss \u2190d\u03b8(e+, e\u2212, e\u2217) 10: Set Loss \u2190Im loss + \u03bb \u00b7 Id loss 11: Update \u03b8 and \u03c6 to minimize Loss 12: end if 13: end for 14: end for output \u03b8 Learning individual policies for every agent can be computationally and statistically prohibitive for large-scale multiagent systems, especially when the number of interaction episodes per agent is small. Moreover, it precludes generalization across the behaviors of such agents. On the other hand, learning a single policy for all agents increases sample ef\ufb01ciency but comes at the cost of reduced modeling \ufb02exibility in simulating diverse agent behaviors. We offset this dichotomy by learning a single conditional policy network. To do so, we \ufb01rst specify a representation function, f\u03b8 : E \u2192Rd, with parameters \u03b8, where E represents the space of episodes. We use this embedding to condition the policy network. Formally, the policy network is denoted by \u03c0\u03c6,\u03b8 : S \u00d7 A \u00d7 E \u2192[0, 1] and \u03c6 are parameters for the function mapping the agent observation and embedding to a distribution over the agent\u2019s actions. The parameters \u03b8 and \u03c6 for the conditional policy network are learned jointly by maximizing the following objective: 1 n n X i=1 Ee1\u223cEi, e2\u223cEi\\e1 \uf8ee \uf8f0 X \u27e8o,a\u27e9\u223ce1 log \u03c0\u03c6,\u03b8(a|o, e2) \uf8f9 \uf8fb (1) For every agent, the objective function samples two distinct episodes e1 and e2. The observation and action pairs from e2 are used to learn an embedding f\u03b8(e2) that conditions the policy network trained on observation and action pairs from e1. The conditional policy network shares statistical strength through a common set of parameters for the policy network and the representation function across all agents. 3.2. Discriminative representations via identi\ufb01cation An intuitive requirement for any representation function learned for a multiagent system is that the embeddings should re\ufb02ect characteristics of an agent\u2019s behavior that distinguish it from other agents. To do so in an unsupervised manner, we propose an objective for agent identi\ufb01cation based on the triplet loss directly in the space of embeddings. To learn a representation for agent i based on interaction episodes, we use the representation function f\u03b8 to compute three sets of embeddings: (i) a positive embedding for an episode e+ \u223cEi involving agent i, (ii) a negative embedding for an episode e\u2212\u223cEj involving a random agent j \u0338= i, and (iii) a reference embedding for an episode e\u2217\u223cEi again involving agent i, but different from e+. Given these embeddings, we de\ufb01ne the triplet loss: d\u03b8(e+, e\u2212, e\u2217) = (1 + exp {\u2225re \u2212ne\u22252 \u2212\u2225re \u2212pe\u22252})\u22122 (2) where pe = f\u03b8(e+), ne = f\u03b8(e\u2212), re = f\u03b8(e\u2217). Intuitively, the loss encourages the positive embedding to be closer to the reference embedding than the negative embedding, which makes the embeddings of the same agent tend to cluster together and be further away from embeddings of other agents. We note that various other notions of distance can also be used. The one presented above corresponding to a squared softmax objective (Hoffer & Ailon, 2015). 3.3. Hybrid generative-discriminative representations Conditional imitation learning encourages f\u03b8 to learn representations that can learn and simulate the entire policy of the agents and agent identi\ufb01cation incentivizes representations that can distinguish between agent policies. Both objectives are complementary, and we combine Eq. (1) and Eq. (2) to get the \ufb01nal objective used for representation learning: 1 n n X i=1 Ee+\u223cEi, e\u2217\u223cEi\\e+ \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 X \u27e8o,a\u27e9\u223ce+ log \u03c0\u03c6,\u03b8(a|o, e\u2217) | {z } imitation \u2212 \u03bb X j\u0338=i Ee\u2212\u223cEj [d\u03b8(e+, e\u2212, e\u2217)] | {z } agent identi\ufb01cation \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb (3) where \u03bb > 0 is a tunable hyperparameter that controls the relative weights of the discriminative and generative terms. The pseudocode for the proposed algorithm is given in Algorithm 1. In experiments, we parameterize the conditional policy \u03c0\u03b8,\u03c6 using neural networks and use stochastic gradient-based methods for optimization. 4. Generalization in MAS Generalization is well-understood for supervised learning\u2014 models that shows similar train and test performance exhibit good generalization. To measure the quality of the learned representations for a multiagent system (MAS), we introduce a graphical formalism for reasoning about agents and their interactions. \fLearning Policy Representations in Multiagent Systems F E A G B C D (a) Agent-Interaction Graph (b) The RoboSumo environment. 0 0 1 0 1 Speaker 1 Listener N E W S target landmark 1 1 0 1 0 0 Speaker 2 (c) The ParticleWorld environment. Figure 1: An example of a graph used for evaluating generalization in a multiagent system (a). Illustrations for the environments used in our experiments: competitive (b) and cooperative (c). 4.1. Generalization across agents & interactions In many scenarios, we are interested in generalization of the policy representation function f\u03b8 across novel agents and interactions in a multiagent system. For instance, we would like f\u03b8 to output useful embeddings for a downstream task, even when evaluated with respect to unseen agents and interactions. This notion of generalization is best understood using agent-interaction graphs (Grover et al., 2018). The agent-interaction graph describes interactions between a set of agent policies P and a set of interaction episodes I through a graph G = (P, I).1 An example graph is shown in Figure 1a. The graph represents a multiagent system consisting of interactions between pairs of agents, and we will especially focus on the interactions involving Alice, Bob, Charlie, and Davis. The interactions could be competitive (e.g., a match between two agents) or cooperative (e.g., two agents communicating for a navigation task). We learn the representation function f\u03b8 on a subset of the interactions, denoted by the solid black edges in Figure 1a. At test time, f\u03b8 is evaluated on some downstream task of interest. The agents and interactions observed at test time can be different from those used for training. In particular, we consider the following cases: Weak generalization.2 Here, we are interested in the generalization performance of the representation function on an unseen interaction between existing agents, all of which are observed during training. This corresponds to the red edge representing the interaction between Alice and Bob in Figure 1a. From the context of an agent-interaction graph, the test graph adds only edges to the train graph. Strong generalization. Generalization can also be evaluated with respect to unseen agents (and their interactions). This corresponds to the addition of agents Charlie and Davis in Figure 1a. Akin to a few shot learning setting, we observe a few of their interactions with existing agents Alice and 1If we have more than two participating agents per interaction episode, we could represent the interactions using a hypergraph. 2Also referred to as intermediate generalization by Grover et al. (2018). Bob (green edges) and generalization is evaluated on unseen interactions involving Charlie and Davis (blue edges). The test graph adds both nodes and edges to the train graph. For brevity, we skip discussion of weaker forms of generalization that involves evaluation of the test performance on unseen episodes of an existing training edge (black edge). 4.2. Generalization across tasks Since the representation function is learned using an unsupervised auxiliary objective, we test its generalization performance by evaluating the usefulness of these embeddings for various kinds downstream tasks described below. Unsupervised. These embeddings can be used for clustering, visualization, and interpretability of agent policies in a low-dimensional space. Such semantic associations between the learned embeddings can be de\ufb01ned for a single agent wherein we expect representations for the same agent based on distinct episodes to be embedded close to each other, or across agents wherein agents with similar policies will have similar embeddings on average. Supervised. Deep neural network representations are especially effective for predictive modeling. In a multiagent setting, the embeddings serve as useful features for learning agent properties and interactions, including assignment of role categories to agents with different skills in a collaborative setting, or prediction of win or loss outcomes of interaction matches between agents in a competitive setting. Reinforcement. Finally, we can use the learned representation functions to improve generalization of the policies learned from a reinforcement signal in competitive and cooperative settings. We design policy networks that, in addition to observations, take embedding vectors of the opposing agents as inputs. The embeddings are computed from the past interactions of the opposing agent either with the agent being trained or with other agents using the representation function (Figure 2). Such embeddings play the role of privileged information and allow us to train a policy network that uses this information to learn faster and generalize better to opponents or cooperators unseen at training time. \fLearning Policy Representations in Multiagent Systems previous interactions \u03c0t \u03c8 f\u03b8 et \u03c0A et\u22121 et\u22121 et\u22121 et\u22121 \u03c0t\u22121 \u03c8 Figure 2: Illustration of the proposed model for optimizing a policy \u03c0\u03c8 that conditions on an embedding of the opponent policy \u03c0A. At time t, the pre-trained representation function f\u03b8 computes the opponent embedding based on a past interaction et\u22121. We optimize \u03c0\u03c8 to maximize the expected rewards in its current interactions et with the opponent. 5. Evaluation methodology & results We evaluate the proposed framework for both competitive and collaborative environments on various downstream machine learning tasks. In particular, we use the RoboSumo and ParticleWorld environments for the competitive and collaborative scenarios, respectively. We consider the embedding objectives in Eq. (1), Eq. (2), and Eq. (3) independently and refer to them as Emb-Im, Emb-Id, and Emb-Hyb respectively. The hyperparameter \u03bb for Emb-Hyb is chosen by grid search over \u03bb \u2208 {0.01, 0.05, 0.1, 0.5} on a held-out set of interactions. In all our experiments, the representation function f\u03b8 is speci\ufb01ed through a multi-layer perceptron (MLP) that takes as input an episode and outputs an embedding of that episode. In particular, the MLP takes as input a single (observation, action) pair to output an intermediate embedding. We average the intermediate embeddings for all (observation, action) pairs in an episode to output an episode embedding. To condition a policy network on the embedding, we simply concatenate the observation fed as input to the network with the embedding. Experimental setup and other details beyond what we state below are deferred to the Appendix. 5.1. The RoboSumo environment For the competitive environment, we use RoboSumo (AlShedivat et al., 2018)\u2014a 3D environment with simulated physics (based on MuJoCo (Todorov et al., 2012)) that allows agents to control multi-legged 3D robots and compete against each other in continuous-time wrestling games (Figure 1b). For our analysis, we train a diverse collection of 25 agents, some of which are trained via self-play and others are trained in pairs concurrently using Proximal Policy Optimization (PPO) algorithm (Schulman et al., 2017). We start with a fully connected agent-interaction graph (clique) of 25 agents. Every edge in this graph corresponds to 10 rollout episodes involving the corresponding agents. Table 1: Intra-inter clustering ratios (IICR) and accuracies for outcome prediction (Acc) for weak (W) and strong (S) generalization on RoboSumo. IICR (W) IICR (S) Acc (W) Acc(S) Emb-Im 0.24 0.23 0.71 0.60 Emb-Id 0.25 0.27 0.67 0.56 Emb-Hyb 0.22 0.21 0.73 0.56 The maximum length (or horizon) of any episode is 500 time steps, after which the episode is declared a draw. To evaluate weak generalization, we sample a connected subgraph for training with approximately 60% of the edges preserved for training, and remaining split equally for validation and testing. For strong generalization, we preserve 15 agents and their interactions with each other for training, and similarly, 5 agents and their within-group interactions each for validation and testing. 5.1.1. EMBEDDING ANALYSIS To evaluate the robustness of the embeddings, we compute multiple embeddings for each policy based on different episodes of interaction at test time. Our evaluation metric is based on the intraand inter-cluster Euclidean distances between embeddings. The intra-cluster distance for an agent is the average pairwise distance between its embeddings computed on the set of test interaction episodes involving the agent. Similarly, the inter-cluster distance is the average pairwise distance between the embeddings of an agent with those of other agents. Let Ti = {t(i) c }ni c=1 denote the set of test interactions involving agent i. We de\ufb01ne the intra-inter cluster ratio (IICR) as: IICR = 1 n Pn i=1 1 n2 i Pni a=1 Pni b=1 \u2225t(i) a \u2212t(i) b \u22252 1 n(n\u22121) n P i=1 n P j\u0338=i 1 ninj ni P a=1 nj P b=1 \u2225t(i) a \u2212t(j) b \u22252 . The intra-inter clustering ratios are reported in Table 1. A ratio less than 1 suggests that there is signal that identi\ufb01es the agent, and the signal is stronger for lower ratios. Even though this task might seem especially suited for the agent identi\ufb01cation objective, we interestingly \ufb01nd that the Emb-Im attains lower clustering ratios than Emb-Id for both weak and strong generalization. Emb-Hyb outperforms both these methods. We qualitatively visualize the embeddings learned using Emb-Hyb by projecting them on the leading principal components, as shown in Figures 3a and 3b for 10 test interaction episodes of 5 randomly selected agents in the weak and strong generalization settings respectively. 5.1.2. OUTCOME PREDICTION We can use these embeddings directly for training a classi\ufb01er to predict the outcome of an episode (win/loss/draw). For classi\ufb01cation, we use an MLP with 3 hidden layers \fLearning Policy Representations in Multiagent Systems first eigenvector second eigenvector third eigenvector (a) RoboSumo: Weak first eigenvector second eigenvector third eigenvector (b) RoboSumo: Strong first eigenvector second eigenvector third eigenvector (c) ParticleWorld: Weak first eigenvector second eigenvector third eigenvector (d) ParticleWorld: Strong Figure 3: Embeddings learned using Emb-Hyb for 10 test interaction episodes of 5 agents projected on the \ufb01rst three principal components for RoboSumo and ParticleWorld. Color denotes agent policy. 0 1000 2000 Iteration 0.25 0.50 0.75 1.00 Win rate Train PPO PPO + Emb-Im PPO + Emb-Id PPO + Emb-Hyb 0 1000 2000 0.2 0.4 0.6 0.8 Test 0 1000 2000 Iteration 0.25 0.50 0.75 1.00 Win rate Train PPO PPO + Emb-online PPO + Emb-of\ufb02ine PPO + Emb-zero PPO + Emb-rand 0 1000 2000 0.25 0.50 0.75 1.00 Test Figure 4: Average win rates of the newly trained agents against 5 training agent and 5 testing agents. The left two charts compare baseline with policies that make use of Emb-Im, Emb-Id, and Emb-Hyb (all computed online). The right two charts compare different embeddings used at evaluation time (all embedding-conditioned policies use Emb-Hyb). At each iteration, win rates were computed based on 50 1-on-1 games. Each agent was trained 3 times, each time from a different random initialization. Shaded regions correspond to 95% CI. of 100 units each and the learning objective minimizes the cross entropy error. The input to the classi\ufb01er are the embeddings of the two agents involved in the episode. The results are reported in Table 1. Again, imitation based methods seem more suited for this task with Emb-Hyb and Emb-Im outperforming other methods for weak and strong generalization respectively. 5.1.3. POLICY OPTIMIZATION Here we ask whether embeddings can be used to improve learned policies in a reinforcement learning setting both in terms of end performance and generalization. To this end, we select 5 training, 5 validation, and 5 testing opponents from the pool of 25 pre-trained agents. Next, we train a new agent with reinforcement learning to compete against the selected 5 training opponents; the agent is trained concurrently against all 5 opponents using a distributed version of PPO algorithm, as described in Al-Shedivat et al. (2018). Throughout training, we evaluate new agents on the 5 testing opponents and record the average win and draw rates. Using this setup, we compare a baseline agent with MLP-based policy with an agent whose policy takes 100dimensional embeddings of the opponents as additional inputs at each time step and uses that information to condition its behavior on the opponent\u2019s representation. The embeddings for each opponent are either computed online, i.e., based on an interaction episode rolled out during training at a previous time step (Figure 2), or of\ufb02ine, i.e., pre-computed before training the new agent using only interactions between the pre-trained opponents. Figure 4 shows the average win rates against the set of training and testing opponents for the baseline and our agents that use different types of embeddings. While every new agent is able to achieve almost 100% win rate against the training opponents, we see that the agents that condition their policies on the opponent\u2019s embeddings perform better on the held-out set of opponents, i.e., generalize better, with the best performance achieved with Emb-Hyb. We also note that embeddings computed of\ufb02ine turn out to lead to better performance than if computed online3. As an ablation test, we also evaluate our agents when they are provided an incorrect embedding (either all zeros, Emb-zero, or an embedding selected for a different random opponent, Emb-rand) and observe that such embeddings lead to a degradation in performance4. 3Perhaps, this is due to differences in the interactions of the opponents between themselves and with the new agent that the embedding network was not able to capture entirely. 4Performance decrease is most signi\ufb01cant for Emb-zero, which is an out-of-distribution all-zeros vector. \fLearning Policy Representations in Multiagent Systems 0 1000 Iteration 0.0 0.5 1.0 Rate Emb-Hyb vs. PPO Win Loss Draw 0 1000 Emb-Hyb vs. Emb-Im 0 1000 Emb-Hyb vs. Emb-Id 0 1000 Emb-Im vs. PPO 0 1000 Emb-Im vs. Emb-Id 0 1000 Emb-Id vs. PPO Figure 5: Win, loss, and draw rates plotted for the \ufb01rst agent in each pair. Each pair of agents was evaluated after each training iteration on 50 1-on-1 games; curves are based on 5 evaluation runs. Shaded regions correspond to 95% CI. 1 2 3 4 PPO + Emb-Hyb 4 PPO + Emb-Id 3 PPO + Emb-Im 2 PPO 1 0.67 0.61 0.57 0.48 0.62 0.44 0.50 0.42 0.55 0.49 0.55 0.36 0.51 0.44 0.36 0.32 Figure 6: Win rates for agents speci\ufb01ed in each row at computed at iteration 1000. Finally, to evaluate strong generalization in the RL setting, we pit the newly trained baseline and agents with embedding-conditional policies against each other. Since the embedding network has never seen the new agents, it must exhibit strong generalization to be useful in such setting. The results are give in Figures 5 and 6. Even though the margin is not very large, the agents that use Emb-Hyb perform the best on average. 5.2. The ParticleWorld environment For the collaborative setting, we evaluate the framework on the ParticleWorld environment for cooperative communication (Mordatch & Abbeel, 2018; Lowe et al., 2017). The environment consists of a continuous 2D grid with 3 landmarks and two kinds of agents collaborating to navigate to a common landmark goal (Figure 1c). At the beginning of every episode, the speaker agent is shown the RGB color of a single target landmark on the grid. The speaker then communicates a \ufb01xed length binary message to the listener agent. Based on the received messages, the listener agent the moves in a particular direction. The \ufb01nal reward, shared across the speaker and listener agents, is the distance of the listener to the target landmark after a \ufb01xed time horizon. The agent-interaction graph for this environment is bipartite with only cross edges between speaker and listener agents. Every interaction edge in this graph corresponds to 1000 rollout episodes where the maximum length of any episode is 25 steps. We pretrain 28 MLP parameterized speaker and listener agent policies. Every speaker learns through communication with only two different listeners Table 2: Intra-inter clustering ratios (IICR) for weak (W) and strong (S) generalization on ParticleWorld. Lower is better. IICR (W) IICR (S) Emb-Im 0.58 0.86 Emb-Id 0.50 0.82 Emb-Hyb 0.54 0.85 Table 3: Average train and test rewards for speaker policies on ParticleWorld. Train Test MADDPG \u221211.66 \u221218.99 MADDPG + Emb-Im \u221211.68 \u221217.75 MADDPG + Emb-Id \u221211.68 \u221217.68 MADDPG + Emb-Hyb \u221211.77 \u221217.20 and vice-versa, giving an extremely sparse agent-interaction graph. We explicitly encoded diversity in these speakers and listener agents by masking bits in the communication channel. In particular, we masked 1 or 2 randomly selected bits for every speaker agent in the graph to give a total of \u00007 1 \u0001 + \u00007 2 \u0001 = 28 distinct speaker agents. Depending on the neighboring speaker agents in the agent-interaction graph, the listener agents also show diversity in the learned policies. The policies are learned using multiagent deep deterministic policy gradients (MADDPG, Lowe et al., 2017). In this environment, the speakers and listeners are tightly coupled. Hence we vary the setup used previously in the competitive scenario. We wish to learn embeddings of listeners based on their interactions with speakers. Since the agent-interaction graph is bipartite, we use the embeddings of listener agents to condition a shared policy network for the respective speaker agents. 5.2.1. EMBEDDING ANALYSIS For the weak generalization setting, we remove an outgoing edge from every listener agent in the original graph to obtain the training graph. In the case of strong generalization, we set aside 7 listener agents (and their outgoing edges) each for validation and testing while the representation function is learned on the remaining 14 listener agents and their interactions. The intra-inter clustering ratios are shown \fLearning Policy Representations in Multiagent Systems in Table 2, and the projections of the embeddings learned using Emb-Hyb are visualized in Figure 3c and Figure 3d for weak and strong generalization respectively. In spite of the high degree of sparsity in the training graph, the intrainter clustering ratio for the test interaction embeddings is less than unity suggesting an agent-speci\ufb01c signal. Emb-id works particularly well in this environment, achieving best results for both weak and strong generalization. 5.2.2. POLICY OPTIMIZATION Here, we are interested in learning speaker agents that can communicate more effectively with a much wider range of listeners given knowledge of their embeddings. Referring back to Figure 2, we learn a policy \u03c0\u03c8 for a speaker agent that conditions on the representation function f\u03b8 for the listener agents. For cooperative communication, we consider interactions with 14 pre-trained listener agents split as 6 training, 4 validation, and 4 test agents.5 Similar to the competitive setting, we compare performance against a baseline speaker agent that does not have access to any privilege information about the listeners. We summarize the results for the best validated models during training and 100 interaction episodes per test listener agent across 5 initializations in Table 3. From the results, we observe that online embedding based methods can generalize better than the baseline methods. The baseline MADDPG achieves the lowest training error, but fails to generalize well enough and incurs a low average reward for the test listener agents. 6. Discussion & Related Work Agent modeling is a well-studied topic within multiagent systems. See Albrecht & Stone (2017) for an excellent recent survey on this subject. The vast majority of literature concerns with learning models for a speci\ufb01c predictive task. Predictive tasks are typically de\ufb01ned over actions, goals, and beliefs of other agents (Stone & Veloso, 2000). In competitive domains such as Poker and Go, such tasks are often integrated with domain-speci\ufb01c heuristics to model opponents and learn superior policies (Rubin & Watson, 2011; Mnih et al., 2015). Similarly, intelligent tutoring systems take into account pedagogical features of students and teachers to accelerate learning of desired behaviors in a collaborative environment (McCalla et al., 2000). In this work, we proposed an approach for modeling agent behavior in multiagent systems through unsupervised representational learning of agent policies. Since we sidestep any domain speci\ufb01c assumptions and learn in an unsupervised manner, our framework learns representations that are 5None of the methods considered were able to learn a nontrivial speaker agent when trained simultaneously with all 28 listener agents. Hence, we simpli\ufb01ed the problem by considering the 14 listener agents that attained the best rewards during pretraining. useful for several downstream tasks. This extends the use of deep neural networks in multiagent systems to applications beyond traditional reinforcement learning and predictive modeling (Mnih et al., 2015; Hoshen, 2017). Both the generative and discriminative components of our framework have been explored independently in prior work. Imitation learning has been extensively studied in the singleagent setting and recent work by Le et al. (2017) proposes an algorithm for imitation in a coordinated multiagent system. Wang et al. (2017) proposed an imitation learning algorithm for learning robust controllers with few expert demonstrations in a single-agent setting that conditions the policy network on an inference network, similar to the encoder in our framework. In another recent work, Li et al. (2017) propose an algorithm for learning interpretable representations using generative adversarial imitation learning. Agent identi\ufb01cation which represents the discriminative term in the learning objective is inspired from triplet losses and Siamese networks that are used for learning representations of data using distance comparisons (Hoffer & Ailon, 2015). A key contribution of this work is a principled methodology for evaluating generalization of representations in multiagent systems based on the graphs of the agent interactions. Graphs are a fundamental abstraction for modeling relational data, such as the interactions arising in multiagent systems (Zhou et al., 2016a;b; Chen et al., 2017; Battaglia et al., 2016; Hoshen, 2017) and concurrent work proposes to learn such graphs directly from data (Kipf et al., 2018). 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1804.01712v1",
+                    "title": "Variational Rejection Sampling",
+                    "abstract": "Learning latent variable models with stochastic variational inference is\nchallenging when the approximate posterior is far from the true posterior, due\nto high variance in the gradient estimates. We propose a novel rejection\nsampling step that discards samples from the variational posterior which are\nassigned low likelihoods by the model. Our approach provides an arbitrarily\naccurate approximation of the true posterior at the expense of extra\ncomputation. Using a new gradient estimator for the resulting unnormalized\nproposal distribution, we achieve average improvements of 3.71 nats and 0.21\nnats over state-of-the-art single-sample and multi-sample alternatives\nrespectively for estimating marginal log-likelihoods using sigmoid belief\nnetworks on the MNIST dataset.",
+                    "authors": "Aditya Grover, Ramki Gummadi, Miguel Lazaro-Gredilla, Dale Schuurmans, Stefano Ermon",
+                    "published": "2018-04-05",
+                    "updated": "2018-04-05",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.AI",
+                        "cs.LG",
+                        "cs.NE"
+                    ],
+                    "main_content": "INTRODUCTION Latent variable models trained using stochastic variational inference can learn complex, high dimensional distributions [Ho\ufb00man et al., 2013, Ranganath et al., 2014]. Learning typically involves maximization of a lower bound to the intractable log-likelihood of the observed data, marginalizing over the latent, unobserved variables. To scale to large datasets, inference is amortized by introducing a recognition model approximating the true posterior over the latent variables, conditioned on the observed data [Dayan et al., 1995, Gershman and Goodman, 2014]. The generative and \u2217equal contribution. Proceedings of the 21st International Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spain. PMLR: Volume 84. Copyright 2018 by the author(s). recognition models are jointly trained and commonly parameterized using deep neural networks. While this provides \ufb02exibility, it also leads to expectations without any closed form expressions in the learning objective and corresponding gradients. The general approach to stochastic optimization of such objectives involves Monte Carlo estimates of the gradients using the variational posterior (a.k.a. the recognition model) as a proposal distribution [Mnih and Rezende, 2016]. A simple feed forward network, however, may not capture the full complexity of the posterior, a di\ufb03culty which shows up in practice as high variance in the gradients estimated with respect to the parameters of the proposal distribution. There is a vast body of prior work in variance reduction for stochastic optimization, including recent work focusing on variational methods for generative modeling. The standard approach is to use score function estimators with appropriate baselines [Glynn, 1990, Williams, 1992, Fu, 2006]. Many continuous distributions are also amenable to reparameterization, which transforms the original problem of taking gradients with respect to the parameters of the proposal to the simpler problem of taking gradients with respect to a deterministic function [Kingma and Welling, 2014, Rezende et al., 2014, Titsias and L\u00b4 azaro-Gredilla, 2014]. Finally, a complementary technique for variance reduction is the use of multisample objectives which compute importance weighted gradient estimates based on multiple samples from the proposal [Burda et al., 2016, Mnih and Rezende, 2016]. We discuss these approaches in Section 2. In this work, we propose a new class of estimators for variational learning based on rejection sampling. The variational rejection sampling approach modi\ufb01es the sampling procedure into a two-step process: \ufb01rst, a proposal distribution (in our case, the variational posterior of a generative model) proposes a sample and then we explicitly accept or reject this sample based arXiv:1804.01712v1  [stat.ML]  5 Apr 2018 \fVariational Rejection Sampling on a novel di\ufb00erentiable accept-reject test. The test is designed to reject samples from the variational posterior that are assigned low likelihoods by the generative model, wherein the threshold for rejection can be controlled based on the available computation. We show how this procedure leads to a modi\ufb01cation of the original variational posterior to a richer family of approximating resampled proposal distributions. The modi\ufb01cation is de\ufb01ned implicitly [Mohamed and Lakshminarayanan, 2016] since the only requirement from the original variational posterior is that it should permit e\ufb03cient sampling. Hence, our soft accept-reject test provides a knob to smoothly interpolate between plain importance sampling with a \ufb01xed variational posterior (no rejections) to obtaining samples from the exact posterior in the limit (with potentially high rejection rate), thereby trading o\ufb00statistical accuracy for computational cost. Further, even though the resampled proposal is unnormalized due to the introduction of an accept-reject test, we can surprisingly derive unbiased gradient estimates with respect to the model parameters that only require the unnormalized density estimates of the resampled proposal, leading to an e\ufb03cient learning algorithm. Empirically, we demonstrate that variational rejection sampling outperforms competing single-sample and multi-sample approaches by 3.71 nats and 0.21 nats respectively on average for estimating marginal log-likelihoods using sigmoid belief networks on the MNIST dataset. 2 BACKGROUND In this section, we present the setup for stochastic optimization of expectations of arbitrary functions with respect to parameterized distributions. We also discuss prior work applicable in the context of variational learning. We use upper-case symbols to denote probability distributions and assume they admit densities on a suitable reference measure, denoted by the corresponding lower-case notation. Consider the following objective: L(\u03b8, \u03c6) = Ez\u223cQ\u03c6 [f\u03b8,\u03c6(z)] (1) where \u03b8 and \u03c6 denote sets of parameters and Q\u03c6 is a parameterized sampling distribution over z which can be discrete or continuous. We will assume that sampling z from Q\u03c6 is e\ufb03cient, and suppress subscript notation in expectations from z \u223cQ\u03c6 to simply Q wherever the context is clear. We are interested in optimizing the expectation of a function f\u03b8,\u03c6 with respect to the sampling distribution Q\u03c6 using gradient methods. In general, f\u03b8,\u03c6 and the density q\u03c6 need not be di\ufb00erentiable with respect to \u03b8 and \u03c6. Such objectives are intractable to even evaluate in general, but unbiased estimates can be obtained e\ufb03ciently using Monte Carlo techniques. The gradients of the objective with respect to \u03b8 are given by: \u2207\u03b8L(\u03b8, \u03c6) = EQ [\u2207\u03b8f\u03b8,\u03c6(z)] . As long as f\u03b8,\u03c6 is di\ufb00erentiable with respect to \u03b8, we can compute unbiased estimates of the gradients using Monte Carlo. There are two primary class of estimators for computing gradients with respect to \u03c6 which we discuss next. Score function estimators. Using the fact that \u2207\u03c6q\u03c6 = q\u03c6\u2207\u03c6 log q\u03c6, the gradients with respect to \u03c6 can be expressed as: \u2207\u03c6L(\u03b8, \u03c6) = EQ [\u2207\u03c6f\u03b8,\u03c6(z)] + EQ [f\u03b8,\u03c6(z)\u2207\u03c6log q\u03c6(z)] . The \ufb01rst term can be e\ufb03ciently estimated using Monte Carlo if f\u03b8,\u03c6 is di\ufb00erentiable with respect to \u03c6. The second term, referred to as the score function estimator or the likelihood-ratio estimator or REINFORCE by di\ufb00erent authors [Fu, 2006, Glynn, 1990, Williams, 1992], requires gradients with respect to the log density of the sampling distribution and can su\ufb00er from large variance [Glasserman, 2013, Schulman et al., 2015]. Hence, these estimators are used in conjunction with control variates (also referred to as baselines). A control variate, c, is any constant or random variable (could even be a function of z if we can correct for its bias) positively correlated with f\u03b8,\u03c6 that reduces the variance of the estimator without introducing any bias: EQ [f\u03b8,\u03c6(z)\u2207\u03c6log q\u03c6(z)] = EQ [(f\u03b8,\u03c6(z) \u2212c)\u2207\u03c6log q\u03c6(z)] . Reparameterization estimators. Many continuous distributions can be reparameterized such that it is possible to obtain samples from the original distribution by applying a deterministic transformation to a sample from a \ufb01xed distribution [Kingma and Welling, 2014, Rezende et al., 2014, Titsias and L\u00b4 azaro-Gredilla, 2014]. For instance, if the sampling distribution is an isotropic Gaussian, Q\u03c6 = N(\u00b5, \u03c32I), then a sample z \u223cQ\u03c6 can be equivalently obtained by sampling \u03f5 \u223cN(0, I) and passing through a deterministic function, z = g\u00b5,\u03c3(\u03f5) = \u00b5 + \u03c3\u03f5. This allows exchanging the gradient and expectation, giving a gradient with respect to \u03c6 after reparameterization as: \u2207\u03c6L(\u03b8, \u03c6) = E\u03f5\u223cS [\u2207zf\u03b8,\u03c6(z)\u2207\u03c6g\u03c6(\u03f5)] where S is a \ufb01xed sampling distribution and z = g\u03c6(\u03f5) is a deterministic transformation. Reparameterized gradient estimators typically have lower variance but are not widely applicable since they require g\u03c6 and f\u03b8,\u03c6 \fAditya Grover, Ramki Gummadi, Miguel L\u00b4 azaro-Gredilla, Dale Schuurmans, Stefano Ermon to be di\ufb00erentiable with respect to \u03c6 and z respectively unlike score function estimators [Glasserman, 2013, Schulman et al., 2015]. Recent work has tried to bridge this gap by reparameterizing continuous relaxations to discrete distributions (called Concrete distributions) that gives low variance, but biased gradient estimates [Maddison et al., 2017, Jang et al., 2017] and deriving gradient estimators that interpolate between score function estimators and reparameterization estimators for distributions that can be simulated using acceptance-rejection algorithms, such as the Gamma and Dirichlet distributions [Ruiz et al., 2016, Naesseth et al., 2017]. Further reductions in the variance of reparameterization estimators is possible as explored in recent work, potentially introducing bias [Roeder et al., 2017, Miller et al., 2017, Levy and Ermon, 2018]. 2.1 Variational learning We can cast variational learning as an objective of the form given in Eq. (1). Consider a generative model that speci\ufb01es a joint distribution p\u03b8(x, z) over the observed variables x and latent variables z respectively, parameterized by \u03b8. We assume the true posterior p\u03b8(z|x) over the latent variables is intractable, and we introduce a variational approximation to the posterior q\u03c6(z|x) represented by a recognition network and parameterized by \u03c6. The parameters of the generative model and the recognition network are learned jointly [Kingma and Welling, 2014, Rezende et al., 2014] by optimizing an evidence lower bound (ELBO) on the marginal log-likelihood of a datapoint x: log p\u03b8(x) \u2265EQ \u0014 log p\u03b8(x, z) q\u03c6(z|x) \u0015 \u225cELBO(\u03b8, \u03c6). (2) Besides reparameterization estimators that were introduced in the context of variational learning, there has been considerable research in the design of control variates (CV) for variational learning using the more broadly applicable score function estimators [Paisley et al., 2012]. In particular, Wingate and Weber [2013] and Ranganath et al. [2014] use simple scalar CV, NVIL proposed input-dependent CV [Mnih and Gregor, 2014], and MuProp combines input-dependent CV with deterministic \ufb01rst-order Taylor approximations to the mean-\ufb01eld approximation of the model [Gu et al., 2016]. Recently, REBAR used CV based on the Concrete distribution to give low variance, unbiased gradient updates [Tucker et al., 2017], which has been subsequently generalized to a more \ufb02exible parametric version in RELAX [Grathwohl et al., 2018]. In a parallel line of work, there is an increasing effort to learn models with more expressive posteriors. Major research in this direction focuses on continuous latent variable models, for e.g., see Gregor et al. [2014, 2015], Salimans et al. [2015], Rezende and Mohamed [2015], Chen et al. [2017], Song et al. [2017], Grover and Ermon [2018] and the references therein. Closely related to the current work is Gummadi [2014], which originally proposed a resampling scheme to improve the richness of the posterior approximation and derived unbiased estimates of gradients for the KL divergence from arbitrary unnormalized posterior approximations. Related work for discrete latent variable models is scarce. Hierarchical models impose a prior over the discrete latent variables to induce dependencies between the variables [Ranganath et al., 2016], which can also be speci\ufb01ed as an undirected model [Kuleshov and Ermon, 2017]. On the theoretical side, random projections of discrete posteriors have been shown to provide tight bounds on the quality of the variational approximation [Zhu and Ermon, 2015, Grover and Ermon, 2016, Hsu et al., 2016]. Multi-sample estimators. Multi-sample objectives improve the family of distributions represented by variational posteriors by trading o\ufb00computational e\ufb03ciency with statistical accuracy. Learning algorithms based on these objectives do not introduce additional parameters but instead draw multiple samples from the variational posterior to reduce the variance in gradient estimates as well as tighten the ELBO. A multi-sample ELBO is given as: log p\u03b8(x) \u2265Ez1,...,zk\u223cQ\u03c6 \" log 1 k k X i=1 p\u03b8(x, zi) q\u03c6(zi|x) # (3) Biased gradient estimators using similar objectives were \ufb01rst used by Raiko et al. [2015] for structured prediction. Burda et al. [2016] showed that a multisample ELBO is a tighter lower bound on the loglikelihood than the ELBO. Further, they derived unbiased gradient estimates for optimizing variational autoencoders trained using the objective in Eq. (3). VIMCO generalized this to discrete latent variable models with arbitrary Monte Carlo objectives using a score function estimator with per-sample control variates [Mnih and Rezende, 2016], which serves as a point of comparison in our experiments. Recently, Naesseth et al. [2018] proposed an importance weighted multisample objective for probabilistic models of dynamical systems based on sequential Monte Carlo. 3 THE VRS FRAMEWORK To motivate variational rejection sampling (VRS), consider the ELBO objective in Eq. (2) for any \ufb01xed \u03b8 and x. This is maximized when the variational posterior matches the true posterior p\u03b8(z|x). However, \fVariational Rejection Sampling (a) Target dist. (b) T, a, KL : \u221e, 1, 18 (c) 10, 0.5, 3.1 (d) 0, 0.2, 0.3 (e) \u22125, 0.01,1e-3 Figure 1: The resampled posterior approximation (b-e) gets closer (in terms of KL divergence) to a target 2D discrete distribution (a) as we decrease the parameter T, which controls the acceptance probability a. The triples shown are T, a, KL divergence to target. in practice, the approximate posterior could be arbitrarily far from the true posterior which we seek to mitigate by rejection sampling. 3.1 The Resampled ELBO (R-ELBO) Consider an alternate sampling distribution for the variational posterior with the density de\ufb01ned below: r\u03b8,\u03c6(z|x, T) \u221dq\u03c6(z|x)a\u03b8,\u03c6(z|x, T) (4) where a\u03b8,\u03c6(z|x, T) \u2208(0, 1] is an acceptance probability function that could depend on \u03b8, \u03c6, and additional parameter(s) T. Unlike p, q, and r, note that a\u03b8,\u03c6(z|x, T) does not represent a density over the latent variables z, but simply a function that maps each possible z to a number between 0-1 and hence, it denotes the probability of acceptance for each z. In order to sample from R\u03b8,\u03c6, we follow a two step sampling procedure de\ufb01ned in Algorithm 1. Hence, computing Monte Carlo expectations with respect to the modi\ufb01ed proposal involves resampling from the original proposal due to an additional accept-reject step. We refer to such sampling distributions as resampled proposal distributions. Algorithm 1 Sampler for R\u03b8,\u03c6(z|x, T) input a\u03b8,\u03c6(z|x, T), Q\u03c6(z|x) output z \u223cR\u03b8,\u03c6(z|x, T) 1: while True do 2: z \u2190sample from proposal Q\u03c6(z|x). 3: Compute acceptance probability a\u03b8,\u03c6(z|x, T) 4: Sample uniform: u \u223cU[0, 1]. 5: if u < a\u03b8,\u03c6(z|x, T) then 6: Output sample z. 7: end if 8: end while The resampled proposal de\ufb01nes a new evidence lower bound on the marginal log-likelihood of x, which we refer to as the \u201cresampled ELBO\u201d, or R-ELBO: log p\u03b8(x) \u2265ER \u0014 log p\u03b8(x, z)ZR(x, T) q\u03c6(z|x)a\u03b8,\u03c6(z|x, T) \u0015 \u225cR-ELBO(\u03b8, \u03c6). (5) where ZR(x, T) = EQ[a\u03b8,\u03c6(z|x, T)] is the (generally intractable) normalization constant for the resampled proposal distribution. To make the resampling framework described above work, we need to de\ufb01ne a suitable acceptance function and derive low variance Monte Carlo gradient estimators with respect to \u03b8 and \u03c6 for the R-ELBO which we discuss next. 3.2 Acceptance probability functions The general intuition behind designing an acceptance probability function is that it should allow for the resampled posterior to come \u201cclose\u201d to the target posterior p\u03b8(z|x) (possibly at the cost of extra computation). While there could be many possible ways of designing such acceptance probability functions, we draw inspiration from rejection sampling [Halton, 1970]. In order to draw samples from a target distribution T (z), a rejection sampler \ufb01rst draws samples from an easy-to-sample distribution z \u223cS(z) with a largeror-equal support, i.e., s(z) > 0 wherever t(z) > 0. Then, provided we have a \ufb01xed, \ufb01nite upper bound M \u2208[1, \u221e) on the likelihood ratio t(z)/s(z), we can obtain samples from the target by accepting samples from s(z) with a probability t(z) Ms(z). The choice of M guarantees that the acceptance probability is less than or equal to 1, and overall probability of any accepted sample z is proportional to t(z) Ms(z)s(z) which gives us z \u223cT (z) as desired. The constant M has to be large enough such that the acceptance probability does not exceed 1, but a very high value of M leads to an increase in computation due to a higher rejection rate. If the target is only known up to a normalization constant, then rejection sampling can be used provided M is large enough to ensure that the acceptance probability never exceeds 1. However, we do not know in general how large M should be and even if we did, it would be computationally infeasible to actually use it in a practical algorithm. A natural approximation that departs from the typical rejection sampler would be to accept proposed samples with probability \fAditya Grover, Ramki Gummadi, Miguel L\u00b4 azaro-Gredilla, Dale Schuurmans, Stefano Ermon min h 1, t(z) Ms(z) i for some M that is no longer guaranteed to dominate the likelihood ratios across the entire state space. In the setting of variational learning, the target corresponds to the true, but intractable posterior that can be speci\ufb01ed up to a normalization constant as p\u03b8(z|x) \u221dp\u03b8(x, z) for any \ufb01xed \u03b8 and x. If Q\u03c6(z|x) denotes the proposal distribution and M(T) is any function of the threshold parameter T, the acceptance probability for the approximate rejection sampler is given by: a\u03b8,\u03c6(z|x, T) = min \u0014 1, p\u03b8(x, z) M(T)q\u03c6(z|x) \u0015 . To get a fully di\ufb00erentiable approximation to the min operator, we consider: a\u03b8,\u03c6(z|x, T) = 1/ max \u0014 1, M(T)q\u03c6(z|x) p\u03b8(x, z) \u0015 \u22481/ \" 1t + \u0012M(T)q\u03c6(z|x) p\u03b8(x, z) \u0013t#1/t where the approximation in the last step holds for large t or when any of the two terms in the max expression dominates the other. For t = 1, we get the exponentiated negative softplus function which we will use to be the acceptance probability function in the remainder of this paper. We leave other approximations to future work. Letting T = \u2212log M, the log probability of acceptance is parameterized as: log a\u03b8,\u03c6(z|x, T) = \u2212log[1 + exp(l\u03b8,\u03c6(z|x, T))] = \u2212[l\u03b8,\u03c6(z|x, T)]+ (6) where l\u03b8,\u03c6(z|x, T) = \u2212log p\u03b8(x, z)+log q\u03c6(z|x)\u2212T and [\u2217]+ denotes the softplus function, i.e., log(1 + e\u2217). Informally, the resampling scheme of Algorithm 1 with the choice of acceptance probability function as in Eq. (6) enforces the following behavior: samples from the approximate posterior that disagree (as measured by the log-likelihoods) with the target posterior beyond a level implied by the corresponding threshold T have an exponentially decaying probability of getting accepted, while leaving the remaining samples with negligible interference from resampling. When the proposed sample z from Q\u03c6 is assigned a small likelihood by p\u03b8, the random variable l\u03b8,\u03c6(z|x, T) is correspondingly large with high probability (and linear in the negative log-likelihood assigned by p\u03b8), resulting in a low acceptance probability. Conversely, when p\u03b8 assigns a high likelihood to z, we get a higher acceptance probability. Furthermore, a large value of the scalar bias T results in an acceptance probability of 1, recovering the regular variational inference setting as a special case. On the other extreme, for a small value of T, we get the behavior of a rejection sampler with high computational cost that is also close to the target distribution in KL divergence. More formally, we have Theorem 1 which shows that the KL divergence can be improved monotonically by decreasing T. However, a smaller value of T would require more aggressive rejections and thereby, more computation. Theorem 1. For \ufb01xed \u03b8, \u03c6, the KL divergence between the approximate and true posteriors, KL(R\u03b8,\u03c6(z|x, T)\u2225P\u03b8(z|x)) is monotonically increasing in T where R\u03b8,\u03c6(z|x, T) is the resampled proposal distribution with the choice of acceptance probability function in Eq. (6). Furthermore, the behavior of the sampler in Algorithm 1 interpolates between the following two extremes: \u2022 As T \u2192 +\u221e, R\u03b8,\u03c6(z|x, T) is equivalent to Q\u03c6(z|x), with perfect sampling e\ufb03ciency for the accept-reject step i.e., a\u03b8,\u03c6(z|x, T) \u21921. \u2022 As T \u2192 \u2212\u221e, R\u03b8,\u03c6(z|x, T) is equivalent to P\u03b8(z|x), with the sampling e\ufb03ciency of a plain rejection sampler i.e., a\u03b8,\u03c6(z|x, T) \u21920 \u2200z. This phenomenon is illustrated in Figure 1 where we approximate an example 2D discrete target distribution on a 5 \u00d7 5 grid, with a uniform proposal distribution plus resampling. With no resampling (T = \u221e), the approximation is far from the target. As T is reduced, Figure 1 demonstrates progressive improvement in the posterior quality both visually as well as via an estimate of the KL divergence from approximation to the target along with an increasing computation cost re\ufb02ected in the lower acceptance probabilities. In summary, we can express the R-ELBO as: R-ELBO(\u03b8, \u03c6) = log p\u03b8(x)\u2212KL(R\u03b8,\u03c6(z|x, T)\u2225P\u03b8(z|x)). (7) Theorem 1 and Eq. (7) give the following corollary. Corollary 1. The R-ELBO gets tighter by decreasing T but more expensive to compute. With an appropriate acceptance probability function, we can therefore traverse the computational-statistical trade o\ufb00for maximum likelihood estimation by adaptively tuning the threshold T based on available computation. 3.3 Gradient estimation The resampled proposal distribution in Eq. (4) is unnormalized with an intractable normalization constant, ZR(x, T) = EQ [a\u03b8,\u03c6(z|x, T)]. The presence of an intractable normalization constant seems challenging for both evaluation and stochastic optimization of \fVariational Rejection Sampling the R-ELBO. Even though the constant cannot be computed in closed form,1 we can nevertheless compute Monte Carlo estimates of its gradients, as we show in Lemma 1 in the appendix. The resulting RELBO gradients are summarized below in Theorem 2. Theorem 2. Let COVR(A(z), B(z)) denote the covariance of the two random variables A(z) and B(z), where z \u223cR\u03b8,\u03c6. Then: \u2022 The R-ELBO gradients with respect to \u03b8: \u2207\u03c6R-ELBO(\u03b8, \u03c6) = COVR (A\u03b8,\u03c6(z|x, T), B\u03b8,\u03c6(z|x, T)) where the covariance is between the following r.v.: A\u03b8,\u03c6(z|x, T) \u225clog p\u03b8(x, z) \u2212log q\u03c6(z|x) \u2212[l\u03b8,\u03c6(z|x, T)]+ B\u03b8,\u03c6(z|x, T) \u225c(1 \u2212\u03c3(l\u03b8,\u03c6(z|x, T))) \u2207\u03c6 log q\u03c6(z|x)). \u2022 The R-ELBO gradients with respect to \u03c6: \u2207\u03b8R-ELBO(\u03b8, \u03c6) = ER [\u2207\u03b8 log p\u03b8(x, z)] \u2212COVR(A\u03b8,\u03c6(z|x, T), \u03c3(l\u03b8,\u03c6(z|x, T))\u2207\u03b8 log p\u03b8(x, z)) where \u03c3(\u2217) denotes the sigmoid function applied to \u2217, i.e., \u03c3(\u2217) = 1/(1 + exp(\u2212\u2217)). In the above expressions, the gradients are expressed as the covariance of two random variables that are a function of the latent variables sampled from the approximate posterior R\u03b8,\u03c6(z|x, T). Hence, we only need samples from R\u03b8,\u03c6 for learning, which can be done using Algorithm 1 followed by Monte Carlo estimation analogous to estimation of the usual ELBO gradients. 4 LEARNING ALGORITHM A practical implementation of variational rejection sampling as shown in Algorithm 2 requires several algorithmic design choices that we discuss in this section. 4.1 Threshold selection heuristic The monotonicity of KL divergence in Theorem 1 suggests that it is desirable to choose a value of T as low as computationally feasible for obtaining the highest accuracy. However, the quality of the approximate posterior, Q\u03c6(z|x) for a \ufb01xed parameter \u03c6 could vary signi\ufb01cantly across di\ufb00erent examples x. This would require making T dependent on x. Although learning T in a parametric way is one possibility, in this work, we restrict attention to a simple estimation based approach that reduces the design choice to a single hyperparameter that can be adjusted to trade extra computation for accuracy. For each \ufb01xed 1Note that Monte Carlo estimates of the partition function can be obtained e\ufb03ciently for evaluation. Algorithm 2 Variational Rejection Sampling input Network architectures for p\u03b8(x, z), q\u03c6(z|x); quantile hyperparameter \u03b3 \u2208(0, 1); initial parameters, \u03b80, \u03c60; threshold update frequency, F; quantile estimation sample count N; Covariance estimate sample count S \u22652, SGD based optimizer, OPT; Dataset {xk}K k=1, Number of epochs: N. output Final estimates, \u03b8, \u03c6. 1: Initialize \u03b8 \u2190\u03b80; \u03c6 \u2190\u03c60; T(x) = +\u221e\u2200x. 2: for e \u2208{1, . . . , N} do 3: if e mod F = 0 then 4: for each x in dataset do 5: Sample ZN \u2190{z1, . . . , zN} \u223cq\u03c6(z|x). 6: T(x) \u2190\u02c6 T N \u03b3 (x, \u03b8, \u03c6), the Monte Carlo estimate of Eq. (8) based on samples ZN. 7: end for 8: end if 9: for each x in dataset do 10: Draw S independent samples {z1, . . . , zS} \u223cR\u03b8,\u03c6(z|x, T(x)) using Algorithm 1. 11: Use Theorem 2 and Eq. (9) with {z1, . . . , zS} to estimate gradients, \u02c6 g\u03b8, \u02c6 g\u03c6. 12: Update \u03b8, \u03c6 \u2190OPT(\u03b8, \u03c6, \u02c6 g\u03b8, \u02c6 g\u03c6). 13: end for 14: end for x, let L\u03b8,\u03c6(x) denote the probability distribution of the scalar random variable, \u2212log p\u03b8(x, z)+log q\u03c6(z|x), where z \u223cQ\u03c6(z|x). Let QL denote the quantile function2 for any given 1-D distribution L. For each quantile parameter \u03b3 \u2208(0, 1], we consider a heuristic family of threshold parameters given x, \u03c6, \u03b8, de\ufb01ned as: T\u03b3(x, \u03b8, \u03c6) \u225cQL\u03b8,\u03c6(x)(\u03b3). (8) For example, for \u03b3 = 0.5, this is the median of L\u03b8,\u03c6(x). Eq. (8) implies that the acceptance probability stays roughly in the range of \u03b3 for most samples. This is due to the fact that the negative log of the acceptance probability, de\ufb01ned in Eq. (6) as [l\u03b8,\u03c6(z|x, T)]+ is positive approximately with probability 1 \u2212\u03b3, an event which is likely to result in a rejection. In Algorithm 2, we compute a Monte Carlo estimate for the threshold and denote the resulting value using N samples as \u02c6 T N \u03b3 (x, \u03b8, \u03c6) (Lines 3-8). This estimation is done independently from the SGD updates, once every F epochs, to save computational cost, and also implies that T is not continuously updated as a function of \u03b8, \u03c6. Technically speaking, this introduces a slight bias in the gradients through their dependence on T, but we ignore this correction since it only happens once every few epochs. 2Recall that for a given CDF F(x), the quantile function is its \u2018inverse\u2019, namely Q(p) = inf{x \u2208R : p \u2264F(x)}. \fAditya Grover, Ramki Gummadi, Miguel L\u00b4 azaro-Gredilla, Dale Schuurmans, Stefano Ermon 4.2 Computing covariance estimates To compute an unbiased Monte Carlo estimate of the covariance terms in the gradients, we need to subtract the mean of at least one random variable while forming the product term. In order to do this in Algorithm 2 (Lines 10-11), we process a \ufb01xed batch of (accepted) samples per gradient update, and for each sample, use all-but-one to compute the mean estimate to be subtracted, similar to the local learning signals proposed in Mnih and Rezende [2016]. This requires generating S \u22652 samples from R\u03b8,\u03c6 simultaneously at each step to be able to compute each gradient. More precisely, the leave-one-out unbiased Monte Carlo estimator for the covariance of two random variables A, B is de\ufb01ned as follows. Let (a1, b1), . . . , (aS, bS) \u223c(A, B) be S independent samples from the joint pair (A, B), and let \u02c6 mA denote the sample mean for A: \u02c6 mA \u225c1 S PS i=1 ai. Then the covariance estimate is given by: [ COVR(A, B) \u225c 1 S \u22121 S X i=1 (ai \u2212\u02c6 mA)bi. (9) 4.3 Hyperparameters and overall algorithm In summary, Algorithm 2 involves the following hyperparameters: S, the number of samples for estimating covariance; \u03b3, the quantile used for setting thresholds; F, the number of epochs between updating T(x). 5 EXPERIMENTAL EVALUATION We evaluated variational rejection sampling (VRS) against competing methods on a diagnostic synthetic experiment and a benchmark density estimation task. 5.1 Diagnostic experiments In this experiment, we consider a synthetic setup that involves \ufb01tting an approximate posterior candidate from a constrained family to a \ufb01xed target distribution that clearly falls outside the approximating family. We restrict attention to training a 1-D parameter, \u03c6 exclusively (i.e., we do not consider optimization over \u03b8 ), and for the non-amortized case (i.e., conditioning on x is not applicable). The target distribution is 1-D, with support on non-negative integers, z \u2208{0, 1, . . .} and denoted as P(z). This distribution, visualized in Figure 2, is obtained by removing the mass on the \ufb01rst c integers of a Poisson distribution with rate \u03bb\u2217> 0. More details are given in the Appendix. The approximate proposal is parameterized as Q\u03c6 \u225cPoi(e\u03c6), where \u03c6 is an unconstrained scalar, and denotes a (unmodi\ufb01ed) Poisson distribution with the (non-negative) rate parameter, e\u03c6. Note that for 0 2 4 6 8 10 12 14 16 18 20 22 24 0.00 0.02 0.04 0.06 0.08 0.10 0.12 target probability mass Figure 2: Target distribution, P. Figure 3: Acceptance probability vs. SGD iteration (a) Error: \u03c6 \u2212\u03c6\u2217 (b) Gradients for \u03c6. Figure 4: VRS learning dynamics. The x-axis shows the number of total samples (both accepted and rejected) at each SGD iteration. (a) Error: \u03c6 \u2212\u03c6\u2217. (b) Gradients for \u03c6. Figure 5: VIMCO learning dynamics. The x-axis shows the number of total samples, which is equal to k times the number of iterations at each SGD iteration. Poi(e\u03c6) to explicitly represent a small mass on z < c would require \u03c6 \u2192\u221e, but this would be a bad \ufb01t for points just above c. As a result, {Q\u03c6} does not contain candidates close to the target distribution in the sense of KL divergence, even a simple resampling modi\ufb01cation could transform the raw proposal Q\u03c6 into a better candidate approximation, R. In Figures 3 and 4 we illustrate the dynamics of SGD using VRS gradients for approximating P. To keep the analysis simple, the threshold T was kept \ufb01xed at a constant value during learning. Figure 3 shows the e\ufb03ciency of the sampler improving as the learning progresses due to a better \ufb01t to the target distribution. Figure 4a shows the di\ufb00erence between the current parameter \u03c6 and \u03c6\u2217= log \u03bb\u2217from the target distribution, quickly converging to 0 as learning proceeds. As a benchmark, we evaluated the dynamics based on VIMCO gradients [Mnih and Rezende, 2016]. Figure 5 suggests that the signal in gradients is too low (i.e., \fVariational Rejection Sampling Table 1: Test NLL (in nats) for MNIST comparing VRS with published results. Lower is better. (a) Baseline results from Tucker et al. [2017] Model / Architecture 200 200-200 NVIL (k = 1) 112.5 99.6 MuProp 111.7 99.07 REBAR (\u03bb = 0.1) 111.7 99 REBAR 111.6 99.8 Concrete (\u03bb = 0.1) 111.3 102.8 VRS (IS, \u03b3 = 0.95) 106.97 96.38 VRS (RS, \u03b3 = 0.95) 106.89 96.30 VRS (IS, \u03b3 = 0.9) 106.71 96.26 VRS (RS, \u03b3 = 0.9) 106.63 96.36 (b) Baseline results from Mnih and Rezende [2016] Model / Architecture 200-200-200 NVIL (k = 1) 95.2 NVIL (k = 2) 93.6 NVIL (k = 5) 93.7 NVIL (k = 10) 93.4 NVIL (k = 50) 96.2 RWS (k = 2) 94.6 RWS (k = 5) 93.4 RWS (k = 10) 93.0 RWS (k = 50) 92.5 VIMCO (k = 2) 93.5 VIMCO (k = 5) 92.8 VIMCO (k = 10) 92.6 VIMCO (k = 50) 91.9 VRS (IS, \u03b3 = 0.95) 92.01 VRS (RS, \u03b3 = 0.95) 91.93 VRS (IS, \u03b3 = 0.9) 92.09 VRS (RS, \u03b3 = 0.9) 91.69 high variance in gradient estimates). This behavior was persistent even with much smaller learning rates and large sample sizes compared to VRS gradients. One explanation is that the VIMCO gradient update for \u03c6 has a term that assigns the same average weight to the entire batch of samples, both good and bad ones (see Eq. (8) in Mnih and Rezende [2016]). In contrast, Algorithm 1 discards rejected samples from contributing to the gradients explicitly. Yet another qualitative aspect that distinguishes VRS gradients from importance weighted multi-sample objective gradients is that Algorithm 1 can dynamically adapt the amount of additional computation spent in resampling based on sample quality, as opposed to being \ufb01xed in advance. 5.2 Generative modeling We trained sigmoid belief networks (SBN) on the binarized MNIST dataset [LeCun et al., 2010]. Following prior work, the SBN benchmark architectures for this task consist of several linear layers of 200 hidden units, and the recognition model has the same architecture in the reverse direction. Training such models is sensitive to choices of hyperparamters, and hence we directly compare VRS with published baselines in Table 1. The hyperparameter details for SBNs trained with VRS are given in the Appendix. With regards to key baseline hyperparameters in Table 1, Concrete and REBAR specify a temperature controlling the degree of relaxation (denoted by \u03bb), whereas multi-sample estimators based on importance weighting specify the number of samples, k to trade-o\ufb00 computation for statistical accuracy. The relevant parameter, \u03b3 in our case is not directly comparable but we report results for values of \u03b3 where empirically, the average number of rejections per training example was somewhere between drawing k = 5 and k = 20 samples for an equivalent importance weighted objective for both \u03b3 = 0.95 and \u03b3 = 0.9 (with the latter requiring more computation). Additionally, we provide two estimators for evaluating the test lower bound for VRS. For the importance sampled (IS) version, we simply evaluate the ELBO using importance sampling with the original posterior, Q\u03c6. The resampled (RS) version, on the other hand uses the resampled proposal, R\u03b8,\u03c6 with the partition function, ZR estimated as a Monte Carlo expectation. From the results, we observe that VRS outperforms other methods, including multi-sample estimators with k as high as 50 that require much greater computation than the VRS models considered. Generally speaking, the RS estimates are better than the corresponding IS estimates, and decreasing \u03b3 improves performance (at the cost of increased computation). 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1803.10937v1",
+                    "title": "Best arm identification in multi-armed bandits with delayed feedback",
+                    "abstract": "We propose a generalization of the best arm identification problem in\nstochastic multi-armed bandits (MAB) to the setting where every pull of an arm\nis associated with delayed feedback. The delay in feedback increases the\neffective sample complexity of standard algorithms, but can be offset if we\nhave access to partial feedback received before a pull is completed. We propose\na general framework to model the relationship between partial and delayed\nfeedback, and as a special case we introduce efficient algorithms for settings\nwhere the partial feedback are biased or unbiased estimators of the delayed\nfeedback. Additionally, we propose a novel extension of the algorithms to the\nparallel MAB setting where an agent can control a batch of arms. Our\nexperiments in real-world settings, involving policy search and hyperparameter\noptimization in computational sustainability domains for fast charging of\nbatteries and wildlife corridor construction, demonstrate that exploiting the\nstructure of partial feedback can lead to significant improvements over\nbaselines in both sequential and parallel MAB.",
+                    "authors": "Aditya Grover, Todor Markov, Peter Attia, Norman Jin, Nicholas Perkins, Bryan Cheong, Michael Chen, Zi Yang, Stephen Harris, William Chueh, Stefano Ermon",
+                    "published": "2018-03-29",
+                    "updated": "2018-03-29",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI",
+                        "stat.ML"
+                    ],
+                    "main_content": "INTRODUCTION Intelligent agents often need to interact with the environment and make rational decisions that optimize for a suitable objective. One such setting that commonly arises is the best arm identi\ufb01cation problem in stochastic multi-armed bandits [Bubeck et al., 2009, Audibert and Bubeck, 2010]. In a multi-armed bandit (MAB) problem, an agent is given a set of n \ufb01nite \u2217primary author. Email: adityag@cs.stanford.edu Proceedings of the 21st International Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spain. PMLR: Volume 84. Copyright 2018 by the author(s). actions (or arms), each associated with a reward drawn from an arm-speci\ufb01c probability distribution. In a pure exploration setting, the goal is to reliably identify the top-k arms while minimizing the exploration cost. This problem has numerous applications, including optimal experimental design. We consider a new variant of this problem where the feedback rewards are received after a delay. Delayed feedback is common in the real-world. For instance, hypothesis testing in science and engineering often suffers from delayed feedback since they involve expensive, time-consuming experiments. In one of the motivating applications of this work we want to search over fast-charging policies for electrochemical batteries to maximize lifetime, overcoming the di\ufb03culties posed due to lengthy experiments. Even within the \ufb01eld of machine learning, \ufb01nding the best hyperparameter settings for a given learning algorithm and dataset can be modeled as a best arm identi\ufb01cation problem involving a non-trivial delay [Jamieson and Talwalkar, 2016]. However, many scenarios of interest are not complete black-boxes during the intermediate time steps before receiving a delayed feedback reward. Depending on the application, we often have access to side-information in the form of partial feedback that can aid decision making. These could be extra measurements such as temperature and remaining capacity while charging batteries in the aforementioned scenario, or learning curves for hyperparameter optimization. In this work, we propose a general-purpose framework for modeling delayed feedback in MAB, and take a deeper dive into several practically relevant instantiations. In particular, we design and analyze algorithms for best arm identi\ufb01cation in the \ufb01xed con\ufb01dence setting where the partial feedback are biased or unbiased estimators of the delayed feedback. Our proposed algorithms adaptively tune the mean and con\ufb01dence estimates wherever the partial feedback reduces the overall uncertainty. We also extend these algorithms to the parallel MAB setting where we are allowed to pull a batch of arms at every time step [Jun et al., 2016]. arXiv:1803.10937v1  [cs.LG]  29 Mar 2018 \fBest arm identi\ufb01cation in multi-armed bandits with delayed feedback Finally, we empirically validate the proposed algorithms on simulated data and real world datasets drawn from two domains. The \ufb01rst corresponds to experimental design for \ufb01nding the optimal charging policy for a battery that maximizes overall lifetime [Moura et al., 2017]. In the second domain, we perform hyperparameter optimization for \ufb01nding the best cut strategy for a standard mixed integer programming solver with performance tested on a benchmark set of problem instances drawn from computational sustainability [Gomes et al., 2008]. Our experiments demonstrate that accounting for partial feedback can reduce the delayed sample complexity on average by 15.6% and 80.8% for sequential MAB over baselines for the two application scenarios respectively. The corresponding average savings over baselines for parallel MAB are 20.7% and 87.6% respectively. 2 BACKGROUND & MODELING FRAMEWORK The chief workhorse of our analysis will be the law of iterated logarithms (LIL) that analyzes the limiting behavior of random walks (sequence of pulls for a given arm in our case) de\ufb01ned over sub-Gaussian random variables [Darling and Robbins, 1967]. Several \ufb01nite LIL bounds have been proposed in the literature; we consider the one proposed by Zhao et al. [2016] which has been shown to outperform others empirically while retaining the same asymptotic behavior. Alternate bounds, such as the one by Jamieson et al. [2014], could also be used with no e\ufb00ect on the theoretical analysis of this work. Lemma 1. Let X(1), X(2), . . . be i.i.d. sub-Gaussian random variables with scale parameter \u03c3 and mean \u00b5. Let \u03c4 be any random variable with domain N. For any c > 1, 2a > c, b > 0, the following holds with probability at least 1 \u22122\u03b6(2a/c)e \u22122b/c: \f \f \f \f \f 1 \u03c4 \u03c4 X l=1 X(l) \u2212\u00b5 \f \f \f \f \f \u2264\u03c3 r a log(logc \u03c4 + 1) + b \u03c4 where \u03b6 denotes the Riemannian zeta function. The constants in Lemma 1 are chosen such that the lemma holds for a target con\ufb01dence. To simplify the notation, we denote the the error probability by \u03b4\u2032 and the right hand side of Lemma 1 by C (\u03c3, \u03c4, \u03b4\u2032) such that the following holds with probability 1 \u2212\u03b4\u2032 for any \u03c4 \u2208N: \f \f \f \f \f 1 \u03c4 \u03c4 X l=1 X(l) \u2212\u00b5 \f \f \f \f \f \u2264C (\u03c3, \u03c4, \u03b4\u2032) . (1) We consider a stochastic multi-armed bandit (MAB) problem characterized by a set of n arms, indexed by i = 1, . . . , n. Each arm is associated with a \ufb01xed, unknown probability distribution with means {\u00b5i}n i=1. We assume that the means are unique. Without loss of generality, assume that the arm indices are sorted as per the means, such that \u00b51 > \u00b52 > . . . > \u00b5n. We are interested in the pure exploration setting, also known as the best arm identi\ufb01cation problem, where the goal of an agent is to identify the top-k arms (with the highest means) with a target con\ufb01dence 1\u2212\u03b4 while minimizing the total time spent on exploration. Exploration in our setting, however, is not the same across the pulls of a given arm. In particular, we assume that each pull of an arm is associated with an unknown (stochastic) delay that contributes to the total exploration time. The presentation in this section assumes a sequential MAB setting where the agent can pull/run only one arm at a given time step; the alternate parallel MAB setting where an agent can control a \u201cbatch\u201d of arms at once is discussed in Section 4 [Perchet et al., 2015, Wu et al., 2015, Jun et al., 2016]. Formally, the stochastic data generating process with delayed feedback can be described as follows. At any given start time ts: 1. Agent chooses an arm i. 2. Nature samples a delay Ds \u22651 from an (unknown) arm speci\ufb01c delay distribution. 3. Nature samples a sequence of partial feedback, (Yi,ts+1, . . . , Yi,ts+Ds) | Ds jointly. The joint distribution of the partial feedback depends on \u00b5i. In general, the delay and partial feedback sequence are unknown to the agent at time ts. At time ts + \u2206where \u2206\u2208[1, Ds], 4. Nature reveals Yi,ts+\u2206to the agent. If \u2206= Ds, the agent goes to step 1. Otherwise, the agent decides whether to continue the current pull (step 4) or start another pull (step 1) in which case any remaining partial feedback for the current pull will not be observed. The agent and nature continue to play the above game until the agent has selected a set of candidate top-k arms. The delay Ds can contribute signi\ufb01cantly to the total time spent on exploration. Under appropriate assumptions however, we can exploit the structure in the partial feedback to signi\ufb01cantly reduce the overall exploration cost of delayed feedback. The data generating process described above is very general and one can make many natural assumptions on the distribution of the partial feedback (Yi,ts+1, \u00b7 \u00b7 \u00b7 , Yi,ts+Ds) | Ds. \fGrover et al. For instance, we can model the following scenarios: \u2022 Full delayed feedback: The partial feedback at the last delay, Yi,ts+Ds is sub-Gaussian with mean \u00b5i and scale parameter \u03c3i. For the intermediate time steps, \u2206\u2208[1, Ds \u22121], we have Yi,ts+\u2206= 0, and hence, we receive no information about \u00b5i at these time steps. \u2022 Incremental partial feedback: The set of partial feedback Yi,ts+\u2206for every time step \u2206\u2208 [1, Ds] consists of mutually independent, subGaussian random variables with mean \u00b5i/Ds and scale parameter \u03c3i/\u221aDs. Hence, the cumulative partial feedback PDs \u2206=1 Yi,ts+\u2206is also sub-Gaussian with mean \u00b5i and scale parameter \u03c3i. \u2022 Unbiased noisy partial feedback: The partial feedback at the last delay, Yi,ts+Ds is sub-Gaussian with mean \u00b5i and scale parameter \u03c3i. For the intermediate time steps, \u2206\u2208[1, Ds \u22121], the set of partial feedback Yi,ts+\u2206| Yi,ts+Ds \u2212Yi,ts+Ds consists of mutually independent, sub-Gaussian random variables with zero mean and scale parameter \u03c3(p) i . \u2022 Biased noisy partial feedback: The partial feedback at the last delay, Yi,ts+Ds is sub-Gaussian with mean \u00b5i and scale parameter \u03c3i. For the intermediated time steps, \u2206\u2208[1, Ds \u22121], the set of partial feedback Yi,ts+\u2206| Yi,ts+Ds \u2212Yi,ts+Ds consists of mutually independent, sub-Gaussian random variables with mean bi and scale parameter \u03c3(p) i . Here, bi is a \ufb01xed, but unknown bias associated with the partial feedback for the arm. Note that the standard MAB setting where we observe the feedback at the immediate next time step is a special case of the full delayed feedback with a constant delay Ds = 1 for every pull. In fact, the algorithms for best arm identi\ufb01cation in the full delayed and incremental partial feedback settings can be derived naturally from the standard MAB algorithms with no delays. Speci\ufb01cally, the agent can simply chose to ignore the time instants at which delayed feedback is unavailable for the full delayed feedback setting. The sample complexity of any such algorithm is hence the number of arm pulls required in the standard MAB setting weighted by the delay of every pull. These settings are still interesting for parallel MAB where information can be shared across arms; we discuss this case in Section 4. The partial feedback settings, however, present an interesting scenario where the agent can extract information from noisy feedback. For such settings, we propose modi\ufb01ed algorithms based on racing-style procedures Algorithm 1 RacingSubroutines function UpdateArmSets(arm sets A, R, S, top k, con\ufb01dence bounds {LCBi, UCBi}i\u2208S) Initialize kt \u2190k \u2212|A|. Update A \u2190A \u222a{i \u2208S | LCBi > max(kt+1) j\u2208S UCBj}. Update R \u2190R \u222a{i \u2208S | UCBi < max(kt) j\u2208S LCBj}. Update S \u2190S\\{R \u222aA}. return A, R, S. end function function GetBatchArms(surviving arms S, counts {Ni, ai}i\u2208S, e\ufb00ective batch size e, limit r) Initialize new arm pulls m \u21900 \u2208Rn. for slot s = {1, \u00b7 \u00b7 \u00b7 , min (e, |S|r)} do Least pulled arm j \u2190arg mini\u2208S:ai\u2264r Ni Update aj \u2190aj + 1. Update mj \u2190mj + 1. Update Nj \u2190Nj + 1. end for return m, {Ni}i\u2208S, {ai}i\u2208S end function typically used for the standard MAB setting [Maron and Moore, 1994]. Typically, racing algorithms maintain three disjoint arm sets: accepted arms A, rejected arms R, and surviving arms S. Initially, all arms are assigned to the surviving set S. Racing procedures uniformly sample arms while removing them from the surviving set based on con\ufb01dence bounds. For convenience, de\ufb01ne the lower con\ufb01dence bounds (LCB) and upper con\ufb01dence bounds (UCB) for every arm i as: LCBi := b \u00b5i \u2212Ci (2) UCBi := b \u00b5i + Ci (3) where b \u00b5i is the empirical mean of the feedback for arm i and the con\ufb01dence bound Ci will depend on the particular racing algorithm under consideration. Let kt := k \u2212|A| be the e\ufb00ective number of top arms remaining to be identi\ufb01ed at a time step t. Each time we receive a feedback reward (full or partial), the racing procedures update these sets based on the rule that any arm in S whose LCB is greater than the UCB of |S| \u2212kt arms is accepted. Similarly, any arm in S whose UCB is less than the LCB of kt arms is rejected. The racing procedure is repeated until S is empty. The pseudocode for the subroutine that updates the arm sets is given in Algorithm 1. 3 SEQUENTIAL MAB In sequential MAB, we assume that the agent can receive (partial) feedback from only a single arm pull at any given time step, e.g., we can only perform one \fBest arm identi\ufb01cation in multi-armed bandits with delayed feedback experiment at a time. We skip a separate discussion on the trivial full feedback (and the related incremental feedback) setting and discuss it only in the context of the noisy feedback settings. For convenience, we denote the partial feedback at the last delay as Xi,ts = Yi,ts+Ds. Here, Xi,ts is a sub-Gaussian random variable with mean \u00b5i and scale parameter \u03c3i. The proofs of all results in this section are given in the Appendix. 3.1 Unbiased noisy partial feedback In this setting, an agent has access to unbiased partial feedback at the intermediate time steps before receiving the full delayed feedback. In the following result, we derive a variation of the \ufb01nite LIL bound for the unbiased partial feedback setting. Proposition 1. Let {Yi,t1+1, Yi,t1+2, . . . , Yi,t1+D1, Yi,t2+1, . . . , Yi,t2+D2, . . .} denote the partial feedback sequences for the pulls of an arm i started at time steps t1, t2, . . . and delays D1, D2, . . .. Then, under the distributional assumptions on the unbiased partial feedback (see Section 2) for any F \u2208N, P \u2208[1, DF ], \u03b4f > 0, \u03b4p > 0, we have with probability 1 \u2212\u03b4f \u2212\u03b4p: \f \f \f \f \f \f 1 F \uf8ee \uf8f0 F \u22121 X f=1 Xi,tf + 1 P P X l=1 Yi,tF +l \uf8f9 \uf8fb\u2212\u00b5i \f \f \f \f \f \f \u2264C (\u03c3i, F, \u03b4f/n) + 1 F C \u0010 \u03c3(p) i , P, \u03b4p/n \u0011 \u2200i \u2208[1, n] (4) where Xi,tf = Yi,tf +Df by de\ufb01nition. At any intermediate time step between the the start and end of the F-th arm pull, Proposition 1 adaptively \u201csplits\u201d the con\ufb01dence bounds pertaining to the full delayed feedback for F steps (\ufb01rst term in the RHS) and the partial delayed feedback for the F-th arm pull (second term in the RHS). Contrast this with the full delayed feedback setting where the following con\ufb01dence bound holds with probability 1 \u2212\u03b4: \f \f \f \f \f \f 1 F \u22121 F \u22121 X f=1 Xi,tf \u2212\u00b5i \f \f \f \f \f \f \u2264C (\u03c3i, F \u22121, \u03b4/n) \u2200i \u2208[1, n] (5) To obtain the same target con\ufb01dence in the two cases above, we constrain \u03b4 = \u03b4f + \u03b4p. Solving for the optimal \u03b4\u2217 f, \u03b4\u2217 p that minimize the RHS of Eq. (4) under the constraint due to \u03b4 corresponds to a convex optimization problem that can be solved in closed form. Comparing the mean estimators in Eq. (4) and Eq. (5), we note that the agent can only use the full delayed feedback up till the (F \u22121)-th arm pull while waiting for the outcome of the F-th arm pull in the latter case while the former dynamically incorporates the partial feedback observed for the F-th arm pull. Algorithm 2 RacingUnbiasedPF (arm parameters {i, \u03c3i, \u03c3(p) i }n i=1, top k, con\ufb01dence \u03b4) 1: Initialize global time step t = 0, surviving S = {i}n i=1, accepted A = {}, rejected R = {}. 2: Initialize per-arm full delayed feedback counter Fi = 0, empirical means \u02c6 \u00b5i = 0, con\ufb01dence bounds LCBi = \u2212\u221e, UCBi = \u221efor all i \u2208S. 3: while S is not empty do 4: while True do 5: Increment t \u2190t + 1. 6: Collect partial feedback Ya,t. 7: Update b \u00b5(p) \u2190(P b \u00b5(p)+Ya,t) (P +1) . 8: Increment P \u2190P + 1. 9: Set C(partial) \u2190C(\u03c3a, Fa + 1, \u03b4\u2217 f/n) + C(\u03c3(p) a ,p,\u03b4\u2217 p/n) Fa+1 . 10: Choose FOrP \u2190arg min \u0000C(\u03c3a, Fa, \u03b4/n), C(partial)\u0001 . 11: Update Ca \u2190C(\u03c3a, Fa, \u03b4/n) if FOrP = F else C(partial). 12: Update b \u00b5a \u2190b \u00b5(f) if FOrP = F else Fab \u00b5(f)+b \u00b5(p) Fa+1 . 13: Update LCBa, UCBa. 14: A, R, S \u2190UpdateArmSets(A, R, S, k, {LCBi, UCBi)}i\u2208S). 15: if P = Da,ta or a \u0338\u2208S then 16: Break \u25b7Pull on termination/elimination 17: end if 18: end while 19: Pull arm a where a \u2190arg mina\u2208S Fa. 20: Initialize start ta \u2190t, partial feedback counter P = 0, partial mean b \u00b5(p) = 0, full mean b \u00b5(f) \u2190b \u00b5i. 21: end while 22: return A Based on the above analysis, we propose a racing algorithm for the unbiased partial feedback setting with the psuedocode given in Algorithm 2. At any intermediate time step, the agent chooses a mean estimator and a con\ufb01dence bound for the current arm (Lines 10-13). The choice corresponds to the tighter con\ufb01dence bound obtained either by optimizing Eq. (4) over \u03b4p, \u03b4f or the one obtained by Eq. (5) where only the full delayed feedback are considered. Thereafter, the agent invokes the racing subroutine that checks whether a surviving arm can be rejected or accepted (Line 14). If the pull has \ufb01nished running or the current arm is itself eliminated (Line 15), the agent pulls a new arm in the next time step which has the least number of full delayed feedback (Line 19). We can make some observations about Algorithm 2. First, we see that an agent adopting the proposed algorithm can never do worse than the alternate racing strategy that considers estimates only based on the full delayed feedback. This is because even at the intermediate time steps, the agent considers the mean estimator corresponding to the smaller of the two con\ufb01dence bounds, which can only reduce the delayed sample complexity of the algorithm. Whenever an arm \fGrover et al. Algorithm 3 BatchRacingFullDF(arm parameters {i, \u03c3i}n i=1, top k, con\ufb01dence \u03b4, batch b, limit r) 1: Initialize global time step t = 0, pull status counts running = 0, surviving arms S = {i}n i=1, accepted arms A = {}, rejected arms R = {}. 2: Initialize per-arm global pull counts Ni = 0, running pull counts ai = 0, full delayed feedback Fi = 0, empirical means \u02c6 \u00b5i = 0, con\ufb01dence bounds LCBi = \u2212\u221e, UCBi = \u221efor all i \u2208S. 3: while S is not empty do 4: if running > 0 then 5: Increment t \u2190t + 1. 6: Collect batch full delayed feedback Y . 7: for all Yh,t \u2208Y do 8: Update b \u00b5h \u2190(Fhb \u00b5(f)+Yh,t)/(Fh+1). 9: Increment Fh \u2190Fh + 1. 10: Update LCBh, UCBh. 11: Decrement ah \u2190ah \u22121. 12: end for 13: if Y is not empty then 14: A, R, S \u2190UpdateArmSets(A, R, S, k, {LCBi, UCBi}i\u2208S). 15: Decrement running \u2190running \u2212|Y |. 16: end if 17: end if 18: Update arms m, counts {Ni, ai}i\u2208S \u2190 GetBatchArms(S, {Ni, ai}i\u2208S, b \u2212running, r). 19: Pull every arm j \u2208m mj times. 20: Update running \u2190running + P j\u2208m mj. 21: end while 22: return A pull has \ufb01nished, the agent also updates the mean and con\ufb01dence interval by an arithmetic averaging over only the full delayed feedback. Using partial feedback is impractical at such time steps since the partial feedback only introduce noise and do not provide any additional information about the true mean. If the maximum possible delay associated with any arm pull is given by Dmax, then we can trivially extend bounds for the sample complexity of racing style procedures [Jamieson and Nowak, 2014] to derive similar bounds on the delayed sample complexity with an extra multiplicative factor of Dmax.1 This is similar to what one would expect from the full delayed feedback setting and is not surprising for Algorithm 2 since in the absence of any additional assumptions, the partial feedback could be completely uninformative and the algorithm will choose to ignore them. We believe domain-speci\ufb01c assumptions about the delay distribution and the noise associated with the partial feedback as a function of time could lead to a tighter analysis 1The delayed sample complexity for an algorithm refers to the total number of time steps (including delays) before termination. and is an interesting direction of future work. The correctness of Algorithm 2 can be summarized below. Theorem 1. Assuming the delay associated with any arm pull is bounded, then Algorithm 2 outputs the top-k arms with probability at least 1 \u2212\u03b4. To get further intuition about the working of Algorithm 2, consider the situation where all arms have been pulled once except one. When the last remaining arm is pulled for the \ufb01rst time, the full delayed feedback setting will necessarily have to wait for the pull to \ufb01nish running before eliminating the arms whereas Algorithm 2 can potentially start eliminating arms right after the \ufb01rst partial delayed feedback is received. 3.2 Biased noisy partial feedback The partial feedback at the intermediate time steps before a full delayed feedback can also correspond to biased estimates of the full delayed feedback. Although the bias for the arms is unknown, it can be estimated empirically based on di\ufb00erences in the full delayed feedback and the partial feedback at the corresponding intermediate time steps. Formally, we assume the bias for a particular arm is an unknown constant bi and derive the following LIL bounds. Proposition 2. Let {Yi,t1+1, Yi,t1+2, . . . , Yi,t1+D1, Yi,t2+1, . . . , Yi,t2+D2 . . .} denote the partial feedback sequences for the pulls of an arm i started at time steps t1, t2, . . . and delays D1, D2, . . . with bias bi. Then, under the distributional assumptions on the partial feedback (see Section 2) for any F \u2208N\\{1}, P \u2208[1, DF ], \u03b4f > 0, \u03b4p > 0, \u03b4b > 0, we have with probability 1 \u2212\u03b4f \u2212\u03b4p \u2212\u03b4b: \f \f \f \f \f \f 1 F \uf8ee \uf8f0 F \u22121 X f=1 Xi,tf + 1 P P X p=1 (Yi,tF +p \u2212Zi,F ) \uf8f9 \uf8fb\u2212\u00b5i \f \f \f \f \f \f \u2264C (\u03c3i, F, \u03b4f/n) + 1 F h C \u0010 \u03c3(p) i , P, \u03b4p/n \u0011 + C \u0010 \u03c3(p) i , F \u22121, \u03b4b/n \u0011i (6) \u2200i \u2208[1, n]where Zi,F = 1 F \u22121 F \u22121 X f=1  PDf \u22121 p=1 Yi,tf +p Df \u22121 \u2212Xi,Df \u22121 ! . Comparing Eq. (6) with Eq. (5) by constraining \u03b4 = \u03b4f + \u03b4p + \u03b4b, we see that the mean estimator takes into account the partial feedback as before but also has a bias correction term. The bias correction term is an empirical average of the biases observed from the past full delayed feedback. This correction has the e\ufb00ect of introducing additional uncertainty (third term in the RHS) and we need at least one full feedback to estimate the bias before we can use the above bound. The corresponding racing algorithm runs similar to Algorithm 2 with the key di\ufb00erence being that the \fBest arm identi\ufb01cation in multi-armed bandits with delayed feedback 101 102 Number of arms (n) 0.80 0.85 0.90 0.95 tpartial/tfull 101 102 103 Number of arms (n) 0.3 0.4 0.5 0.6 0.7 tpartial/tfull 101 102 103 Delay (d) 0.54 0.56 0.58 0.60 0.62 tpartial/tfull 101 102 Number of arms (n) 0.6 0.7 0.8 0.9 1.0 tpartial/tfull (a) Number of arms Bounded means 101 102 103 Number of arms (n) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 tpartial/tfull (b) Number of arms Free means 101 102 103 Delay (d) 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 tpartial/tfull (c) Delay Figure 1: Synthetic experiments evaluating performance. Top: sequential. Bottom: parallel. Lower is better. mean estimator corresponds to the minimum of the con\ufb01dence bounds in Eq. (5) and Eq. (6), where the RHS of Eq. (6) is speci\ufb01ed for the optimal \u03b4\u2217 f, \u03b4\u2217 p, \u03b4\u2217 b minimizing the expression under the constraint due to \u03b4. We defer the pseudocode for this setting to the Appendix (see Algorithm 4). 4 PARALLEL MAB In parallel MAB, an agent has the additional ability to \u201caccumulate\u201d bulk information by controlling a batch of arm pulls. We extend the (b, r) setting proposed in Jun et al. [2016] where the agent is allowed to run at most b arm pulls in parallel at any given time step with an upper limit r on the number of pulls of each arm. Even the full delayed feedback setting becomes interesting, as the agent can exploit information from arm pulls which have \ufb01nished running in parallel to accept/reject delayed arm pulls that are still running thereby avoiding the pitfalls of long delays. The pseudocode for the proposed batch racing algorithm with full delayed feedback is given in Algorithm 3. At every time step, an agent pulls a batch of arms with the least pull count Ni that obeys the (b, r) constraints (Lines 18-19). Whenever we obtain at least one full delayed feedback, we can update our arm sets as per the racing criteria (Lines 13-15). The algorithms for the noisy partial feedback settings discussed in Section 3 can be extended for parallel MAB in a similar manner and are skipped here to keep the presentation clean. The theoretical analysis of the batch MAB setting in Jun et al. [2016] builds on the analysis of standard MAB in ways independent of the choice of LIL bounds and hence, a merged analysis for delayed batch MAB using the LIL bounds for delayed feedback (as in Propositions 1 and 2) suggests a reduction factor of b in the corresponding upper bounds. 5 EXPERIMENTS We empirically validated the proposed algorithms on a simulated setting and two real world datasets. All experiments use an error probability of \u03b4 = 0.05 and we observed that in each case, the algorithm obtains the desired con\ufb01dence level empirically. For the parallel MAB setting, we set b = r = 10. 5.1 Simulated data We performed an ablation study of the proposed algorithms for sequential and parallel MAB under di\ufb00erent settings of delayed feedback. All experiments were repeated for 100 random runs such that the standard errors are vanishingly small and the number of top arms to be identi\ufb01ed, k is set to 0.2n. We quantify improvement as the ratio (=tpartial/tfull) of the time taken by Algorithm 2 or its parallel MAB extension (i.e., tpartial) and the time taken by a full delayed feedback racing procedure (i.e., tfull). We evaluate performance as a function of the following problem parameters. Number of arms. To analyze the di\ufb00erence in performance as a function of the number of arms (n), we further consider two distribution of means. \fGrover et al. 10\u22123 10\u22122 10\u22121 100 Scale parameter, \u03c3(p) 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 tpartial/tfull (a) Sequential 10\u22123 10\u22122 10\u22121 100 Scale parameter, \u03c3(p) 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 tpartial/tfull (b) Parallel Figure 2: Experiments on battery charging. In the bounded means case, we set the means of the arms as \u00b5i = c \u2212(i/n)\u02dc c for any choice of constants c and \u02dc c > 0. Hence, the range of the means does not vary with n. In Figure 1a, we observe that accounting for unbiased partial feedback can give gains of up to 25% and 40% for the sequential and parallel MAB when the number of arms is low. The gains are reduced when the number of arms is large, which suggests that partial feedback is less advantageous in scenarios where a large number of full pulls are required for disambiguating very closely spaced means. In the free means case, we set the means of the arms as \u00b5i = c \u2212\u02dc ci for any choice of constants c and \u02dc c > 0. Here, the range of the means increases with n. From the results in Figure 1b, we observe that the gains due to partial feedback improve as the number of arms increases. This suggests that when the relative separation in means between the arms is \ufb01xed, Algorithm 2 and its parallel MAB extension quickly eliminate arms with extreme means (very high or very low) unlike the racing algorithms that wait for full delayed feedback. Delay. Here, we \ufb01x n = 100 and vary the delay of the arms. For all settings of the delay in Figure 1c, Algorithm 2 and its parallel MAB extension require a signi\ufb01cantly lower fraction of the time with the lowest ratios observed to be 0.59 and 0.57 for sequential and parallel MAB respectively. While we did not see much variation in improvements for sequential MAB, the improvements are better for longer delays in the case of parallel MAB. 5.2 Policy search for fast battery charging For any given battery chemistry, the charging (and discharging) policy has a signi\ufb01cant impact on the lifetime of the cells. However, a single run of a particular policy however takes months to complete since every cell needs to be repeatedly charged and discharged until the end of its lifetime. Hence, delayed feedback can signi\ufb01cantly slow down the search procedure. The true, unknown reward for any arm (charging policy) is stochastic and corresponds to the lifetime of the battery [Harris et al., 2017, Baumh\u00a8 ofer et al., 2014, Schuster et al., 2015].2 We model the search for the best charging policy for the Li-ion battery chemistry as a best arm identi\ufb01cation problem in a stochastic MAB with n = 40 arms, k = 1. The true mean cycle life, cell-to-cell variances, and delays are obtained from a battery charging simulator [Moura et al., 2017, Perez et al., 2016]. While a battery cell undergoes charging and discharging, we can additionally monitor key indicators such as voltage, temperature, and internal resistance. Predictive models of lifetime based on these factors is an active area of research, and can serve the purpose of partial feedback estimator [Burns et al., 2013, Dubarry et al., 2017]. We assume the existence of such an estimator and test the robustness of our algorithm by evaluating the relative improvements obtained from Algorithm 2 on varying the noise \u03c3(p) i associated with the partial feedback. The results are shown in Figure 2. When the estimator is \u201ctrustworthy\u201d (low \u03c3(p) i ), we can achieve improvements of up to 35% in the number of experiments required. As expected, the gains diminish for poorer models of partial feedback in which case the algorithm can choose to ignore the noisy feedback. 5.3 Hyperparameter optimization for mixed integer programming The CPLEX solver3 for mixed integer programming has a host of hyperparameters, including options to switch on or o\ufb00di\ufb00erent cut strategies employed by the solver during the search process. We model the task of \ufb01nding the best cut strategy as a stochastic MAB problem with n = 32 arms (i.e., cut strategies), 2Formally, the lifetime of the cell is de\ufb01ned to be the number of cycles until a battery reaches 80% of its original capacity at which point a battery is considered dead. 3https://www.ibm.com/software/commerce/ optimization/cplex-optimizer/index.html \fBest arm identi\ufb01cation in multi-armed bandits with delayed feedback k = 1. The performance is measured on CORLAT, a benchmark set of 2, 000 (maximization) mixed integer linear programming instances derived from real world data used for the construction of a wildlife corridor for grizzly bears in the Northern Rockies region [Gomes et al., 2008, Hutter et al., 2010]. The true mean for each arm is the average of lower bounds attained by the cut strategy on the feasible instances in the dataset under speci\ufb01ed time and resource constraints per instance (10 seconds on 1 core). Every pull of an arm corresponds to running a cut strategy on a sampled problem instance. Instead of waiting for the solver to completely solve (or time out) a sampled problem instance, we can save computation by using partial feedback about the search process. In particular, the solver outputs the best integral lower bound (LB) and real valued upper bound (UB) found after executing each cut during search. The \ufb01nal output of the solver is the best lower bound. To obtain an unbiased partial feedback estimator, we use a training subset of 500 instances to learn a linear model that predicts the \ufb01nal lower bound for a given input instance based on the intermediate lower and upper bounds. The best arm identi\ufb01cation algorithms are tested on the remaining instances in the dataset. Conditioned on a problem instance, the uncertainty associated with the partial feedback, \u03c3(p) i is given by (UB \u2212LB)/2 and shrinks with an increase in the time steps elapsed. Note that the delays are not \ufb01xed and depend on both the cut strategy and the problem instance under consideration. We directly report the \ufb01nal results: the percentage reduction in time taken by the unbiased partial feedback scenarios over full delayed feedback is 80.8% and 87.6% for sequential and parallel MAB respectively stressing the importance of partial feedback for this particular application scenario. 6 RELATED WORK Early work in pure exploration is attributed to Bechhofer [1958] and Paulson [1964] who studied this problem in the context of optimal experimental design. Modern day literature can be categorized into either the \ufb01xed budget or the \ufb01xed con\ufb01dence settings. Algorithms for the \ufb01xed budget setting strive to maximize the probability of identifying the top-k arms [Audibert and Bubeck, 2010, Bubeck et al., 2013, Kaufmann et al., 2015]. In the \ufb01xed con\ufb01dence setting, which is the one we consider in this paper, the goal is to minimize the number of pulls to attain a target con\ufb01dence [Maron and Moore, 1994, Bubeck et al., 2009]. See Gabillon et al. [2012] for a uni\ufb01ed treatment of the two settings. Algorithms for the \ufb01xed con\ufb01dence setting can be broadly classi\ufb01ed into racing style procedures which sample arms uniformly and eliminate sub-optimal arms [Maron and Moore, 1994, Even-Dar et al., 2002] and the UCB/LUCB style procedures which adaptively sample arms without explicit elimination. We direct the reader to the excellent survey by Jamieson and Nowak [2014] that summarizes the major advancements in the analysis of the sample complexity of these algorithms. Algorithmic generalizations of the best arm identi\ufb01cation include top-k identi\ufb01cation [Heidrich-Meisner and Igel, 2009] and the parallel MAB settings for batch arm pulls [Perchet et al., 2015, Jun et al., 2016, Wu et al., 2015] among others. While the delayed feedback framework we propose is novel to the pure exploration problem, online learning with delays has been studied previously in the regret minimization setting [Weinberger and Ordentlich, 2002, Joulani et al., 2013, Desautels et al., 2014]. In particular, algorithms designed particularly for hyperparameter optimization have enjoyed great success. Krueger et al. [2015] proposes a modi\ufb01ed cross-validation procedure performed on increasing subsets of data coupled with a sequential testing strategy to eliminate the poor parameter con\ufb01gurations early on. Jamieson and Talwalkar [2016] and Li et al. [2017] recently proposed algorithms for hyperparameter optimization based on non-stochastic MAB. Here, the arms correspond to hyperparameter con\ufb01gurations, and a pull is equivalent to observing a \ufb01xed sequence of losses. For many real-world problems, we have access to a shared structure across arms that makes the overall problem amenable to Bayesian optimization techniques [Snoek et al., 2012, Eggensperger et al., 2013, Snoek et al., 2015, Feurer et al., 2015, McIntire et al., 2016b,a]. Combining the LIL bounds we proposed for noisy partial feedback with Bayesian multi-armed bandits [Srinivas et al., 2010, Krause and Ong, 2011, Ho\ufb00man et al., 2014] is a promising extension we are pursuing for our on-going real world application relating to e\ufb03cient search of fast charging policies for Li-ion battery cells [Ermon et al., 2012]. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1803.10459v4",
+                    "title": "Graphite: Iterative Generative Modeling of Graphs",
+                    "abstract": "Graphs are a fundamental abstraction for modeling relational data. However,\ngraphs are discrete and combinatorial in nature, and learning representations\nsuitable for machine learning tasks poses statistical and computational\nchallenges. In this work, we propose Graphite, an algorithmic framework for\nunsupervised learning of representations over nodes in large graphs using deep\nlatent variable generative models. Our model parameterizes variational\nautoencoders (VAE) with graph neural networks, and uses a novel iterative graph\nrefinement strategy inspired by low-rank approximations for decoding. On a wide\nvariety of synthetic and benchmark datasets, Graphite outperforms competing\napproaches for the tasks of density estimation, link prediction, and node\nclassification. Finally, we derive a theoretical connection between message\npassing in graph neural networks and mean-field variational inference.",
+                    "authors": "Aditya Grover, Aaron Zweig, Stefano Ermon",
+                    "published": "2018-03-28",
+                    "updated": "2019-05-15",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG",
+                        "cs.NE",
+                        "cs.SI"
+                    ],
+                    "main_content": "Introduction Latent variable generative modeling is an effective approach for unsupervised representation learning of highdimensional data (Loehlin, 1998). In recent years, representations learned by latent variable models parameterized by deep neural networks have shown impressive performance on many tasks such as semi-supervised learning and structured prediction (Kingma et al., 2014; Sohn et al., 2015). However, these successes have been largely restricted to speci\ufb01c data modalities such as images and speech. In particular, it is challenging to apply current deep generative models for large scale graph-structured data which arise in a wide variety of domains in physical sciences, information sciences, and social sciences. To effectively model the relational structure of large graphs for deep learning, prior works have proposed to use graph 1Department of Computer Science, Stanford University, USA. Correspondence to: Aditya Grover <adityag@cs.stanford.edu>. Proceedings of the 36 th International Conference on Machine Learning, Long Beach, California, PMLR 97, 2019. Copyright 2019 by the author(s). neural networks (Gori et al., 2005; Scarselli et al., 2009; Bruna et al., 2013). A graph neural network learns nodelevel representations by parameterizing an iterative message passing procedure between nodes and their neighbors. The tasks which have bene\ufb01ted from graph neural networks, including semi-supervised learning (Kipf & Welling, 2017) and few shot learning (Garcia & Bruna, 2018), involve encoding an input graph to a \ufb01nal output representation (such as the labels associated with the nodes). The inverse problem of learning to decode a hidden representation into a graph, as in the case of a latent variable generative model, is a pressing challenge that we address in this work. We propose Graphite, a latent variable generative model for graphs based on variational autoencoding (Kingma & Welling, 2014). Speci\ufb01cally, we learn a directed model expressing a joint distribution over the entries of adjacency matrix of graphs and latent feature vectors for every node. Our framework uses graph neural networks for inference (encoding) and generation (decoding). While the encoding is straightforward and can use any existing graph neural network, the decoding of these latent features to reconstruct the original graph is done using a multi-layer iterative procedure. The procedure starts with an initial reconstruction based on the inferred latent features, and iteratively re\ufb01nes the reconstructed graph via a message passing operation. The iterative re\ufb01nement can be ef\ufb01ciently implemented using graph neural networks. In addition to the Graphite model, we also contribute to the theoretical understanding of graph neural networks by deriving equivalences between message passing in graph neural networks with mean-\ufb01eld inference in latent variable models via kernel embeddings (Smola et al., 2007; Dai et al., 2016), formalizing what has thus far has been largely speculated empirically to the best of our knowledge (Yoon et al., 2018). In contrast to recent works focussing on generation of small graphs e.g., molecules (You et al., 2018; Li et al., 2018), the Graphite framework is particularly suited for representation learning on large graphs. Such representations are useful for several downstream tasks. In particular, we demonstrate that representations learned via Graphite outperform competing approaches for graph representation learning empirically for the tasks of density estimation (over entire graphs), link prediction, and semi-supervised node classi\ufb01cation on synthetic and benchmark datasets. arXiv:1803.10459v4  [stat.ML]  15 May 2019 \fGraphite: Iterative Generative Modeling of Graphs 2. Preliminaries Throughout this work, we assume that all probability distributions admit absolutely continuous densities on a suitable reference measure. Consider a weighted undirected graph G = (V, E) where V and E denote index sets of nodes and edges respectively. Additionally, we denote the (optional) feature matrix associated with the graph as X \u2208Rn\u00d7m for an m-dimensional signal associated with each node, for e.g., these could refer to the user attributes in a social network. We represent the graph structure using a symmetric adjacency matrix A \u2208Rn\u00d7n where n = |V | and the entries Aij denote the weight of the edge between node i and j. 2.1. Weisfeiler-Lehman algorithm The Weisfeiler-Lehman (WL) algorithm (Weisfeiler & Lehman, 1968; Douglas, 2011) is a heuristic test of graph isomorphism between any two graphs G and G\u2032. The algorithm proceeds in iterations. Before the \ufb01rst iteration, we label every node in G and G\u2032 with a scalar isomorphism invariant initialization (e.g., node degrees). That is, if G and G\u2032 are assumed to be isomorphic, then an isomorphism invariant initialization is one where the matching nodes establishing the isomorphism in G and G\u2032 have the same labels (a.k.a. messages). Let H(0) = [h(l) 1 , h(l) 2 , \u00b7 \u00b7 \u00b7 , h(l) n ]T denote the vector of initializations for the nodes in the graph at iteration l \u2208N\u222a0. At every iteration l > 0, we perform a relabelling of nodes in G and G\u2032 based on a message passing update rule: H(l) \u2190hash \u0010 AH(l\u22121)\u0011 (1) where A denotes the adjacency matrix of the corresponding graph and hash : Rn \u2192Rn is any suitable hash function e.g., a non-linear activation. Hence, the message for every node is computed as a hashed sum of the messages from the neighboring nodes (since Aij \u0338= 0 only if i and j are neighbors). We repeat the process for a speci\ufb01ed number of iterations, or until convergence. If the label sets for the nodes in G and G\u2032 are equal (which can be checked using sorting in O(n log n) time), then the algorithm declares the two graphs G and G\u2032 to be isomorphic. The \u201ck-dim\u201d WL algorithm extends the 1-dim algorithm above by simultaneously passing messages of length k (each initialized with some isomorphism invariant scheme). A positive test for isomorphism requires equality in all k dimensions for nodes in G and G\u2032 after the termination of message passing. This algorithmic test is a heuristic which guarantees no false negatives but can give false positives wherein two non-isomorphic graphs can be falsely declared isomorphic. Empirically, the test has been shown to fail on some regular graphs but gives excellent performance on real-world graphs (Shervashidze et al., 2011). 2.2. Graph neural networks Intuitively, the WL algorithm encodes the structure of the graph in the form of messages at every node. Graph neural networks (GNN) build on this observation and parameterize an unfolding of the iterative message passing procedure which we describe next. A GNN consists of many layers, indexed by l \u2208N with each layer associated with an activation \u03b7l and a dimensionality dl. In addition to the input graph A, every layer l \u2208N of the GNN takes as input the activations from the previous layer H(l\u22121) \u2208Rn\u00d7dl\u22121, a family of linear transformations Fl : Rn\u00d7n \u2192Rn\u00d7n, and a matrix of learnable weight parameters Wl \u2208Rdl\u22121\u00d7dl and optional bias parameters Bl \u2208Rn\u00d7dl. Recursively, the layer wise propagation rule in a GNN is given by: H(l) \u2190\u03b7l \uf8eb \uf8edBl + X f\u2208Fl f(A)H(l\u22121)Wl \uf8f6 \uf8f8 (2) with the base cases H(0) = X and d0 = m. Here, m is the feature dimensionality. If there are no explicit node features, we set H(0) = In (identity) and d0 = n. Several variants of graph neural networks have been proposed in prior work. For instance, graph convolutional networks (GCN) (Kipf & Welling, 2017) instantiate graph neural networks with a permutation equivariant propagation rule: H(l) \u2190\u03b7l \u0010 Bl + \u02dc AH(l\u22121)Wl \u0011 (3) where \u02dc A = D\u22121/2AD\u22121/2 is the symmetric diagonalization of A given the diagonal degree matrix D (i.e., Dii = P (i,j)\u2208E Aij), and same base cases as before. Comparing the above with the WL update rule in Eq. (1), we can see that the activations for every layer in a GCN are computed via parameterized, scaled activations (messages) of the previous layer being propagated over the graph, with the hash function implicitly speci\ufb01ed using an activation function \u03b7l. Our framework is agnostic to instantiations of message passing rule of a graph neural network in Eq. (2), and we use graph convolutional networks for experimental validation due to the permutation equivariance property. For brevity, we denote the output H for the \ufb01nal layer of a multi-layer graph neural network with input adjacency matrix A, node feature matrix X, and parameters \u27e8W, B\u27e9as H = GNN\u27e8W,B\u27e9(A, X), with appropriate activation functions and linear transformations applied at each hidden layer of the network. 3. Generative Modeling via Graphite For generative modeling of graphs, we are interested in learning a parameterized distribution over adjacency matri\fGraphite: Iterative Generative Modeling of Graphs Figure 1. Latent variable model for Graphite. Observed evidence variables in gray. ces A. In this work, we restrict ourselves to modeling graph structure only, and any additional information in the form of node features X is incorporated as conditioning evidence. In Graphite, we adopt a latent variable approach for modeling the generative process. That is, we introduce latent variable vectors Zi \u2208Rk and evidence feature vectors Xi \u2208Rm for each node i \u2208{1, 2, \u00b7 \u00b7 \u00b7 , n} along with an observed variable for each pair of nodes Aij \u2208R. Unless necessary, we use a succinct representation Z \u2208Rn\u00d7k, X \u2208Rn\u00d7m, and A \u2208Rn\u00d7n for the variables henceforth. The conditional independencies between the variables can be summarized in the directed graphical model (using plate notation) in Figure 1. We can learn the model parameters \u03b8 by maximizing the marginal likelihood of the observed adjacency matrix conditioned on X: max \u03b8 log p\u03b8(A|X) = log Z Z p\u03b8(A, Z|X)dZ (4) Here, p(Z|X) is a \ufb01xed prior distribution over the latent features of every node e.g., isotropic Gaussian. If we have multiple graphs in our dataset, we maximize the expected log-likelihoods over all the corresponding adjacency matrices. We can obtain a tractable, stochastic evidence lower bound (ELBO) to the above objective by introducing a variational posterior q\u03c6(Z|A, X) with parameters \u03c6: log p\u03b8(A|X) \u2265Eq\u03c6(Z|A,X) \u0014 log p\u03b8(A, Z|X) q\u03c6(Z|A, X) \u0015 (5) The lower bound is tight when the variational posterior q\u03c6(Z|A, X) matches the true posterior p\u03b8(Z|A, X) and hence maximizing the above objective optimizes for the parameters that de\ufb01ne the best approximation to the true posterior within the variational family (Blei et al., 2017). We now discuss parameterizations for specifying q\u03c6(Z|A, X) (i.e., encoder) and p\u03b8(A|Z, X) (i.e., decoder). Encoding using forward message passing. Typically we use the mean \ufb01eld approximation for de\ufb01ning the variational family and hence: q\u03c6(Z|A, X) \u2248 n Y i=1 q\u03c6i(Zi|A, X) (6) Additionally, we would like to make distributional assumptions on each variational marginal density q\u03c6i(Zi|A, X) such that it is reparameterizable and easy-to-sample, such that the gradients w.r.t. \u03c6i have low variance (Kingma & Welling, 2014). In Graphite, we assume isotropic Gaussian variational marginals with diagonal covariance. The parameters for the variational marginals q\u03c6i(Z|A, X) are speci\ufb01ed using a graph neural network: \u00b5, \u03c3 = GNN\u03c6(A, X) (7) where \u00b5 and \u03c3 denote the vector of means and standard deviations for the variational marginals {q\u03c6i(Zi|A, X)}n i=1 and \u03c6 = {\u03c6i}n i=1 are the full set of variational parameters. Decoding using reverse message passing. For specifying the observation model p\u03b8(A|Z, X), we cannot directly use a graph neural network since we do not have an input graph for message passing. To sidestep this issue, we propose an iterative two-step approach that alternates between de\ufb01ning an intermediate graph and then gradually re\ufb01ning this graph through message passing. Formally, given a latent matrix Z and an input feature matrix X, we iterate over the following sequence of operations: b A = ZZT \u2225Z\u22252 + 11T , (8) Z\u2217= GNN\u03b8( b A, [Z|X]) (9) where the second argument to the GNN is a concatenation of Z and X. The \ufb01rst step constructs an intermediate weighted graph b A \u2208Rn\u00d7n by applying an inner product of Z with itself and adding an additional constant of 1 to ensure entries are non-negative. And the second step performs a pass through a parameterized graph neural network. We can repeat the above sequence to gradually re\ufb01ne the feature matrix Z\u2217. The \ufb01nal distribution over graph parameters is obtained using an inner product step on Z\u2217\u2208Rn\u00d7k\u2217akin to Eq. (8), where k\u2217\u2208N is determined via the network architecture. For ef\ufb01cient sampling, we assume the observation model factorizes: p\u03b8(A|Z, X) = n Y i=1 n Y j=1 p(i,j) \u03b8 (Aij|Z\u2217). (10) The distribution over the individual edges can be expressed as a Bernoulli or Gaussian distribution for unweighted and real-valued edges respectively. E.g., the edge probabilities for an unweighted graph are given as sigmoid(Z\u2217Z\u2217T ). \fGraphite: Iterative Generative Modeling of Graphs Table 1. Mean reconstruction errors and negative log-likelihood estimates (in nats) for autoencoders and variational autoencoders respectively on test instances from six different generative families. Lower is better. Erdos-Renyi Ego Regular Geometric Power Law Barabasi-Albert GAE 221.79 \u00b1 7.58 197.3 \u00b1 1.99 198.5 \u00b1 4.78 514.26 \u00b1 41.58 519.44 \u00b1 36.30 236.29 \u00b1 15.13 Graphite-AE 195.56 \u00b1 1.49 182.79 \u00b1 1.45 191.41 \u00b1 1.99 181.14 \u00b1 4.48 201.22 \u00b1 2.42 192.38 \u00b1 1.61 VGAE 273.82 \u00b1 0.07 273.76 \u00b1 0.06 275.29 \u00b1 0.08 274.09 \u00b1 0.06 278.86 \u00b1 0.12 274.4 \u00b1 0.08 Graphite-VAE 270.22 \u00b1 0.15 270.70 \u00b1 0.32 266.54 \u00b1 0.12 269.71 \u00b1 0.08 263.92 \u00b1 0.14 268.73 \u00b1 0.09 3.1. Scalable learning & inference in Graphite For representation learning of large graphs, we require the encoding and decoding steps in Graphite to be computationally ef\ufb01cient. On the surface, the decoding step involves inner products of potentially dense matrices Z, which is an O(n2k) operation. Here, k is the dimension of the per-node latent vectors Zi used to de\ufb01ne b A. For any intermediate decoding step as in Eq. (8), we propose to offset this expensive computation by using the associativity property of matrix multiplications for the message passing step in Eq. (9). For notational brevity, consider the simpli\ufb01ed graph propagation rule for a GNN: H(l) \u2190\u03b7l \u0010 b AH(l\u22121)\u0011 where b A is de\ufb01ned in Eq. (8). Instead of directly taking an inner product of Z with itself, we note that the subsequent operation involves another matrix multiplication and hence, we can perform right multiplication instead. If dl and dl\u22121 denote the size of the layers H(l) and H(l\u22121) respectively, then the time complexity of propagation based on right multiplication is given by O(nkdl\u22121 + ndl\u22121dl). The above trick sidesteps the quadratic n2 complexity for decoding in the intermediate layers without any loss in statistical accuracy. The \ufb01nal layer however still involves an inner product with respect to Z\u2217between potentially dense matrices. However, since the edges are generated independently, we can approximate the loss objective by performing a Monte Carlo evaluation of the reconstructed adjacency matrix parameters in Eq. (10). By adaptively choosing the number of entries for Monte Carlo approximation, we can trade-off statistical accuracy for computational budget. 4. Experimental Evaluation We evaluate Graphite on tasks involving entire graphs, nodes, and edges. We consider two variants of our proposed framework: the Graphite-VAE, which corresponds to a directed latent variable model as described in Section 3 and Graphite-AE, which corresponds to an autoencoder trained to minimize the error in reconstructing an input adjacency matrix. For unweighted graphs (i.e., A \u2208{0, 1}n\u00d7n), the reconstruction terms in the objectives for both GraphiteVAE and Graphite-AE minimize the negative cross entropy between the input and reconstructed adjacency matrices. For weighted graphs, we use the mean squared error. Additional hyperparameter details are described in Appendix B. 4.1. Reconstruction & density estimation In the \ufb01rst set of tasks, we evaluate learning in Graphite based on held-out reconstruction losses and log-likelihoods estimated by the learned Graphite-VAE and Graphite-AE models respectively on a collection of graphs with varying sizes. In direct contrast to modalities such as images, graphs cannot be straightforwardly reduced to a \ufb01xed number of vertices for input to a graph convolutional network. One simplifying modi\ufb01cation taken by Bojchevski et al. (2018) is to consider only the largest connected component for evaluating and optimizing the objective, which we appeal to as well. Thus by setting the dimensions of Z\u2217to a maximum number of vertices, Graphite can be used for inference tasks over entire graphs with potentially smaller sizes by considering only the largest connected component. We create datasets from six graph families with \ufb01xed, known generative processes: the Erdos-Renyi, ego-nets, random regular graphs, random geometric graphs, random Power Law Tree and Barabasi-Albert. For each family, 300 graph instances were sampled with each instance having 10 \u221220 nodes and evenly split into train/validation/test instances. As a benchmark comparison, we compare against the Graph Autoencoder/Variational Graph Autoencoder (GAE/VGAE) (Kipf & Welling, 2016). The GAE/VGAE models consist of an encoding procedure similar to Graphite. However, the decoder has no learnable parameters and reconstruction is done solely through an inner product operation (such as the one in Eq. (8)). The mean reconstruction errors and the negative loglikelihood results on a test set of instances are shown in Table 1. Both Graphite-AE and Graphite-VAE outperform AE and VGAE signi\ufb01cantly on these tasks, indicating the usefulness of learned decoders in Graphite. \fGraphite: Iterative Generative Modeling of Graphs Table 2. Citation network statistics Nodes Edges Node Features Labels Cora 2708 5429 1433 7 Citeseer 3327 4732 3703 6 Pubmed 19717 44338 500 3 4.2. Link prediction The task of link prediction is to predict whether an edge exists between a pair of nodes (Loehlin, 1998). Even though Graphite learns a distribution over graphs, it can be used for predictive tasks within a single graph. In order to do so, we learn a model for a random, connected training subgraph of the true graph. For validation and testing, we add a balanced set of positive and negative (false) edges to the original graph and evaluate the model performance based on the reconstruction probabilities assigned to the validation and test edges (similar to denoising of the input graph). In our experiments, we held out a set of 5% edges for validation, 10% edges for testing, and train all models on the remaining subgraph. Additionally, the validation and testing sets also each contain an equal number of non-edges. Datasets. We compared across standard benchmark citation network datasets: Cora, Citeseer, and Pubmed with papers as nodes and citations as edges (Sen et al., 2008). The node-level features correspond to the text attributes in the papers. The dataset statistics are summarized in Table 2. Baselines and evaluation metrics. We evaluate performance based on the Area Under the ROC Curve (AUC) and Average Precision (AP) metrics. We evaluated GraphiteVAE and Graphite-AE against the following baselines: Spectral Clustering (SC) (Tang & Liu, 2011), DeepWalk (Perozzi et al., 2014), node2vec (Grover & Leskovec, 2016), and GAE/VGAE (Kipf & Welling, 2016). SC, DeepWalk, and node2vec do not provide the ability to incorporate node features while learning embeddings, and hence we evaluate them only on the featureless datasets. Results. The AUC and AP results (along with standard errors) are shown in Table 3 and Table 4 respectively averaged over 50 random train/validation/test splits. On both metrics, Graphite-VAE gives the best performance overall. Graphite-AE also gives good results, generally outperforming its closest competitor GAE. Qualitative evaluation. We visualize the embeddings learned by Graphite and given by a 2D t-SNE projection (Maaten & Hinton, 2008) of the latent feature vectors (given as rows for Z with \u03bb = 0.5) on the Cora dataset in Figure 2. Even without any access to label information for (a) Graphite-AE (b) Graphite-VAE Figure 2. t-SNE embeddings of the latent feature vectors for the Cora dataset. Colors denote labels. the nodes during training, the name models are able to cluster the nodes (papers) as per their labels (paper categories). 4.3. Semi-supervised node classi\ufb01cation Given labels for a subset of nodes in an underlying graph, the goal of this task is to predict the labels for the remaining nodes. We consider a transductive setting, where we have access to the test nodes (without their labels) during training. Closest approach to Graphite for this task is a supervised graph convolutional network (GCN) trained end-to-end. We consider an extension of this baseline, wherein we augment the GCN objective with the Graphite objective and a hyperparameter to control the relative importance of the two terms in the combined objective. The parameters \u03c6 for the encoder are shared across these two objectives, with an additional GCN layer for mapping the encoder output to softmax probabilities over the requisite number of classes. All parameters are learned jointly. \fGraphite: Iterative Generative Modeling of Graphs Table 3. Area Under the ROC Curve (AUC) for link prediction (* denotes dataset with features). Higher is better. Cora Citeseer Pubmed Cora* Citeseer* Pubmed* SC 89.9 \u00b1 0.20 91.5 \u00b1 0.17 94.9 \u00b1 0.04 DeepWalk 85.0 \u00b1 0.17 88.6 \u00b1 0.15 91.5 \u00b1 0.04 node2vec 85.6 \u00b1 0.15 89.4 \u00b1 0.14 91.9 \u00b1 0.04 GAE 90.2 \u00b1 0.16 92.0 \u00b1 0.14 92.5 \u00b1 0.06 93.9 \u00b1 0.11 94.9 \u00b1 0.13 96.8 \u00b1 0.04 VGAE 90.1 \u00b1 0.15 92.0 \u00b1 0.17 92.3 \u00b1 0.06 94.1 \u00b1 0.11 96.7 \u00b1 0.08 95.5 \u00b1 0.13 Graphite-AE 91.0 \u00b1 0.15 92.6 \u00b1 0.16 94.5 \u00b1 0.05 94.2 \u00b1 0.13 96.2 \u00b1 0.10 97.8 \u00b1 0.03 Graphite-VAE 91.5 \u00b1 0.15 93.5 \u00b1 0.13 94.6 \u00b1 0.04 94.7 \u00b1 0.11 97.3 \u00b1 0.06 97.4 \u00b1 0.04 Table 4. Average Precision (AP) scores for link prediction (* denotes dataset with features). Higher is better. Cora Citeseer Pubmed Cora* Citeseer* Pubmed* SC 92.8 \u00b1 0.12 94.4 \u00b1 0.11 96.0 \u00b1 0.03 DeepWalk 86.6 \u00b1 0.17 90.3 \u00b1 0.12 91.9 \u00b1 0.05 node2vec 87.5 \u00b1 0.14 91.3 \u00b1 0.13 92.3 \u00b1 0.05 GAE 92.4 \u00b1 0.12 94.0 \u00b1 0.12 94.3 \u00b1 0.5 94.3 \u00b1 0.12 94.8 \u00b1 0.15 96.8 \u00b1 0.04 VGAE 92.3 \u00b1 0.12 94.2 \u00b1 0.12 94.2 \u00b1 0.04 94.6 \u00b1 0.11 97.0 \u00b1 0.08 95.5 \u00b1 0.12 Graphite-AE 92.8 \u00b1 0.13 94.1 \u00b1 0.14 95.7 \u00b1 0.06 94.5 \u00b1 0.14 96.1 \u00b1 0.12 97.7 \u00b1 0.03 Graphite-VAE 93.2 \u00b1 0.13 95.0 \u00b1 0.10 96.0 \u00b1 0.03 94.9 \u00b1 0.13 97.4 \u00b1 0.06 97.4 \u00b1 0.04 Table 5. Classi\ufb01cation accuracies (* denotes dataset with features). Baseline numbers from Kipf & Welling (2017). Cora* Citeseer* Pubmed* SemiEmb 59.0 59.6 71.1 DeepWalk 67.2 43.2 65.3 ICA 75.1 69.1 73.9 Planetoid 75.7 64.7 77.2 GCN 81.5 70.3 79.0 Graphite 82.1 \u00b1 0.06 71.0 \u00b1 0.07 79.3 \u00b1 0.03 Results. The classi\ufb01cation accuracy of the semisupervised models is given in Table 5. We \ufb01nd that Graphitehybrid outperforms the competing models on all datasets and in particular the GCN approach which is the closest baseline. Recent work in Graph Attention Networks shows that extending GCN by incoporating attention can boost performance on this task (Veli\u02c7 ckovi\u00b4 c et al., 2018). Using GATs in place of GCNs for parameterizing Graphite could yield similar performance boost in future work. 5. Theoretical Analysis In this section, we derive a theoretical connection between message passing in graph neural networks and approximate inference in related undirected graphical models. 5.1. Kernel embeddings We \ufb01rst provide a brief background on kernel embeddings. A kernel de\ufb01nes a notion of similarity between pairs of objects (Sch\u00a8 olkopf & Smola, 2002; Shawe-Taylor & Cristianini, 2004). Let K : Z \u00d7 Z \u2192R be the kernel function de\ufb01ned over a space of objects, say Z. With every kernel function K, we have an associated feature map \u03c8 : Z \u2192H where H is a potentially in\ufb01nite dimensional feature space. Kernel methods can be used to specify embeddings of distributions of arbitrary objects (Smola et al., 2007; Gretton et al., 2007). Formally, we denote these functional mappings as T\u03c8 : P \u2192H where P speci\ufb01es the space of all distributions on Z. These mappings, referred to as kernel embeddings of distributions, are de\ufb01ned as: T\u03c8(p) := EZ\u223cp[\u03c8(Z)] (11) for any p \u2208P. We are particularly interested in kernels with feature maps \u03c8 that de\ufb01ne injective embeddings, i.e., for any pair of distributions p1 and p2, we have T\u03c8(p1) \u0338= T\u03c8(p2) if p1 \u0338= p2. For injective embeddings, we can compute functionals of any distribution by directly applying a corresponding function on its embedding. Formally, for every function O : P \u2192Rd, d \u2208N and injective embedding T\u03c8, there exists a function \u02dc O\u03c8 : H \u2192Rd such that: O(p) = \u02dc O\u03c8(T\u03c8(p)) \u2200p \u2208P. (12) Informally, we can see that the operator \u02dc O\u03c8 can be de\ufb01ned as the composition of O with the inverse of T\u03c8. \fGraphite: Iterative Generative Modeling of Graphs 2 3 1 (a) Input graph with edge set E = {(1, 2), (1, 3)}. X2 Z2 X3 Z3 X1 Z1 A12 A23 A13 (b) Latent variable model G satisfying Property 1 with A12 \u0338= 0, A23 = 0, A13 \u0338= 0. Figure 3. Interpreting message passing in Graph Neural Networks via Kernel Embeddings and Mean-\ufb01eld inference 5.2. Connections with mean-\ufb01eld inference Locality preference for representational learning is a key inductive bias for graphs. We formulate this using an (undirected) graphical model G over X, A, and {Z1, \u00b7 \u00b7 \u00b7 , Zn}. As in a GNN, we assume that X and A are observed and specify conditional independence structure in a conditional distribution over the latent variables, denoted as r(Z1, \u00b7 \u00b7 \u00b7 , Zn|A, X). We are particularly interested in models that satisfy the following property. Property 1. The edge set E de\ufb01ned by the adjacency matrix A is an undirected I-map for the distribution r(Z1, \u00b7 \u00b7 \u00b7 , Zn|A, X). In words, the above property implies that according to the conditional distribution over Z, any individual Zi is independent of all other Zj when conditioned on A, X, and the neighboring latent variables of node i as determined by the edge set E. See Figure 3 for an illustration. A mean-\ufb01eld (MF) approximation for G approximates the conditional distribution r(Z1, \u00b7 \u00b7 \u00b7 , Zn|A, X) as: r(Z1, \u00b7 \u00b7 \u00b7 , Zn|A, X) \u2248 n Y i=1 q\u03c6i(Zi|A, X) (13) where \u03c6i denotes the set of parameters for the i-th variational marginal. These parameters are optimized by minimizing the KL-divergence between the variational and the true conditional distributions: min \u03c61,\u00b7\u00b7\u00b7 ,\u03c6n KL  n Y i=1 q\u03c6i (Zi|A, X)\u2225r(Z1, \u00b7 \u00b7 \u00b7 , Zn|A, X) ! (14) Using standard variational arguments (Wainwright et al., 2008), we know that the optimal variational marginals assume the following functional form: q\u03c6i(Zi|A, X) = OMF G \u0000Zi, {q\u03c6j}j\u2208N (i) \u0001 (15) where N(i) denotes the neighbors of Zi in G and O is a function determined by the \ufb01xed point equations that depends on the potentials associated with G. Importantly, the above functional form suggests that the optimal marginals in mean \ufb01eld inference are locally consistent that they are only a function of the neighboring marginals. An iterative algorithm for mean-\ufb01eld inference is to perform message passing over the underlying graph until convergence. With an appropriate initialization at l = 0, the updated marginals at iteration l \u2208N are given as: q(l) \u03c6i (Zi|A, X) = OMF G \u0010 Zi, {q(l\u22121) \u03c6j }j\u2208N (i) \u0011 . (16) We will sidestep deriving O, and instead use the kernel embeddings of the variational marginals to directly reason in the embedding space. That is, we assume we have an injective embedding for each marginal q\u03c6i given by \u00b5i = EZi\u223cq\u03c6i [\u03c8(Zi)] for some feature map \u03c8 : Rk \u2192Rk\u2032 and directly use the equivalence established in Eq. (12) iteratively. For mean-\ufb01eld inference via message passing as in Eq. (16), this gives us the following recursive expression for iteratively updating the embeddings at iteration l \u2208N: \u00b5(l) i = \u02dc OMF \u03c8,G \u0010 {\u00b5(l\u22121) j }j\u2208N (i) \u0011 (17) with an appropriate base case for \u00b5(0) i . We then have the following result: Theorem 2. Let G be any undirected latent variable model such that the conditional distribution r(Z1, \u00b7 \u00b7 \u00b7 , Zn|A, X) expressed by the model satis\ufb01es Property 1. Then there exists a choice of \u03b7l, Fl, Wl, and Bl such that for all {\u00b5(l\u22121) i }n i=1, the GNN propagation rule in Eq. (2) is computationally equivalent to updating {\u00b5(l\u22121) i }n i=1 via a \ufb01rst order approximation of Eq. (17). Proof. See Appendix A. While \u03b7l and Fl are typically \ufb01xed beforehand, the parameters Wl, and Bl are directly learned from data in practice. Hence we have shown that a GNN is a good model for computation with respect to latent variable models that attempt to capture inductive biases relevant to graphs, i.e., ones where the latent feature vector for every node is conditionally independent from everything else given the feature vectors of its neighbors (and A, X). Note that such a graphical model would satisfy Property 1 but is in general different \fGraphite: Iterative Generative Modeling of Graphs from the posterior speci\ufb01ed by the one in Figure 1. However if the true (but unknown) posterior on the latent variables for the model proposed in Figure 1 could be expressed as an equivalent model satisfying the desired property, then Theorem 2 indeed suggests the use of GNNs for parameterizing variational posteriors, as we do so in the case of Graphite. 6. Discussion & Related Work Our framework effectively marries probabilistic modeling and representation learning on graphs. We review some of the dominant prior works in these \ufb01elds below. Probabilistic modeling of graphs. The earliest probabilistic models of graphs proposed to generate graphs by creating an edge between any pair of nodes with a constant probability (Erd\u00a8 os & R\u00b4 enyi, 1959). Several alternatives have been proposed since; e.g., the small-world model generates graphs that exhibit local clustering (Watts & Strogatz, 1998), the Barabasi-Albert models preferential attachment wherein high-degree nodes are likely to form edges with newly added nodes (Barabasi & Albert, 1999), the stochastic block model is based on inter and intra community linkages (Holland et al., 1983) etc. We direct the interested reader to prominent surveys on this topic (Newman, 2003; Mitzenmacher, 2004; Chakrabarti & Faloutsos, 2006). Representation learning on graphs. For representation learning on graphs, there are broadly three kinds of approaches: matrix factorization, random walk based approaches, and graph neural networks. We include a brief discussion on the \ufb01rst two kinds in Appendix C and refer the reader to Hamilton et al. (2017b) for a recent survey. Graph neural networks, a collective term for networks that operate over graphs using message passing, have shown success on several downstream applications, e.g., (Duvenaud et al., 2015; Li et al., 2016; Kearnes et al., 2016; Kipf & Welling, 2017; Hamilton et al., 2017a) and the references therein. Gilmer et al. (2017) provides a comprehensive characterization of these networks in the message passing setup. We used Graph Convolution Networks, partly to provide a direct comparison with GAE/VGAE and leave the exploration of other GNN variants for future work. Latent variable models for graphs. Hierarchical Bayesian models parameterized by deep neural networks have been recently proposed for graphs (Hu et al., 2017; Wang et al., 2017). Besides being restricted to single graphs, these models are limited since inference requires running expensive Markov chains (Hu et al., 2017) or are task-speci\ufb01c (Wang et al., 2017). Johnson (2017) and Kipf et al. (2018) generate graphs as latent representations learned directly from data. In contrast, we are interested in modeling observed (and not latent) relational structure. Finally, there has been a fair share of recent work for generation of special kinds of graphs, such as parsed trees of source code (Maddison & Tarlow, 2014) and SMILES representations for molecules (Olivecrona et al., 2017). Several deep generative models for graphs have recently been proposed. Amongst adversarial generation approaches, Wang et al. (2018) and Bojchevski et al. (2018) model local graph neighborhoods and random walks on graphs respectively. Li et al. (2018) and You et al. (2018) model graphs as sequences and generate graphs via autoregressive procedures. Adversarial and autoregressive approaches are successful at generating graphs, but do not directly allow for inferring latent variables via encoders. Latent variable generative models have also been proposed for generating small molecular graphs (Jin et al., 2018; Samanta et al., 2018; Simonovsky & Komodakis, 2018). These methods involve an expensive decoding procedure that limits scaling to large graphs. Finally, closest to our framework is the GAE/VGAE approach (Kipf & Welling, 2016) discussed in Section 4. Pan et al. (2018) extends this approach with an adversarial regularization framework but retain the inner product decoder. Our work proposes a novel multi-step decoding mechanism based on graph re\ufb01nement. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1705.08868v2",
+                    "title": "Flow-GAN: Combining Maximum Likelihood and Adversarial Learning in Generative Models",
+                    "abstract": "Adversarial learning of probabilistic models has recently emerged as a\npromising alternative to maximum likelihood. Implicit models such as generative\nadversarial networks (GAN) often generate better samples compared to explicit\nmodels trained by maximum likelihood. Yet, GANs sidestep the characterization\nof an explicit density which makes quantitative evaluations challenging. To\nbridge this gap, we propose Flow-GANs, a generative adversarial network for\nwhich we can perform exact likelihood evaluation, thus supporting both\nadversarial and maximum likelihood training. When trained adversarially,\nFlow-GANs generate high-quality samples but attain extremely poor\nlog-likelihood scores, inferior even to a mixture model memorizing the training\ndata; the opposite is true when trained by maximum likelihood. Results on MNIST\nand CIFAR-10 demonstrate that hybrid training can attain high held-out\nlikelihoods while retaining visual fidelity in the generated samples.",
+                    "authors": "Aditya Grover, Manik Dhar, Stefano Ermon",
+                    "published": "2017-05-24",
+                    "updated": "2018-01-03",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI",
+                        "cs.NE",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Highly expressive parametric models have enjoyed great success in supervised learning, where learning objectives and evaluation metrics are typically well-speci\ufb01ed and easy to compute. On the other hand, the learning objective for unsupervised settings is less clear. At a fundamental level, the idea is to learn a generative model that minimizes some notion of divergence with respect to the data distribution. Minimizing the Kullback-Liebler divergence between the data distribution and the model, for instance, is equivalent to performing maximum likelihood estimation (MLE) on the observed data. Maximum likelihood estimators are asymptotically statistically ef\ufb01cient, and serve as natural objectives for learning prescribed generative models (Mohamed and Lakshminarayanan 2016). In contrast, an alternate principle that has recently attracted much attention is based on adversarial learning, where the objective is to generate data indistinguishable from the training data. Adversarially learned models such as generative adversarial networks (GAN; (Goodfellow et al. 2014)) can sidestep specifying an explicit density for any data point and belong to the class of implicit generative models (Diggle and Gratton 1984). Copyright c \u20dd2018, Association for the Advancement of Arti\ufb01cial Intelligence (www.aaai.org). All rights reserved. The lack of characterization of an explicit density in GANs is however problematic for two reasons. Several application areas of deep generative models rely on density estimates; for instance, count based exploration strategies based on density estimation using generative models have recently achieved state-of-the-art performance on challenging reinforcement learning environments (Ostrovski et al. 2017). Secondly, it makes the quantitative evaluation of the generalization performance of such models challenging. The typical evaluation criteria based on ad-hoc sample quality metrics (Salimans et al. 2016; Che et al. 2017) do not address this issue since it is possible to generate good samples by memorizing the training data, or missing important modes of the distribution, or both (Theis, Oord, and Bethge 2016). Alternatively, density estimates based on approximate inference techniques such as annealed importance sampling (AIS; (Neal 2001; Wu et al. 2017)) and non-parameteric methods such as kernel density estimation (KDE; (Parzen 1962; Goodfellow et al. 2014)) are computationally slow and crucially rely on assumptions of a Gaussian observation model for the likelihood that could lead to misleading estimates as we shall demonstrate in this paper. To sidestep the above issues, we propose Flow-GANs, a generative adversarial network with a normalizing \ufb02ow generator. A Flow-GAN generator transforms a prior noise density into a model density through a sequence of invertible transformations. By using an invertible generator, FlowGANs allow us to tractably evaluate exact likelihoods using the change-of-variables formula and perform exact posterior inference over the latent variables while still permitting ef\ufb01cient ancestral sampling, desirable properties of any probabilistic model that a typical GAN would not provide. Using a Flow-GAN, we perform a principled quantitative comparison of maximum likelihood and adversarial learning on benchmark datasets viz. MNIST and CIFAR-10. While adversarial learning outperforms MLE on sample quality metrics as expected based on strong evidence in prior work, the log-likelihood estimates of adversarial learning are orders of magnitude worse than those of MLE. The difference is so stark that a simple Gaussian mixture model baseline outperforms adversarially learned models on both sample quality and held-out likelihoods. Our quantitative analysis reveals that the poor likelihoods of adversarial learning can be explained as a result of an ill-conditioned Jacobian maarXiv:1705.08868v2  [cs.LG]  3 Jan 2018 \ftrix for the generator function suggesting a mode collapse, rather than over\ufb01tting to the training dataset. To resolve the dichotomy of perceptually good-looking samples at the expense of held-out likelihoods in the case of adversarial learning (and vice versa in the case of MLE), we propose a hybrid objective that bridges implicit and prescribed learning by augmenting the adversarial training objective with an additional term corresponding to the loglikelihood of the observed data. While the hybrid objective achieves the intended effect of smoothly trading-off the two goals in the case of CIFAR-10, it has a regularizing effect on MNIST where it outperforms MLE and adversarial learning on both held-out likelihoods and sample quality metrics. Overall, this paper makes the following contributions: 1. We propose Flow-GANs, a generative adversarial network with an invertible generator that can perform ef\ufb01cient ancestral sampling and exact likelihood evaluation. 2. We propose a hybrid learning objective for Flow-GANs that attains good log-likelihoods and generates highquality samples on MNIST and CIFAR-10 datasets. 3. We demonstrate the limitations of AIS and KDE for loglikelihood evaluation and ranking of implicit models. 4. We analyze the singular value distribution for the Jacobian of the generator function to explain the low loglikelihoods observed due to adversarial learning. 2 Preliminaries We begin with a review of maximum likelihood estimation and adversarial learning in the context of generative models. For ease of presentation, all distributions are w.r.t. any arbitrary x \u2208Rd, unless otherwise speci\ufb01ed. We use upper-case to denote probability distributions and assume they all admit absolutely continuous densities (denoted by the corresponding lower-case notation) on a reference measure dx. Consider the following setting for learning generative models. Given some data X = {xi \u2208Rd}m i=1 sampled i.i.d. from an unknown probability density pdata, we are interested in learning a probability density p\u03b8 where \u03b8 denotes the parameters of a model. Given a parameteric family of models M, the typical approach to learn \u03b8 \u2208M is to minimize a notion of divergence between Pdata and P\u03b8. The choice of divergence and the optimization procedure dictate learning, leading to the following two objectives. 2.1 Maximum likelihood estimation In maximum likelihood estimation (MLE), we minimize the Kullback-Liebler (KL) divergence between the data distribution and the model distribution. Formally, the learning objective can be expressed as: min \u03b8\u2208M KL(Pdata, P\u03b8) = Ex\u223cPdata \u0014 log pdata(x) p\u03b8(x) \u0015 Since pdata is independent of \u03b8, the above optimization problem can be equivalently expressed as: max \u03b8\u2208M Ex\u223cPdata [log p\u03b8(x)] (1) Hence, evaluating the learning objective for MLE in Eq. (1) requires the ability to evaluate the model density p\u03b8(x). Models that provide an explicit characterization of the likelihood function are referred to as prescribed generative models (Mohamed and Lakshminarayanan 2016). 2.2 Adversarial learning A generative model can be learned to optimize divergence notions beyond the KL divergence. A large family of divergences can be conveniently expressed as: max \u03c6\u2208F Ex\u223cP\u03b8 [h\u03c6(x)] \u2212Ex\u223cPdata \u0002 h\u2032 \u03c6(x) \u0003 (2) where F denotes a set of parameters, h\u03c6 and h\u2032 \u03c6 are appropriate real-valued functions parameterized by \u03c6. Different choices of F, h\u03c6 and h\u2032 \u03c6 can lead to a variety of fdivergences such as Jenson-Shannon divergence and integral probability metrics such as the Wasserstein distance. For instance, the GAN objective proposed by Goodfellow et al. (2014) can also be cast in the form of Eq. (2) below: max \u03c6\u2208F Ex\u223cP\u03b8 [log (1 \u2212D\u03c6(x))] + Ex\u223cPdata [D\u03c6(x)] (3) where \u03c6 denotes the parameters of a neural network function D\u03c6. We refer the reader to (Nowozin, Cseke, and Tomioka 2016; Mescheder, Nowozin, and Geiger 2017b) for further details on other possible choices of divergences. Importantly, a Monte Carlo estimate of the objective in Eq. (2) requires only samples from the model. Hence, any model that allows tractable sampling can be used to evaluate the following minimax objective: min \u03b8\u2208M max \u03c6\u2208F Ex\u223cP\u03b8 [h\u03c6(x)] \u2212Ex\u223cPdata \u0002 h\u2032 \u03c6(x) \u0003 . (4) As a result, even differentiable implicit models which do not provide a characterization of the model likelihood1 but allow tractable sampling can be learned adversarially by optimizing minimax objectives of the form given in Eq. (4). 2.3 Adversarial learning of latent variable models From a statistical perspective, maximum likelihood estimators are statistically ef\ufb01cient asymptotically (under some conditions) and hence minimizing the KL divergence is a natural objective for many prescribed models (Huber 1967). However, not all models allow for a well-de\ufb01ned, tractable, and easy-to-optimize likelihood. For example, exact likelihood evaluation and sampling are tractable in directed, fully observed models such as Bayesian networks and autoregressive models (Larochelle and Murray 2011; Oord, Kalchbrenner, and Kavukcuoglu 2016). Hence, they are usually trained by maximum likelihood. Undirected models, on the other hand, provide only unnormalized likelihoods and are sampled from using expensive Markov chains. Hence, they are usually learned by approximating the likelihood using methods such as contrastive divergence (Carreira-Perpinan and Hinton 2005) and pseudolikelihood (Besag 1977). The likelihood is generally intractable to compute in latent variable models (even directed 1This could be either due to computational intractability in evaluating likelihoods or because the likelihood is ill-de\ufb01ned. \fones) as it requires marginalization. These models are typically learned by optimizing a stochastic lower bound to the log-likelihood using variational Bayes approaches (Kingma and Welling 2014). Directed latent variable models allow for ef\ufb01cient ancestral sampling and hence these models can also be trained using other divergences, e.g., adversarially (Mescheder, Nowozin, and Geiger 2017a; Mao et al. 2017; Song, Zhao, and Ermon 2017). A popular class of latent variable models learned adversarially consist of generative adversarial networks (GAN; (Goodfellow et al. 2014)). GANs comprise of a pair of generator and discriminator networks. The generator G\u03b8 : Rk \u2192Rd is a deterministic function differentiable with respect to the parameters \u03b8. The function takes as input a source of randomness z \u2208Rk sampled from a tractable prior density p(z) and transforms it to a sample G\u03b8(z) through a forward pass. Evaluating likelihoods assigned by a GAN is challenging because the model density p\u03b8 is speci\ufb01ed only implicitly using the prior density p(z) and the generator function G\u03b8. In fact, the likelihood for any data point is ill-de\ufb01ned (with respect to the Lesbegue measure over Rn) if the prior distribution over z is de\ufb01ned over a support smaller than the support of the data distribution. GANs are typically learned adversarially with the help of a discriminator network. The discriminator D\u03c6 : Rd \u2192R is another real-valued function that is differentiable with respect to a set of parameters \u03c6. Given the discriminator function, we can express the functions h and h\u2032 in Eq. (4) as compositions of D\u03c6 with divergence-speci\ufb01c functions. For instance, the Wasserstein GAN (WGAN; (Arjovsky, Chintala, and Bottou 2017)) optimizes the following objective: min \u03b8 max \u03c6\u2208F Ex\u223cPdata [D\u03c6(x)] \u2212Ez\u223cPz [D\u03c6(G\u03b8(z))] (5) where F is de\ufb01ned such that D\u03c6 is 1-Lipschitz. Empirically, GANs generate excellent samples of natural images (Radford, Metz, and Chintala 2015), audio signals (Pascual, Bonafonte, and Serr` a 2017), and of behaviors in imitation learning (Ho and Ermon 2016; Li, Song, and Ermon 2017). 3 Flow Generative Adversarial Networks As discussed above, generative adversarial networks can tractably generate high-quality samples but have intractable or ill-de\ufb01ned likelihoods. Monte Carlo techniques such as AIS and non-parameteric density estimation methods such as KDE get around this by assuming a Gaussian observation model p\u03b8(x|z) for the generator.2 This assumption alone is not suf\ufb01cient for quantitative evaluation since the marginal likelihood of the observed data, p\u03b8(x) = R p\u03b8(x, z)dz in this case would be intractable as it requires integrating over all the latent factors of variation. This would then require approximate inference (e.g., Monte Carlo or variational methods) which in itself is a computational challenge for highdimensional distributions. To circumvent these issues, we propose \ufb02ow generative adversarial networks (Flow-GAN). 2The true observation model for a GAN is a Dirac delta distribution, i.e., p\u03b8(x|z) is in\ufb01nite when x = G\u03b8(z) and zero otherwise. A Flow-GAN consists of a pair of generator-discriminator networks with the generator speci\ufb01ed as a normalizing \ufb02ow model (Dinh, Krueger, and Bengio 2014). A normalizing \ufb02ow model speci\ufb01es a parametric transformation from a prior density p(z) : Rd \u2192R+ 0 to another density over the same space, p\u03b8(x) : Rd \u2192R+ 0 where R+ 0 is the set of nonnegative reals. The generator transformation G\u03b8 : Rd \u2192 Rd is invertible, such that there exists an inverse function f\u03b8 = G\u22121 \u03b8 . Using the change-of-variables formula and letting z = f\u03b8(x), we have: p\u03b8(x) = p(z) \f \f \f \fdet\u2202f\u03b8(x) \u2202x \f \f \f \f (6) where \u2202f\u03b8(x) \u2202x denotes the Jacobian of f\u03b8 at x. The above formula can be applied recursively over compositions of many invertible transformations to produce a complex \ufb01nal density. Hence, we can evaluate and optimize for the loglikelihood assigned by the model to a data point as long as the prior density is tractable and the determinant of the Jacobian of f\u03b8 evaluated at x can be ef\ufb01ciently computed. Evaluating the likelihood assigned by a Flow-GAN model in Eq. (6) requires overcoming two major challenges. First, requiring the generator function G\u03b8 to be reversible imposes a constraint on the dimensionality of the latent variable z to match that of the data x. Thereafter, we require the transformations between the various layers of the generator to be invertible such that their overall composition results in an invertible G\u03b8. Secondly, the Jacobian of high-dimensional distributions can however be computationally expensive to compute. If the transformations are designed such that the Jacobian is an upper or lower triangular matrix, then the determinant can be easily evaluated as the product of its diagonal entries. We consider two such family of transformations. 1. Volume preserving transformations. Here, the Jacobian of the transformations have a unit determinant. For example, the NICE model consists of several layers performing a location transformation (Dinh, Krueger, and Bengio 2014). The top layer is a diagonal scaling matrix with nonzero log determinant. 2. Non-volume preserving transformations. The determinant of the Jacobian of the transformations is not necessarily unity. For example, in Real-NVP, layers performs both location and scale transformations (Dinh, Sohl-Dickstein, and Bengio 2017). For brevity, we direct the reader to Dinh, Krueger, and Bengio (2014) and Dinh, Sohl-Dickstein, and Bengio (2017) for the speci\ufb01cations of NICE and Real-NVP respectively. Crucially, both volume preserving and non-volume preserving transformations are invertible such that the determinant of the Jacobian can be computed tractably. 3.1 Learning objectives In a Flow-GAN, the likelihood is well-de\ufb01ned and computationally tractable for exact evaluation of even expressive volume preserving and non-volume preserving transformations. Hence, a Flow-GAN can be trained via maximum likelihood estimation using Eq. (1) in which case the discriminator is redundant. Additionally, we can perform ancestral \f(a) MLE (b) ADV (c) Hybrid Figure 1: Samples generated by Flow-GAN models with different objectives for MNIST (top) and CIFAR-10 (bottom). sampling just like a regular GAN whereby we sample a random vector z \u223cPz and transform it to a model generated sample via G\u03b8 = f \u22121 \u03b8 . This makes it possible to learn a Flow-GAN using an adversarial learning objective (for example, the WGAN objective in Eq. (5)). A natural question to ask is why should one use adversarial learning given that MLE is statistically ef\ufb01cient asymptotically (under some conditions). Besides dif\ufb01culties that could arise due to optimization (in both MLE and adversarial learning), the optimality of MLE holds only when there is no model misspeci\ufb01cation for the generator i.e., the true data distribution Pdata is a member of the parametric family of distributions under consideration (White 1982). This is generally not the case for high-dimensional distributions, and hence the choice of the learning objective becomes largely an empirical question. Unlike other models, a Flow-GAN allows both maximum likelihood and adversarial learning, and hence we can investigate this question experimentally. 3.2 Evaluation metrics and experimental setup Our criteria for evaluation is based on held-out loglikelihoods and sample quality metrics. We focus on natural images since they allow visual inspection as well as quanti\ufb01cation using recently proposed metrics. A \u201cgood\u201d generative model should generalize to images outside the training data and assign high log-likelihoods to held-out data. The Inception and MODE scores are standard quantitative measures of the quality of generated samples of natural images for labelled datasets (Salimans et al. 2016; Che et al. 2017). The Inception scores are computed as: exp (Ex\u2208P\u03b8[KL(p(y|x)\u2225p(y)]) where x is a sample generated by the model, p(y|x) is the softmax probability for the labels y assigned by a pretrained classi\ufb01er for x, and p(y) is the overall distribution of labels in the generated samples (as predicted by the pretrained classi\ufb01er). The intuition is that the conditional distribution p(y|x) should have low entropy for good looking images while the marginal distribution p(y) has high entropy to ensure sample diversity. Hence, a generative model can perform well on this metric if the KL divergence between the two distributions (and consequently, the Inception score for the generated samples) is large. The MODE score given below modi\ufb01es the Inception score to take into account the distribution of labels in the training data, p\u2217(y): exp (Ex\u2208P\u03b8[KL(p(y|x)\u2225p\u2217(y)] \u2212KL(p\u2217(y)\u2225p(y))) . We compare learning of Flow-GANs using MLE and adversarial learning (ADV) for the MNIST dataset of handwritten digits (LeCun, Cortes, and Burges 2010) and the CIFAR-10 dataset of natural images (Krizhevsky and Hinton 2009). The normalizing \ufb02ow generator architectures are chosen to be NICE (Dinh, Krueger, and Bengio 2014) and Real-NVP (Dinh, Sohl-Dickstein, and Bengio 2017) for MNIST and CIFAR-10 respectively. We \ufb01x the Wasserstein distance as the choice of the divergence being optimized by ADV (see Eq. (5)) with the Lipschitz constraint over the critic imposed by penalizing the norm of the gradient with respect to the input (Arjovsky, Chintala, and Bottou 2017; Gulrajani et al. 2017). The discriminator is based on the DCGAN architecture (Radford, Metz, \fand Chintala 2015). The above choices are among the current state-of-the-art in maximum likelihood estimation and adversarial learning and greatly stabilize GAN training. Further experimental setup details are provided in Appendix A. The code for reproducing the results is available at https://github.com/ermongroup/flow-gan. 0.0 0.5 1.0 1e5 generator iterations 3 2 1 0 1e3 NLL (in nats) 0.0 0.5 1.0 1e5 generator iterations 3 2 1 0 1e3 NLL (in nats) train nll val nll 0.00 0.25 0.50 0.75 1.00 1e5 generator iterations 0.0 0.2 0.4 0.6 0.8 1.0 1e5 NLL (in nats) 0 5 10 15 20 wgan loss 0.00 0.25 0.50 0.75 1.00 1e5 generator iterations 0.0 0.2 0.4 0.6 0.8 1.0 1e5 NLL (in nats) train nll val nll wgan loss 0 1 2 1e5 generator iterations 4 6 8 10 NLL (in bits per dim) 0 1 2 1e5 generator iterations 4 6 8 10 NLL (in bits per dim) train nll val nll (a) MLE 0 1 2 1e5 generator iterations 103 104 105 106 NLL (in bits per dim) 0 100 200 300 400 wgan loss 0 1 2 1e5 generator iterations 103 104 105 106 NLL (in bits per dim) train nll val nll wgan loss (b) ADV Figure 2: Learning curves for negative log-likelihood (NLL) evaluation on MNIST (top, in nats) and CIFAR (bottom, in bits/dim). Lower NLLs are better. 3.3 Evaluation results Log-likelihood. The log-likelihood learning curves for Flow-GAN models learned using MLE and ADV are shown in Figure 2a and Figure 2b respectively. Following convention, we report the negative log-likelihoods (NLL) in nats for MNIST and bits/dimension for CIFAR-10. MLE. In Figure 2a, we see that normalizing \ufb02ow models attain low validation NLLs (blue curves) after few gradient updates as expected because it is explicitly optimizing for the MLE objective in Eq. (1). Continued training however could lead to over\ufb01tting as the train NLLs (red curves) begin to diverge from the validation NLLs. ADV. Surprisingly, ADV models show a consistent increase in validation NLLs as training progresses as shown in Figure 2b (for CIFAR-10, the estimates are reported on a log scale!). Based on the learning curves, we can disregard over\ufb01tting as an explanation since the increase in NLLs is observed even on the training data. The training and validation NLLs closely track each other suggesting that ADV models are not simply memorizing the training data. Comparing the left vs. right panels in Figure 2, we see that the log-likelihoods attained by ADV are orders of magnitude worse than those attained by MLE after suf\ufb01cient training. Finally, we note that the WGAN loss (green curves) does not correlate well with NLL estimates. While the WGAN loss stabilizes after few iterations of training, the NLLs continue to increase. This observation is in contrast to prior work showing the loss to be strongly correlated with sample quality metrics (Arjovsky, Chintala, and Bottou 2017). Sample quality. Samples generated from MLE and ADVbased models with the best MODE/Inception are shown in Figure 1a and Figure 1b respectively. ADV models signi\ufb01cantly outperform MLE with respect to the \ufb01nal MODE/Inception scores achieved. Visual inspection of samples con\ufb01rms the observations made on the based of the sample quality metrics. Curves monitoring the sample quality metrics at every training iteration are given in Appendix B. 3.4 Gaussian mixture models The above experiments suggest that ADV can produce excellent samples but assigns low likelihoods to the observed data. However, a direct comparison of ADV with the loglikelihoods of MLE is unfair since the latter is explicitly optimizing for the desired objective. To highlight that generating good samples at the expense of low likelihoods is not a challenging goal, we propose a simple baseline. We compare the adversarially learned Flow-GAN models that achieves the highest MODE/Inception score with a Gaussian Mixture Model consisting of m isotropic Gaussians with equal weights centered at each of the m training points as the baseline Gaussian Mixture Model (GMM). The bandwidth hyperparameter, \u03c3, is the same for each of the mixture components and optimized for the lowest validation NLL by doing a line search in (0, 1]. We show results for CIFAR10 in Figure 3. Our observations below hold for MNIST as well; results deferred to Appendix C. We overload the y-axis in Figure 3 to report both NLLs and sample quality metrics. The horizontal maroon and cyan dashed lines denote the best attainable MODE/Inception scores and corresponding validation NLLs respectively attained by the adversarially learned Flow-GAN model. The GMM can clearly attain better sample quality metrics since it is explicitly over\ufb01tting to the training data for low values of the bandwidth parameter (any \u03c3 for which the red curve is above the maroon dashed line). Surprisingly, the simple GMM also outperforms the adversarially learned model with respect to NLLs attained for several values of the bandwidth parameter (any \u03c3 for which the blue curve is below the cyan dashed line). Bandwidth parameters for which GMM models outperform the adversarially learned model on both loglikelihoods and sample quality metrics are highlighted using the green shaded area. We show samples from the GMM in the appendix. Hence, a trivial baseline that is memorizing the training data can generate high quality samples and better held-out log-likelihoods, suggesting that the loglikelihoods attained by adversarial training are very poor. \f0.0 0.1 0.2 0.3 bandwidth  101 102 103 104 validation NLL (in bits per dim) 2 3 4 5 6 7 8 9 inception scores 0.0 0.1 0.2 0.3 bandwidth  101 102 103 104 validation NLL (in bits per dim) NLL of ADV model with best inception score Best Inception Score of ADV model Figure 3: Gaussian Mixture Models outperform adversarially learned models on both held-out log-likelihoods and sampling metrics on CIFAR-10 (green shaded region). 4 Hybrid learning of Flow-GANs In the previous section, we observed that adversarially learning Flow-GANs models attain poor held-out log-likelihoods. This makes it challenging to use such models for applications requiring density estimation. On the other hand, FlowGANs learned using MLE are \u201cmode covering\u201d but do not generate high quality samples. With a Flow-GAN, it is possible to trade-off the two goals by combining the learning objectives corresponding to both these inductive principles. Without loss of generality, let V (G\u03b8, D\u03c6) denote the minimax objective of any GAN model (such as WGAN). The hybrid objective of a Flow-GAN can be expressed as: min \u03b8 max \u03c6 V (G\u03b8, D\u03c6) \u2212\u03bbEx\u223cPdata [log p\u03b8(x)] (7) where \u03bb \u22650 is a hyperparameter for the algorithm. By varying \u03bb, we can interpolate between plain adversarial training (\u03bb = 0) and MLE (very high \u03bb). We summarize the results from MLE, ADV, and Hybrid for log-likelihood and sample quality evaluation in Table 1 and Table 2 for MNIST and CIFAR-10 respectively. The tables report the test log-likelihoods corresponding to the best validated MLE and ADV models and the highest MODE/Inception scores observed during training. The samples generated by models with the best MODE/Inception scores for each objective are shown in Figure 1c. While the results on CIFAR-10 are along expected lines, the hybrid objective interestingly outperforms MLE and ADV on both test log-likelihoods and sample quality metrics in the case of MNIST. One potential explanation for this is that the ADV objective can regularize MLE to generalize to the test set and in turn, the MLE objective can stabilize the optimization of the ADV objective. Hence, the hybrid objective in Eq. (7) can smoothly balance the two objectives using the tunable hyperparameter \u03bb, and in some cases such as MNIST, the performance on both tasks could improve as a result of the hybrid objective. Table 1: Best MODE scores and test negative log-likelihood estimates for Flow-GAN models on MNIST. Objective MODE Score Test NLL (in nats) MLE 7.42 \u22123334.56 ADV 9.24 \u22121604.09 Hybrid (\u03bb = 0.1) 9.37 \u22123342.95 Table 2: Best Inception scores and test negative loglikelihood estimates for Flow-GAN models on CIFAR-10. Objective Inception Score Test NLL (in bits/dim) MLE 2.92 3.54 ADV 5.76 8.53 Hybrid (\u03bb = 1) 3.90 4.21 5 Interpreting the results Our \ufb01ndings are in contrast with prior work which report much better log-likelihoods for adversarially learned models with a standard generator architecture based on annealed importance sampling (AIS; (Wu et al. 2017)) and kernel density estimation (KDE; (Goodfellow et al. 2014)). These methods rely on approximate inference techniques for log-likelihood evaluation and make assumptions about a Gaussian observation model which does not hold for GANs. Since Flow-GANs allow us to compute exact loglikelihoods, we can evaluate the quality of approximation made by AIS and KDE for density estimation of invertible generators. For a detailed description of the methods, we refer the reader to prior work (Neal 2001; Parzen 1962). We consider the MNIST dataset where these methods have been previously applied to by Wu et al. (2017) and Goodfellow et al. (2014) respectively. Since both AIS and KDE inherently rely on the samples generated, we evaluate these methods for the MLE, ADV, and Hybrid FlowGAN model checkpoints corresponding to the best MODE scores observed during training. In Table 3, we observe that both AIS and KDE produce estimates of log-likelihood that are far from the ground truth, accessible through the exact Flow-GAN log-likelihoods. Even worse, the ranking of log-likelihood estimates for AIS (ADV>Hybrid>MLE) and KDE (Hybrid>MLE>ADV) do not obey the relative rankings of the Flow-GAN estimates (MLE>Hybrid>ADV). 5.1 Explaining log-likelihood trends In order to explain the variation in log-likelihoods attained by various Flow-GAN learning objectives, we investigate the distribution of the magnitudes of singular values for the Jacobian matrix of several generator functions, G\u03b8 for MNIST in Figure 4 evaluated at 64 noise vectors z randomly sampled from the prior density p(z). The x-axis of the \ufb01gure shows the singular value magnitudes on a log scale and for each singular value s, we show the corresponding cumulative distribution function value on the y-axis which signi\ufb01es the fraction of singular values less than s. The results on CIFAR-10 in Appendix D show a similar trend. The Jacobian is a good \ufb01rst-order approximation of the generator function locally. In Figure 4, we observe that the \fTable 3: Comparison of inference techniques for negative log-likelihood estimation of Flow-GAN models on MNIST. Objective Flow-GAN NLL AIS KDE MLE -3287.69 -2584.40 -167.10 ADV 26350.30 -2916.10 -3.03 Hybrid -3121.53 -2703.03 -205.69 singular value distribution for the Jacobian of an invertible generator learned using MLE (orange curves) is concentrated in a narrow range, and hence the Jacobian matrix is well-conditioned and easy to invert. In the case of invertible generators learned using ADV with Wasserstein distance (green curves) however, the spread of singular values is very wide, and hence the Jacobian matrix is ill-conditioned. The average log determinant of the Jacobian matrices for MLE, ADV, and Hybrid models are \u22124170.34, \u221215588.34, and \u22125184.40 respectively which translates to the trend ADV<Hybrid<MLE. This indicates that the ADV models are trying to squish a sphere of unit volume centered at a latent vector z to a very small volume in the observed space x. Tiny perturbations of training as well as held-out datapoints can hence manifest as poor log-likelihoods. In spite of not being limited in the representational capacity to cover the entire space of the data distribution (the dimensions of z (i.e., k) and x (i.e., d) match for invertible generators), ADV prefers to learn a distribution over a smaller support. The Hybrid learning objective (blue curves), however, is able to correct for this behavior, and the distribution of singular value magnitudes matches closely to that of MLE. We also considered variations involving the standard DCGAN architectures with k = d minimizing the Wasserstein distance (red curves) and Jenson-Shannon divergence (purple curves). The relative shift in distribution of singular value magnitudes to lower values is apparent even in these cases. 6 Discussion Any model which allows for ef\ufb01cient likelihood evaluation and sampling can be trained using maximum likelihood and adversarial learning. This line of reasoning has been explored to some extent in prior work that combine the objectives of prescribed latent variable models such as VAEs (maximizing an evidence lower bound on the data) with adversarial learning (Larsen et al. 2015; Mescheder, Nowozin, and Geiger 2017a; Srivastava et al. 2017). However, the bene\ufb01ts of such procedures do not come for \u201cfree\u201d since we still need some form of approximate inference to get a handle on the log-likelihoods. This could be expensive, for instance combining a VAE with a GAN introduces an additional inference network that increases the overall model complexity. Our approach sidesteps the additional complexity due to approximate inference by considering a normalizing \ufb02ow model. The trade-off made by a normalizing \ufb02ow model is that the generator function needs to be invertible while other generative models such as VAEs have no such requirement. On the positive side, we can tractably evaluate exact log-likelihoods assigned by the model for any data point. Normalizing \ufb02ow models have been previously used 10 45 10 37 10 29 10 21 10 13 10 5 magnitude of singular value, s 0.0 0.2 0.4 0.6 0.8 1.0 P(S<s) Hybrid (NICE) MLE (NICE) ADV (NICE, WD) ADV (DCGAN-784,WD) ADV (DCGAN-784,JSD) Figure 4: CDF of the singular values magnitudes for the Jacobian of the generator functions trained on MNIST. in the context of maximum likelihood estimation of fully observed and latent variable models (Dinh, Krueger, and Bengio 2014; Rezende and Mohamed 2015; Kingma, Salimans, and Welling 2016; Dinh, Sohl-Dickstein, and Bengio 2017). The low dimensional support of the distributions learned by adversarial learning often manifests as lack of sample diversity and is referred to as mode collapse. In prior work, mode collapse is detected based on visual inspection or heuristic techniques (Goodfellow 2016; Arora and Zhang 2017). Techniques for avoiding mode collapse explicitly focus on stabilizing GAN training such as (Metz et al. 2016; Che et al. 2017; Mescheder, Nowozin, and Geiger 2017b) rather than quantitative methods based on likelihoods. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1702.08484v2",
+                    "title": "Boosted Generative Models",
+                    "abstract": "We propose a novel approach for using unsupervised boosting to create an\nensemble of generative models, where models are trained in sequence to correct\nearlier mistakes. Our meta-algorithmic framework can leverage any existing base\nlearner that permits likelihood evaluation, including recent deep expressive\nmodels. Further, our approach allows the ensemble to include discriminative\nmodels trained to distinguish real data from model-generated data. We show\ntheoretical conditions under which incorporating a new model in the ensemble\nwill improve the fit and empirically demonstrate the effectiveness of our\nblack-box boosting algorithms on density estimation, classification, and sample\ngeneration on benchmark datasets for a wide range of generative models.",
+                    "authors": "Aditya Grover, Stefano Ermon",
+                    "published": "2017-02-27",
+                    "updated": "2017-12-22",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction A variety of deep generative models have shown promising results on tasks spanning computer vision, speech recognition, natural language processing, and imitation learning (Poon and Domingos 2011; Oord, Kalchbrenner, and Kavukcuoglu 2016; Kingma and Welling 2014; Goodfellow et al. 2014; Zhao, Song, and Ermon 2017; Li, Song, and Ermon 2017). These parametric models differ from each other in their ability to perform various forms of tractable inference, learning algorithms, and objectives. Despite signi\ufb01cant progress, existing generative models cannot \ufb01t complex distributions with a suf\ufb01ciently high degree of accuracy, limiting their applicability and leaving room for improvement. In this paper, we propose a technique for ensembling (imperfect) generative models to improve their overall performance. Our meta-algorithm is inspired by boosting, a technique used in supervised learning to combine weak classi\ufb01ers (e.g., decision stumps or trees), which individually might not perform well on a given classi\ufb01cation task, into a more powerful ensemble. The boosting algorithm will attempt to learn a classi\ufb01er to correct for the mistakes made by reweighting the original dataset, and repeat this procedure recursively. Under some conditions on the weak classi\ufb01ers\u2019 effectiveness, this procedure can drive the (training) error to zero (Freund, Schapire, and Abe 1999). Boosting can also be thought as a feature learning algorithm, where at each round a new feature is learned by training a classi\ufb01er on a Copyright c \u20dd2018, Association for the Advancement of Arti\ufb01cial Intelligence (www.aaai.org). All rights reserved. reweighted version of the original dataset. In practice, algorithms based on boosting perform extremely well in machine learning competitions (Caruana and Niculescu-Mizil 2006). We show that a similar procedure can be applied to generative models. Given an initial generative model that provides an imperfect \ufb01t to the data distribution, we construct a second model to correct for the error, and repeat recursively. The second model is also a generative one, which is trained on a reweighted version of the original training set. Our meta-algorithm is general and can construct ensembles of any existing generative model that permits (approximate) likelihood evaluation such as fully-observed belief networks, sum-product networks, and variational autoencoders. Interestingly, our method can also leverage powerful discriminative models. Speci\ufb01cally, we train a binary classi\ufb01er to distinguish true data samples from \u201cfake\u201d ones generated by the current model and provide a principled way to include this discriminator in the ensemble. A prior attempt at boosting density estimation proposed a sum-of-experts formulation (Rosset and Segal 2002). This approach is similar to supervised boosting where at every round of boosting we derive a reweighted additive estimate of the boosted model density. In contrast, our proposed framework uses multiplicative boosting which multiplies the ensemble model densities and can be interpreted as a product-of-experts formulation. We provide a holistic theoretical and algorithmic framework for multiplicative boosting contrasting with competing additive approaches. Unlike prior use cases of product-of-experts formulations, our approach is black-box, and we empirically test the proposed algorithms on several generative models from simple ones such as mixture models to expressive parameteric models such as sum-product networks and variational autoencoders. Overall, this paper makes the following contributions: 1. We provide theoretical conditions for additive and multiplicative boosting under which incorporating a new model is guaranteed to improve the ensemble \ufb01t. 2. We design and analyze a \ufb02exible meta-algorithmic boosting framework for including both generative and discriminative models in the ensemble. 3. We demonstrate the empirical effectiveness of our algorithms for density estimation, generative classi\ufb01cation, and sample generation on several benchmark datasets. arXiv:1702.08484v2  [cs.LG]  22 Dec 2017 \f2 Unsupervised boosting Supervised boosting provides an algorithmic formalization of the hypothesis that a sequence of weak learners can create a single strong learner (Schapire and Freund 2012). Here, we propose a framework that extends boosting to unsupervised settings for learning generative models. For ease of presentation, all distributions are with respect to any arbitrary x \u2208Rd, unless otherwise speci\ufb01ed. We use upper-case symbols to denote probability distributions and assume they all admit absolutely continuous densities (denoted by the corresponding lower-case notation) on a reference measure dx. Our analysis naturally extends to discrete distributions, which we skip for brevity. Formally, we consider the following maximum likelihood estimation (MLE) setting. Given some data points X = {xi \u2208Rd}m i=1 sampled i.i.d. from an unknown distribution P, we provide a model class Q parameterizing the distributions that can be represented by the generative model and minimize the Kullback-Liebler (KL) divergence with respect to the true distribution: min Q\u2208Q DKL(P\u2225Q). (1) In practice, we only observe samples from P and hence, maximize the log-likelihood of the observed data X. Selecting the model class for maximum likelihood learning is nontrivial; MLE w.r.t. a small class can be far from P, whereas a large class poses the risk of over\ufb01tting in the absence of suf\ufb01cient data, or even under\ufb01tting due to dif\ufb01culty in optimizing non-convex objectives that frequently arise due to the use of latent variable models, neural networks, etc. The boosting intuition is to greedily increase model capacity by learning a sequence of weak intermediate models {ht \u2208Ht}T t=0 that can correct for mistakes made by previous models in the ensemble. Here, Ht is a prede\ufb01ned model class (such as Q) for ht. We defer the algorithms pertaining to the learning of such intermediate models to the next section, and \ufb01rst discuss two mechanisms for deriving the \ufb01nal estimate qT from the individual density estimates at each round, {ht}T t=0. 2.1 Additive boosting In additive boosting, the \ufb01nal density estimate is an arithmetic average of the intermediate models: qT = T X t=0 \u03b1t \u00b7 ht where 0 \u2264\u03b1t \u22641 denote the weights assigned to the intermediate models. The weights are re-normalized at every round to sum to 1 which gives us a valid probability density estimate. Starting with a base model h0, we can express the density estimate after a round of boosting recursively as: qt = (1 \u2212\u02c6 \u03b1t) \u00b7 qt\u22121 + \u02c6 \u03b1t \u00b7 ht where \u02c6 \u03b1t denotes the normalized weight for ht at round t. We now derive conditions on the intermediate models that guarantee \u201cprogress\u201d in every round of boosting. Theorem 1. Let \u03b4t KL(ht, \u02c6 \u03b1t) = DKL(P\u2225Qt\u22121) \u2212DKL(P\u2225Qt) denote the reduction in KL-divergence at the tth round of additive boosting. The following conditions hold: 1. Suf\ufb01cient: If EP h log ht qt\u22121 i \u22650, then \u03b4t KL(ht, \u02c6 \u03b1t) \u22650 for all \u02c6 \u03b1t \u2208[0, 1]. 2. Necessary: If \u2203\u02c6 \u03b1t \u2208(0, 1] such that \u03b4t KL(ht, \u02c6 \u03b1t) \u22650, then EP h ht qt\u22121 i \u22651. Proof. In Appendix A.1. The suf\ufb01cient and necessary conditions require that the expected log-likelihood and likelihood respectively of the current intermediate model, ht are better-or-equal than those of the combined previous model, qt\u22121 under the true distribution when compared using density ratios. Next, we consider an alternative formulation of multiplicative boosting for improving the model \ufb01t to an arbitrary data distribution. 2.2 Multiplicative boosting In multiplicative boosting, we factorize the \ufb01nal density estimate as a geometric average of T + 1 intermediate models {ht}T t=0, each assigned an exponentiated weight \u03b1t: qT = QT t=0 h\u03b1t t ZT where the partition function ZT = R QT t=0 h\u03b1t t dx. Recursively, we can specify the density estimate as: \u02dc qt = h\u03b1t t \u00b7 \u02dc qt\u22121 (2) where \u02dc qt is the unnormalized estimate at round t. The base model h0 is learned using MLE. The conditions on the intermediate models for reducing KL-divergence at every round are stated below. Theorem 2. Let \u03b4t KL(ht, \u03b1t) = DKL(P\u2225Qt\u22121) \u2212DKL(P\u2225Qt) denote the reduction in KL-divergence at the tth round of multiplicative boosting. The following conditions hold: 1. Suf\ufb01cient: If EP [log ht] \u2265 log EQt\u22121[ht], then \u03b4t KL(ht, \u03b1t) \u22650 for all \u03b1t \u2208[0, 1]. 2. Necessary: If \u2203\u03b1t \u2208(0, 1] such that \u03b4t KL(ht, \u03b1t) \u22650, then EP [log ht] \u2265EQt\u22121[log ht]. Proof. In Appendix A.2. In contrast to additive boosting, the conditions above compare expectations under the true distribution with expectations under the model distribution in the previous round, Qt\u22121. The equality in the conditions holds for \u03b1t = 0, which corresponds to the trivial case where the current intermediate model is ignored in Eq. (2). For other valid \u03b1t, the non-degenerate version of the suf\ufb01cient inequality guarantees progress towards the true data distribution. Note that the intermediate models increase the overall capacity of the ensemble at every round. As we shall demonstrate later, we \ufb01nd models \ufb01t using multiplicative boosting to outperform their additive counterparts empirically suggesting the conditions in Theorem 2 are easier to ful\ufb01ll in practice. \fAlgorithm 1 GenBGM(X = {xi}m i=1, T rounds) Initialize d0(xi) = 1/m for all i = 1, 2, . . . , m. Obtain base generative model h0. Set (unnormalized) density estimate \u02dc q0 = h0 for t = 1, 2, . . . , T do Choose \u03b2t and update dt using Eq. (4). Train generative model ht to maximize Eq. (3). Choose \u03b1t. Set density estimate \u02dc qt = h\u03b1t t \u00b7 \u02dc qt\u22121. end for Estimate ZT = R \u02dc qT dx. return qT = \u02dc qT /ZT . From the necessary condition, we see that a \u201cgood\u201d intermediate model ht assigns a better-or-equal log-likelihood under the true distribution as opposed to the model distribution, Qt\u22121. This condition suggests two learning algorithms for intermediate models which we discuss next. 3 Boosted generative models In this section, we design and analyze meta-algorithms for multiplicative boosting of generative models. Given any base model which permits (approximate) likelihood evaluation, we provide a mechanism for boosting this model using an ensemble of generative and/or discriminative models. 3.1 Generative boosting Supervised boosting algorithms such as AdaBoost typically involve a reweighting procedure for training weak learners (Freund and Schapire 1995). We can similarly train an ensemble of generative models for unsupervised boosting, where every subsequent model performs MLE w.r.t a reweighted data distribution Dt: max ht EDt[log ht] (3) where dt \u221d \u0012 p qt\u22121 \u0013\u03b2t (4) and \u03b2t \u2208[0, 1] is the reweighting coef\ufb01cient at round t. Note that these coef\ufb01cients are in general different from the model weights \u03b1t that appear in Eq. (2). Proposition 1. If we can maximize the objective in Eq. (3) optimally, then \u03b4t KL(ht, \u03b1t) \u22650 for any \u03b2t \u2208[0, 1] with the equality holding for \u03b2t = 0. Proof. In Appendix A.3. While the objective in Eq. (3) can be hard to optimize in practice, the target distribution becomes easier to approximate as we reduce the reweighting coef\ufb01cient. For the extreme case of \u03b2t = 0, the reweighted data distribution is simply uniform. There is no free lunch however, since a low \u03b2t results in a slower reduction in KL-divergence leading to a computational-statistical trade-off. The pseudocode for the corresponding boosting metaalgorithm, referred to as GenBGM, is given in Algorithm 1. Algorithm 2 DiscBGM(X = {xi}m i=1, T rounds, f-div) Initialize d0(xi) = 1/m for all i = 1, 2, . . . , m. Obtain base generative model h0. Set (unnormalized) density estimate \u02dc q0 = h0 for t = 1, 2, . . . , T do Generate negative samples from qt\u22121 Optimize rt to maximize RHS in Eq. (5). Set ht = [f \u2032]\u22121 (rt). Choose \u03b1t. Set density estimate \u02dc qt = h\u03b1t t \u00b7 \u02dc qt\u22121. end for Estimate ZT = R \u02dc qT dx. return qT = \u02dc qT /ZT . In practice, we only observe samples from the true data distribution, and hence, approximate p based on the empirical data distribution which is de\ufb01ned to be uniform over the dataset X. At every subsequent round, GenBGM learns an intermediate model that maximizes the log-likelihood of data sampled from a reweighted data distribution. 3.2 Discriminative boosting A base generative model can be boosted using a discriminative approach as well. Here, the intermediate model is speci\ufb01ed as the density ratio obtained from a binary classi\ufb01er. Consider the following setup: we observe an equal number of samples drawn i.i.d. from the true data distribution (w.l.o.g. assigned the label y = +1) and the model distribution in the previous round Qt\u22121 (label y = 0). De\ufb01nition 1. Let f : R+ \u2192R be any convex, lower semicontinuous function satisfying f(1) = 0. The f-divergence between P and Q is de\ufb01ned as, Df(P\u2225Q) = R q\u00b7f (p/q) dx. Notable examples include the Kullback-Liebler (KL) divergence, Hellinger distance, and the Jenson-Shannon (JS) divergence among many others. The binary classi\ufb01er in discriminative boosting maximizes a variational lower bound on any f-divergence at round t: Df (P\u2225Qt\u22121) \u2265sup rt\u2208Rt \u0000EP [rt] \u2212EQt\u22121[f \u22c6(rt)] \u0001 . (5) where f \u22c6denotes the Fenchel conjugate of f and rt : Rd \u2192 domf \u22c6parameterizes the classi\ufb01er. Under mild conditions on f (Nguyen, Wainwright, and Jordan 2010), the lower bound in Eq. (5) is tight if r\u22c6 t = f \u2032 (p/qt\u22121). Hence, a solution to Eq. (5) can be used to estimate density ratios. The density ratios naturally \ufb01t into the multiplicative boosting framework and provide a justi\ufb01cation for the use of objectives of the form Eq. (5) for learning intermediate models as formalized in the proposition below. Proposition 2. For any given f-divergence, let r\u22c6 t denote the optimal solution to Eq. (5) in the tth round of boosting. Then, the model density at the end of the boosting round matches the true density if we set \u03b1t = 1 and ht = [f \u2032]\u22121 (r\u22c6 t ) where [f \u2032]\u22121 denotes the inverse of the derivative of f. Proof. In Appendix A.4. \f(a) Target (b) Base model Figure 1: The mixture of Gaussians setup showing (a) true density and (b) base (misspeci\ufb01ed) model. The pseudocode for the corresponding meta-algorithm, DiscBGM is given in Algorithm 2. At every round, we train a binary classi\ufb01er to optimize the objective in Eq. (5) for a chosen f-divergence. As a special case, the negative of the cross-entropy loss commonly used for binary classi\ufb01cation is also a lower bound on an f-divergence. While Algorithm 2 is applicable for any f-divergence, we will focus on crossentropy henceforth to streamline the discussion. Corollary 1. Consider the (negative) cross-entropy objective maximized by a binary classi\ufb01er: sup ct\u2208Ct EP [log ct] + EQt\u22121[log(1 \u2212ct)]. (6) If a binary classi\ufb01er ct trained to optimize Eq. (6) is Bayes optimal, then the model density after round t matches the true density if we set \u03b1t = 1 and ht = ct/1\u2212ct. Proof. In Appendix A.5. In practice, a classi\ufb01er with limited capacity trained on a \ufb01nite dataset will not generally be Bayes optimal. The above corollary, however, suggests that a good classi\ufb01er can provide a \u2018direction of improvement\u2019, in a similar spirit to gradient boosting for supervised learning (Freund and Schapire 1995). Additionally, if the intermediate model distribution ht obtained using the above corollary satis\ufb01es the conditions in Theorem 2, it is guaranteed to improve the \ufb01t. The weights \u03b1t \u2208[0, 1] can be interpreted as our con\ufb01dence in the classi\ufb01cation estimates, akin to the step size used in gradient descent. While in practice we heuristically assign weights to the intermediate models, the greedy optimum value of these weights at every round is a critical point for \u03b4t KL (de\ufb01ned in Theorem 2). For example, in the extreme case where ct is uninformative, i.e., ct \u22610.5, then \u03b4t KL(ht, \u03b1t) = 0 for all \u03b1t \u2208[0, 1]. If ct is Bayes optimal, then \u03b4t KL attains a maxima when \u03b1t = 1 (Corollary 1). 3.3 Hybrid boosting Intermediate models need not be exclusively generators or discriminators; we can design a boosting ensemble with any combination of generators and discriminators. If an intermediate model is chosen to be a generator, we learn a generative model using MLE after appropriately reweighting the data points. If a discriminator is used to implicitly specify an intermediate model, we set up a binary classi\ufb01cation problem. Table 1: Average test NLL for mixture of Gaussians. Model NLL (in nats, with std. error) Base model 4.69 \u00b1 0.01 Add model 4.64 \u00b1 0.02 GenBGM 4.58 \u00b1 0.10 DiscBGM-NCE 4.42 \u00b1 0.01 DiscBGM-HD 4.35 \u00b1 0.01 3.4 Regularization In practice, we want boosted generative models (BGM) to generalize to data outside the training set X. Regularization in BGMs is imposed primarily in two ways. First, every intermediate model can be independently regularized by incorporating explicit terms in the learning objective, early stopping based on validation error, heuristics such as dropout, etc. Moreover, restricting the number of rounds of boosting is another effective mechanism for regularizing BGMs. Fewer rounds of boosting are required if the intermediate models are suf\ufb01ciently expressive. 4 Empirical evaluation Our experiments are designed to demonstrate the superiority of the proposed boosting meta-algorithms on a wide variety of generative models and tasks. A reference implementation of the boosting meta-algorithms is available at https://github.com/ermongroup/bgm. Additional implementation details for the experiments below are given in Appendix B. 4.1 Multiplicative vs. additive boosting A common pitfall with learning parameteric generative models is model misspeci\ufb01cation with respect to the true underlying data distribution. For a quantitative and qualitative understanding of the behavior of additive and multiplicative boosting, we begin by considering a synthetic setting for density estimation on a mixture of Gaussians. Density estimation on synthetic dataset. The true data distribution is a equi-weighted mixture of four Gaussians centered symmetrically around the origin, each having an identity covariance matrix. The contours of the underlying density are shown in Figure 1a. We observe 1, 000 training samples drawn independently from the data distribution (shown as black dots in Figure 2), and the task is to learn this distribution. The test set contains 1, 000 samples from the same distribution. We repeat the process 10 times for statistical signi\ufb01cance. As a base (misspeci\ufb01ed) model, we \ufb01t a mixture of two Gaussians to the data; the contours for an example instance are shown in Figure 1b. We compare multiplicative and additive boosting, each run for T = 2 rounds. For additive boosting (Add), we extend the algorithm proposed by Rosset and Segal (2002) setting \u02c6 \u03b10 to unity and doing a line search over \u02c6 \u03b11, \u02c6 \u03b12 \u2208[0, 1]. For Add and GenBGM, the intermediate \f(a) Add model (1) (b) Add model (2) (c) GenBGM (1) (d) GenBGM (2) (e) DiscBGM-NCE (1) (f) DiscBGM-NCE (2) (g) DiscBGM-HD (1) (h) DiscBGM-HD (2) Figure 2: Multiplicative boosting algorithms such as GenBGM (c-d) and DiscBGM with negative cross-entropy (e-f) and Hellinger distance (g-h) outperform additive boosting (a-b) in correcting for model misspeci\ufb01cation. Numbers in parenthesis indicate boosting round t. models are mixtures of two Gaussians as well. The classi\ufb01ers for DiscBGM are multi-layer perceptrons with two hidden layers of 100 units each and ReLU activations, trained to maximize f-divergences corresponding to the negative cross-entropy (NCE) and Hellinger distance (HD) using the Adam optimizer (Kingma and Welling 2014). The test negative log-likelihood (NLL) estimates are listed in Table 1. Qualitatively, the contour plots for the estimated densities after every boosting round on a sample instance are shown in Figure 2. Multiplicative boosting algorithms outperform additive boosting in correcting for model misspeci\ufb01cation. GenBGM initially leans towards maximizing coverage, whereas both versions of DiscBGM are relatively more conservative in assigning high densities to data points away from the modes. Heuristic model weighting strategies. The multiplicative boosting algorithms require as hyperparameters the number of rounds of boosting and weights assigned to the intermediate models. For any practical setting, these hyperparameters are speci\ufb01c to the dataset and task under consideration and should be set based on cross-validation. While automatically setting model weights is an important direction for future work, we propose some heuristic weighting strategies. Speci\ufb01cally, the unity heuristic assigns a weight of 1 to every model in the ensemble, the uniform heuristic assigns a weight of 1/(T + 1) to every model, and the decay heuristic assigns as a weight of 1/2t to the tth model in the ensemble. In Figure 3, we observe that the performance of the algorithms is sensitive to the weighting strategies. In particular, DiscBGM produces worse estimates as T increases for the \u201cuniform\u201d (red) strategy. The performance of GenBGM also degrades slightly with increasing T for the \u201cunity\u201d (green) strategy. Notably, the \u201cdecay\u201d (cyan) strategy achieves stable performance for both the algorithms. Intuitively, this 0 1 2 3 4 5 T 4.2 4.4 4.6 4.8 5.0 NLL (in nats) Base Unity Uniform Decay (a) GenBGM 0 1 2 3 4 5 T 4.2 4.4 4.6 4.8 5.0 NLL (in nats) (b) DiscBGM-NCE Figure 3: Train (dashed curves) and test (bold curves) NLL (in nats) for weighting heuristics on mixture of Gaussians. T is the number of rounds of boosting. The base model is shown as a black cross at T = 0. heuristic follows the rationale of reducing the step size in gradient based stochastic optimization algorithms, and we expect this strategy to work better even in other settings. However, this strategy could potentially result in slower convergence as opposed to the unity strategy. Density estimation on benchmark datasets. We now evaluate the performance of additive and multiplicative boosting for density estimation on real-world benchmark datasets (Van Haaren and Davis 2012). We consider two generative model families: mixture of Bernoullis (MoB) and sum-product networks (Poon and Domingos 2011). While our results for multiplicative boosting with sum-product networks (SPN) are competitive with the state-of-the-art, the goal of these experiments is to perform a robust comparison of boosting algorithms as well as demonstrate their applicability to various model families. We set T = 2 rounds for additive boosting and GenBGM. \fTable 2: Experimental results for density estimation. Negative log-likelihoods reported in nats. Lower is better with best performing models in bold. Overall, multiplicative boosting outperforms additive boosting and baseline models speci\ufb01ed as Mixture of Bernoullis (MoB, middle columns) and Sum Product Networks (SPN, right columns). Dataset #vars MoB Base Add GenBGM DiscBGM SPN Base Add GenBGM DiscBGM Accidents 111 42.23 43.19 41.43 34.51 31.08 29.92 29.55 28.09 Retail 135 11.27 12.24 11.20 10.91 14.94 11.27 11.21 10.88 Pumsbstar 163 55.67 55.91 50.66 34.93 26.70 25.00 25.00 23.69 DNA 180 99.42 100.37 99.23 98.45 92.60 86.93 87.79 86.63 Kosarek 190 11.72 12.57 12.41 11.13 12.71 10.97 10.73 10.67 Ad 1556 63.13 63.73 63.19 54.79 19.19 18.12 18.14 17.82 Table 3: Experimental results for classi\ufb01cation. Prediction accuracy for predicting one variable given the rest. Higher is better with best performing models in bold. Multiplicative boosting again outperforms additive boosting and baseline models speci\ufb01ed as Mixture of Bernoullis (MoB, middle columns) and Sum Product Networks (SPN, right columns). Dataset #test MoB Base Add GenBGM DiscBGM SPN Base Add GenBGM DiscBGM Accidents 283,161 0.8395 0.8393 0.8473 0.9043 0.9258 0.9266 0.9298 0.9416 Retail 595,080 0.9776 0.9776 0.9776 0.9792 0.9780 0.9790 0.9789 0.9791 Pumsb-star 399,676 0.8461 0.8501 0.8819 0.9267 0.9599 0.9610 0.9611 0.9636 DNA 213,480 0.7517 0.7515 0.7531 0.7526 0.7799 0.7817 0.7828 0.7811 Kosarek 1,268,250 0.9817 0.9816 0.9818 0.9831 0.9824 0.9838 0.9838 0.9838 Ad 763,996 0.9922 0.9923 0.9818 0.9927 0.9982 0.9981 0.9982 0.9982 Since DiscBGM requires samples from the model density at every round, we set T = 1 to ensure computational fairness such that the samples can be obtained ef\ufb01ciently from the base model sidestepping running expensive Markov chains. Model weights are chosen based on cross-validation. The results on density estimation are reported in Table 2. Since multiplicative boosting estimates are unnormalized, we use importance sampling to estimate the partition function. When the base model is MoB, the Add model underperforms and is often worse than even the baseline model for the best performing validated non-zero model weights. GenBGM consistently outperforms Add and improves over the baseline model in a most cases (4/6 datasets). DiscBGM performs the best and convincingly outperforms the baseline, Add, and GenBGM on all datasets. For results on SPNs, the boosted models all outperform the baseline. GenBGM again edges out Add models (4/6 datasets), whereas DiscBGM models outperform all other models on all datasets. These results demonstrate the usefulness of boosted expressive model families, especially the DiscBGM approach, which performs the best, while GenBGM is preferable to Add. 4.2 Applications of generative models Classi\ufb01cation. Here, we evaluate the performance of boosting algorithms for classi\ufb01cation. Since the datasets above do not have any explicit labels, we choose one of the dimensions to be the label (say y). Letting x\u00af y denote the remaining dimensions, we can obtain a prediction for y as, p(y = 1|x\u00af y) = p(y = 1, x\u00af y) p(y = 1, x\u00af y) + p(y = 0, x\u00af y) which is ef\ufb01cient to compute even for unnormalized models. We repeat the above procedure for all the variables predicting one variable at a time using the values assigned to the remaining variables. The results are reported in Table 3. When the base model is a MoB, we observe that the Add approach could often be worse than the base model whereas GenBGM performs slightly better than the baseline (4/6 datasets). The DiscBGM approach consistently performs well, and is only outperformed by GenBGM for two datasets for MoB. When SPNs are used instead, both Add and GenBGM improve upon the baseline model while DiscBGM again is the best performing model on all but one dataset. Sample generation. We compare boosting algorithms based on their ability to generate image samples for the binarized MNIST dataset of handwritten digits (LeCun, Cortes, and Burges 2010). We use variational autoencoders (VAE) as the base model (Kingma and Welling 2014). While any suf\ufb01ciently expressive VAE can generate impressive examples, we design the experiment to evaluate the model complexity approximated as the number of learnable parameters. Ancestral samples obtained by the baseline VAE model are shown in Figure 4a. We use the evidence lower bound (ELBO) as a proxy for approximately evaluating the marginal log-likelihood during learning. The conventional approach to improving the performance of a latent variable model is to increase its representational capacity by adding hidden layers (Base + depth) or increasing the number of hidden units in the existing layers (Base + width). These lead to a marginal improvement in sample quality as seen in Figure 4b and Figure 4c. In contrast, boosting makes steady improvements in sample quality. We start with a VAE with much fewer parameters and generate samples using a hybrid boosting GenDiscBGM sequence VAE\u2192CNN\u2192VAE (Figure 4d) . The discriminator used is a convolutional neural network (CNN) (LeCun and Bengio 1995) trained to maximize the negative cross-entropy. We then generate samples using independent \f(a) Base VAE (200-100) (b) Base + depth (200-100-100) (c) Base + width (300-100) (d) GenDiscBGM (100-50) Figure 4: The boosted model (d) demonstrates how ensembles of weak learners can generate sharper samples, compared to naively increasing model capacity (a-c). Note that we show samples of binarized digits and not mean values for the pixels. VAE hidden layer architecture given in parenthesis. Markov chain Monte Carlo (MCMC) runs. The boosted sequences generate sharper samples than all baselines in spite of having similar model capacity. 5 Discussion and related work In this work, we revisited boosting, a class of metaalgorithms developed in response to a seminal question: Can a set of weak learners create a single strong learner? Boosting has offered interesting theoretical insights into the fundamental limits of supervised learning and led to the development of algorithms that work well in practice (Schapire 1990; Freund, Schapire, and Abe 1999; Friedman 2002; Caruana and Niculescu-Mizil 2006). Our work provides a foundational framework for unsupervised boosting with connections to prior work discussed below. Sum-of-experts. Rosset and Segal (2002) proposed an algorithm for density estimation using Bayesian networks similar to gradient boosting. These models are normalized and easy to sample, but are generally outperformed by multiplicative formulations for correcting for model misspeci\ufb01cation, as we show in this work. Similar additive approaches have been used for improving approximate posteriors for speci\ufb01c algorithms for variational inference (Guo et al. 2016; Miller, Foti, and Adams 2017) and generative adversarial networks (Tolstikhin et al. 2017). For a survey on variations of additive ensembling for unsupervised settings, refer to the survey by Bourel and Ghattas (2012). Product-of-experts. Our multiplicative boosting formulation can be interpreted as a product-of-experts approach, which was initially proposed for feature learning in energy based models such as Boltzmann machines. For example, the hidden units in a restricted Boltzmann machine can be interpreted as weak learners performing MLE. If the number of weak learners is \ufb01xed, they can be ef\ufb01ciently updated in parallel but there is a risk of learning redundant features (Hinton 1999; Hinton 2002). Weak learners can also be added incrementally based on the learner\u2019s ability to distinguish observed data and model-generated data (Welling, Zemel, and Hinton 2002). Tu (2007) generalized the latter to boost arbitrary probabilistic models; their algorithm is a special case of DiscBGM with all \u03b1\u2019s set to 1 and the discriminator itself a boosted classi\ufb01er. DiscBGM additionally accounts for imperfections in learning classi\ufb01ers through \ufb02exible model weights. Further, it can include any classi\ufb01er trained to maximize any f-divergence. Related techniques such as noise-contrastive estimation, ratio matching, and score matching methods can be cast as minimization of Bregman divergences, akin to DiscBGM with unit model weights (Gutmann and Hirayama 2011). A non-parametric algorithm similar to GenBGM was proposed by Di Marzio and Taylor (2004) where an ensemble of weighted kernel density estimates are learned to approximate the data distribution. In contrast, our framework allows for both parametric and non-parametric learners and uses a different scheme for reweighting data points than proposed in the above work. Unsupervised-as-supervised learning. The use of density ratios learned by a binary classi\ufb01er for estimation was \ufb01rst proposed by Friedman, Hastie, and Tibshirani (2001) and has been subsequently applied elsewhere, notably for parameter estimation using noise-contrastive estimation (Gutmann and Hyv\u00a8 arinen 2010) and sample generation in generative adversarial networks (GAN) (Goodfellow et al. 2014). While GANs consist of a discriminator distinguishing real data from model generated data similar to DiscBGM for a suitable f-divergence, they differ in the learning objective for the generator (Nowozin, Cseke, and Tomioka 2016). The generator of a GAN performs an adversarial minimization of the same objective the discriminator maximizes, whereas DiscBGM uses the likelihood estimate of the base generator (learned using MLE) and the density ratios derived from the discriminator(s) to estimate the model density for the ensemble. Limitations and future work. In the multiplicative boosting framework, the model density needs to be speci\ufb01ed only up to a normalization constant at any given round of boosting. Additionally, while many applications of generative modeling such as feature learning and classi\ufb01cation can sidestep computing the partition function, if needed it can be estimated using techniques such as Annealed Impor\ftance Sampling (Neal 2001). Similarly, Markov chain Monte Carlo methods can be used to generate samples. The lack of implicit normalization can however be limiting for applications requiring fast log-likelihood evaluation and sampling. In order to sidestep this issue, a promising direction for future work is to consider boosting of normalizing \ufb02ow models (Dinh, Krueger, and Bengio 2014; Dinh, SohlDickstein, and Bengio 2017; Grover, Dhar, and Ermon 2018). These models specify an invertible multiplicative transformation from one distribution to another using the change-of-variables formula such that the resulting distribution is self-normalized and ef\ufb01cient ancestral sampling is possible. The GenBGM algorithm can be adapted to normalizing \ufb02ow models whereby every transformation is interpreted as a weak learner. The parameters for every transformation can be trained greedily after suitable reweighting resulting in a self-normalized boosted generative model. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1607.00653v1",
+                    "title": "node2vec: Scalable Feature Learning for Networks",
+                    "abstract": "Prediction tasks over nodes and edges in networks require careful effort in\nengineering features used by learning algorithms. Recent research in the\nbroader field of representation learning has led to significant progress in\nautomating prediction by learning the features themselves. However, present\nfeature learning approaches are not expressive enough to capture the diversity\nof connectivity patterns observed in networks. Here we propose node2vec, an\nalgorithmic framework for learning continuous feature representations for nodes\nin networks. In node2vec, we learn a mapping of nodes to a low-dimensional\nspace of features that maximizes the likelihood of preserving network\nneighborhoods of nodes. We define a flexible notion of a node's network\nneighborhood and design a biased random walk procedure, which efficiently\nexplores diverse neighborhoods. Our algorithm generalizes prior work which is\nbased on rigid notions of network neighborhoods, and we argue that the added\nflexibility in exploring neighborhoods is the key to learning richer\nrepresentations. We demonstrate the efficacy of node2vec over existing\nstate-of-the-art techniques on multi-label classification and link prediction\nin several real-world networks from diverse domains. Taken together, our work\nrepresents a new way for efficiently learning state-of-the-art task-independent\nrepresentations in complex networks.",
+                    "authors": "Aditya Grover, Jure Leskovec",
+                    "published": "2016-07-03",
+                    "updated": "2016-07-03",
+                    "primary_cat": "cs.SI",
+                    "cats": [
+                        "cs.SI",
+                        "cs.LG",
+                        "stat.ML"
+                    ],
+                    "main_content": "INTRODUCTION Many important tasks in network analysis involve predictions over nodes and edges. In a typical node classi\ufb01cation task, we are interested in predicting the most probable labels of nodes in a network [33]. For example, in a social network, we might be interested in predicting interests of users, or in a protein-protein interaction network we might be interested in predicting functional labels of proteins [25, 37]. Similarly, in link prediction, we wish to Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro\ufb01t or commercial advantage and that copies bear this notice and the full citation on the \ufb01rst page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior speci\ufb01c permission and/or a fee. Request permissions from permissions@acm.org. KDD \u201916, August 13 17, 2016, San Francisco, CA, USA c \u20dd2016 Copyright held by the owner/author(s). Publication rights licensed to ACM. ISBN 978-1-4503-4232-2/16/08...$15.00 DOI: http://dx.doi.org/10.1145/2939672.2939754 predict whether a pair of nodes in a network should have an edge connecting them [18]. Link prediction is useful in a wide variety of domains; for instance, in genomics, it helps us discover novel interactions between genes, and in social networks, it can identify real-world friends [2, 34]. Any supervised machine learning algorithm requires a set of informative, discriminating, and independent features. In prediction problems on networks this means that one has to construct a feature vector representation for the nodes and edges. A typical solution involves hand-engineering domain-speci\ufb01c features based on expert knowledge. Even if one discounts the tedious effort required for feature engineering, such features are usually designed for speci\ufb01c tasks and do not generalize across different prediction tasks. An alternative approach is to learn feature representations by solving an optimization problem [4]. The challenge in feature learning is de\ufb01ning an objective function, which involves a trade-off in balancing computational ef\ufb01ciency and predictive accuracy. On one side of the spectrum, one could directly aim to \ufb01nd a feature representation that optimizes performance of a downstream prediction task. While this supervised procedure results in good accuracy, it comes at the cost of high training time complexity due to a blowup in the number of parameters that need to be estimated. At the other extreme, the objective function can be de\ufb01ned to be independent of the downstream prediction task and the representations can be learned in a purely unsupervised way. This makes the optimization computationally ef\ufb01cient and with a carefully designed objective, it results in task-independent features that closely match task-speci\ufb01c approaches in predictive accuracy [21, 23]. However, current techniques fail to satisfactorily de\ufb01ne and optimize a reasonable objective required for scalable unsupervised feature learning in networks. Classic approaches based on linear and non-linear dimensionality reduction techniques such as Principal Component Analysis, Multi-Dimensional Scaling and their extensions [3, 27, 30, 35] optimize an objective that transforms a representative data matrix of the network such that it maximizes the variance of the data representation. Consequently, these approaches invariably involve eigendecomposition of the appropriate data matrix which is expensive for large real-world networks. Moreover, the resulting latent representations give poor performance on various prediction tasks over networks. Alternatively, we can design an objective that seeks to preserve local neighborhoods of nodes. The objective can be ef\ufb01ciently optimized using stochastic gradient descent (SGD) akin to backpropogation on just single hidden-layer feedforward neural networks. Recent attempts in this direction [24, 28] propose ef\ufb01cient algorithms but rely on a rigid notion of a network neighborhood, which results in these approaches being largely insensitive to connectivity patterns unique to networks. Speci\ufb01cally, nodes in networks arXiv:1607.00653v1  [cs.SI]  3 Jul 2016 \fcould be organized based on communities they belong to (i.e., homophily); in other cases, the organization could be based on the structural roles of nodes in the network (i.e., structural equivalence) [7, 10, 36]. For instance, in Figure 1, we observe nodes u and s1 belonging to the same tightly knit community of nodes, while the nodes u and s6 in the two distinct communities share the same structural role of a hub node. Real-world networks commonly exhibit a mixture of such equivalences. Thus, it is essential to allow for a \ufb02exible algorithm that can learn node representations obeying both principles: ability to learn representations that embed nodes from the same network community closely together, as well as to learn representations where nodes that share similar roles have similar embeddings. This would allow feature learning algorithms to generalize across a wide variety of domains and prediction tasks. Present work. We propose node2vec, a semi-supervised algorithm for scalable feature learning in networks. We optimize a custom graph-based objective function using SGD motivated by prior work on natural language processing [21]. Intuitively, our approach returns feature representations that maximize the likelihood of preserving network neighborhoods of nodes in a d-dimensional feature space. We use a 2nd order random walk approach to generate (sample) network neighborhoods for nodes. Our key contribution is in de\ufb01ning a \ufb02exible notion of a node\u2019s network neighborhood. By choosing an appropriate notion of a neighborhood, node2vec can learn representations that organize nodes based on their network roles and/or communities they belong to. We achieve this by developing a family of biased random walks, which ef\ufb01ciently explore diverse neighborhoods of a given node. The resulting algorithm is \ufb02exible, giving us control over the search space through tunable parameters, in contrast to rigid search procedures in prior work [24, 28]. Consequently, our method generalizes prior work and can model the full spectrum of equivalences observed in networks. The parameters governing our search strategy have an intuitive interpretation and bias the walk towards different network exploration strategies. These parameters can also be learned directly using a tiny fraction of labeled data in a semisupervised fashion. We also show how feature representations of individual nodes can be extended to pairs of nodes (i.e., edges). In order to generate feature representations of edges, we compose the learned feature representations of the individual nodes using simple binary operators. This compositionality lends node2vec to prediction tasks involving nodes as well as edges. Our experiments focus on two common prediction tasks in networks: a multi-label classi\ufb01cation task, where every node is assigned one or more class labels and a link prediction task, where we predict the existence of an edge given a pair of nodes. We contrast the performance of node2vec with state-of-the-art feature learning algorithms [24, 28]. We experiment with several real-world networks from diverse domains, such as social networks, information networks, as well as networks from systems biology. Experiments demonstrate that node2vec outperforms state-of-the-art methods by up to 26.7% on multi-label classi\ufb01cation and up to 12.6% on link prediction. The algorithm shows competitive performance with even 10% labeled data and is also robust to perturbations in the form of noisy or missing edges. Computationally, the major phases of node2vec are trivially parallelizable, and it can scale to large networks with millions of nodes in a few hours. Overall our paper makes the following contributions: 1. We propose node2vec, an ef\ufb01cient scalable algorithm for feature learning in networks that ef\ufb01ciently optimizes a novel network-aware, neighborhood preserving objective using SGD. 2. We show how node2vec is in accordance with established u  s3  s2  s1  s4  s8  s9  s6  s7  s5  BFS  DFS  Figure 1: BFS and DFS search strategies from node u (k = 3). principles in network science, providing \ufb02exibility in discovering representations conforming to different equivalences. 3. We extend node2vec and other feature learning methods based on neighborhood preserving objectives, from nodes to pairs of nodes for edge-based prediction tasks. 4. We empirically evaluate node2vec for multi-label classi\ufb01cation and link prediction on several real-world datasets. The rest of the paper is structured as follows. In Section 2, we brie\ufb02y survey related work in feature learning for networks. We present the technical details for feature learning using node2vec in Section 3. In Section 4, we empirically evaluate node2vec on prediction tasks over nodes and edges on various real-world networks and assess the parameter sensitivity, perturbation analysis, and scalability aspects of our algorithm. We conclude with a discussion of the node2vec framework and highlight some promising directions for future work in Section 5. Datasets and a reference implementation of node2vec are available on the project page: http://snap.stanford.edu/node2vec. 2. RELATED WORK Feature engineering has been extensively studied by the machine learning community under various headings. In networks, the conventional paradigm for generating features for nodes is based on feature extraction techniques which typically involve some seed hand-crafted features based on network properties [8, 11]. In contrast, our goal is to automate the whole process by casting feature extraction as a representation learning problem in which case we do not require any hand-engineered features. Unsupervised feature learning approaches typically exploit the spectral properties of various matrix representations of graphs, especially the Laplacian and the adjacency matrices. Under this linear algebra perspective, these methods can be viewed as dimensionality reduction techniques. Several linear (e.g., PCA) and non-linear (e.g., IsoMap) dimensionality reduction techniques have been proposed [3, 27, 30, 35]. These methods suffer from both computational and statistical performance drawbacks. In terms of computational ef\ufb01ciency, eigendecomposition of a data matrix is expensive unless the solution quality is signi\ufb01cantly compromised with approximations, and hence, these methods are hard to scale to large networks. Secondly, these methods optimize for objectives that are not robust to the diverse patterns observed in networks (such as homophily and structural equivalence) and make assumptions about the relationship between the underlying network structure and the prediction task. For instance, spectral clustering makes a strong homophily assumption that graph cuts will be useful for classi\ufb01cation [29]. Such assumptions are reasonable in many scenarios, but unsatisfactory in effectively generalizing across diverse networks. Recent advancements in representational learning for natural language processing opened new ways for feature learning of discrete objects such as words. In particular, the Skip-gram model [21] aims to learn continuous feature representations for words by optimizing a neighborhood preserving likelihood objective. The algorithm \fproceeds as follows: It scans over the words of a document, and for every word it aims to embed it such that the word\u2019s features can predict nearby words (i.e., words inside some context window). The word feature representations are learned by optmizing the likelihood objective using SGD with negative sampling [22]. The Skip-gram objective is based on the distributional hypothesis which states that words in similar contexts tend to have similar meanings [9]. That is, similar words tend to appear in similar word neighborhoods. Inspired by the Skip-gram model, recent research established an analogy for networks by representing a network as a \u201cdocument\u201d [24, 28]. The same way as a document is an ordered sequence of words, one could sample sequences of nodes from the underlying network and turn a network into a ordered sequence of nodes. However, there are many possible sampling strategies for nodes, resulting in different learned feature representations. In fact, as we shall show, there is no clear winning sampling strategy that works across all networks and all prediction tasks. This is a major shortcoming of prior work which fail to offer any \ufb02exibility in sampling of nodes from a network [24, 28]. Our algorithm node2vec overcomes this limitation by designing a \ufb02exible objective that is not tied to a particular sampling strategy and provides parameters to tune the explored search space (see Section 3). Finally, for both node and edge based prediction tasks, there is a body of recent work for supervised feature learning based on existing and novel graph-speci\ufb01c deep network architectures [15, 16, 17, 31, 39]. These architectures directly minimize the loss function for a downstream prediction task using several layers of non-linear transformations which results in high accuracy, but at the cost of scalability due to high training time requirements. 3. FEATURE LEARNING FRAMEWORK We formulate feature learning in networks as a maximum likelihood optimization problem. Let G = (V, E) be a given network. Our analysis is general and applies to any (un)directed, (un)weighted network. Let f : V \u2192Rd be the mapping function from nodes to feature representaions we aim to learn for a downstream prediction task. Here d is a parameter specifying the number of dimensions of our feature representation. Equivalently, f is a matrix of size |V | \u00d7 d parameters. For every source node u \u2208V , we de\ufb01ne NS(u) \u2282V as a network neighborhood of node u generated through a neighborhood sampling strategy S. We proceed by extending the Skip-gram architecture to networks [21, 24]. We seek to optimize the following objective function, which maximizes the log-probability of observing a network neighborhood NS(u) for a node u conditioned on its feature representation, given by f: max f X u\u2208V log Pr(NS(u)|f(u)). (1) In order to make the optimization problem tractable, we make two standard assumptions: \u2022 Conditional independence. We factorize the likelihood by assuming that the likelihood of observing a neighborhood node is independent of observing any other neighborhood node given the feature representation of the source: Pr(NS(u)|f(u)) = Y ni\u2208NS(u) Pr(ni|f(u)). \u2022 Symmetry in feature space. A source node and neighborhood node have a symmetric effect over each other in feature space. Accordingly, we model the conditional likelihood of every source-neighborhood node pair as a softmax unit parametrized by a dot product of their features: Pr(ni|f(u)) = exp(f(ni) \u00b7 f(u)) P v\u2208V exp(f(v) \u00b7 f(u)). With the above assumptions, the objective in Eq. 1 simpli\ufb01es to: max f X u\u2208V \u0014 \u2212log Zu + X ni\u2208NS(u) f(ni) \u00b7 f(u) \u0015 . (2) The per-node partition function, Zu = P v\u2208V exp(f(u) \u00b7 f(v)), is expensive to compute for large networks and we approximate it using negative sampling [22]. We optimize Eq. 2 using stochastic gradient ascent over the model parameters de\ufb01ning the features f. Feature learning methods based on the Skip-gram architecture have been originally developed in the context of natural language [21]. Given the linear nature of text, the notion of a neighborhood can be naturally de\ufb01ned using a sliding window over consecutive words. Networks, however, are not linear, and thus a richer notion of a neighborhood is needed. To resolve this issue, we propose a randomized procedure that samples many different neighborhoods of a given source node u. The neighborhoods NS(u) are not restricted to just immediate neighbors but can have vastly different structures depending on the sampling strategy S. 3.1 Classic search strategies We view the problem of sampling neighborhoods of a source node as a form of local search. Figure 1 shows a graph, where given a source node u we aim to generate (sample) its neighborhood NS(u). Importantly, to be able to fairly compare different sampling strategies S, we shall constrain the size of the neighborhood set NS to k nodes and then sample multiple sets for a single node u. Generally, there are two extreme sampling strategies for generating neighborhood set(s) NS of k nodes: \u2022 Breadth-\ufb01rst Sampling (BFS) The neighborhood NS is restricted to nodes which are immediate neighbors of the source. For example, in Figure 1 for a neighborhood of size k = 3, BFS samples nodes s1, s2, s3. \u2022 Depth-\ufb01rst Sampling (DFS) The neighborhood consists of nodes sequentially sampled at increasing distances from the source node. In Figure 1, DFS samples s4, s5, s6. The breadth-\ufb01rst and depth-\ufb01rst sampling represent extreme scenarios in terms of the search space they explore leading to interesting implications on the learned representations. In particular, prediction tasks on nodes in networks often shuttle between two kinds of similarities: homophily and structural equivalence [12]. Under the homophily hypothesis [7, 36] nodes that are highly interconnected and belong to similar network clusters or communities should be embedded closely together (e.g., nodes s1 and u in Figure 1 belong to the same network community). In contrast, under the structural equivalence hypothesis [10] nodes that have similar structural roles in networks should be embedded closely together (e.g., nodes u and s6 in Figure 1 act as hubs of their corresponding communities). Importantly, unlike homophily, structural equivalence does not emphasize connectivity; nodes could be far apart in the network and still have the same structural role. In real-world, these equivalence notions are not exclusive; networks commonly exhibit both behaviors where some nodes exhibit homophily while others re\ufb02ect structural equivalence. We observe that BFS and DFS strategies play a key role in producing representations that re\ufb02ect either of the above equivalences. In particular, the neighborhoods sampled by BFS lead to embeddings that correspond closely to structural equivalence. Intuitively, \fwe note that in order to ascertain structural equivalence, it is often suf\ufb01cient to characterize the local neighborhoods accurately. For example, structural equivalence based on network roles such as bridges and hubs can be inferred just by observing the immediate neighborhoods of each node. By restricting search to nearby nodes, BFS achieves this characterization and obtains a microscopic view of the neighborhood of every node. Additionally, in BFS, nodes in the sampled neighborhoods tend to repeat many times. This is also important as it reduces the variance in characterizing the distribution of 1-hop nodes with respect the source node. However, a very small portion of the graph is explored for any given k. The opposite is true for DFS which can explore larger parts of the network as it can move further away from the source node u (with sample size k being \ufb01xed). In DFS, the sampled nodes more accurately re\ufb02ect a macro-view of the neighborhood which is essential in inferring communities based on homophily. However, the issue with DFS is that it is important to not only infer which node-to-node dependencies exist in a network, but also to characterize the exact nature of these dependencies. This is hard given we have a constrain on the sample size and a large neighborhood to explore, resulting in high variance. Secondly, moving to much greater depths leads to complex dependencies since a sampled node may be far from the source and potentially less representative. 3.2 node2vec Building on the above observations, we design a \ufb02exible neighborhood sampling strategy which allows us to smoothly interpolate between BFS and DFS. We achieve this by developing a \ufb02exible biased random walk procedure that can explore neighborhoods in a BFS as well as DFS fashion. 3.2.1 Random Walks Formally, given a source node u, we simulate a random walk of \ufb01xed length l. Let ci denote the ith node in the walk, starting with c0 = u. Nodes ci are generated by the following distribution: P(ci = x | ci\u22121 = v) = ( \u03c0vx Z if (v, x) \u2208E 0 otherwise where \u03c0vx is the unnormalized transition probability between nodes v and x, and Z is the normalizing constant. 3.2.2 Search bias \u03b1 The simplest way to bias our random walks would be to sample the next node based on the static edge weights wvx i.e., \u03c0vx = wvx. (In case of unweighted graphs wvx = 1.) However, this does not allow us to account for the network structure and guide our search procedure to explore different types of network neighborhoods. Additionally, unlike BFS and DFS which are extreme sampling paradigms suited for structural equivalence and homophily respectively, our random walks should accommodate for the fact that these notions of equivalence are not competing or exclusive, and real-world networks commonly exhibit a mixture of both. We de\ufb01ne a 2nd order random walk with two parameters p and q which guide the walk: Consider a random walk that just traversed edge (t, v) and now resides at node v (Figure 2). The walk now needs to decide on the next step so it evaluates the transition probabilities \u03c0vx on edges (v, x) leading from v. We set the unnormalized transition probability to \u03c0vx = \u03b1pq(t, x) \u00b7 wvx, where \u03b1pq(t, x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 p if dtx = 0 1 if dtx = 1 1 q if dtx = 2 t  x2  x1  v  x3  \u03b1=1  \u03b1=1/q  \u03b1=1/q  \u03b1=1/p  u  s3  s2  s1  s4  s7  s6  s5  BFS  DFS  v  \u03b1=1  \u03b1=1/q  \u03b1=1/q  \u03b1=1/p  x2  x3  t  x1  Figure 2: Illustration of the random walk procedure in node2vec. The walk just transitioned from t to v and is now evaluating its next step out of node v. Edge labels indicate search biases \u03b1. and dtx denotes the shortest path distance between nodes t and x. Note that dtx must be one of {0, 1, 2}, and hence, the two parameters are necessary and suf\ufb01cient to guide the walk. Intuitively, parameters p and q control how fast the walk explores and leaves the neighborhood of starting node u. In particular, the parameters allow our search procedure to (approximately) interpolate between BFS and DFS and thereby re\ufb02ect an af\ufb01nity for different notions of node equivalences. Return parameter, p. Parameter p controls the likelihood of immediately revisiting a node in the walk. Setting it to a high value (> max(q, 1)) ensures that we are less likely to sample an alreadyvisited node in the following two steps (unless the next node in the walk had no other neighbor). This strategy encourages moderate exploration and avoids 2-hop redundancy in sampling. On the other hand, if p is low (< min(q, 1)), it would lead the walk to backtrack a step (Figure 2) and this would keep the walk \u201clocal\u201d close to the starting node u. In-out parameter, q. Parameter q allows the search to differentiate between \u201cinward\u201d and \u201coutward\u201d nodes. Going back to Figure 2, if q > 1, the random walk is biased towards nodes close to node t. Such walks obtain a local view of the underlying graph with respect to the start node in the walk and approximate BFS behavior in the sense that our samples comprise of nodes within a small locality. In contrast, if q < 1, the walk is more inclined to visit nodes which are further away from the node t. Such behavior is re\ufb02ective of DFS which encourages outward exploration. However, an essential difference here is that we achieve DFS-like exploration within the random walk framework. Hence, the sampled nodes are not at strictly increasing distances from a given source node u, but in turn, we bene\ufb01t from tractable preprocessing and superior sampling ef\ufb01ciency of random walks. Note that by setting \u03c0v,x to be a function of the preceeding node in the walk t, the random walks are 2nd order Markovian. Bene\ufb01ts of random walks. There are several bene\ufb01ts of random walks over pure BFS/DFS approaches. Random walks are computationally ef\ufb01cient in terms of both space and time requirements. The space complexity to store the immediate neighbors of every node in the graph is O(|E|). For 2nd order random walks, it is helpful to store the interconnections between the neighbors of every node, which incurs a space complexity of O(a2|V |) where a is the average degree of the graph and is usually small for realworld networks. The other key advantage of random walks over classic search-based sampling strategies is its time complexity. In particular, by imposing graph connectivity in the sample generation process, random walks provide a convenient mechanism to increase the effective sampling rate by reusing samples across different source nodes. By simulating a random walk of length l > k we can generate k samples for l \u2212k nodes at once due to the Marko\fOperator Symbol De\ufb01nition Average \u229e [f(u) \u229ef(v)]i = fi(u)+fi(v) 2 Hadamard \u22a1 [f(u) \u22a1f(v)]i = fi(u) \u2217fi(v) Weighted-L1 \u2225\u00b7 \u2225\u00af 1 \u2225f(u) \u00b7 f(v)\u2225\u00af 1i = |fi(u) \u2212fi(v)| Weighted-L2 \u2225\u00b7 \u2225\u00af 2 \u2225f(u) \u00b7 f(v)\u2225\u00af 2i = |fi(u) \u2212fi(v)|2 Table 1: Choice of binary operators \u25e6for learning edge features. The de\ufb01nitions correspond to the ith component of g(u, v). vian nature of the random walk. Hence, our effective complexity is O \u0000l k(l\u2212k) \u0001 per sample. For example, in Figure 1 we sample a random walk {u, s4, s5, s6, s8, s9} of length l = 6, which results in NS(u) = {s4, s5, s6}, NS(s4) = {s5, s6, s8} and NS(s5) = {s6, s8, s9}. Note that sample reuse can introduce some bias in the overall procedure. However, we observe that it greatly improves the ef\ufb01ciency. 3.2.3 The node2vec algorithm Algorithm 1 The node2vec algorithm. LearnFeatures (Graph G = (V, E, W), Dimensions d, Walks per node r, Walk length l, Context size k, Return p, In-out q) \u03c0 = PreprocessModi\ufb01edWeights(G, p, q) G\u2032 = (V, E, \u03c0) Initialize walks to Empty for iter = 1 to r do for all nodes u \u2208V do walk = node2vecWalk(G\u2032, u, l) Append walk to walks f = StochasticGradientDescent(k, d, walks) return f node2vecWalk (Graph G\u2032 = (V, E, \u03c0), Start node u, Length l) Inititalize walk to [u] for walk_iter = 1 to l do curr = walk[\u22121] Vcurr = GetNeighbors(curr, G\u2032) s = AliasSample(Vcurr, \u03c0) Append s to walk return walk The pseudocode for node2vec, is given in Algorithm 1. In any random walk, there is an implicit bias due to the choice of the start node u. Since we learn representations for all nodes, we offset this bias by simulating r random walks of \ufb01xed length l starting from every node. At every step of the walk, sampling is done based on the transition probabilities \u03c0vx. The transition probabilities \u03c0vx for the 2nd order Markov chain can be precomputed and hence, sampling of nodes while simulating the random walk can be done ef\ufb01ciently in O(1) time using alias sampling. The three phases of node2vec, i.e., preprocessing to compute transition probabilities, random walk simulations and optimization using SGD, are executed sequentially. Each phase is parallelizable and executed asynchronously, contributing to the overall scalability of node2vec. node2vec is available at: http://snap.stanford.edu/node2vec. 3.3 Learning edge features The node2vec algorithm provides a semi-supervised method to learn rich feature representations for nodes in a network. However, we are often interested in prediction tasks involving pairs of nodes instead of individual nodes. For instance, in link prediction, we predict whether a link exists between two nodes in a network. Since our random walks are naturally based on the connectivity structure between nodes in the underlying network, we extend them to pairs of nodes using a bootstrapping approach over the feature representations of the individual nodes. Given two nodes u and v, we de\ufb01ne a binary operator \u25e6over the corresponding feature vectors f(u) and f(v) in order to generate a representation g(u, v) such that g : V \u00d7 V \u2192Rd\u2032 where d\u2032 is the representation size for the pair (u, v). We want our operators to be generally de\ufb01ned for any pair of nodes, even if an edge does not exist between the pair since doing so makes the representations useful for link prediction where our test set contains both true and false edges (i.e., do not exist). We consider several choices for the operator \u25e6such that d\u2032 = d which are summarized in Table 1. 4. EXPERIMENTS The objective in Eq. 2 is independent of any downstream task and the \ufb02exibility in exploration offered by node2vec lends the learned feature representations to a wide variety of network analysis settings discussed below. 4.1 Case Study: Les Mis\u00e9rables network In Section 3.1 we observed that BFS and DFS strategies represent extreme ends on the spectrum of embedding nodes based on the principles of homophily (i.e., network communities) and structural equivalence (i.e., structural roles of nodes). We now aim to empirically demonstrate this fact and show that node2vec in fact can discover embeddings that obey both principles. We use a network where nodes correspond to characters in the novel Les Mis\u00e9rables [13] and edges connect coappearing characters. The network has 77 nodes and 254 edges. We set d = 16 and run node2vec to learn feature representation for every node in the network. The feature representations are clustered using kmeans. We then visualize the original network in two dimensions with nodes now assigned colors based on their clusters. Figure 3(top) shows the example when we set p = 1, q = 0.5. Notice how regions of the network (i.e., network communities) are colored using the same color. In this setting node2vec discovers clusters/communities of characters that frequently interact with each other in the major sub-plots of the novel. Since the edges between characters are based on coappearances, we can conclude this characterization closely relates with homophily. In order to discover which nodes have the same structural roles we use the same network but set p = 1, q = 2, use node2vec to get node features and then cluster the nodes based on the obtained features. Here node2vec obtains a complementary assignment of node to clusters such that the colors correspond to structural equivalence as illustrated in Figure 3(bottom). For instance, node2vec embeds blue-colored nodes close together. These nodes represent characters that act as bridges between different sub-plots of the novel. Similarly, the yellow nodes mostly represent characters that are at the periphery and have limited interactions. One could assign alternate semantic interpretations to these clusters of nodes, but the key takeaway is that node2vec is not tied to a particular notion of equivalence. As we show through our experiments, these equivalence notions are commonly exhibited in most real-world networks and have a signi\ufb01cant impact on the performance of the learned representations for prediction tasks. 4.2 Experimental setup Our experiments evaluate the feature representations obtained through node2vec on standard supervised learning tasks: multilabel classi\ufb01cation for nodes and link prediction for edges. For both tasks, we evaluate the performance of node2vec against the following feature learning algorithms: \fFigure 3: Complementary visualizations of Les Mis\u00e9rables coappearance network generated by node2vec with label colors re\ufb02ecting homophily (top) and structural equivalence (bottom). \u2022 Spectral clustering [29]: This is a matrix factorization approach in which we take the top d eigenvectors of the normalized Laplacian matrix of graph G as the feature vector representations for nodes. \u2022 DeepWalk [24]: This approach learns d-dimensional feature representations by simulating uniform random walks. The sampling strategy in DeepWalk can be seen as a special case of node2vec with p = 1 and q = 1. \u2022 LINE [28]: This approach learns d-dimensional feature representations in two separate phases. In the \ufb01rst phase, it learns d/2 dimensions by BFS-style simulations over immediate neighbors of nodes. In the second phase, it learns the next d/2 dimensions by sampling nodes strictly at a 2-hop distance from the source nodes. We exclude other matrix factorization approaches which have already been shown to be inferior to DeepWalk [24]. We also exclude a recent approach, GraRep [6], that generalizes LINE to incorporate information from network neighborhoods beyond 2-hops, but is unable to ef\ufb01ciently scale to large networks. In contrast to the setup used in prior work for evaluating samplingbased feature learning algorithms, we generate an equal number of samples for each method and then evaluate the quality of the obtained features on the prediction task. In doing so, we discount for performance gain observed purely because of the implementation language (C/C++/Python) since it is secondary to the algorithm. Thus, in the sampling phase, the parameters for DeepWalk, LINE and node2vec are set such that they generate equal number of samples at runtime. As an example, if K is the overall sampling budget, then the node2vec parameters satisfy K = r \u00b7 l \u00b7 |V |. In the optimization phase, all these benchmarks optimize using SGD with two key differences that we correct for. First, DeepWalk uses hierarchical sampling to approximate the softmax probabilities with an objective similar to the one use by node2vec. However, hierarchical softmax is inef\ufb01cient when compared with negative sampling [22]. Hence, keeping everything else the same, we switch to negative sampling in DeepWalk which is also the de facto approximation in node2vec and LINE. Second, both node2vec and DeepWalk have Algorithm Dataset BlogCatalog PPI Wikipedia Spectral Clustering 0.0405 0.0681 0.0395 DeepWalk 0.2110 0.1768 0.1274 LINE 0.0784 0.1447 0.1164 node2vec 0.2581 0.1791 0.1552 node2vec settings (p,q) 0.25, 0.25 4, 1 4, 0.5 Gain of node2vec [%] 22.3 1.3 21.8 Table 2: Macro-F1 scores for multilabel classi\ufb01cation on BlogCatalog, PPI (Homo sapiens) and Wikipedia word cooccurrence networks with 50% of the nodes labeled for training. a parameter for the number of context neighborhood nodes to optimize for and the greater the number, the more rounds of optimization are required. This parameter is set to unity for LINE, but since LINE completes a single epoch quicker than other approaches, we let it run for k epochs. The parameter settings used for node2vec are in line with typical values used for DeepWalk and LINE. Speci\ufb01cally, we set d = 128, r = 10, l = 80, k = 10, and the optimization is run for a single epoch. We repeat our experiments for 10 random seed initializations, and our results are statistically signi\ufb01cant with a p-value of less than 0.01.The best in-out and return hyperparameters were learned using 10-fold cross-validation on 10% labeled data with a grid search over p, q \u2208{0.25, 0.50, 1, 2, 4}. 4.3 Multi-label classi\ufb01cation In the multi-label classi\ufb01cation setting, every node is assigned one or more labels from a \ufb01nite set L. During the training phase, we observe a certain fraction of nodes and all their labels. The task is to predict the labels for the remaining nodes. This is a challenging task especially if L is large. We utilize the following datasets: \u2022 BlogCatalog [38]: This is a network of social relationships of the bloggers listed on the BlogCatalog website. The labels represent blogger interests inferred through the metadata provided by the bloggers. The network has 10,312 nodes, 333,983 edges, and 39 different labels. \u2022 Protein-Protein Interactions (PPI) [5]: We use a subgraph of the PPI network for Homo Sapiens. The subgraph corresponds to the graph induced by nodes for which we could obtain labels from the hallmark gene sets [19] and represent biological states. The network has 3,890 nodes, 76,584 edges, and 50 different labels. \u2022 Wikipedia [20]: This is a cooccurrence network of words appearing in the \ufb01rst million bytes of the Wikipedia dump. The labels represent the Part-of-Speech (POS) tags inferred using the Stanford POS-Tagger [32]. The network has 4,777 nodes, 184,812 edges, and 40 different labels. All these networks exhibit a fair mix of homophilic and structural equivalences. For example, we expect the social network of bloggers to exhibit strong homophily-based relationships; however, there might also be some \u201cfamiliar strangers\u201d, i.e., bloggers that do not interact but share interests and hence are structurally equivalent nodes. The biological states of proteins in a protein-protein interaction network also exhibit both types of equivalences. For example, they exhibit structural equivalence when proteins perform functions complementary to those of neighboring proteins, and at other times, they organize based on homophily in assisting neighboring proteins in performing similar functions. The word cooccurence network is fairly dense, since edges exist between words cooccuring in a 2length window in the Wikipedia corpus. Hence, words having the same POS tags are not hard to \ufb01nd, lending a high degree of homophily. At the same time, we expect some structural equivalence \f0.0 0.2 0.4 0.6 0.8 1.0 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Micro-F1 score BlogCatalog 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Macro-F1 score 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 PPI (Homo Sapiens) 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0 0.35 0.40 0.45 0.50 0.55 0.60 Wikipedia 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Spectral Clustering DeepWalk LINE node2vec Figure 4: Performance evaluation of different benchmarks on varying the amount of labeled data used for training. The x axis denotes the fraction of labeled data, whereas the y axis in the top and bottom rows denote the Micro-F1 and Macro-F1 scores respectively. DeepWalk and node2vec give comparable performance on PPI. In all other networks, across all fractions of labeled data node2vec performs best. in the POS tags due to syntactic grammar patterns such as nouns following determiners, punctuations succeeding nouns etc. Experimental results. The node feature representations are input to a one-vs-rest logistic regression classi\ufb01er with L2 regularization. The train and test data is split equally over 10 random instances. We use the Macro-F1 scores for comparing performance in Table 2 and the relative performance gain is over the closest benchmark. The trends are similar for Micro-F1 and accuracy and are not shown. From the results, it is evident we can see how the added \ufb02exibility in exploring neighborhoods allows node2vec to outperform the other benchmark algorithms. In BlogCatalog, we can discover the right mix of homophily and structural equivalence by setting parameters p and q to low values, giving us 22.3% gain over DeepWalk and 229.2% gain over LINE in Macro-F1 scores. LINE showed worse performance than expected, which can be explained by its inability to reuse samples, a feat that can be easily done using the random walk methods. Even in our other two networks, where we have a mix of equivalences present, the semi-supervised nature of node2vec can help us infer the appropriate degree of exploration necessary for feature learning. In the case of PPI network, the best exploration strategy (p = 4, q = 1) turns out to be virtually indistinguishable from DeepWalk\u2019s uniform (p = 1, q = 1) exploration giving us only a slight edge over DeepWalk by avoiding redudancy in already visited nodes through a high p value, but a convincing 23.8% gain over LINE in Macro-F1 scores. However, in general, the uniform random walks can be much worse than the exploration strategy learned by node2vec. As we can see in the Wikipedia word cooccurrence network, uniform walks cannot guide the search procedure towards the best samples and hence, we achieve a gain of 21.8% over DeepWalk and 33.2% over LINE. For a more \ufb01ne-grained analysis, we also compare performance while varying the train-test split from 10% to 90%, while learning parameters p and q on 10% of the data as before. For brevity, we summarize the results for the Micro-F1 and Macro-F1 scores graphically in Figure 4. Here we make similar observations. All methods signi\ufb01cantly outperform Spectral clustering, DeepWalk outperforms LINE, node2vec consistently outperforms LINE and achieves large improvement over DeepWalk across domains. For example, we achieve the biggest improvement over DeepWalk of 26.7% on BlogCatalog at 70% labeled data. In the worst case, the search phase has little bearing on learned representations in which case node2vec is equivalent to DeepWalk. Similarly, the improvements are even more striking when compared to LINE, where in addition to drastic gain (over 200%) on BlogCatalog, we observe high magnitude improvements upto 41.1% on other datasets such as PPI while training on just 10% labeled data. 4.4 Parameter sensitivity The node2vec algorithm involves a number of parameters and in Figure 5a, we examine how the different choices of parameters affect the performance of node2vec on the BlogCatalog dataset using a 50-50 split between labeled and unlabeled data. Except for the parameter being tested, all other parameters assume default values. The default values for p and q are set to unity. We measure the Macro-F1 score as a function of parameters p and q. The performance of node2vec improves as the in-out parameter p and the return parameter q decrease. This increase in performance can be based on the homophilic and structural equivalences we expect to see in BlogCatalog. While a low q encourages outward exploration, it is balanced by a low p which ensures that the walk does not go too far from the start node. We also examine how the number of features d and the node\u2019s neighborhood parameters (number of walks r, walk length l, and neighborhood size k) affect the performance. We observe that performance tends to saturate once the dimensions of the representations reaches around 100. Similarly, we observe that increasing the number and length of walks per source improves performance, \f\u22123 \u22122 \u22121 0 1 2 3 log2 p 0.16 0.18 0.20 0.22 0.24 0.26 0.28 Macro-F1 score \u22123 \u22122 \u22121 0 1 2 3 log2 q 0.16 0.18 0.20 0.22 0.24 0.26 0.28 3 4 5 6 7 8 9 log2 d 0.16 0.18 0.20 0.22 0.24 0.26 0.28 6 8 10 12 14 16 18 20 Number of walks per node, r 0.16 0.18 0.20 0.22 0.24 0.26 0.28 Macro-F1 score 30 40 50 60 70 80 90 100110 Length of walk, l 0.16 0.18 0.20 0.22 0.24 0.26 0.28 8 10 12 14 16 18 20 Context size, k 0.16 0.18 0.20 0.22 0.24 0.26 0.28 (a) \u22123 \u22122 \u22121 0 1 2 3 log2 p 0.16 0.18 0.20 0.22 0.24 0.26 0.28 Macro-F1 score \u22123 \u22122 \u22121 0 1 2 3 log2 q 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Fraction of missing edges 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Macro-F1 score 6 8 10 12 14 16 18 20 Number of walks per node, r 0.16 0.18 0.20 0.22 0.24 0.26 0.28 Macro-F1 score 30 40 50 60 70 80 90 100110 Length of walk, l 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Fraction of additional edges 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Macro-F1 score (b) Figure 5: (a). Parameter sensitivity (b). Perturbation analysis for multilabel classi\ufb01cation on the BlogCatalog network. which is not surprising since we have a greater overall sampling budget K to learn representations. Both these parameters have a relatively high impact on the performance of the method. Interestingly, the context size, k also improves performance at the cost of increased optimization time. However, the performance differences are not that large in this case. 4.5 Perturbation Analysis For many real-world networks, we do not have access to accurate information about the network structure. We performed a perturbation study where we analyzed the performance of node2vec for two imperfect information scenarios related to the edge structure in the BlogCatalog network. In the \ufb01rst scenario, we measure performace as a function of the fraction of missing edges (relative to the full network). The missing edges are chosen randomly, subject to the constraint that the number of connected components in the network remains \ufb01xed. As we can see in Figure 5b(top), the decrease in Macro-F1 score as the fraction of missing edges increases is roughly linear with a small slope. Robustness to missing edges in the network is especially important in cases where the graphs are evolving over time (e.g., citation networks), or where network construction is expensive (e.g., biological networks). In the second perturbation setting, we have noisy edges between randomly selected pairs of nodes in the network. As shown in Figure 5b(bottom), the performance of node2vec declines slightly faster initially when compared with the setting of missing edges, however, the rate of decrease in Macro-F1 score gradually slows down over time. Again, the robustness of node2vec to false edges is useful in several situations such as sensor networks where the measurements used for constructing the network are noisy. 4.6 Scalability To test for scalability, we learn node representations using node2vec with default parameter values for Erdos-Renyi graphs with increasing sizes from 100 to 1,000,000 nodes and constant average degree 1 2 3 4 5 6 7 log10 nodes 0 1 2 3 4 log10 time (in seconds) sampling + optimization time sampling time Figure 6: Scalability of node2vec on Erdos-Renyi graphs with an average degree of 10. of 10. In Figure 6, we empirically observe that node2vec scales linearly with increase in number of nodes generating representations for one million nodes in less than four hours. The sampling procedure comprises of preprocessing for computing transition probabilities for our walk (negligibly small) and simulation of random walks. The optimization phase is made ef\ufb01cient using negative sampling [22] and asynchronous SGD [26]. Many ideas from prior work serve as useful pointers in making the sampling procedure computationally ef\ufb01cient. We showed how random walks, also used in DeepWalk [24], allow the sampled nodes to be reused as neighborhoods for different source nodes appearing in the walk. Alias sampling allows our walks to generalize to weighted networks, with little preprocessing [28]. Though we are free to set the search parameters based on the underlying task and domain at no additional cost, learning the best settings of \fScore De\ufb01nition Common Neighbors | N(u) \u2229N(v) | Jaccard\u2019s Coef\ufb01cient |N (u)\u2229N (v)| |N (u)\u222aN (v)| Adamic-Adar Score P t\u2208N (u)\u2229N (v) 1 log|N (t)| Preferential Attachment | N(u) | \u00b7 | N(v) | Table 3: Link prediction heuristic scores for node pair (u, v) with immediate neighbor sets N(u) and N(v) respectively. our search parameters adds an overhead. However, as our experiments con\ufb01rm, this overhead is minimal since node2vec is semisupervised and hence, can learn these parameters ef\ufb01ciently with very little labeled data. 4.7 Link prediction In link prediction, we are given a network with a certain fraction of edges removed, and we would like to predict these missing edges. We generate the labeled dataset of edges as follows: To obtain positive examples, we remove 50% of edges chosen randomly from the network while ensuring that the residual network obtained after the edge removals is connected, and to generate negative examples, we randomly sample an equal number of node pairs from the network which have no edge connecting them. Since none of feature learning algorithms have been previously used for link prediction, we additionally evaluate node2vec against some popular heuristic scores that achieve good performance in link prediction. The scores we consider are de\ufb01ned in terms of the neighborhood sets of the nodes constituting the pair (see Table 3). We test our benchmarks on the following datasets: \u2022 Facebook [14]: In the Facebook network, nodes represent users, and edges represent a friendship relation between any two users. The network has 4,039 nodes and 88,234 edges. \u2022 Protein-Protein Interactions (PPI) [5]: In the PPI network for Homo Sapiens, nodes represent proteins, and an edge indicates a biological interaction between a pair of proteins. The network has 19,706 nodes and 390,633 edges. \u2022 arXiv ASTRO-PH [14]: This is a collaboration network generated from papers submitted to the e-print arXiv where nodes represent scientists, and an edge is present between two scientists if they have collaborated in a paper. The network has 18,722 nodes and 198,110 edges. Experimental results. We summarize our results for link prediction in Table 4. The best p and q parameter settings for each node2vec entry are omitted for ease of presentation. A general observation we can draw from the results is that the learned feature representations for node pairs signi\ufb01cantly outperform the heuristic benchmark scores with node2vec achieving the best AUC improvement on 12.6% on the arXiv dataset over the best performing baseline (Adamic-Adar [1]). Amongst the feature learning algorithms, node2vec outperforms both DeepWalk and LINE in all networks with gain up to 3.8% and 6.5% respectively in the AUC scores for the best possible choices of the binary operator for each algorithm. When we look at operators individually (Table 1), node2vec outperforms DeepWalk and LINE barring a couple of cases involving the Weighted-L1 and Weighted-L2 operators in which LINE performs better. Overall, the Hadamard operator when used with node2vec is highly stable and gives the best performance on average across all networks. 5. DISCUSSION AND"
+                }
+            ],
+            "Anuroop Sriram": [
+                {
+                    "url": "http://arxiv.org/abs/2311.00341v2",
+                    "title": "The Open DAC 2023 Dataset and Challenges for Sorbent Discovery in Direct Air Capture",
+                    "abstract": "New methods for carbon dioxide removal are urgently needed to combat global\nclimate change. Direct air capture (DAC) is an emerging technology to capture\ncarbon dioxide directly from ambient air. Metal-organic frameworks (MOFs) have\nbeen widely studied as potentially customizable adsorbents for DAC. However,\ndiscovering promising MOF sorbents for DAC is challenging because of the vast\nchemical space to explore and the need to understand materials as functions of\nhumidity and temperature. We explore a computational approach benefiting from\nrecent innovations in machine learning (ML) and present a dataset named Open\nDAC 2023 (ODAC23) consisting of more than 38M density functional theory (DFT)\ncalculations on more than 8,400 MOF materials containing adsorbed $CO_2$ and/or\n$H_2O$. ODAC23 is by far the largest dataset of MOF adsorption calculations at\nthe DFT level of accuracy currently available. In addition to probing\nproperties of adsorbed molecules, the dataset is a rich source of information\non structural relaxation of MOFs, which will be useful in many contexts beyond\nspecific applications for DAC. A large number of MOFs with promising properties\nfor DAC are identified directly in ODAC23. We also trained state-of-the-art ML\nmodels on this dataset to approximate calculations at the DFT level. This\nopen-source dataset and our initial ML models will provide an important\nbaseline for future efforts to identify MOFs for a wide range of applications,\nincluding DAC.",
+                    "authors": "Anuroop Sriram, Sihoon Choi, Xiaohan Yu, Logan M. Brabson, Abhishek Das, Zachary Ulissi, Matt Uyttendaele, Andrew J. Medford, David S. Sholl",
+                    "published": "2023-11-01",
+                    "updated": "2023-11-27",
+                    "primary_cat": "cond-mat.mtrl-sci",
+                    "cats": [
+                        "cond-mat.mtrl-sci",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Annual anthropogenic carbon emissions reached nearly 36 billion tonnes in 2020, and the atmospheric carbon dioxide concentration has increased \u223c50% since preindustrial times to approximately 420 ppm.1 Rising CO2 levels have motivated the development of carbon capture and sequestration (CCS) technologies to combat the effects of emissions on global climate change.2 Direct air capture (DAC) is an emerging technology with the potential for distributed capture and negative emissions.3 DAC operates at ambient conditions and avoids impurities that are common for point source capture of CO2, but the low concentration of CO2 requires the movement of large volumes of air and strong adsorption of CO2.4 Many current DAC absorbents, such as liquid amines and solid alkali hydroxides, strongly bind CO2 through chemisorption, requiring energy-intensive regeneration of the sorbent.5,6 Metal-organic frameworks (MOFs) are a promising class of alternative sorbent materials for DAC allowing regeneration at relatively low temperatures. In contrast to sorbents such as alkali hydroxides, MOFs are modular, flexible, and highly tunable, and they possess remarkably high porosities, low densities, and long-range order.7 The chemical tunability and long-range order make MOFs worthy of highthroughput computational screening studies. Computational materials design is a promising strategy for DAC sorbents.8 Design of efficient DAC processes may require tailoring of materials to the specifics of the air temperature and humidity conditions in a given environment or the temperature/pressure swings that are required to keep energy consumption low.9 This is particularly true of DAC processes that seek to leverage air movement and energy content of existing systems such as heating, ventilation, and air conditioning.10 The consideration of humidity is particularly important, since dehumidifying air requires significant energy input, the presence of H2O can result in competitive adsorption even at low relative humidities, and humidity can in some cases cause adsorbent degradation over time.11\u201314 The availability of large datasets of MOFs and other solid sorbent materials can facilitate identification of specific materials or chemical moieties that are well suited for the specific conditions of a given DAC process.15,16 High-throughput computational studies and machine learning (ML) techniques are already a common practice in the screening and discovery of MOFs and other reticular materials.12,17\u201325 There are several large databases of MOF26\u201331 and zeolite structures32 and multiple computational toolkits33\u201337 and ML models23,38\u201343 to analyze and predict the adsorption properties of these materials. However, there are several key limitations to the existing body of work. First, because of the computational costs involved, many studies rely on empirical force field (FF) models for predicting adsorption properties. Inaccuracies associated with FFs can lead to both qualitative and quantitative inconsistencies in the prediction of material performance, particularly in the case of open-metal sites (OMS) or defects where covalent bonding or complexation occurs.17,44\u201349 There are several large databases of density functional theory (DFT) calculations for MOF materials,27,31 but to date these are focused only on the MOF structure and do not include adsorption data. Second, many existing databases and studies of CO2 adsorption focus only on adsorption of CO2, neglecting the possibility of competition with H2O.11,50\u201353 Failure to consider competitive adsorption will strongly limit the ability to predict materials for practical DAC processes, where bicomponent CO2/H2O isotherms are required. Accurately modeling H2O adsorption with classical FFs is challenging due to the complex physical properties of water.54\u201357 Third, many computational databases and studies focus on hypothetical materials,28,30,58,59 which leads to practical challenges in the synthesis and experimental testing of new predicted materials. Finally, most datasets are restricted to pristine materials. In reality, MOFs will contain a wide range of defects that may govern their adsorption properties under practical conditions.60,61 New materials can also be created by inserting defects in MOFs via so-called defect-engineering.62 Large datasets of high-quality DFT simulations of 2 \fmixed CO2 and H2O adsorption on realistic pristine and defective MOFs are needed to address these limitations. ML is also a well-established approach in the discovery of MOFs and other nanoporous materials. ML models have been applied to directly predict adsorption properties and isotherms of MOFs based on their physical and chemical structure.23,41\u201343,63\u201365 Descriptors based on the porosity, chemical constituents, and energy landscape of probe adsorbates in MOFs have been combined with a range of regression and classification models to provide predictions of gas loadings,23,42,66,67 Henry\u2019s constants,40,68 and temperature-dependent isotherms.63,64 Neural networks have been used to predict MOF properties and perform inverse design tasks to identify MOF materials with high thermal stability58,69 and strong or selective CO2 adsorption.43,59,65 ML models have also been trained to provide insight into synthesizability and stability of MOFs and zeolites.70\u201374 However, the training data required for many of these properties, such as adsorption isotherms, are generated using classical FFs, which have been shown to exhibit systematic errors.45 Efforts to train ML models that can directly emulate DFT data for MOFs are more limited.43 The ability to use ML models to directly replace FFs in MOFs has the potential to enhance many of the prior efforts. In this work, we introduce the Open DAC 2023 (ODAC23) dataset to address these challenges. The dataset consists of adsorption energies for CO2, H2O, and mixtures thereof on \u223c8K MOFs, amounting to a total of \u223c176K adsorption energies and \u223c38M single-point calculations (Fig. 1). All calculations were performed using DFT with the PBE+D3 exchange correlation functional, ensuring that covalent and electrostatic interactions are treated quantum mechanically and van der Waals interactions are included with well-established empirical accuracy. Approximately 76K adsorption energies involve MOFs that have missing linker defects, providing a route to predicting the role of defects. The dataset is used to train and evaluate state-of-the-art ML models for the prediction of adsorption energies and atomic forces using approaches developed for the Open Catalyst Project.75 In addition, we include several out-of-domain datasets taken from the extended CoRE MOF database58,76 to evaluate the ability of the trained models to generalize to unseen topologies and linker chemistries. We expect that this dataset and the associated infrastructure will accelerate the development of MOF materials for DAC by providing a common dataset that far exceeds the size of any currently available dataset, establishing welldefined standards and benchmarks for development of new ML models, and providing accessible pre-trained ML models that enable routine prediction of mixed CO2 and H2O adsorption on MOFs at an accuracy that approaches DFT. The ODAC23 dataset is publicly available at the OpenDAC website i. All of our trained ML models and training code are available in the OCP repository ii. Scope and Structure of the ODAC23 Dataset Enormous numbers of hypothetical MOF structures exist, as illustrated by the hypothetical MOF database (hMOF) of Wilmer et al., which contains 138,000 structures.28 Several other MOF databases have been developed, including the Topologically Based Crystal Constructor (ToBaCCo) database of 13,512 MOFs with 41 unique topologies developed by Col\u00b4 on et al.30 Perhaps most importantly, Chung et al. developed the Computation-Ready, Experimental (CoRE) MOF database26 and its 2019 expansion27 from experimentally synthesized structures in the Cambridge Structural Database (CSD).77 The CoRE MOF database has been the foundation of many studies and extensions, including assignment of DFT-derived point charges,78 more thorough cleaning by removal of structures with misbonded or overlapping atoms,79 and the QMOF database of DFT-derived properties of many CoRE MOF structures.31 ihttps://open-dac.github.io/ iihttps://github.com/Open-Catalyst-Project/ ocp 3 \fFigure 1: Materials, adsorbates, tasks, and potential applications of the ODAC23 dataset. Images are randomly sampled from the dataset. The Open DAC dataset uses the CoRE MOF 2019 work as a starting point. This approach is beneficial because the data are readily available and the origin of each MOF in the database in an experimentally reported synthesis partially addresses concerns surrounding practicality when considering candidate MOFs for experimental testing. The CoRE MOF database has also been shown to be more chemically diverse than larger databases of hypothetical materials, which is beneficial for training transferable and generalizable ML models.80 The CoRE MOF 2019-ASR database contains 12,020 unique structures with accessible data. We only consider MOFs that contain fewer than 1,000 atoms in the unit cell due to computational cost. MOFs with a pore limiting diameter (PLD) of less than 3.3 \u02da A are excluded because a CO2 molecule (kinetic diameter of 3.3 \u02da A) may experience kinetic limitations in entering such small pores.27 With these limitations, 8,803 MOFs serve as our starting point for DFT relaxation. We used the Perdew-Burke-Ernzerhof functional81 with a D3 dispersion correction82,83 (PBE-D3) for all calculations. The generalized gradient approximation (GGA) approach was chosen over more accurate methods such as hybrid functionals or coupled cluster techniques because of the size and diversity of the dataset. Nazarian et al. showed that several different functionals and dispersion corrections perform similarly when making structural and partial charge predictions on a chemically diverse set of MOFs.78 We did not include a Hubbard U correction. Without this correction, PBE systematically overpredicts binding energies on openmetal sites, but U values are empirical and are difficult to find for every metal type.84 We ran calculations as spin polarized to capture spin effects associated with open metal sites. Our work ultimately seeks to push the baseline description of MOFs for DAC from classical FFs to the PBE-D3 level of theory, so we prioritized consistency across a very large number of calculations rather than absolute accuracy. The ODAC23 dataset consists of complete relaxation trajectories of CO2, H2O, and mixtures of CO2 and H2O on MOF structures derived from the CoRE MOF database. We include two classes of MOF frameworks: pristine frameworks and defective structures with missing linker defects systematically added.85 Pristine MOF structures are obtained from the CoRE MOF database without further modification. Approximately 66% of the pristine MOFs 4 \finclude frameworks with open metal sites. To test generalizability, we also included 114 \u201cultrastable\u201d MOFs from Nandy et al. created by fragmenting and recombining linkers and nodes from the original CoRE MOF database.76 The final dataset includes a total of 4,942 pristine MOFs and 3,470 defective MOFs with defect concentrations ranging from 1-16%. The MOFs contain a diverse set of 57 metals, with Zn, Cu, and Cd being the most common, and include a mix of monometallic (89%), bimetallic (10.7%), and trimetallic (< 1%) frameworks. The abundance of various metals is provided in Fig. S1, and the most common linkers are listed in Table S2. The adsorbates were initially placed using classical FFs and Monte Carlo sampling, with \u223c2-6 placements per framework. The selection of MOFs and adsorption configurations included in the final set are established by pragmatic constraints and practical considerations. In total, the dataset consists of over 170K converged adsorption energies and nearly 40M single point calculations, corresponding to over 400M core-hours of compute time. Details are provided in the Methods section. The Open DAC 2023 (ODAC23) dataset has been designed to allow training of ML models to approximate DFT calculations, similar to previous work in heterogeneous catalysis (OC20 and OC22).75,86 We use the same three task definitions used in the OC20 work. These tasks are briefly summarized below, and we refer the reader to the OC20 paper75 for more detailed descriptions. In each task, the input structure is a unit cell periodic in all directions containing a MOF with one or more adsorbates. The ground truth targets of forces, energies, and relaxed structures were all calculated using DFT. For energy targets, we used a non-relaxed adsorption energy: \u02dc Eads = Esystem \u2212EMOF \u2212nCO2ECO2 \u2212nH2OEH2O (1) where Esystem is the energy of the MOF and adsorbates, EMOF is the energy of the relaxed MOF structure without an adsorbate, ni is the number of adsorbate i and Ei is the energy of adsorbate i in the gas phase. The tilde on \u02dc Eads denotes that Esystem is not necessarily a relaxed structure. In specific cases where Esystem is relaxed, the tilde is dropped and the adsorption energy is denoted as Eads. More details are provided in the Methods section. The energies of these MOF+adsorbate structures were used to train models for three tasks: 1. Structure to Total Energy and Forces (S2EF) takes a structure as input and predicts \u02dc Eads of the system as well the force on each atom. This task is analogous to training a force field for all atoms in the system. 2. Initial Structure to Relaxed Energy (IS2RE) takes an initial guess structure as input and predicts Eads of its relaxed structure. This task is analogous to predicting an adsorption energy from an initial structure. 3. Initial Structure to Relaxed Structure (IS2RS) takes an initial guess structure as input and predicts the relaxed position of each atom. This task is analogous to geometry optimization. The S2EF task is the most general, and an S2EF model can be used to complete the IS2RS and IS2RE tasks. The dataset is organized by task and train/test splits. For each task, the data is split into a training set, testing set, and validation set. These in-domain (id) sets are randomly sampled from the full dataset derived from CoRE MOF, but are stratified by MOF framework to ensure that all defective structures are in the same set as the pristine structure from which they are generated. Four out-of-domain (ood) sets are included. The \u201cbig\u201d ood set corresponds to MOFs from CoRE with over 500 atoms in their unit cell (testing the ability to generalize to larger structures). The \u201clinker\u201d, \u201ctopology\u201d ood sets contain linkers and topologies not included in the training data, selected from MOFs in the ultrastable MOF dataset of Nandy et al.76 The \u201clinker and topology\u201d ood set contains MOFs from the ultrastable MOF dataset that contain both unseen linkers and topologies. The number of 5 \fTable 1: Overview of ODAC23 dataset organised by dataset split, number of MOF frameworks, and number of DFT calculations. Split # Pristine MOFs # Defective MOFs # Total MOFs # Total DFT Relaxations # Total DFT Single Points train 4,537 3,287 7,824 162,224 35,871,295 val 121 71 192 3,998 839,565 test-id 120 93 213 4,669 973,515 test-ood (big) 66 19 85 1,768 381,219 test-ood (linker) 28 0 28 1,182 287,125 test-ood (topology) 55 0 55 1,612 472,256 test-ood (linker & topology) 15 0 15 579 158,773 Total 4,942 3,470 8,412 176,032 38,983,748 MOF structures and DFT calculations in each set is provided in Table 1, and a more detailed breakdown based on adsorbate type is provided in S1. Fig. 2 illustrates this detailed distribution across adsorbate types split by task. Further details are described in the Methods section. Identification of Selective CO2 Adsorption Sites We used our DFT calculations to directly search for MOFs that are potentially interesting for DAC following the criteria suggested by Findley and Sholl12 that the adsorption energy of CO2 is < \u22120.5 eV (with our sign convention, more negative binding energies correspond to more favorable binding) and that the adsorption energy of CO2 needs to be more favorable than that for H2O. Materials not satisfying the first criterion are unlikely to bind sufficient quantities of CO2 at the dilute concentrations relevant for DAC, and materials not satisfying the second criterion are likely to adsorb far more water from air than CO2. In the following analysis, we compared the lowest adsorption energy of all computed configurations for each MOF + adsorbate case. We neglected cases with |Eads/(nCO2 +nH2O)| > 2 eV because we suspect these cases are unphysical. Fig. 3a and b compare the CO2 and H2O adsorption energies in each pristine and defective MOF from our DFT calculations. As expected, most of the MOFs bind water more favorably than CO2. However, 135 of the 5,079 pristine MOFs bind CO2 strongly and have higher affinity for CO2 than for H2O. The top 10 pristine MOFs identified by our DFT calculations with the highest values of |Eads(CO2) \u2212Eads(H2O)| are tabulated in Table S3. Several screenings of the CoRE MOF database for CO2 capture in the presence of water have been conducted previously.87,88 Here, we compare our promising MOFs with two previous studies where the adsorption energies of CO2 and H2O in CoRE MOFs are available. Findley and Sholl performed a similar screening of CoRE MOFs using FF methods, finding no cases that satisfied the criteria stated above.12 The observation that our DFT calculations of analogous quantities identified many interesting materials suggests that the generic FFs used previously are insufficiently accurate. Kancharlapall and Snurr recently screened the CoRE MOF 2019 database with a combination of FF and DFT calculations, using somewhat different selection criteria.89 Kancharlapall and Snurr also found that FF-based calculations failed to identify MOFs that satisfy our criteria. They further analyzed a subset of their most promising structures using DFT, with a slightly different workflow than we use for ODAC23. We find that 17 materials identified by Kancharlapall and Snurr also appear in the ODAC23 dataset, though we find that 7 of these materials bind H2O more strongly than CO2 and the remaining 10 MOFs bind CO2 weakly (Eads(CO2) \u2265\u22120.5 eV), indicating that they may not be promising for DAC. In addition to considering the adsorption of single CO2 and H2O molecules, we also used DFT to to probe the co-adsorption of CO2 and H2O in MOFs. With the resulting co-adsorption energies, we computed the adsorbate-adsorbate interaction energies associated with removing both molecules from the co-adsorbed state, denoted E1st inter mol, for each 6 \fFigure 2: Distribution of the number of MOF+adsorbate DFT calculations for the (a) S2EF and (b) IS2RS/IS2RE tasks on a logarithmic scale. The horizontal lines emphasize the size of the dataset. MOF using equation (6). For the 10 MOFs listed in Table S3 there are three distinct scenarios for this quantity. In a simple case like ZIDBEV, E1st inter mol = 0 eV is small relative to the single molecule adsorption energies, so co-adsorption can be approximated in a simple way as separate adsorption of the two molecules. For MOFs with negative adsorbateadsorbate interaction energies like IMAGAG (E1st inter mol = \u22120.64 eV), co-adsorption of CO2 and H2O is strongly favored relative to adsorption of the individual molecules. Positive adsorbate-adsorbate interaction values such as those seen for IPIDUH (E1st inter mol = 1.04 eV) and TUGTAR (E1st inter mol = 0.51 eV) indicate the co-adsorption is much less favorable than adsorption of isolated molecules. In some cases the first adsorbate-adsorbate interaction energies are strongly nonzero (e.g. KOQLUZ, E1st inter mol = \u22122.31 eV), suggesting that rearrangement of the MOF structure occurred in the co-adsorbed case that was not observed for the individual adsorbed molecules. For the CO2 + 2H2O configurations, we also computed the second adsorbate-adsorbate interaction energy using equation (7). This energy is small or negative for all of the 10 promising MOFs listed in Table S3. One example, LEWZET, shows an extremely negative second adsorbate-adsorbate interaction energy of \u22125.48 eV; this occurs because of significant distortion in the relaxed MOF that occurs due to adsorption of a second water molecule. We note that these effects cannot be explored in existing FF-based searches of MOFs, which assume that the MOF structure is unperturbed by adsorbates. It would be challenging, however, to draw in depth conclusions about selec7 \fFigure 3: Parity plots showing DFT-calculated CO2 and H2O adsorption energies in (a) pristine and (b) defective MOFs. (c-f) MOF examples with common features of the promising MOFs. tion of MOFs from a limited number of DFT calculations. The complexities associated with the changes in MOF frameworks during coadsorption and the challenges with sampling the many possible placements of co-adsorbed states both point to the need to be able to derive FFs or ML models that allow rapid assessment of large numbers of states to provide a thorough description of co-adsorption. Our results also include the first large collection of adsorbed molecules in defective MOFs relaxed with DFT. The cell volume of most of the MOFs decreased after introducing defects (Fig. S2a). From the 3,628 defective MOFs, we found 107 defective MOFs with CO2 adsorption energy greater than water (Fig. 3b). The top 10 defective MOFs, ranked in the same way as the pristine materials, are listed in Table S4. Defects play an important role in the adsorption of water and CO2. For example, pristine TIDLID has adsorption energy of \u22121.10 eV for CO2 and \u22120.52 eV for H2O (Fig. S2b), but defective TIDLID was no longer considered promising because the porous structure collapsed and the PLD was smaller than 3.3 \u02da A (Fig. S2c). The defect concentration was not strongly correlated with the difference in adsorption energies associated with the presence of defects (Fig. S3). The average differences of CO2 adsorption energy were nearly zero for all defect concentrations, and adding defects to MOFs resulted in slightly more favorable water adsorption on average. However, the effect of defects on adsorption energies differs greatly from case to case. In Fig. 4a to d, defects in QOVSOL resulted in more favorable H2O adsorption and less favorable CO2 adsorption, making it no longer a promising cadidate for DAC. On the other hand, our calculations with defective MOFs show that the defects in some of these materials can create interesting adsorption environments for DAC. We found multiple cases where pristine MOFs would not be selected based on the criteria defined above, but the defective material is a promising candidate. Fig. 4e to h show one example of POLDUQ. Our observations are broadly consistent with previous experimental and simulation results for 8 \fFigure 4: Examples showing different impacts of the defects in MOFs. The defects generated are shown in red squares. Negative impact of defects on DAC (a-d): Defective QOVSOL with a defect concentration of 0.12 shows less favorable CO2 adsorption (a and c) and stronger H2O adsorption (b and d). Positive impact of defects on DAC (e-g): The H2O adsorption is slightly more favorable in defective POLDUQ with a defect concentration of 0.06 (f and h), but the CO2 adsorption in much stronger at the defect site (e and g). CO2 adsorption in UiO-66,90,91 and enhanced CO2 adsorption in Cu-BTC due to water coordinated to OMS.92 Although defects are capped with water or hydroxyl groups in most cases, it is also possible for defects to create OMSs. The diversity of possibilities illustrates the need for accurate and efficient methods to rapidly explore the many configurations and effects that can exist in defective MOF structures. It is interesting to ask what motifs or attributes give MOFs adsorption energies that are favorable for DAC. Previous research has suggested several characteristics of good candidates for this application. Boyd et al. identified three favorable characteristics: parallel aromatic rings with spacing of approximately 7 \u02da A, metal-oxygen-metal bridges, and open-metal sites.11 The presence of uncoordinated N atoms has also been proposed as a contributing factor to strong CO2 adsorption.93,94 We examined these four characteristics (Fig. 3c-f) in our list of promising MOFs: 224 of the 241 of the promising MOFs can be characterized by at least one of these characteristics, confirming their importance. The ODAC23 dataset contains 251 pristine and 267 defective MOFs with an amine functional group. Of these, 7 MOFs (2 pristine and 5 defective) were found to be promising. Structure files of the promising MOFs and the code for promising MOF analysis are available in our open-source repository on GitHubiii. Although the structures in the CoRE MOF set were derived from experiments, it is important to be cautious in concluding that every structure in this dataset is in fact a real material. In developing the CoRE MOF 2019 database, automatic cleaning procedures iiihttps://github.com/Open-Catalyst-Project/ odac-data/tree/main/promising_mof 9 \fTable 2: 5 pristine MOFs suitable for synthesis on the basis of ODAC23 calculations and manual evaluation of original synthesis reports. MOF Eads(CO2) Eads(H2O) PLD LCD Metal Characteristics Exp. CO2 Loading (mmol/g) # of Citations OMS PAR M-O-M Uncoordinated N 150 mbar 1 bar ODIXEG -0.94 -0.24 7.80 10.4 Zn \u2713 \u2713 5695 QOVSOL -0.93 -0.63 3.67 6.21 Cd \u2713 \u2713 0.1 (298 K) 0.2 (298 K)96 3597 QEFNAQ -0.57 -0.32 4.72 6.03 Cu \u2713 \u2713 0.4 (293 K) 1.0 (293 K)98 27299 FECXES -0.64 -0.39 6.59 10.83 Cu \u2713 \u2713 1.6 (273 K) 6.3 (273 K)100 56100 DITYOW -0.60 -0.36 4.79 4.86 Cu \u2713 \u2713 52101 Table 3: 5 defective MOFs suitable for synthesis on the basis of ODAC23 calculations and manual evaluation of original synthesis reports. MOF Defect conc. Eads(CO2) Eads(H2O) PLD LCD Metal Characteristics Exp. CO2 Loading (mmol/g) # of Citations OMS PAR M-O-M Uncoordinated N 150 mbar 1 bar POLDUQ 0.06 -0.70 -0.36 5.09 5.27 Cu \u2713 \u2713 12102 CUGVUW 0.16 -1.14 -0.82 3.41 5.64 Cu \u2713 \u2713 24103 PEPKOL 0.08 -0.62 -0.35 3.46 3.92 Ni \u2713 444104 SUJNUH 0.12 -0.93 -0.68 6.62 7.08 Cu \u2713 1.3 (195 K) 2.2 (195 K)105 77105 LUYHAP 0.16 -0.58 -0.37 8.39 12.35 Cu \u2713 3.1 (298 K),106 2.5 (296 K),107 5.2 (270 K)107 158106 were applied to experimentally reported crystal structures, including the removal of solvent molecules and the resolution of partial occupancies. Although this procedure was generally effective, there are cases where it was too aggressive. We observed linker removal and wrong partial occupancies in a number of the MOFs listed above. Charge-balancing ions were also removed for MOFs denoted \u2018charged\u2019 in Table S4. For each MOF listed above, we manually compared the MOF structures retrieved from the CoRE MOF 2019 database and the original publications. From this analysis, we curated a selection of promising MOFs that are completely charge neutral and where the CoRE MOF structure is fully consistent with the original experimental data. On the basis of current DFT data and manual analysis, we expect these to be the most promising MOFs for experimental synthesis and testing. These MOFs are listed in Tables 2 and 3. The tables include the number of times the original synthesis report has been cited, since this has been suggested as a proxy for the ease of synthesis/reuse of a material, and the tables indicate which of the four promising MOF characteristics mentioned above appear in each material. The available CO2 adsorption isotherms from experimental measurements of these MOFs show relatively strong CO2 adsorption at low partial pressures,108 which is consistent with the implications of our calculations. Evaluation of the Accuracy of Classical Force Fields Our large library of DFT calculations allowed us to further investigate the accuracy of existing classical FFs against our DFT calculations. We focus here on the energy of interaction between adsorbed molecules and MOFs, since this is the key calculation underlying previous high throughput assessments of MOFs for CO2 adsorption. Specifically, we considered a \u201cstandard\u201d FF for adsorption in MOFs that combines the UFF4MOF,109\u2013111 TraPPE,112 and SPC/E113 FFs for atoms in the MOF, CO2, and H2O, respectively. Coulombic interactions were defined using DDEC point charges assigned to MOF atoms from our DFT calculations.114 Further technical details are provided in the Methods section. We computed the interaction energy for 51,478 DFT-relaxed MOF + adsorbate systems using the FF and DFT. This is analogous to the energies in the S2EF task. Using interaction energies for this comparison rather than adsorption energies is consistent with previous FF-based studies that assume framework rigidity.14,28,115,116 The ODAC23 dataset also includes information on MOF deformation associated with the presence of adsorbates, and future work could explore how accurately existing FFs for MOF atoms describe these effects. The results of our FF calculations and comparisons to DFT interaction energies are shown 10 \fin Fig. 5. All structures in this comparison contained only one adsorbate molecule (either CO2 or H2O), and we omit 226 structures with DFT interaction energies outside the range of [-2, 2] eV since we suspect these structures are unphysical. We also omit 716 structures with reasonable DFT energies because their FF predictions also fall ouside of [-2, 2] eV. This is done to avoid heavily skewing the subsequent discussion and is revisited at the end of this analysis. Fig. 5a shows that in many cases the difference between the classical FF and DFT is less than 0.25 eV and that many of the DFT results can be described as physisorption. Van der Waals (vdW) interactions dominate within the physisorption regime of \u22120.5 \u2264EDFT int \u22640 eV, and if interaction energies are restricted to this range then the mean absolute error (MAE, or simple error) between FF and DFT energies is 0.06 eV. This indicates that the physics-based FFs we tested are quite well adapted to predict the interaction energy when physisorption is dominant. The results in Fig. 5b\u2013d provide a less promising view of the classical FF. The error between the FF and DFT calculations scales approximately linearly with the DFT energy outside the physisorption regime, showing that the FF predicts a physisorption energy even when DFT indicates that chemisorption is occurring. The minima in these graphs around \u22120.4 eV again indicate that the FF is only capable of accurately predicting physisorption. In the chemisorption regime from \u20132 to \u20130.5 eV, the MAEs for CO2 and H2O are 0.29 eV and 0.39 eV, respectively. Fig. 5b shows the number of points and average error as a function of DFT interaction energy. Although relatively few points outside the physisorption regime exist, the FF interaction energy errors increase drastically with the magnitude of the interaction energy. Many interesting chemistries that are beneficial for DAC occur due to chemisorption (e.g., CO2 binding more strongly than H2O). These cases would be missed by a classical FF that is unable to model chemisorption. There are also many instances in which the FF energy prediction is substantially larger than the DFT-calculated energy. We attribute these to cases involving chemisorption where the adsorbate is close to the framework and therefore returns very large Lennard-Jones energies. That is, the FF exhibits unstable behavior here because very slight changes in geometry cause large spikes in energy predictions. An additional takeaway from Fig. 5 is that H2O is significantly more challenging to model than CO2. This is consistent with the fact that physics-based water models are complex and are themselves the subject of a rich body of literature.117 We found that the error in interaction energy calculations within the [\u22122, 2] eV domain involving H2O (0.19 eV) was more than triple that for CO2 (0.05 eV). The vast majority of unstable FF calculations involved H2O and not CO2. Selecting and implementing an appropriate water model is a non-trivial task that further complicates the use of classical FFs for material screening. Finally, there are a number of cases where the FFs predict very large interaction energies, with the maximum error being 187.2 eV. These cases typically correspond to dissociative adsorption, where the FF is not an appropriate model. Fig. S5 presents the binned FF errors as a function of the DFT interaction energy for all configurations with a DFT interaction energy in [-2, 2] eV, irrespective of whether the FF interaction energy falls within this range. Comparison with Fig. 5b shows that the 716 cases with reasonable DFT energies but unreasonable FF energies drastically increase the error, and that catastrophic failures (e.g. errors >10 eV) begin to dominate when the DFT adsorption energies are stronger than 1 eV. The large errors cause the FF MAE for all structures to be quite large at 0.28 eV. If the MAE is calculated only for cases where the FF interaction energy is in the range of [-2, 2] eV, then the classical FF performs reasonably well with an interaction energy MAE of 0.11 eV across 50,536 calculations. Overall, the results indicate that the FF performs well for physisorption, but fails to capture strong chemical interactions that are likely critical for DAC. 11 \fFigure 5: Comparison of adsorbate interaction energies calculated with FFs and DFT. (a) Histogram of energy differences between FF and DFT for 29,644 CO2 calculations (red) and 20,892 H2O calculations (blue). (b) Binned errors and DFT interaction energy distributions split by adsorbate. (c,d) Absolute difference between FF and DFT plotted versus DFT interaction energy for CO2 and H2O, respectively. Training and Analysis of Machine Learning Models We begin by training and benchmarking models for the S2EF task, since it is the most general. We tested six graph neural network (GNN) architectures for this task: SchNet,118 DimeNet++,119 PaiNN,120 GemNet-OC,121 eSCN,122 and EquiformerV2.123 We chose models that performed well on the OC20 and OC22 benchmarks since those datasets and tasks are most similar to ours. These models use GNNs containing equivariant or non-equivariant operations to compute energies and forces. All models were trained to minimize the following objective function for forces and energies: L = \u03bbE X i | \u02c6 Ei \u2212Ei| + \u03bbF X i,j 1 3Ni | \u02c6 Fij \u2212Fij|p (2) where the loss coefficients \u03bbE and \u03bbF are used to trade-off the force and energy losses. Ei and \u02c6 Ei are, respectively, the ground truth and predicted energies of system i, and Fij and \u02c6 Fij are, respectively, the ground truth and predicted forces for the j-th atom in system i. The number of atoms in system i is denoted by Ni. p is the order of the norm \u2013 SchNet and DimeNet++ used p = 1, while the other models used p = 2. 12 \fWe used the same model sizes as those used for OC20 (Table S5). To prevent overfitting due to the smaller size of the data set, we adjusted the weight decay for each model. We also slightly adjusted the initial learning rates, batch sizes, learning rate schedules, and the loss coefficients \u03bbE and \u03bbF. All error metrics are reported for test sets that were not included in the training and optimization process. Additional information can be found in the Methods section. The results of all ML models on the S2EF task are presented in Table S6, revealing that GemNet-OC, eSCN, and EquiformerV2 have the best performance. Fig. 6 shows a radar plot comparing these models, indicating thatEquiformerV2 (large) achieved the best results for both forces and energies, with a force MAE of 8.20 meV/\u02da A and energy MAE of 0.15 eV on the in-domain test set. The eSCN and GemNet-OC models also performed well, with force MAEs of less than 10 meV/\u02da A and energy MAEs of under 0.17 eV. The models\u2019 relative performance was consistent with their performance on the OC20 and OC22 datasets, suggesting that improvements in model architecture generalize to various materials datasets. Next, we consider how the models generalize to out-of-domain test sets. The results in Table S6 and Fig. 6 demonstrate that the EquiformerV2 (large) model outperforms the other models on most metrics for all out-ofdomain sets. The ML models show only a slight decrease in performance on the test-ood(b) and test-ood(l) sets, suggesting that they generalize well to larger graphs or to new linker chemistry. However, the energy predictions for the testood(t) and test-ood(lt) sets are substantially worse than the test-id set, although the force errors are similar to the other test sets. This could be due to errors in long-range vdW interactions for unseen topologies, since this is the main contribution that varies with topology. We also analyze the performance of the models on the more complex chemical environments of OMSs and defects. OMSs are significant for DAC as they can enable stronger CO2 adsorption.46 Classical FFs are known to be less accurate for MOFs with OMSs as they can cause high polarization in adsorbed molecules.46,124 Tables S7 and S8 compare GemNet-OC, eSCN, and EquiformerV2 on different subsets of the test-id split. Table S7 shows the performance across pristine MOFs with and without OMSs, and Table S8 compares the performance of the same models on pristine and defective structures. The ML models have similar force MAEs on the OMS and non-OMS sets, as well as the pristine and defective sets. However, the energy MAEs are lower for MOFs without OMSs or defects. This may be due to the stronger and more complex interactions at OMSs, or may be related to the relative abundance of different types of examples within the dataset. Fig. 7a analyzes the binned error for MOFs with and without OMSs, indicating that errors are slightly higher for OMS-containing MOFs in the chemisorption regime, suggesting that the ML models perform slightly worse at predicting the more complex chemical interactions at OMS sites. A direct comparison between classical FFs and ML models is not feasible because the architecture of the FFs makes it challenging to relax framework atoms. However, we can compare the S2EF adsorption energy errors to the interaction energy errors from FFs to gain insight, since both evaluate the ability to describe interactions between frameworks and adsorbates. We did this with 1,391 relaxed single-adsorbate configurations in the test-id set, which is a subset of the 50,536 structures that excludes all systems used in ML model training. For this reason, energy errors reported in this section may vary slightly from those in the evaluation of the accuracy of classical force fields. The energy MAE for EquiformerV2 (large) for these systems was 0.10 eV, while the MAE for the FF interaction energies on the same structures was 0.49 eV. It is clear that, on average, the best ML models outperform the classical FF models, even when only focusing on relaxed single-adsorbate geometries. However, a more detailed analysis reveals that the large FF error occurs due to a small number of large failures. The maximum force field error is 67.66 eV, compared to a maximum error of 1.23 eV for the EquiformerV2 (large) model. If 13 \fFigure 6: Radar plots for S2EF (a) energy and (b) force MAEs, (c) IS2RE energy MAEs, and (d) IS2RS AFbT for the top three best models \u2013 GemNet-OC (red), eSCN (blue), and EquiformerV2 (large, except in (c) where the lighter model is shown) (cyan). Dashed lines correspond to the relaxation approach for IS2RE; all other models are direct predictions. Axes correspond to different inand out-of-domain test sets, and are aligned so that the best result is closest to the origin of the plot in all cases. the analysis is restricted to the cases where force fields predict interaction energies in the range of [-2, 2] eV, the average errors are quite comparable, with MAEs of 0.10 eV for both. In the regime where adsorption energies range from 0.5 to 0 eV and physisorption is expected to be dominant, the FF performance becomes comparable to that of ML, with an MAE of 0.10 eV for the FFs and 0.09 eV for the ML models. A detailed analysis is provided in Fig. 7b, which indicates that ML models exhibit consistently lower errors in the chemisorption regime, in contrast to FF models, which fail for chemisorption. Given the importance of chemisorption in selective CO2 capture at low concentrations, this finding supports the need for ML models for DAC. See Fig. S4 for errors in the repulsive region. Next, we move to the IS2RE and IS2RS tasks, which evaluate the ability of ML models to directly predict the relaxed adsorption energy (IS2RE) and structure (IS2RS) from an initial guess of framework and adsorbate positions. The IS2RE task only predicts energy and is evaluated with the energy MAE (similar to S2EF) and the \u201cenergy within threshold\u201d 14 \fFigure 7: Binned errors and relative density of the number of points (solid lines) as a function of DFT adsorption energy for (a) ML predicted adsorption energies on open metal site (OMS) (red) and non-OMS (blue) and (b) interaction energies predicted by FFs (magenta) and corresponding adsorption energies predicted by ML (green) models. Compared to FFs, ML models are significantly more accurate in the chemisorption regime, and are comparable in the physisorption regime. Positive adsorption energies are omitted from the plot because they are rare and likely unphysical; plots with the full range of adsorption energies are provided in Fig. S4. (EwT) which evaluates the fraction of predictions within 0.02 eV of the DFT energy. The IS2RE task can be solved by training ML models to directly predict the relaxed adsorption energy from the initial structure (the direct method), or by running a structure relaxation with an S2EF model (the relaxation method). In the case of the relaxation approach, the task is identical to IS2RS, where the energy of the final structure is used as the IS2RE prediction. However, the metrics used to evaluate the IS2RS task are significantly different, since the goal is to compare structures. The metrics used are the average distance within threshold (ADwT), force below threshold (FbT), and average force below threshold (AFbT), with details provided in the Methods. Evaluating the IS2RS models is quite expensive since it requires performing a DFT single-point for each of the predicted relaxed structures. Therefore, we only evaluated the best 4 models (GemNetOC, eSCN, EquiformerV2, and EquiformerV2 (large)) and only computed DFT single-point energies on 500 randomly selected structures from each test split. For the IS2RE task, any S2EF model can be used for the indirect approach, so we evaluated all six S2EF models from this work using the model to perform structure relaxations with each model. The resulting structures are also used for the IS2RS task. In addition, we selected the best three models \u2013 GemNet-OC, SCN, and EquiformerV2 \u2013 and retrained them for the direct approach, with settings identical to the corresponding S2EF models unless otherwise noted. Fig. 6 and Table S9 show the results for the IS2RE task on each of the test splits. On the test-id set, the direct methods obtain an energy MAE around 0.18 eV and an EwT of over 10%. The relaxation approach with older S2EF models like SchNet, DimeNet++, and PaiNN perform worse than direct methods, while newer methods such as GemNetOC, eSCN, EquiformerV2, and EquiformerV2 (large) are marginally better than direct approaches. Similar to the S2EF task, we find that the performance of the ML models degrades marginally on the test-ood(b) or testood(l) datasets, while they degrade significantly on the test-ood(t) and test-ood(lt) datasets. This is true for both direct and relaxation-based approaches. Fig. 6 and Table S10 show the IS2RS results on each test split. The ADwT results are reasonably high for the test-id and testood(b) sets but degrade significantly for testood(l) and test-ood(t) sets. eowever, the results 15 \fon the DFT-based metrics (FbT and AFbT) indicate that the models achieve relaxed structures consistent with what would be obtained from DFT < 1% of the time in all cases (and 0% in many cases). This inconsistency between ADwT and (A)FbT has also been observed for OC2075 and indicates that the models need significant improvement to achieve the level of accuracy needed to replace DFT for the prediction of relaxed structures. However, the fact that the models are able to predict the energies of relaxed structures with reasonable accuracy in the IS2RE task is an encouraging sign, since the state of the art for high throughput MOF screening with force field is to assume that the structures are rigid. This assumption becomes particularly questionable in the case of defective MOFs or strong adsorption, indicating the need for models capable of accounting for relaxation effects. 250 500 1000 2000 4000 8000 Number of MOFs in training 8 12 16 Force MAE [meV/\u00c5] Model Gemnet-OC eSCN Equiformer V2 Figure 8: Force MAE on the test-id set for the top 3 S2EF models when trained on different amounts of training data. The lines show scaling laws obtained by fitting a line between log of the force MAE and log of the number of training MOFs for each model. It is clear that the ML models presented here demonstrate significant promise compared to the standard classical FF models. However, there are also obvious deficiencies. One advantage of ML models is that they tend to improve with more data. In particular, scaling laws for deep learning models relate model performance to a parameter like the number of model parameters or size of the training dataset. Scaling laws have helped to choose the optimal model and training parameters in several domains.125\u2013127 Fig. 8 shows the scaling laws for the ODAC23 dataset size, comparing the force MAEs of different models as a function of the number of MOFs in the training data. Consistent with previous work in other domains, we observe a power-law relationship between force MAE and the number of MOFs. This implies that we can continue to improve the performance of these models by including more training data. It is also interesting to note that equivariant models like EquiformerV2 and eSCN have better scaling properties than GemNet-OC, matching the findings of Batzner et al.128 This indicates that the use of more sophisticated model architectures is a promising route forward. Based on these scaling laws, a much larger number of MOFs would be required to achieve force MAEs of 3 meV/\u02da A (approaching the numerical error of DFT). An alternative strategy common in deep learning is to leverage similar datasets. This has proven useful in the Open Catalyst Project models,129 and we plan to explore this approach in future work. Another possible strategy is to develop model architectures that are tailored for the DAC application. In particular, the strong performance of FFs in the weak-binding regime suggests that incorporating information on vdW interactions into the model130 or \u2206-ML131 models may be promising strategies. Ultimately, we expect that improved model architectures, advanced transfer learning, and joint training techniques may provide a route to leveraging physical knowledge and other large atomistic datasets to improve performance on ODAC23, although we leave this as future work.132,133 16 \fImpact and Future Outlook The results of this study provide the most comprehensive DFT dataset of CO2 and H2O adsorption in MOFs available. Analysis of the resulting DFT calculations has shown that, contrary to the findings from FF-based studies, there are numerous MOF-based adsorption sites with strong and selective CO2 adsorption. A direct comparison of the DFT results to classical FFs provides the most comprehensive perspective to date on the accuracy of FFs. The results reveal that the FFs work well in cases where vdW interactions dominate but fail when stronger bonding is involved. These findings demonstrate that highthroughput screening with methods capable of treating chemisorption and framework distortion will be required to identify MOFs that can strongly and selectively bind CO2 under humid conditions. In addition, the work provides a benchmark for state-of-the-art ML models for CO2 and H2O adsorption in MOFs. The results indicate that the best performing GNN models, such as EquiformerV2, are capable of predicting adsorption energies with average errors of \u223c0.15 0.3 eV, and forces with errors of \u223c5-10 meV/\u02da A. Comparison with a classical FF shows that these ML models are more accurate outside the regime of vdW interactions. This, coupled with the importance of strong binding in identifying selective CO2 adsorption sites, suggests that these ML models have the potential to replace classical FFs as the standard approach in high-throughput MOF screening for DAC and other applications in separations and catalysis. Moving forward, it will be important to critically evaluate and improve ML models and associated datasets so that they can be applied to other steps in the computational sorbent selection process. For example, grand canonical Monte Carlo simulations are critical for predicting adsoprtion isotherms. The models here are untested for this task since they have not seen configurations with higher molecular loadings. Testing and improving the models will facilitate calculation of full single and multicomponent isotherms with accuracies that approach DFT. This is especially critical for the case of bicomponent CO2/H2O isotherms that are needed to predict the behavior of MOF materials in DAC process models. The complex mixture of vdW, hydrogen, and covalent bonding in H2O makes it difficult to accurately predict these bicomponent isotherms with existing methods, but the ML models presented here provide a promising foundation for future developments. Methods ODAC23 Dataset Generation A workflow diagram with details on the dataset generation workflow is provided in Fig. S6, and more details are provided in the sub-sections below. Structure relaxations DFT relaxations used the PBE exchange\u2013correlation functional81 with a D3 dispersion correction82 including Becke-Johnson damping and with spin polarization.83 Relaxations were performed with conjugate gradient methods with a step size of 0.01, and Gaussian smearing was used with a width of 0.2 eV. A plane wave cutoff energy of 600 eV to minimize effects of Pulay stress and a precision of 10\u22125 eV were used. All simulations were performed in the Vienna Ab Initio Simulation Package (VASP) v5 software with a 1x1x1 k-point grid.134 We relaxed all 8,803 CoRE MOF pristine structures using DFT as described above before generating defective structures and placing adsorbate molecules, and a total of 5,079 MOFs converged. DFT convergence failures are due to a variety of issues. For example, Chen and Manz identified several failure modes in CoRE MOF input files beyond overlapping atoms (3.5% of all screened structures), including isolated atoms (7.8%), misbonded hydrogens (1.3%), and over/underbonded carbons (15.3%).79 Examples of VASP convergence issues were large systems that took too long or ran out of memory (\u223c10% of screened structures) and numerical errors pertaining to 17 \fHamiltonian diagonalization. We noticed several converged structures with very high initial formation energies (>3 eV/atom). All initial inputs of converged structures were thus screened for overlapping atoms resulting from imperfect solvent removal processes and partial occupancies in the CoRE work. We used the published list of effective atomic radii by Chen and Manz for atom typing; a structure failed if any atom pairs were less than half the sum of their respective atomic radii apart.79 In total, 161 structures failed and were excluded from further analysis due to overlapping atoms and unphysically large initial formation energies. Defective MOF generation We expanded the pristine set of MOFs from CoRE MOF by introducing missing linker defects using the methods introduced recently by Yu et al.85 This approach requires identification of the linker and nodes in each MOF, a task completed using the algorithm MOFid developed by Bucior et al.135 Out of 5,079 pristine MOFs that converged in our DFT calculations, we successfully identified the nodes and linkers of 4,780 MOFs. In each MOF we created structures with different defect concentrations from 0.01 to 0.16, where the defect concentration is defined as the number of removed linkers divided by the total number of linkers. For MOFs that have multiple types of linkers, we generated corresponding defective structures by removing one kind of linker at a time. OMSs were capped using either a water molecule if the removed linker is charge neutral or hydroxyl(s) if the removed linker was charged to create structures that have no overall charge. In total, 16,358 distinct structures were generated and relaxed by DFT, and 6,340 of them converged. We only kept the relaxed structures with PLD > 3.3 \u02da A, and the final set of defective MOFs contained 3,470 frameworks. Adsorbate placement In each relaxed MOF (either pristine or defective) structure, we placed an adsorbate(s) using non-bonded pairwise interactions defined by one of the classical FFs by the RASPA 2.0 package.33 FF parameters for framework atoms and adsorbates (CO2 and H2O) were defined by the United Force Field (UFF)109 and TraPPEUnited Atom FF,136 respectively. Specifically, we adopted the rigid TIP5P model for H2O molecules.137,138 The Lorentz-Berthelot mixing rules and a tail correction with a cutoff radius of 14 \u02da A were used to define the LennardJones interactions between MOFs and adsorbates. Coulombic interactions were considered when partial charges of the framework atoms were available by the DDEC method. We collected configurations of [MOF+CO2] or [MOF+H2O] from every 10,000 Monte Carlo cycles with the same translation, rotation, and reinsertion probabilities. We took two approaches to ensure that structures do not have duplicated positions and exhibit diversity in structures: (i) energy matching and (ii) random sampling. The energy matching approach notes that different non-bonded interaction energies will correspond to different configurations. Starting from the minimum observed energy, we sampled configurations in 5 kJ/mol intervals until the non-bonded interaction energy reached a threshold (\u221215 kJ/mol and \u22125 kJ/mol for CO2 and H2O, respectively). If the minimum energy was greater than the threshold, we included only the configuration with the minimum energy. No configuration was added for cases where the minimum energy was > 0 kJ/mol. This resulted in having 0-9 adsorbate placements for each MOF structure, leading to a diverse collection of more than 10,000 MOF+adsorbate configurations per adsorbate by the energy matching approach. For random sampling, we randomly chose 2 configurations from the collection of 10,000 cycles and added these configurations to the set selected from energy matching, leading to more than 16,000 MOF+adsorbate configurations per adsorbate by the random sampling approach. Several MOF structures were further excluded from the dataset because their pore size shrunk during the relaxation, making it impossible for RASPA to place an adsorbate in their pores. We manually added 158 converged [MOF+H2O] configurations to position water molecules closer 18 \fto OMSs. This was done for MOF structures where a CO2 molecule was near OMSs without nearby water or when they were identified as promising but with fewer than 4 H2O placements. For co-adsorption cases, we used a similar strategy to obtain configurations. Coadsorption studies include the following examples: [1CO2+1H2O] and [1CO2+2H2O]. In each study, we inserted all of the participating molecules into each empty MOF structure. Since we are interested in the behavior of CO2 in the presence of water, we discarded configurations where the distance between the centersof-mass for any pair of adsorbate molecules was greater than 5 \u02da A. For MOF structures whose primitive cells were too small to place multiple adsorbates in the pores, the primitive cell was repeated to form a bigger supercell. Whether a supercell was used to save the configuration can be found on GitHub.iv Both energy matching and random sampling strategies were applied to multi-adsorbate configurations. The energy threshold was set to be \u22125 kJ/mol for all molecule combinations in case of energy matching approach. After placing adsorbates, we perform DFT structure relaxations on each adsorbate-MOF configuration. We used the same DFT settings as the MOF relaxations but with fixed unit cell parameters. Out-of-Domain MOF selection The ODAC23 dataset contains four out-ofdomain (OOD) test sets in addition to the indomain test set to evaluate the ability of ML models trained on the ODAC23 dataset to new topologies, new linker chemistries, and to larger MOFs. The test-ood (big) or test-ood(b) test split only contains MOFs with over 500 atoms in their unit cells. Testing on this set allows us to assess how well our models generalize to larger MOFs than those contained in the training set. The other three OOD test sets were designed to study how our ML models generalize to new ivhttps://github.com/Open-Catalyst-Project/ odac-data/tree/main/supercell_info.csv chemistries and topologies not present in CoreMOF. To create these splits, we sampled structures from the \u201cultrastable MOF database\u201d developed by Nandy et al.76 To create our OOD test sets, we selected the ultrastable MOFs with less than 500 atoms, and contained either novel linkers or topologies not present in the rest of our dataset. This allowed us to create three OOD test sets: the test-ood (linker) set contains novel linkers but known topologies, the test-ood (topology) set contains novel topologies but known linkers, and the test-ood (linker & topology) set contains both novel linkers and novel topologies. We abbreviate these three sets as test-ood(l), test-ood(t), and test-ood(lt) respectively. We used the MOFid library135 to identify the organic linkers and topologies. We believe that the inclusion of these OOD sets, which are biased to a property not related to the DAC application, provides a useful test of the generalizability of our trained ML models. Energy Definitions We defined three energy definitions for analysis of our work. Throughout this section, EB A denotes the total energy of a system of interest A calculated by a method B. If not specifically noted, B defaults to DFT. Energies are a function of atomic coordinates (C) either from DFT relaxation (rrelax C ) or from a single-point DFT calculation (rsingle C ). Adsorption energy The adsorption energy can be defined as: Eads = Esystem(rrelax system) \u2212EMOF(rrelax MOF) \u2212nCO2ECO2(rrelax CO2 ) \u2212nH2OEH2O(rrelax H2O ) (3) where Esystem is the DFT energy of the MOF + adsorbate system, EMOF is the reference DFT energy of the relaxed standalone MOF, nCO2 and nH2O denote the number of CO2 and H2O molecules in the system respectively, and ECO2 and EH2O are the gas phase energies of the corresponding molecules. In equation (3), the structure of the MOF in the system and the standalone MOF are from 19 \fseparate DFT relaxations. When a supercell was created during adsorbate placement, the reference energy EMOF was computed by performing an additional DFT relaxation on the supercell without the adsorbate. The inclusion of adsorbate molecules during relaxation broke framework symmetry and resulted in lower energy empty MOF configurations in a small number of cases. We conducted a second round of relaxations on these empty MOFs and successfully found lower energy states for 690 pristine and 625 defective MOFs. These lower energy states were used as the reference energy for all adsorption energy calculations. We removed all configurations where the adsorption energy was found to be < \u22122 eV per adsorbate. We also define \u02dc Eads for which we obtained the total energy of the current MOF + adsorbate configuration instead of seeking its relaxed state. This can be expressed as: \u02dc Eads = Esystem(rsingle system) \u2212EMOF(rrelax MOF) \u2212nCO2ECO2(rrelax CO2 ) \u2212nH2OEH2O(rrelax H2O ) (4) \u02dc Eads indicates how far the current state of a MOF + adsorbate system is from its reference state and is used as one of the main targets in our ML studies. In the case that \u02dc Eads is computed from the single-point DFT calculation of a relaxed structure (i.e. rsingle system = rrelax system) it is equivalent to Eads. Interaction energy The interaction energy is defined as: Eint = Esystem(rrelax system) \u2212EMOF(rrelax system) \u2212nCO2ECO2(rrelax system) \u2212nH2OEH2O(rrelax sytem) (5) where Eint was calculated either by DFT (EDFT int ) or the classical FF (EFF int ). Interaction energies calculations were performed only on the relaxed MOF + adsorbate configurations using single-point DFT. For simplicity, interaction energies were computed only in single adsorption cases, thus nCO2 +nH2O = 1 in equation (5). Adsorbate-adsorbate interaction energy The adsorbate-adsorbate interaction energy quantifies interactions between adsorbates in co-adsorption cases and is defined as: E1st inter mol = Eads(CO2 + H2O) \u2212Eads(CO2) \u2212Eads(H2O) (6) E2nd inter mol = Eads(CO2 + 2H2O) \u2212Eads(CO2 + H2O) \u2212Eads(H2O) (7) where the number of each adsorbate is shown in parentheses. The first adsorbateadsorbate interaction energy (E1st inter mol) shows the adsorbate-adsorbate interactions between CO2 and H2O and the second adsorbateadsorbate interaction energy (E2nd inter mol) shows the adsorbate-adsorbate interactions induced by introducing a second H2O molecule. Evaluation Metrics For all machine learning models we use the same evaluation metrics used for OC20. We briefly describe the metrics used for each task in this section, but refer the reader to the OC20 paper75 for more details. Structure to Total Energy and Forces (S2EF) : The S2EF task is evaluated on the accuracy of force and adsorption energy predictions through the following metrics. For these metrics E \u2261\u02dc Eads computed by equation 4. \u2022 Energy MAE: Mean absolute error between the predicted energy and the ground truth energy : EMAE = 1 N X i | \u02c6 Ei \u2212Ei|, (8) where Ei and \u02c6 Ei are the ground truth and predicted energies of system i and N is the total number of systems. \u2022 Force MAE: Mean absolute error between 20 \fpredicted and ground truth DFT forces: FMAE = 1 N X i 1 Ni X j \u2225\u02c6 Fij \u2212Fij\u22251, (9) where Fij and \u02c6 Fij are the predicted and ground truth forces on the j-th atom of system i and Ni is the number of atoms in system i. \u2022 Force Cos: Cosine similarity between the predicted and ground truth forces. \u2022 Energy and forces within threshold (EFwT): The fraction of energies and forces that are respectively within 0.02 eV and 0.03 eV/\u02da A of the ground truth DFT values. Initial Structure to Relaxed Energy (IS2RE) : The IS2RE task is evaluated on the accuracy of relaxed energy predictions using the following metrics. For these metrics E \u2261Eads computed by equation 3. \u2022 Energy MAE: Mean absolute error between predicted energy and the ground truth DFT energy of the relaxed state. \u2022 Energy within Threshold (EwT): The fraction of energies within 0.02 eV of the DFT relaxed energy. Initial Structure to Relaxed Structure (IS2RS) : The IS2RS task is evaluated on whether the predicted relaxed structure is close to a local minimum in the energy landscape using the following metrics. \u2022 Average Distance within Threshold (ADwT): Distance within Threshold (DwT) is the percentage of structures with an atom position MAE below a threshold \u03b2. ADwT averages DwT across thresholds ranging from \u03b20 = 0.01 \u02da A to \u03b21 = 0.5 \u02da A in increments of 0.001 \u02da A. \u2022 Force below Threshold (FbT): Percentage of relaxed structures with maximum DFT calculated per-atom force magnitudes below a threshold of \u03b1 = 50 meV/\u02da A. This is only computed for structures that satisfy the DwT criterion with \u03b2 = 0.5 \u02da A. \u2022 Average Force below Threshold (AFbT): FbT averaged over a range of thresholds: \u03b10 = 10 meV/\u02da A to \u03b11 = 400 meV/\u02da A in increments of 1 meV/\u02da A. As the systems in ODAC23 do not contain any fixed atoms, per-atom metrics like Force MAE, ADwT, FbT and AFbT are computed over all atoms. Note that a new single point DFT calculation is required to evaluate FbT and AFbT on a given data point. Classical Force Fields All classical FF calculations in this work used the readily available MOF extension to the ubiquitous UFF force field (UFF4MOF)110,111 in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS).36 Topology files were generated using LAMMPS Interface.139 CO2 and H2O molecules were described using the TraPPE112 and SPC/E113 models, respectively. The SPC/E model was chosen to avoid challenges related to the geometry of massless sites in newer models such as TIP5P,137 which was used for adsorbate placement but can be cumbersome to place in high-throughput FF calculations. Electrostatic interactions were described using DDEC framework point charges provided as part of the ODAC23 dataset,114 and long-range interactions were computed using an Ewald summation with a force tolerance of 10\u22125 kcal/mol/\u02da A. The cutoff for all pairwise interactions was 12.5 \u02da A. Periodic boundary conditions were applied in all calculations, and tail corrections were not applied. Code for FF calculations is available in our open-source repository on GitHubv. ML Models Various ML FF models have been proposed for molecular and material tasks over the last vhttps://github.com/Open-Catalyst-Project/ odac-data/tree/main/force_field 21 \ffew years.118,120\u2013123,140\u2013142 Here, we benchmark a subset of the state-of-the-art models on our tasks. All of our models were implemented using PyTorch143 and the code is available in our open-source repository on GitHubvi. For S2EF, we trained SchNet,118 DimeNet++,119 GemNet-OC,142 PaiNN,140 eSCN,122 and EquiformerV2123 models. We trained 2 versions of the EquiformerV2 model \u2013 a small 31M parameter model and a large 153M parameter model. The list of models used is summarized in Table S5.Edges were computed on-the-fly using a nearest-neighbor search with a cutoff of 8 \u02da A and a maximum of 50 neighbors for SchNet, DimeNet++ and PaiNN, and a maximum of 20 neighbors for eSCN and EquiformerV2. GemNet-OC uses different cutoffs for different types of interaction triplets and quadruplets. These S2EF models can then be used to run machine learning relaxations to solve the IS2RE and IS2RS task. We benchmarked the top performing S2EF models \u2013 GemNet-OC, eSCN and EquiformerV2 \u2013 to run these ML relaxations using the L-BFGS optimizer for 125 steps or until the magnitude of the predicted forces on each atom was less than 0.05 eV/\u02da A. IS2RE can also be solved by directly predicting the energy from the initial system, which we call direct IS2RE prediction. We trained GemNet-OC, eSCN and EquiformerV2 models on the direct IS2RE task. Acknowledgement The authors acknowledge Larry Zitnick (Meta), Joe Spisak (Meta), and Julius Kusuma (Meta) for helpful discussions about the project, Muhammed Shuaibi (Meta) for feedback on dataset construction, and Kyle Michel (Meta) for his help with the compute infrastructure necessary for running DFT calculations. Notice of Copyright : This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The publisher acknowledges the US government license to provide public access under the DOE Public Access Plan (http://energy.gov/downloads/doevihttps://github.com/Open-Catalyst-Project/ocp public-access-plan)"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2206.13654v1",
+                    "title": "Wav2Vec-Aug: Improved self-supervised training with limited data",
+                    "abstract": "Self-supervised learning (SSL) of speech representations has received much\nattention over the last few years but most work has focused on languages and\ndomains with an abundance of unlabeled data. However, for many languages there\nis a shortage even in the unlabeled data which limits the effectiveness of SSL.\nIn this work, we focus on the problem of applying SSL to domains with limited\navailable data by leveraging data augmentation for Wav2Vec 2.0 pretraining.\nFurther, we propose improvements to each component of the model which result in\na combined relative word error rate (WER) improvement of up to 13% compared to\nWav2Vec 2.0 on Librispeech test-clean / other.",
+                    "authors": "Anuroop Sriram, Michael Auli, Alexei Baevski",
+                    "published": "2022-06-27",
+                    "updated": "2022-06-27",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL"
+                    ],
+                    "main_content": "Introduction The use of self-supervised learning (SSL) for learning representations from unlabeled speech has received much attention over the last few years [1,2,3,4,5]. However, much of this research assumes that a large amount of unlabeled audio data is available, which is not always the case for rare languages or for specialized domains. The performance of SSL for speech with limited unlabeled data has been less studied. Data augmentation has proven to be an effective strategy for supervised learning [6,7] when the amount of labeled data is limited. Kharitonov et al. [8] showed that data augmentation bene\ufb01ts Contrastive Predictive Coding (CPC; [1]) when a limited amount of unlabeled data is available. CPC is light-weight selfsupervised learning algorithm which uses past context to make predictions about the future, and we show that data augmentation also works well for stronger transformer based bi-directional models such as Wav2Vec 2.0 [9,10]. We also propose improvements to each component of the Wav2Vec 2.0 model that are useful for both the small data setting as well as the large data settings. First, we replace some of the convolutional layers of the feature encoder in Wav2Vec 2.0 with improved layers based on light and dynamic convolutions [11]. Second, we replace the transformer part of the model with conformer layers, similar to [12, 13]. Finally, we add MLP projections on top of the context and latent vectors of Wav2Vec 2.0 inspired by self-supervised learning algorithms in the computer vision literature [14,15]. Together, these changes improve the performance on downstream ASR task by up to 13% (relative) WER compared to a Wav2Vec 2.0 baseline. We call this model Wav2Vec-Aug. 2. Background Our model is based on Wav2Vec 2.0 (W2V2; [9,10]) which maps raw audio x \u2208X to a latent feature representation c1, . . . , cT . The Wav2Vec 2.0 model consists of a convolutional feature encoder f : X 7\u2192Z that \ufb01rst maps the input audio to a latent speech representation z1, . . . zT . This latent representation is then input to a Transformer model g : Z 7\u2192C to output the context representations c1, . . . , cT [16,17]. Each zt represents about 25ms of audio strided by 20ms and the Transformer architecture follows the BERT model [18,19]. The model architecture is shown in Figure 1. In the pre-training phase, latent representations are discretized to q1, . . . , qT with a quantization module Z 7\u2192Q to represent the targets in the objective. The quantization module uses a Gumbel softmax [20] to choose entries from G = 2 codebooks with V = 320 entries each and the chosen entries are concatenated to obtain q [16,21,22]. The model is trained to identify the true quantized latent qt using ct for each masked time-step within a set of K = 100 distractors Qt sampled from other masked time steps. 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Data Augmentation Prior research has shown that data augmentation on input audio can be helpful for supervised learning [6,7], particularly in the small data setting. This has been con\ufb01rmed for self-supervised learning using the CPC algorithm [8]. In this work, we experiment with three different data augmentation methods for self-supervised learning: additive augmentation, pitch shift and reverberation. We have also experimented with speed perturbation, but found that it does not improve accuracy. Additive augmentation involves adding a noise signal x\u2032 to the input signal x. The noise signal can be chosen uniformly at random from a large collection of audio signals. The chosen noise signal is added at an SNR value of s \u223cUniform(s0, s1), for given hyperparameters s0 and s1. Pitch shift involves raising or lowering the pitch of the input audio by some random factor f. In our model, we sample the pitch shift factor from a gaussian distribution: f \u223cN(0, \u03c3p), where \u03c3p is a hyperparameter. Reverberation involves simulating far-\ufb01eld speech by convolving the input audio signal with a randomly generated room impulse response (RIR). The room size parameter r was chosen randomly by sampling a number r\u2032 \u223cN(0, \u03c3r), and then setting r = min(|r\u2032|, 100). During pre-training, for each data sample, we choose arXiv:2206.13654v1  [cs.CL]  27 Jun 2022 \fwhether or not to apply each augmentation method independently with a probability p. This ensures that every combination of augmentations is applied to some training example. We \ufb01rst duplicate the input audio x into source audio x(s) and target audio x(t). We then use x(s) to generate the context vectors, and x(t) to generate the target latent representations qt. In our experiments, we found that it is bene\ufb01cial to apply different augmentations to the source and target audios. We used the WavAugment toolkit1 [8] to implement all augmentation methods. 4. Architectural Improvements 4.1. Lightweight and Dynamic Convolution The Wav2Vec 2.0 model uses a purely convolutional feature encoder to extract features from the input audio signal. Wu et al. [11] introduced lightweight and dynamic convolutional layers that have been shown to outperform standard convolutions for a range of tasks.2 Lightweight convolutions are depth-wise separable convolutions [23] that share weights across groups of m-channels (m is treated as a hyper-parameter). Dynamic convolutions build on lightweight convolutions by dynamically computing the convolution kernels as a function of the input at a given timestep. These layers are computationally more ef\ufb01cient than transformer layers, making them suitable for the feature encoder, which operates over many time steps. We replace the \ufb01nal k convolutional layers of the feature encoder with lightweight or dynamic convolutions. We \ufb01nd that k = 2 leads to improved performance for both light and dynamic convolutions (Section 6). 4.2. Conformer The transformer part of the Wav2Vec 2.0 model uses a series of multi-head self-attention blocks [18]. Recently, Conformer models [12] have been shown to outperform the original transformer architecture for both supervised and self-supervised speech recognition [12,13]. The Conformer architecture consists of a series of multihead self attentions, depth-wise convolution and feed-forward layers. This combination helps the Conformer make effective use of both global and local interactions which leads to improved performance. 4.3. Context and Target MLPs Our \ufb01nal improvement is to modify the computation of the loss values by adding multi-layer perceptrons to the network [14,15]. Speci\ufb01cally, we introduce two multi-layer perceptrons (MLPs), dubbed ContextMLP and TargetMLP, which are applied to the context c and latent vectors q, respectively: c\u2032 = CMLP(c) q\u2032 = TMLP(q) The original Wav2Vec 2.0 model was trained to identify the true qt from c within a set of K distractors \u02dc q using a contrastive loss. The inputs to the contrastive loss are cosine similarities between the context vector c with qt and \u02dc q. In our model, 1https://github.com/facebookresearch/WavAugment 2We refer to non-separable convolutions as standard convolutions. we apply the CMLP to c, and the TMLP to qt and \u02dc q before computing the cosine similarities. This allows the network to learn more complex, non-linear similarity functions than a simple cosine similarity. Concurrent to our work, Wu et al. [24] has also incorporated the use of Context MLPs in a Wav2Vec 2.0 style model. 5. Experimental Setup We implement our changes on top of the Wav2Vec 2.0 Base model architecture [9], by modifying its various components. We use the same training setup as the Wav2Vec 2.0 Base model, except that we reduce the effective batch size by training models on 48 GPUs instead of the 64 GPUS used in [9]. We implemented our model in the fairseq library [25] and use the Librispeech dataset [26] for all of our experiments. Pre-training our model for 400K steps takes roughly 2.5 days with 64 V100 GPUs. We test our model with varying amounts of pre-training and \ufb01ne-tuning data by creating subsets of the Librispeech dataset. For pre-training, we consider a subset of 50 hours randomly sampled from the clean-100h set, the clean-100h set and the full set (960 hours). For \ufb01ne-tuning, we use the 1 hour split of Libri-light [27] (LS-Lab-1H) and clean-100h (LS-Lab-100H). We pre-train models for 150K, 200K or 400K steps for the 50 hour, 100 hour and full datasets, respectively and follow the same procedure for \ufb01ne-tuning and beam search as [9]. After pre-training, we \ufb01ne-tune each model on the labeled datasets with Connectionist Temporal Classi\ufb01cation [28]. After \ufb01ne-tuning, we decode the \ufb01ne-tuned models using the wordlevel 4-gram language model (LM) from [26] using the beam search decoder from [29]. We use a Bayesian Optimization3 procedure to \ufb01nd the best decoding hyper-parameters over 128 trials with the following search space: LM weight ([0, 8]), word score ([\u22125, 5]), and silence score ([\u22125, 5]). We use a beam size of 500 for the hyperparameter search. After \ufb01nding the optimal hyperparameters, we increase the beam size to 1, 500 to obtain the \ufb01nal results. 6. Results 6.1. Performance of Wav2Vec-Aug We \ufb01rst compare the performance of Wav2Vec-Aug with all of the proposed modi\ufb01cations (data augmentation during pretraining, replacing the last two feature encoder layers with dynamic convolutions, context and target MLPs, and conformer blocks) with Wav2Vec 2.0 on different amounts of pre-training and \ufb01ne-tuning data. Table 1 shows that our approach can obtain up to a 13% relative WER improvement over the baseline on test-clean/other depending on the amount of available data. In the rest of this section, we examine each component of our approach in more detail. 6.2. Data Augmentation We measure the effect of data augmentation vs. no data augmentation for different amounts of pre-training data when we use one hour of labeled data. For additive augmentation, we used the MUSAN dataset [30] to draw the noise samples. Table 2 shows results for different settings of the data augmentation hyperparameters. Based on these ablations, we set the augmentation parameters in our \ufb01nal model to s0 = 10, s1 = 15 for additive 3https://github.com/facebook/Ax \fTable 1: Comparison of our improvements to the original Wav2vec 2.0 model on the Librispeech benchmark for different amounts of training pre-training and \ufb01ne-tuning data. The Wav2Vec-Aug model obtains 9-13% improvement in WER compared to Wav2Vec 2.0 on the test datasets. Pre-train data Fine-tune data Model WER dev-clean dev-other test-clean test-other 50H 1H Wav2vec 2.0 15.73 30.23 16.14 31.96 Wav2Vec-Aug 13.37 25.57 14.04 28.03 100H 1H Wav2vec 2.0 10.50 20.66 10.82 21.52 Wav2Vec-Aug 9.17 18.15 9.81 19.45 100H Wav2vec 2.0 3.82 11.97 4.50 12.31 Wav2Vec-Aug 3.65 10.34 4.34 11.37 960H 1H Wav2vec 2.0 6.03 12.17 6.22 12.45 Wav2Vec-Aug 5.17 10.39 5.71 11.42 100H Wav2vec 2.0 2.86 8.17 3.46 8.27 Wav2Vec-Aug 2.53 7.14 3.04 7.21 Augmentation Probability WER 32 33 34 35 36 37 0.00 0.25 0.50 0.75 Figure 2: WER on dev-other for different augmentation probabilities. All models were pre-trained on the 50 hour subset with additive, pitch and reverb augmentations with s0 = 10, s1 = 15, \u03c3p = 50, and \u03c3r = 60. Note that these experiments use a beam size of 5 and are not directly comparable to other results. augmentation, \u03c3p = 50 as the standard deviation for pitch shift augmentation, and \u03c3r = 60 as the room size standard deviation for reverberation. Figure 2 shows the WERs for different augmentation probabilities when all three augmentation methods are used with these parameters. We obtain the best results with an augmentation probability of p = 0.5. Figure 3 shows results of using data augmentation with these hyperparameters with different amounts of pre-training data using the same model architecture. It is clear that data augmentation is most helpful in the low-data regime, improving WER by almost 11% relative to the baseline model when only 50 hours of data is available for pre-training. These improvements begin to taper off as the size of the pre-training dataset is increased. 6.3. Lightweight and Dynamic Convolution Table 3 shows the results of replacing either the last two or the last four layers in the feature encoder with lightweight or dynamic convolutional layers. The Wav2Vec 2.0 model uses convolutional layers with 512 channels in the feature encoder. To keep the number of parameters constant across experiments, Num Pre-training hours  WER 10 20 30 40 50 100 500 1000 No Data Aug Data Aug Figure 3: Effect of data augmentation during pre-training for different amounts of pre-training data. We show WER on the devother. The blue line shows the baseline model with no augmentation, while the red line shows results with the best augmentation probability p = 0.5. All models were \ufb01ne-tuned on the 1 hour labeled subset of Librispeech. we increased the number of channels to 640 when the last two layers are changed, and to 608 when the last four layers are changed. We used 8 attention heads in the lightweight and dynamic convolution layers, and a dropout rate of 0.1. Replacing the last two layers yields a roughly 5% relative improvement in WER on the dev-other validation set for both lightweight and dynamic convolutions, when \ufb01ne-tuned on the 100 hour subset of Librispeech. Replacing more layers appears to be detrimental to the performance, which indicates that lightweight / dynamic convolutions are most useful in the \ufb01nal layers where the number of timesteps is relatively small. 6.4. Conformer Our next experiment replaces the multi-head attention layers in the Transformer model with Conformer layers. Our model contains 14 Conformer blocks with an embedding dimension of 512, and 8 attention heads. The resulting model has the same number of parameters as the Wav2vec 2.0 model. Table 3 shows that the Conformer block improves performance by 10% relative \fTable 2: WER on dev-other for different data augmentation hyper-parameters for pre-training. All models were pre-trained on the 50 hour subset and then \ufb01ne-tuned on the 1 hour labeled set. Note that these experiments use a beam size of 5 and are not directly comparable to other results. Augmentation Aug. Params Aug. Prob WER \u2013 \u2013 0.0 36.69 Additive s0 = 10, s1 = 15 0.5 33.37 Pitch \u03c3p = 20 0.5 33.72 \u03c3p = 50 0.5 33.56 \u03c3p = 100 0.5 34.07 Reverb \u03c3r = 10 0.5 38.30 \u03c3r = 25 0.5 36.87 \u03c3r = 60 0.5 36.02 Table 3: WER on the dev-other dataset showing the effects of our proposed architectural improvements. All models were pretrained on the full Librispeech dataset and \ufb01ne-tuned on the 1 hour or 100 hour labeled sets. Each architectural improvement is applied independently to the Wav2vec 2.0-Base model. Modi\ufb01cation Dev-other WER LS-Lab-1H LS-Lab-100H Wav2vec 2.0 12.17 8.17 + Light Conv (last 2) 11.73 7.80 + Light Conv (last 4) 12.09 8.34 + Dyn Conv (last 2) 11.64 7.79 + Dyn Conv (last 4) 12.17 8.23 + Conformer 11.34 7.32 + CMLP (2 layers) 11.43 8.04 + TMLP (2 layers) 11.50 8.12 + CMLP + TMLP 2 layers 11.36 7.98 3 layers 10.76 7.82 4 layers 10.46 7.71 when \ufb01ne-tuned on the 100 hour labeled subset. 6.5. Context and Target MLPs Our \ufb01nal ablation experiments involve adding context and target MLPs to the wav2vec 2.0 model. We experimented with adding each MLP independently, and also the two MLPs together. For the latter case, we also report experiments with different number of MLP layers. These MLPs contain a batch normalization in each layer, with ReLU activation functions. The results shown in Table 3 indicate that adding either MLP is helpful by itself, with the context MLP being more bene\ufb01cial than the target MLP. Including both MLPs is more bene\ufb01cial than adding either MLP by itself. Table 3 also shows that increasing the number of hidden layers in the MLPs helps. Our best model contains 4 hidden layers in each MLP and obtains a 15% lower WER when \ufb01ne-tuned on the 1 hour dataset, and 6% when \ufb01netuned on the 100 hour dataset. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2203.09697v1",
+                    "title": "Towards Training Billion Parameter Graph Neural Networks for Atomic Simulations",
+                    "abstract": "Recent progress in Graph Neural Networks (GNNs) for modeling atomic\nsimulations has the potential to revolutionize catalyst discovery, which is a\nkey step in making progress towards the energy breakthroughs needed to combat\nclimate change. However, the GNNs that have proven most effective for this task\nare memory intensive as they model higher-order interactions in the graphs such\nas those between triplets or quadruplets of atoms, making it challenging to\nscale these models. In this paper, we introduce Graph Parallelism, a method to\ndistribute input graphs across multiple GPUs, enabling us to train very large\nGNNs with hundreds of millions or billions of parameters. We empirically\nevaluate our method by scaling up the number of parameters of the recently\nproposed DimeNet++ and GemNet models by over an order of magnitude. On the\nlarge-scale Open Catalyst 2020 (OC20) dataset, these graph-parallelized models\nlead to relative improvements of 1) 15% on the force MAE metric for the S2EF\ntask and 2) 21% on the AFbT metric for the IS2RS task, establishing new\nstate-of-the-art results.",
+                    "authors": "Anuroop Sriram, Abhishek Das, Brandon M. Wood, Siddharth Goyal, C. Lawrence Zitnick",
+                    "published": "2022-03-18",
+                    "updated": "2022-03-18",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "physics.comp-ph",
+                        "stat.ML"
+                    ],
+                    "main_content": "INTRODUCTION Graph Neural Networks (GNNs) (Gori et al., 2005; Zhou et al., 2020) have emerged as the standard architecture of choice for modeling atomic systems, with a wide range of applications from protein structure prediction to catalyst discovery and drug design (Sch\u00fctt et al., 2017b; Gilmer et al., 2017; J\u00f8rgensen et al., 2018; Zitnick et al., 2020; Sch\u00fctt et al., 2017a; Xie & Grossman, 2018). These models operate on graph-structured inputs, where nodes of the graph represent atoms, and edges represent bonds or atomic neighbors. Despite their widespread success and the availability of large molecular datasets, training massive GNNs (with up to billions of parameters) is an important but under-explored area. Success of similarly large models in computer vision, natural language processing, and speech recognition (Shoeybi et al., 2020; Huang et al., 2019; Brown et al., 2020; Zhai et al., 2021) suggests that scaling up GNNs could yield signi\ufb01cant performance gains. Most previous approaches to scaling GNNs have focused on scaling small models (with up to a few million parameters) to massive graphs. These methods generally assume that we are working with a large, \ufb01xed graph, leading to the development of methods like neighborhood sampling (Jangda et al., 2021; Zheng et al., 2021) (Jia et al., 2020; Ma et al., 2019; Tripathy et al., 2020). These methods do not apply to atomic simulation datasets that contain millions of smaller graphs where it is necessary to consider the entire graph for prediction. Our focus is on the complementary problem of scaling to very large models for a dataset of many moderately-sized graphs (\u223c1k nodes, 500k edges). Another limitation of existing methods is that they focus on scaling simple GNN architectures such as Graph Convolutional Networks (GCNs) that only represent lower-order interactions i.e., representations for nodes and edges, that are then updated by passing messages between neighboring nodes. In practice, the most successful GNNs used for atomic systems also model higher-order interactions between atoms, such as the interactions between triplets or quadruplets of atoms (Klicpera et al., 2020a; 2021; Liu et al., 2022). These interactions are necessary to capture the geometry of the underlying system, critical in making accurate predictions. Scaling such GNN architectures is challenging because even moderately-sized graphs can contain a large number of higher-order interactions. For example, a single graph with 1k nodes could contain several million triplets of atoms. 1 arXiv:2203.09697v1  [cs.LG]  18 Mar 2022 \fPublished as a conference paper at ICLR 2022 In this paper, we introduce Graph Parallelism, an approach to scale up such GNNs with higher-order interactions to billions of parameters, by splitting up the input graph across multiple GPUs. We benchmark our approach by scaling up two recent GNN architectures \u2013 DimeNet++ (Klicpera et al., 2020a) and GemNet-T (Klicpera et al., 2021) \u2013 on the Open Catalyst (OC20) dataset (Chanussot* et al., 2021). The OC20 dataset, aimed at discovering new catalyst materials for renewable energy storage, consists of 134M training examples spanning a wide range of adsorbates and catalyst materials. A GNN that can accurately predict per-atom forces and system energies on OC20 has the potential to replace the computationally expensive quantum chemistry calculations based on Density Functional Theory (DFT) that are currently the bottleneck in computational catalysis. Our large-scale, graph-parallelized models lead to relative improvements of 1) 15% for predicting forces on the force MAE metric (S2EF task), and 2) 21% on the AFbT metric for predicting relaxed structures (IS2RS task), establishing new state-of-the-art results on this dataset. 2 GRAPH PARALLELISM 2.1 EXTENDED GRAPH NETS Battaglia et al. (2018) introduced a framework called Graph Network (GN) that provides a general abstraction for many popular Graph Neural Networks (GNNs) operating on edge and node representations of graphs. We build on their work and de\ufb01ne the Extended Graph Network (EGN) framework to include GNNs that also operate on higher order terms like triplets or quadruplets of nodes. In the Graph Network (GN) framework, a graph is de\ufb01ned as a 3-tuple G = (u, V, E), where u represents global attributes about the entire graph; V = {vi}i=1:N v is the set of all nodes, with vi representing the attributes of node i; and E = {(ek, rk, sk)}k=1:N e is the set of all edges where ek represents the attributes for the edge from node sk to rk. A GNN then contains a series of GN blocks that iteratively operate on the input graph, updating the various representations. In our Extended Graph Network (EGN) framework, a graph is de\ufb01ned as a 4-tuple G = (u, V, E, T), where u, V, and E are de\ufb01ned as in the Graph Network, and T = {(tm, em1, em2, . . .)} is the set of higher-order interaction terms involving edges indexed by m1, m2, . . .. As a concrete example, consider an atomic system represented as a graph in this framework with the nodes representing the atoms and the edges representing atomic neighbors. The node attributes vi and edge attributes ek could represent the atom\u2019s atomic numbers and distances between atoms respectively. The higher order interactions could represent triplets of atoms, i.e., pairs of neighboring edges with tm representing the bond angle, which is the angle between edges that share a common node. Finally, the global attribute u can represent the energy of the system. For clarity of exposition, we will limit our discussion to triplets in the rest of the paper, but higher order interactions can be handled in a similar manner. We denote these triplets as (tm, em1, em2). In the EGN framework, GNNs then contain a series of EGN blocks that iteratively update the graph. Each EGN block consists of several update and aggregation functions that are applied to transform an input graph (u, V, E, T) into an output graph (u\u2032, V \u2032, E\u2032, T \u2032). Each EGN block starts by updating the highest order interactions, which are then aggregated before updating the next highest order interaction. For instance, the triplet representations are \ufb01rst updated (using TripletUpdate function) and then aggregated (TripletAggr) at each edge. These aggregated representations are then used to update the edge representation (EdgeUpdate). Next, the edges going into a node are aggregated and used to update the node representations, and so on. This is illustrated in \ufb01gure 1a. Many GNNs can be cast in the EGN framework using appropriate update and aggregation functions. 2.2 GRAPH PARALLELISM FOR EXTENDED GRAPH NETS Training large EGNs can be challenging even on moderately sized graphs because of the large memory footprint required in storing and updating the representation for each triplet, edge, and node. In many applications, the number of edges is one or two orders of magnitude larger than the number of nodes, while the number of triplets is one or two orders of magnitude larger than the number of edges. Thus, storing and updating the triplet representations is often the bottleneck in terms of GPU memory and compute. Many recent methods such as (Klicpera et al., 2020a; 2021) overcome this 2 \fPublished as a conference paper at ICLR 2022 Triplet Update Triplet  Aggr. Edge Update Node Update Global Update Edge  Aggr. Node  Aggr. vi t\u2032 m ek t\u2032 m vi e\u2032 k e\u2032 k t\u2032 m v\u2032 i e\u2032 k t\u2032 m v\u2032 i (a) Forward computation of an EGN block. First, each triplet representation is updated (TripletUpdate function), followed by aggregation of these updated representations (TripletAggr). Next, these aggregated values are used to update edge representations (EdgeUpdate), followed by edge aggregation (EdgeAggr), and \ufb01nally node and global updates. All Reduce Processor 0 Processor 1 Processor 2 Processor 3 All Reduce All Reduce Triplet Update and Local  Triplet Aggregation Global Triplet  Aggregation Edge Update and Local  Edge Aggregation Global Edge  Aggregation Node Update and Local  Node Aggregation  Global Node  Aggregation Global Update (b) Distributed computation of an EGN block. The graph is split up among the different GPUs such that processing unit p contains its subset of triplets T (p) in memory, along with all edges E and nodes V . The triplet attributes are updated in parallel, followed by a triplet aggregation to locally update the edge attributes. Next, an allreduce operation is performed to update all edge attributes globally. We continue this process to update the node and global attributes. problem by using a very low-dimensional representation for the triplets. However, we found this to signi\ufb01cantly reduce model capacity leading to under\ufb01tting for some applications with large training datasets. It is necessary to overcome these memory limitations for better generalization. One way to avoid the memory limits is to distribute the computation across multiple GPUs. For a graph neural network, a natural choice to distribute the computation is by splitting the graph. The update functions in an EGN are easy to apply in parallel since they are applied independently for each triplet, edge, or node. It is substantially more challenging to apply the aggregations in parallel. To simplify parallel aggregation, we restrict ourselves to aggregation functions that are commutative and associative. This is not limiting in practice since most popular GNN architectures use sum or mean aggregations which can be implemented in this manner. We will now describe the implementation of the distributed EGN block. Suppose we have access to P processing units that we wish to split the graph computation over. Each unit p is responsible for computing the updates to a subset of triplets, edges and nodes, that we denote by T (p), E(p), and V (p), respectively. At the beginning of computation, we split the graph so that the processing unit p contains its subset of triplets T (p) in memory, along with the entire set of edges E and nodes V . During the forward pass, we \ufb01rst update each set of triplets T (p) in parallel to obtain T \u2032(p), followed by a local triplet aggregation. Next, an all-reduce operation is performed on these locally aggregated triplet representations to obtain globally aggregated triplet representations. These globally aggregated representations are then used to update edges in parallel, and the process is repeated for nodes and ultimately for the global graph level representation. Figure 1b shows this. 3 \fPublished as a conference paper at ICLR 2022 In this framework, the highest order interaction attributes are never communicated across processors. Therefore, the communication time is bound by the number of lower order node interactions. In our example, the triplet representations are not communicated, while the edge and node representations are communicated once per EGN block, making the total communication cost equal to O(NvDv + NeDe), where Dv and De are the dimensions of node and edge representations. This makes it possible to work with a large number of triplets and large triplet representations. In section 3, we show concrete instantiations of this framework for two contemporary GNNs. 3 GRAPH PARALLELISM FOR ATOMIC SIMULATIONS In this section, we present two concrete examples of using GNNs for the problem of predicting the energy and the forces for an atomic system, modeled as a graph whose nodes represent the atoms and whose edges represent the atoms\u2019 neighbors. The GNN takes such a graph as input and predicts the energy of the entire system as well as a 3D force vector on each atom. Two paradigms have been proposed in the literature, that we call energy-centric and force-centric approaches. These approaches differ in how they estimate forces and whether they are energy conserving, which is an important physical property. We begin by describing the components shared by both approaches, followed by one recent model in each paradigm. 3.1 INPUTS AND OUTPUTS The inputs to the network are 3D positions xi \u2208R3 and atomic number zi for each atom i \u2208 {1 . . . n}. The outputs are the per-atom forces fi \u2208R3 and the energy E \u2208R of the entire structure. The distance between atoms i and j is denoted as dij = ||xi \u2212xj||. If edges (i, j) and (j, k) exist, then (i, j, k) de\ufb01nes a triplet in the graph and we denote the angle between the edges as \u03b1kj,ji. The input graph is constructed with each atom t (target) as a node and the edges representing the atom\u2019s neighbors s \u2208Nt where Nt contains all atoms s (source) within a distance \u03b4, which is treated as a hyperparameter. Each edge has a corresponding message mst that passes information from source atom s to target atom t. 3.2 ESTIMATING FORCES AND ENERGY As previously stated, there are two paradigms for estimating the energy and forces for an atomic system: energy-centric and force-centric. In energy-centric models, the model \ufb01rst computes the energy E by applying a forward pass of the GNN: E = GNN(x, z), where x, and z represent the atomic positions and atomic numbers respectively. The forces are then computed as the negative gradient of the energy with respect to atomic positions by using backpropagation: f = \u2212\u2207xE. In force-centric models, the energy and forces are both computed directly during the forward propagation: E, F = GNN(x, z) where F represents the matrix of all atomic forces. Force-centric models tend to be more ef\ufb01cient in terms of computation time and memory usage compared to the energy-centric models. However, energy-centric models guarantee that the forces are energy conserving which is an important physical property satis\ufb01ed by atomic systems. In this work, we demonstrate the bene\ufb01ts of scaling up GNNs in both paradigms. 3.3 ENERGY-CENTRIC MODEL: DIMENET++ DimeNet++ (Klicpera et al., 2020a) is a recently proposed energy-centric model for atomic systems. In this model, the edges are represented by a feature representation mji, which are iteratively updated using both directional information (via bond angles) as well as the interatomic distances. The edges are initially represented using a radial basis function (RBF) representation e(ji) RBF of their lengths dji. The triplets are represented using a spherical basis function (SBF) expansion a(kj,ji) SBF of the distances dkj as well as bond angles \u03b1(kj,ji). In each block, the messages are updated as: m\u2032 ji = fupdate(mji, X k\u2208Nj\\{i} fint(mkj, e(ji) RBF , a(kj,ji) SBF )) (1) 4 \fPublished as a conference paper at ICLR 2022 where the interaction function fint corresponds to the TripletUpdate function, the summation corresponds to the TripletAggr function, and the update function fupdate to the EdgeUpdate function respectively. The interaction function consists of a Hadamard product of the embeddings, followed by a multi-layer perceptron. To speed up these computations, the embeddings are projected down to a smaller dimension before computing interactions, and later projected back up. The updated messages are then fed as input to an output block that sums them up per atom i to obtain a per-atom representation: hi = P j m\u2032 ji (EdgeAggr function). These are then transformed by another MLP to obtain the node representation v\u2032 i (NodeUpdate function). Thus, DimeNet++ is a case of EGN, and we closely follow the recipe from Sec. 2.2 to parallelize it. 3.4 FORCE-CENTRIC MODEL: GEMNET GemNet (Klicpera et al., 2021) extends DimeNet++ in a number of ways, including the addition of quadruplets as well as a force-centric version. Here, we focus only on the force-centric GemNet-T model since it was recently shown to obtain state-of-the-art results on the OC20 dataset1. GemNet-T largely follows the structure of the DimeNet++ model, but includes some modi\ufb01cations. First, GemNet-T uses a bilinear layer instead of the Hadamard product for the interaction function. This is made ef\ufb01cient by optimally choosing the order of operations in the bilinear function to minimize computation. Second, GemNet-T maintains an explicit embedding for each atom that is \ufb01rst updated by aggregating the directional embeddings involving that atom, similar to DimeNet++. Next, the updated atom embedding is used to update each of the edge embeddings. This creates a second edge update function, EdgeUpdate\u2032, that is run after the node embeddings are updated. Third, GemNet makes use of symmetric message passing, that is the messages mji and mij that are on the same edge, but in different directions, are coupled. In a parallel implementation, this step requires an additional all-reduce step since messages mji and mij could be on different processors. Thus, GemNet-T largely follows the EGN framework, with a few minor deviations from the standard formulation. The distributed EGN implementation described in section 2.2 can be used for the GemNet-T as well, but with additional communication steps to account for the second edge update function and symmetric message passing. 4 EXPERIMENTS In this section, we present the results of our scaling experiments on the Open Catalyst 2020 (OC20) dataset (Chanussot* et al., 2021). The OC20 dataset contains over 130 million atomic structures used to train models for predicting forces and energies during structure relaxations. We report results for three tasks: 1) Structure to Energy and Forces (S2EF) that involves predicting energy and forces for a given structure; 2) Initial Structure to Relaxed Energy (IS2RE) that involves predicting the relaxed energy for a given initial structure; and 3) Initial Structure to Relaxed Structure (IS2RS) which involves performing a structure relaxation using the predicted forces. DimeNet++ and GemNet-T are the current state-of-the-art energy-centric and force-centric models respectively. 4.1 EXPERIMENTAL SETUP DimeNet++-XL. Our DimeNet++ model consists of B = 4 interaction blocks, with a hidden dimension of H = 2048, an output block dimension of D = 1536, and intermediate triplet dimension of T = 256. This model has about 240M parameters, which is over 20\u00d7 larger than the DimeNet++large model used in (Chanussot* et al., 2021). We call this model DimeNet++-XL. The model was trained with the AdamW optimizer (Kingma & Ba, 2015; Loshchilov & Hutter, 2019) starting with an initial learning rate of 10\u22124, that was multiplied by 0.8 whenever the validation error plateaus. The model was trained with an effective batch size of 128 on 256 Volta 32GB GPUs with a combination of data parallel and graph parallel training: each graph was split over 4 GPUs with data parallel training across groups of 4 GPUs. 1opencatalystproject.org/leaderboard.html 5 \fPublished as a conference paper at ICLR 2022 Model #Params Training GPU-days S2EF Test Energy MAE (eV)\u2193 Force MAE (eV/A)\u2193 Force Cos\u2191 EFwT\u2191 ForceNet-Large (Hu et al., 2021) 34.8M 194 2.2628 0.03115 0.5195 0.01% DimeNet++-Large (Klicpera et al., 2020a) 10.8M 1600 31.5409 0.03132 0.5440 0.00% SpinConv (Shuaibi et al., 2021) 8.9M 76 0.3363 0.02966 0.5391 0.45% GemNet-T (Klicpera et al., 2021) 31M 47 0.2924 0.02422 0.6162 1.20% GemNet-XL 300M 1962 0.2701 0.02040 0.6603 1.81% Table 1: Experimental results on the S2EF task comparing our GemNet-XL to the top entries on the Open Catalyst leaderboard, showing metrics averaged across the 4 test datasets. Model #Params Training IS2RS Test Dataset AFbT\u2191 ADwT\u2191 FbT\u2191 SpinConv (Shuaibi et al., 2021) 8.9M S2EF-All 16.67% 53.62% 0.05% DimeNet++ (Klicpera et al., 2020a) 1.8M S2EF 20M + MD 17.15% 47.72% 0.15% DimeNet++-large (Klicpera et al., 2020a) 10.8M S2EF-All 21.82% 51.68% 0.40% GemNet-T (Klicpera et al., 2021) 31M S2EF-All 27.60% 58.68% 0.70% DimeNet++-XL 240M S2EF 20M + MD 33.44% 59.21% 1.25% GemNet-XL 300M S2EF-All 30.82% 62.65% 0.90% Table 2: Results on the IS2RS task comparing our models to the top entries on the Open Catalyst leaderboard, showing metrics averaged across the 4 test datasets. The DimeNet++ and DimeNet++XL models were trained on the S2EF 20M + MD dataset, that contains additional molecular dynamics data and has been shown to be helpful for the IS2RS task (Chanussot* et al., 2021). GemNet-XL. Our GemNet model consists of B = 6 interaction blocks, with an edge embedding size of E = 1536, triplet embedding size of T = 384 and embedding dimension of the bilinear layer of B = 192. We found that it was bene\ufb01cial to use a small atom embedding size of A = 128, much smaller than the previous SOTA GemNet model2. This model has roughly 300M parameters, which is about 10\u00d7 larger than the previous SOTA model. However, since we reduced the atom embedding dimension and increased the edge and triplet dimensions, the total amount of compute and memory usage is signi\ufb01cantly larger. We call this model GemNet-XL. We followed the same training procedure as with DimeNet++, except for a starting learning rate of 2 \u00d7 10\u22124. 4.2 STRUCTURE TO ENERGY AND FORCES (S2EF) The Structure to Energy and Forces task takes an atomic structure as input and predicts the energy of the entire structure and per-atom forces. The S2EF task has four metrics: the energy and force Mean Absolute Error (MAE), the Force Cosine Similarity, and the Energy and Forces within a Threshold (EFwT). EFwT indicates the percentage of energy and force predictions below a preset threshold. Table 1 compares the top models on the Open Catalyst Project leaderboard1 with the GemNetXL model. GemNet-XL obtains a roughly 16% lower force MAE and an 8% lower energy MAE relative to the previous state-of-the-art. Further, GemNet-XL improves the EFwT metric more than 50% relative to the previous best, although the value is still very small. These results indicate that model scaling is bene\ufb01cial for the S2EF task. 4.3 INITIAL STRUCTURE TO RELAXED STRUCTURE (IS2RS) The Initial Structure to Relaxed Structure (IS2RS) task involves taking an initial atomic structure and predicting the atomic structure that minimizes the energy. This is performed by iteratively predicting the forces on each atom and then using these forces to update the atomic positions. This process is repeated until convergence or 200 iterations. There are three metrics for this task: Average Distance within Threshold (ADwT), that measures the fraction of \ufb01nal atomic positions within a distance threshold of the ground truth; Forces below Threshold (FbT), which measures whether a true energy minimum was found (i.e., forces are smaller than a preset threshold); and, the Average Forces below Threshold (AFbt), which averages the FbT over several thresholds. Table 2 shows the results on the IS2RS task, comparing our models with the top few models on the Open Catalyst leaderboard. Both the DimeNet++-XL and GemNet-XL models outperform all 2discuss.opencatalystproject.org/t/new-gemnet-dt-code-results-model-weights/102 6 \fPublished as a conference paper at ICLR 2022 Model Approach IS2RE Test Energy MAE (EV)\u2193 EwT\u2191 SpinConv (Shuaibi et al., 2021) Relaxation 0.4343 7.90% GemNet-T (Klicpera et al., 2021) Relaxation 0.3997 9.86% Noisy Nodes (Godwin et al., 2022) Direct 0.4728 6.50% 3D-Graphormer (Ying et al., 2021) Direct 0.4722 6.10% GemNet-XL Relaxation 0.3712 11.13% GemNet-XL-FT Direct 0.4623 5.60% Table 3: Results on the IS2RE task comparing our GemNet-XL to the top entries on the Open Catalyst leaderboard, showing metrics averaged across the 4 test datasets. #Blocks Node Dim Edge Dim Trip Dim Bil Dim Params #GP GPUs #GP+DP GPUs 3 1280 768 128 64 125M 1 32 4 1536 1024 192 96 245M 2 64 6 1792 1184 288 160 480M 4 128 8 2320 1302 512 288 960M 8 256 Table 4: Model hyperparameters for the scaling analysis. \u201c#GP GPUs\u201d denotes the number of GPUs over which the graph is distributed over for pure graph parallel training on a single node. \u201c#GP+DP GPUs\u201d denotes the total number of GPUs used to train with graph parallel training together with 32-way data parallel training. existing models. The DimeNet++-XL model obtains a relative improvement of 53% on the AFbT metric, and more than triples the FbT metric compared to the DimeNet-large model. The GemNetXL model obtains similar improvements compared to the smaller GemNet-T model. These results underscore the importance of model scaling for this task. 4.4 INITIAL STRUCTURE TO RELAXED ENERGY (IS2RE) The Initial Structure to Relaxed Energy (IS2RE) task takes an initial atomic structure and attempts to predict the energy of the structure after it has been relaxed. Two approaches can be taken to address this problem, the direct and the relaxation approaches (Chanussot* et al., 2021). In the direct approach, we treat this task as a regression problem, and train a model to directly estimate the relaxed energy for a given atomic structure. The relaxation approach \ufb01rst estimates the relaxed structure (IS2RS task) after which the energy is estimated using the relaxed structure as input. Relaxation approaches typically outperform the direct approaches, though they are generally two orders of magnitude slower during inference time due to the need to estimate the relaxed structure. Table 3 compares our GemNet-XL model to the top three models from the Open Catalyst Project leaderboard. There are two metrics for the IS2RE task: energy Mean Absolute Error (MAE) and the Energy within Threshold (EwT) which measures the percentage of time the predicted energy is within a threshold of the true energy. Table 3 shows that the GemNet-XL model obtains a roughly 8% lower energy MAE and a 12% higher EwT compared to the previous best, which is the smaller GemNet-T model, demonstrating the bene\ufb01ts of scaling up IS2RE models. Since direct IS2RE models are very fast at inference time, there are applications where they are more useful than relaxation based approaches. It is possible to convert a trained S2EF model into a direct IS2RE model by \ufb01ne-tuning it on the IS2RE training data. We \ufb01netune our GemNet-XL model in this manner for 5 epochs, starting with an initial learning rate of 3 \u00d7 10\u22125 that is exponentially decayed by multiplying with 0.95 at the end of each epoch. The resuting model \u2013 GemNet-XL-FT \u2013 obtains a relative improvement of \u223c2% on energy MAE compared to 3D-Graphormer (Ying et al., 2021), the current state-of-the-art direct approach (Table 3). 4.5 SCALING ANALYSIS Weak scaling studies the effect on the throughput when computation is scaled proportional to the number of processors. We study weak scaling ef\ufb01ciency in terms of scaling the model size propor7 \fPublished as a conference paper at ICLR 2022 1 2 4 8 ... 32 64 128 256 Number of GPUs 0 20 40 60 80 100 Weak Scaling Graph Parallel Graph + Data Parallel (a) Weak scaling ef\ufb01ciency. 100 101 102 Number of GPUs 102 103 TeraFLOPs per second Graph Parallel Pipeline Parallel Graph & Pipeline Parallel (b) Comparison of graph and pipeline parallel training. Figure 2: Left: Weak scaling ef\ufb01ciency measured by scaling the model size proportional to the number of GPUs. Right: Comparing graph parallelism with pipeline parallelism. Solid blue, green and red curves show the scaling performance of graph, pipeline, and graph+pipeline parallel training. Dashed lines show the same with 32-way data parallel training. We do not show the results of training the largest models with pipeline parallelism as those runs ran out of GPU memory. tional to the number of GPUs. For these experiments, we use 4 different GemNet-T models, ranging from 120M parameters to nearly 1B parameters (see Table 4). We train increasingly larger models on multiple GPUs that do not \ufb01t on a single GPU. Figure 2a shows that we are able to obtain a scaling ef\ufb01ciency of roughly 79% with 8 GPUs for our largest model with nearly a billion parameters. This shows that our graph parallel training can be scaled up to billion parameter GemNet-T models while obtaining reasonably good scaling performance. Figure 2a further shows the scaling ef\ufb01ciency for each model combining graph parallelism with 32-way data parallelism. Pure data parallel training with 32 GPUs obtains a scaling ef\ufb01ciency of 75% for the smallest model, showing the effect of network communication and load imbalance between the GPUs. Combining graph and data parallel training with 256 GPUs only reduces the scaling ef\ufb01ciency to 47% for the largest model compared to the 1 GPU case, suggesting the graph parallelism is promising for training extremely large models on hundreds of GPUs. Finally, \ufb01gure 2b shows the raw performance of running these models on V100 GPUs in terms of TeraFLOPs per second as a function of the number of GPUs. On a single GPU, the 120M parameter GemNet-T sustains 32 TeraFLOPs or roughly 25% of the theoretical peak FLOPS for a single GPU, and is thus a strong baseline. With 256 GPUs, the largest model sustains 3.5 PetaFLOPs. Figure 2b compares graph and pipeline parallelism (Huang et al., 2019) showing that graph parallelism outperforms pipeline parallelism for the models that we consider. Since each graph in the training data contains different numbers of nodes, edges and triplets, load balancing across GPUs is dif\ufb01cult for pipeline parallelism. Graph parallelism is able to overcome this problem since the nodes, edges and triplets of a given batch are always distributed evenly across the GPUs, helping it outperform pipeline parallelism. It is possible, however, that pipeline parallelism might outperform graph parallelism for very deep GNNs since inter-GPU communication overhead for pipeline parallelism is independent of the number of blocks. Figure 2b also shows results with graph and pipeline parallelism combined, indicating that these methods are complementary to each other. 5 RELATED WORK GNNs for simluating atomic systems Many GNN based approaches have been proposed for the task of estimating atomic properties such as (Sch\u00fctt et al., 2017b; Gilmer et al., 2017; J\u00f8rgensen et al., 2018; Sch\u00fctt et al., 2017a; 2018; Xie & Grossman, 2018; Qiao et al., 2020; Klicpera et al., 2020b), where the atoms are represented by nodes and neighboring atoms are connected by edges. An early approach for force estimation was the SchNet model Sch\u00fctt et al. (2017a), which computed forces using only the distance between atoms without the use of angular information. SchNet proposed the use of differentiable edge \ufb01lters which enabled constructing energy-conserving models by estimating forces as the gradient of the energy. Subsequent work (Klicpera et al., 2020a;b; 2021; Liu et al., 2022) has extended on the SchNet model by adding bond angles and dihedral angles, 8 \fPublished as a conference paper at ICLR 2022 which has resulted in improved performance. These models make use of higher order interactions among nodes which make them highly compute and memory intensive. An alternate approach for estimating forces is to directly regress the forces as an output of the network. While this approach does not enforce energy conservation or rotational equivariance, recent work (Hu et al., 2021) has shown that such models can still accurately predict forces. Distributed GNN Training. Research on distributed GNN training has focused on the regime of training small models on a single, large graph, for instance by sampling local neighborhoods around nodes to create mini-batches (Jangda et al., 2021; Zhang et al., 2020; Zhu et al., 2019). While these approaches can scale to very large graphs, they are not suitable for the task of modeling atomic systems where it is important to consider the entire graph for predicting the energy and forces. An alternate line of work, that is more similar to ours, keeps the entire graph in memory by ef\ufb01ciently partitioning the graph among multiple nodes (Jia et al., 2020; Ma et al., 2019; Tripathy et al., 2020). These methods still operate in a single graph regime that is partitioned ahead of time. Thus the focus of that work is on \ufb01nding ef\ufb01cient partitions of the graph, which is not applicable to our problem since we operate on millions of graphs. Further, these works do not train very large GNNs, or GNNs that operate on higher-order interactions (e.g. triplets), that are important for atomic systems. Model Parallelism. methods focus on training large models that do not \ufb01t entirely on one GPU (even with a batch size of 1). GPipe (Huang et al., 2019) splits different sequences of layers into different processors, and splits each training mini-batch into micro-batches to avoid idle time per GPU. Megatron-LM (Shoeybi et al., 2020) splits the model breadth-wise, where Transformer layer weights are partitioned across multiple GPUs to distribute computation. We see model and graph parallelism as complementary approaches that can be combined to train even larger models. 6 DISCUSSION We presented Graph Parallelism, a new method for training large GNNs for modeling atomic simulations, where modeling higher-order interactions between atoms (triplets / quadruplets) is critical. We showed that training larger DimeNet++ and GemNet-T models can yield signi\ufb01cant improvements on the OC20 dataset. Although we demonstrated graph parallelism for just two GNNs, it is broadly applicable to a host of message-passing GNNs, including equivariant networks, that can be cast in the GN / EGN framework (Sec. 2.1) by appropriately picking update and aggregate functions. Further, it is possible to combine graph parallelism with model parallel methods such as GPipe (Huang et al., 2019) to train even larger models, which could yield further improvements, as brie\ufb02y explored in Sec 4.5. For force-centric GNNs, it should be possible to use graph parallel for \u2018breadth-wise\u2019 scaling i.e., to split higher-order computations (e.g. triplets) across GPUs, and GPipe for \u2018depth-wise\u2019 scaling i.e., to scale to larger number of message-passing blocks, sequentially split across GPUs. For energy-centric GNNs e.g., DimeNet++, this combination is less obvious since these models require an additional backward pass through the network to compute forces as gradients of energy with respect to atomic positions. Energy-centric GNNs are common for atomic systems because they enforce the physical constraint of energy conservation. As we demonstrate with our DimeNet++-XL experiments, graph parallelism is applicable to energy-centric GNNs. We see scaling to large model sizes as a necessary (but not suf\ufb01cient) step to effectively model atomic simulations from large, diverse datasets of adsorbates and catalysts. Further progress may require marrying large scale with ways to better capture 3D geometry and physical priors. The carbon emissions from training large deep learning models are non-trivial and the work we have presented here is no exception (Strubell et al., 2019; Schwartz et al., 2019). We estimate that training our GemNet-XL model with Tesla v100 32 GB GPUs on cloud resources in the US ranges from 3490 8052 kg of CO2 eq. (Lacoste et al., 2019). In the worst case, the emissions are roughly equivalent to 16 round trip \ufb02ights from Los Angeles to New York. Assuming that the training time is \ufb01xed, emissions largely depend on the carbon intensity of the energy generation in a given region and the percentage of emissions offset with other investments by the resource provider. When choosing compute resources we recommend evaluating the stated carbon offset commitments and if possible, consider running experiments in regions that have more sustainable energy generation. The compute resources we utilized for this paper were committed to be 100% offset. 9 \fPublished as a conference paper at ICLR 2022"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2101.04909v2",
+                    "title": "COVID-19 Prognosis via Self-Supervised Representation Learning and Multi-Image Prediction",
+                    "abstract": "The rapid spread of COVID-19 cases in recent months has strained hospital\nresources, making rapid and accurate triage of patients presenting to emergency\ndepartments a necessity. Machine learning techniques using clinical data such\nas chest X-rays have been used to predict which patients are most at risk of\ndeterioration. We consider the task of predicting two types of patient\ndeterioration based on chest X-rays: adverse event deterioration (i.e.,\ntransfer to the intensive care unit, intubation, or mortality) and increased\noxygen requirements beyond 6 L per day. Due to the relative scarcity of\nCOVID-19 patient data, existing solutions leverage supervised pretraining on\nrelated non-COVID images, but this is limited by the differences between the\npretraining data and the target COVID-19 patient data. In this paper, we use\nself-supervised learning based on the momentum contrast (MoCo) method in the\npretraining phase to learn more general image representations to use for\ndownstream tasks. We present three results. The first is deterioration\nprediction from a single image, where our model achieves an area under receiver\noperating characteristic curve (AUC) of 0.742 for predicting an adverse event\nwithin 96 hours (compared to 0.703 with supervised pretraining) and an AUC of\n0.765 for predicting oxygen requirements greater than 6 L a day at 24 hours\n(compared to 0.749 with supervised pretraining). We then propose a new\ntransformer-based architecture that can process sequences of multiple images\nfor prediction and show that this model can achieve an improved AUC of 0.786\nfor predicting an adverse event at 96 hours and an AUC of 0.848 for predicting\nmortalities at 96 hours. A small pilot clinical study suggested that the\nprediction accuracy of our model is comparable to that of experienced\nradiologists analyzing the same information.",
+                    "authors": "Anuroop Sriram, Matthew Muckley, Koustuv Sinha, Farah Shamout, Joelle Pineau, Krzysztof J. Geras, Lea Azour, Yindalon Aphinyanaphongs, Nafissa Yakubova, William Moore",
+                    "published": "2021-01-13",
+                    "updated": "2021-01-25",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction The SARS COV-2 infection (COVID-19) has caused a surge of patients with respiratory illness presenting to emergency departments worldwide throughout 2020. When patients arrive at the emergency department, hospitals need to evaluate their risk of deterioration for effective clinical decision making and resource allocation. This is especially important during a pandemic when hospital resources are strained. Initial patient assessment often includes imaging studies, and these initial assessments can be used for resource planning within the hospital. Chest X-ray radiography (CXR) is a well-established imaging examination for patients with symptoms of shortness of breath and fever [1]. Although traditionally used for diagnosis, the use of CXR has expanded to COVID deterioration prediction [2, 3, 4] and is a standard component of COVID patient health assessments. *Equal contribution. arXiv:2101.04909v2  [cs.CV]  25 Jan 2021 \fA PREPRINT JANUARY 26, 2021 Deep learning methods for image-based diagnosis are a standard tool used in radiology research [5, 6, 7, 8, 9, 10] and have been applied to COVID-19 [2, 3, 11, 12, 13, 14, 15]. Most deep learning methods for radiology rely on the collection of large numbers of labeled radiology images for supervised training, which introduces a number of constraints. First, the collection of large, labeled, training sets is expensive [7], and can only be accomplished by well-funded research organizations. Second, it can be dif\ufb01cult to assign labels for many radiology tasks. Radiologists are often tasked to give impressions of images and radiology reports are expected to convey nuance and uncertainty with regards to the imaging \ufb01ndings. This can create downstream uncertainty in label assignment [16]. Third and related, the requirement for large datasets often restricts the use of deep learning methods to established and well-understood pathologies. Thus far, large public X-ray datasets have been collected at major research centers over time spans as long as a decade [17, 16, 18]. The availability of such data is essential to facilitate new research into machine learning methods for improving image-based diagnosis. However these datasets do not capture data from emerging diseases, such as COVID-19. At this time, most hospital centers have not collected enough COVID-19 patient data at the scale required to train deep learning representations. Technological and privacy considerations prevent data pooling between many research centers. One common approach to dealing with small amounts of labeled data is to apply transfer learning [19], where a model is pretrained in a supervised fashion on a large, labeled dataset (e.g., ImageNet), then \ufb01netuned on the task of interest. However, transfer learning can lead to poor performance if the tasks are too different, and the model is not able to learn the features necessary for the transfer task during the pretraining step. Recently, new self-supervised methods which rely on contrastive losses have been shown to generate representations that are as good for classi\ufb01cation as those generated using purely supervised methods [20, 21, 22, 23]. The advantage of contrastive loss functions is that they are able to achieve feature extraction independent of labels or tasks associated with the pretraining dataset. These features are then used for training a classi\ufb01er in the \ufb01ne-tuning stage with the target data. In this paper, we study the applicability of self-supervised learning to the task of COVID-19 deterioration prediction. We pretrain a model using momentum contrast (MoCo) [22, 23] on two large, public chest X-ray datasets, MIMIC-CXR-JPG [18, 24, 25] and CheXpert [16], and then use the pretrained model as a feature extractor for the downstream task of prediction COVID-19 patient outcomes. Concurrent to our work, Sowrirajan et al. [26] have shown that MoCo pre-training can be used to learn representations from chest X-rays that are useful for identifying the presence of pleural effusions. We study the effectiveness of this approach on three downstream tasks: 1) adverse event prediction from single images (SIP), oxygen requirements prediction from single images (ORP), and adverse event prediction from multiple images (MIP). We \ufb01nd that, when using the DenseNet architecture [27, 6, 10], self-supervised training achieved higher areas under receiver operating characteristic curve (AUC) on a hold-out test set than supervised training for predicting an adverse event at all time points. Self-supervised pretrained models achieved higher AUC values for the oxygen requirements prediction task at early time points. We then show that performance for predicting adverse events can be further improved by making predictions based on image sequences with a Transformer model. Finally, we characterize the performance of our single-image models vs. the previous COVID-GMIC model [2] and our multi-image models vs. human radiologists in a pilot clinical study. In combination with this paper, we also open source our code and models pretrained on public data at https://github.com/facebookresearch/CovidPrognosis for other researchers interested in using them for \ufb01ne-tuning on their own X-ray datasets. 2 Datasets 2.1 Public Chest X-ray Datasets (Pretraining Data) We used two datasets for pretraining our models [18, 24, 25] and CheXpert [16]. The MIMIC-CXR dataset consists of 377,110 chest X-ray images corresponding to 227,835 radiographic studies performed at the Beth Israel Deaconess Medical Center in Boston, MA. The CheXpert dataset consists of 224,316 chest radiographs of 65,240 patients compiled from chest radiographic studies performed at the Stanford Hospital. The datasets did not contain any COVID-19 patients during the time of our study as they were collected before the COVID-19 pandemic. Imaging \ufb01ndings included a variety of chest conditions such as atelectasis, cardiomegaly, consolidation, edema, and pleural effusion, based on a labeler designed for the CheXpert dataset [18, 16]. We did not use these \ufb01ndings in our study beyond initial validation of the training pipeline. 2.2 COVID-19 Dataset and Deterioration Labels (Fine-tuning Data) For \ufb01ne-tuning, we used an extended version of the NYU COVID dataset previously described [2], containing 26,838 X-ray radiographs from 4,914 patients. COVID-19 prognosis can be de\ufb01ned by a wide array of factors. Our goal 2 \fA PREPRINT JANUARY 26, 2021 in this study is to predict patient deterioration that will require physical transfers throughout the hospital or altered equipment usage. As an example, one type of deterioration would be a transfer to the ICU. During the transfer process, a COVID-positive patient is removed from isolation, heightening the risk of intra-hospital infection. By locating at-risk patients closer to the ICU from the beginning, the the transfer time can be reduced, correspondingly reducing the risk of intra-hospital infection. Additionally, there and limited number of intensive care beds available per hospital. Predicting the need for these beds is critical to hospital resource management. Another form of deterioration is increased oxygen utilization. Normally, patients are administered oxygen via the nasal cannula. When a patient\u2019s oxygen requirements exceeds 6 L per day, oxygen administration through the nasal cannula is no longer possible and delivery must be done via a mask. Formally, our study considers two label classes: 1) adverse events [2, 3] and 2) increased oxygen requirements. An \u201cadverse event\u201d consists of any of the three events: transfer to the intensive care unit (ICU), intubation, or mortality. We labeled each image with whether the patient developed any adverse event within 24, 48, 72 or 96 hours of the scan. We obtained these labels by occurrences of adverse events recorded in the patient\u2019s anonymized electronic health records data shared by NYU, following previous practice [2]. We de\ufb01ne \u201cincreased oxygen utilization\u201d as an event where a patient requires more than 6 L of oxygen in a day. Similarly to the case of adverse events, we labeled each image in the dataset with whether the patient required increased oxygen on within 24, 48, 72, or 96 hours of the scan. 2.3 De\ufb01nition of Prediction Tasks The goal of our machine learning models is to predict deterioration labels from either single chest radiograph or sequences of chest radiographs. The \ufb01rst task is to predict adverse events from a single chest radiograph. This constitutes a multi-class classi\ufb01cation setting for ICU transfer, intubation, mortality, and \u201cAny\u201d adverse event at 24, 48, 72, 96, and \u201cAny\u201d time windows (20 total labels). We call this task the Single Image Prediction (SIP) task. The second task is to predict increased oxygen requirements from a single chest radiograph. This also constitutes a multi-class classi\ufb01cation setting with increased oxygen at 24, 48, 72, 96, and \u201cAny\u201d time windows (5 total labels). We call this the Oxygen Requirement Prediction (ORP) task. Our \ufb01nal task is to predict adverse events from a sequence of radiographs. This task is of particular interest, as it more closely aligns with how radiologists interpret images. Radiologists use relative changes in images to identify a patient\u2019s trajectory. For this task, we considered the same multi-class classi\ufb01cation problem as in the SIP setting. We call this task the Multiple Image Prediction (MIP) task. 2.4 Data Splits Table 1 shows the counts of scans and patients from the NYU COVID data in each split for each task. The tasks vary in counts due to varying selection criteria used for each task. For the SIP task, we followed the previous standard [2] in including chest radiographs of patients collected in the emergency department and excluding chest radiographs of patients who had already experienced any of the three adverse events, as they had already deteriorated. This enables us to directly compare our results to those in [2]. For the ORP task, we only included chest radiographs of patients collected in the emergency department, and did not apply any other exclusion criteria because we considered the oxygen requirement prediction to be independent of the occurrence of any adverse event. In the MIP task, the goal was to predict deterioration of patients in the emergency department and during any subsequent hospital admissions. Therefore, for this task we only excluded chest radiographs collected after a patient had experienced any adverse event. Table 1: Subsplits for prediction tasks. Task # Scans # Patients {train, val} test total {train, val} test total SIP 5,617 770 6,387 2,943 718 3,661 MIP 17,915 2,205 20,120 2,774 438 3,212 ORP 15,370 5,711 21,081 2,887 1,096 3,983 Figure 1 shows the number of adverse events in the SIP and MIP splits as a function of time. We note the labels are necessarily monotonically increasing due to the fact that adverse events at 48 hours include all events at 24 hours, etc. We did not make any special adjustments to our models to take advantage of this aspect, but this could be a topic for future investigation. 3 \fA PREPRINT JANUARY 26, 2021 24 48 72 96 Adverse Event Time (hours) 200 400 600 800 1000 1200 Adverse Event Counts Adverse Events Over Time ICU (MIP) Intubated (MIP) Mortality (MIP) ICU (SIP) Intubated (SIP) Mortality (SIP) Figure 1: Distribution of adverse events with respect to the time from chest X-ray acquisition for the single-image (SIP) and multiple image (MIP) prediction tasks. 3 Methods Here we describe the models and training process. All of our proposed models were based on the DenseNet-121 architecture [27], which has been previously applied to chest radiographs [6, 10, 3]. We begin by describing our self-supervised pretraining procedure, then follow with the \ufb01ne-tuning procedure for the SIP and ORP tasks. Lastly, we describe the development of a new transformer based architecture that we applied to the MIP task. 3.1 Self-Supervised Pretraining using Momentum Contrast Learning The contrastive loss framework is as follows: a deep neural network is constructed for mapping an image to a latent space [20]. The neural network is trained to minimize a contrastive loss. The contrastive loss is constructed such that similar images are mapped to vectors that are closer to each other (as measured by a contrastive loss function) while dissimilar images are mapped to vectors that are further apart. Figure 2 shows an schematic of the contrastive loss training procedure. base image xq encoder rq xk momentum encoder rk L(rq, rk) Aug. 1 Aug. 2 Figure 2: Diagram for momentum contrast training. A base image is transformed via two random augmentations (Aug. 1 and Aug. 2) into images xq and xk. xq is passed through an encoder network, while xk is passed through a momentum encoder network. The representations generated by each network are then passed into a contrastive loss that promotes similarity between the representations rq and rk. Each training step begins by selecting a base image, x, from a training dataset of unlabeled images. Then, two different augmentations are chosen at random and applied to the base image separately to generate two augmented images, the query image xq and the key image xk. These images are passed through two different neural networks, called the encoder and the momentum encoder respectively, to generate representation rq for xq and rk for xk. 4 \fA PREPRINT JANUARY 26, 2021 The goal of the contrastive loss is to identify that rq and rk come from the same underlying image in the presence of divergent augmentations. In practice contrastive losses usually require very large batch sizes due to the fact that many negative examples (i.e., values for rq,j and rk,i where rk,i is not from the same image) are necessary to achieve strong performance [21]. MoCo [22] is a recent contrastive loss method that avoids the need for large batch sizes by maintaining a queue of representations. Within the queue, the model stores K examples of rk,i for i \u2208[1, ..., K]. The model is then asked to identify which of the K examples is the matching one for rq. This can be modeled mathematically via the InfoNCE contrastive loss function [28]: L (rq, rk) = \u2212log exp (rq \u00b7 rk+/\u03c4) PK i=1 exp (rq \u00b7 rk,i/\u03c4) , (1) where \u03c4 is a temperature hyperparameter and K is the number of currently stored representations. The gradient from the contrastive loss is backpropagated to the encoder network, and then the momentum encoder is updated via a momentum update [22]. The momentum update forces the momentum encoder to change more slowly than the encoder network, which helps stabilize training. 3.2 Single Image Prediction (SIP) and Oxygen Requirement Prediction (ORP) Models The SIP and ORP tasks were described in Section 2.3. We extend the MoCo encoder model to an image classi\ufb01er model by appending a linear classi\ufb01er to the end of the encoder pipeline. We then \ufb01ne-tune the model on the COVID image dataset, reshaping input images to a 224 \u00d7 224 grid and applying random horizontal and vertical \ufb02ipping during training. We used the Adam optimizer [29] with a cosine annealing learning rate decay [30]. 3.3 Multiple Image Prediction (MIP) Model For the MIP task, we propose a new model that takes a sequence of X-ray images (x0, ..., xn) along with their scan times (t0, ..., tn) relative to the \ufb01nal scan as inputs and predicts the likelihood of adverse events occurring after the \ufb01nal scan. The overall model structure is shown in Figure 3. The scan times are represented as the number of hours from the \ufb01nal scan time. Thus, if a patient had two previous scans done, e.g., 50 hours and 20 hours before the \ufb01nal scan, then the scan times will be represented as (t0 = \u221250, t1 = \u221220, t2 = 0). The \ufb01nal scan time tn is always 0. During the forward pass, each image xi is \ufb01rst passed to the MoCo encoder model in parallel to obtain an image representation hi. Separately, each relative scan time ti is passed to a Continuous Position Embedding (CPE) module to learn a time embedding ei. The CPE module, described in more detail below, maps each time point to a different embedding. The two representations hi and ei for each i are then concatenated together and then projected to a lower dimension using a fully connected layer. The full sequence of images is then input to a transformer network [31]. The transformer network maps an input sequence to an output sequence of the same length using a self-attention mechanism. The output of the transformer is the sum-pooled to obtain a hidden representation f that contains aggregated information from all images. This hidden representation f is input to a linear classi\ufb01er. Similar to the SIP task, the weights of the MoCo encoder are initialized to the pretrained weight values while the weights of the transformer and the linear classi\ufb01er are randomly initialized. The entire network is \ufb01ne-tuned jointly on a labeled dataset to mimize the binary cross entropy loss. Continuous Positional Embedding (CPE) The CPE, inspired by the positional embedding (PE) from [31], is designed to map each time point to a d-dimensional vector representation. The CPE has the same functional form as the PE, but it can take continuous values within a certain range as input. Formally, the CPE maps a relative scan time s to a vector e as: e(t,2i) = sin \u0010 t/100002i/d\u0011 , e(t,2i+1) = cos \u0010 t/100002i/d\u0011 , (2) where t \u2208[0, 360). We only include X-ray scans taken within the last 360 hours (15 days) of the end exam since the COVID-19 infection is unlikely to last longer than 15 days. DropImage Regularizer To reduce over\ufb01tting and to improve the robustness of the MIP model, we apply a masking based regularizer that we call the DropImage regularizer. The DropImage regularizer, when applied to a an image 5 \fA PREPRINT JANUARY 26, 2021 Figure 3: Schematic of the Multiple Image Prediction (MIP) model. The MIP model takes a sequence of images along with their relative scan times as input and outputs a binary prediction for each adverse event. The MoCo encoder model is applied to each image to learn image representations. These image representations are concatenated with a time embedding obtained from the relative scan times and then projected to a lower dimension and then input to a transformer network to aggregate information from all images. The output of the transformer is pooled together and input to a linear classi\ufb01er. sequence, drops a subset of images chosen independently at random. The \ufb01nal image is never removed. DropImage encourages learning more robust models that can make good predictions even if some of the past images were missing. Further, DropImage can also be viewed as a data augmentation scheme. 4 Experiments 4.1 Pretraining We applied both supervised and self-supervised pretraining procedures. Our supervised pretraining was similar to that for CheXpert [16]. For supervised pretraining, we trained the DenseNet-121 model with the Adam optimizer [29] with a learning rate of 10\u22123, weight decay of 10\u22125, and batch size of 64 on the MIMIC-JPG dataset [24]. Data augmentation included interpolation to a 224 \u00d7 224 grid and random vertical/horizontal \ufb02ipping. We pretrained for 10 epochs, decaying the learning rate by a factor of 10 each epoch. We pretrained our self-supervised model using the momentum contrast training described in Section 3.1. For data augmentation, we used random cropping and interpolation to a 224 \u00d7 224 grid, random horizontal/vertical \ufb02ipping, and random Gaussian noise addition. We investigated a number of augmentation strategies for pretraining the MoCo models, including noise additions, af\ufb01ne transformations, color transformations, and X-ray acquisition simulation, but in the end we found that these provided little bene\ufb01t beyond those in the original MoCo paper [22]. After the augmentations, we applied histogram normalization. The histogram normalization procedure preserves the relative values of the pixels by interpolating along a shifted version of the image pixel value histogram while constraining the resulting pixel values to be in the speci\ufb01ed target range. The augmentations are described in further detail in Appendix A. 6 \fA PREPRINT JANUARY 26, 2021 We tuned the following hyperparameters during the pretraining phase: learning rate, MoCo latent feature dimension size, and the queue size. We searched over a logarithmic scale of values, varying the learning rate within 10{\u22122,\u22121,0}, and MoCo feature dimensions within {64, 128, 256}. The queue size was \ufb01xed at 65,536. We used a batch size of 128 for each of 8 GPUs, the largest we could achieve in initial testing, accumulating gradients using PyTorch\u2019s DistributedDataParallel framework [32, 22]. We selected hyperparameters based a cross-validation analysis on the downstream tasks. We optimized models using stochastic gradient descent with momentum [33], using 0.9 as the momentum term and a weight decay parameter of 10\u22124. Pretraining took approximately four days using eight 16 GB Nvidia V100 GPUs. 4.2 Example Downstream Task Outputs Figure 4 illustrates the frameworks for each of the prediction tasks over the succeeding 24 hours. Figure 4a shows an example of SIP predictions from a patient with increased lung opacity. In this case the patient did not suffer any adverse event in the next 24 hours, but ultimately suffered all three adverse events within 72 hours. Figure 4b shows an example of ORP predictions from a patient that required increased oxygeen within 24 hours. In Figure 4c, a sequence of chest X-rays with increasing lung opacity is used by the MIP model to predict COVID deterioration for a patient. The images were taken 49 hours apart. For the case of Figure 4c, the patient was transferred to the ICU, intubated, and suffered a mortality within 24 hours. SIP Predictions ICU24: 0.844 Int24: 0.997 Mor24: 0.559 (a) ORP Predictions >6L24: 0.768 (b) MIP Predictions ICU24: 0.784 Int24: 0.782 Mor24: 0.965 (c) Figure 4: Example model outputs for ICU (ICU24), intubation (Int24) mortality (Mor24), and oxygen greater than 6 L per day (>6L24) prediction tasks, all at 24 hours. (a) Example SIP outputs based on a single image with evident increased lung opacity. In this case the patient did not suffer any adverse event in the next 24 hours, but would ultimately suffer all three adverse events within 72 hours. (b) Example ORP output based on a single image. This patient required greater than 6 L per day of oxygen within 24 hours. (c) Example MIP outputs. Both images were taken from the same patient with 49 hours of separation. Increased lung opacity is observable in the second (later) image. The patient suffered all three adverse events within 24 hours. 4.3 Single-Image Task Ablations We considered both self-supervised and supervised pretraining for each of our single-image tasks (SIP and ORP). Data splits for these tasks are shown in Table 1. Following previous work [16, 13, 3], we compare models via their areas under receiver operator characteristic curve (AUC). In order to facilitate a straightforward comparisons, our pretraining in this section for both supervised and MoCo only used the MIMIC dataset. Before \ufb01ne-tuning for the SIP and ORP tasks, the weights of the encoder are initialized to the pretrained weight values while the weights of the linear classi\ufb01er are randomly reinitialized with a uniform distribution [34]. We considered three different \ufb01ne-tuning ablations. The \ufb01rst ablation (CL) \ufb01xed the MoCo encoder (including BatchNorm statistics) and only trained the new linear classi\ufb01er [22]. The second ablation (FT) allowed the entire model to train, not just the classi\ufb01er. The \ufb01nal ablation (FT RA) allowed the entire model to train and also incorporated further random augmentations (rotation, X-shear, Y-shear, and translations). The CL model was \ufb01ne-tuned for 5 epochs. The FT models were \ufb01ne-tuned for 20 epochs, and the FT RA models were \ufb01ne-tuned for 40 epochs. We also show a \u201cScratch\u201d ablation, which is a DenseNet model with random initialization [34] trained only on the COVID data (i.e., no pretraining). 7 \fA PREPRINT JANUARY 26, 2021 We tuned each ablation using a strati\ufb01ed 5-fold cross validation of the {train,val} split of the data. We tuned learning rate in the range 10{\u22124,\u22123,\u22122,\u22121} and optimized over all MoCo/supervised-pretrained models for model selection. We also tuned whether to use the Adam [29] or SGD [33] optimizers. All \ufb01ne-tunings used the cosine annealing learning rate decay [30]. After the cross-validation, we selected the best model for each ablation and applied a bootstrap analysis to estimate performance characteristics on the test set. To test for signi\ufb01cance between methods we applied 1,000 bootstrap iterations to the test set and computed the difference in AUC between the best MoCo method and the best supervised pretraining method. Methods where the best MoCo AUC is signi\ufb01cantly higher than the best supervised AUC are indicated with *. Our tables show results in two rows for each method. The top row shows the holdout test AUC, while the bottom row shows percentile bootstrap 95% con\ufb01dence intervals. In some cases we have statistical signi\ufb01cance with overlapping con\ufb01dence intervals\u2013this is because the two methods are correlated, an aspect captured in the bootstrapped AUC differences. We chose this non-parametric test over the standard method of DeLong [35, 36] to maintain a consistent presentation, but in a separate analysis found the DeLong AUC difference test generally matched our non-parametric test. Table 2: SIP AUCs Across Ablations (95% CI: Bootstrap) AUC of Any Adverse Event Prediction 24 hours 48 hours 72 hours 96 hours Scratch FT (No PT) 0.616 0.645 0.651 0.667 (0.550, 0.679) (0.588, 0.701) (0.598, 0.705) (0.620, 0.715) Supervised PT CL 0.668 0.690 0.681 0.692 (0.599, 0.738) (0.625, 0.753) (0.628, 0.730) (0.645, 0.743) Supervised PT FT 0.672 0.697 0.687 0.702 (0.610, 0.739) (0.639, 0.754) (0.631, 0.737) (0.653, 0.750) Supervised PT FT RA 0.658 0.694 0.686 0.703 (0.591, 0.722) (0.638, 0.747) (0.635, 0.736) (0.658, 0.748) Moco PT CL 0.691 0.716 0.711 0.730 (0.627, 0.748) (0.662, 0.766) (0.660, 0.758) (0.686, 0.775) Moco PT FT 0.686 0.719 0.726* 0.742* (0.619, 0.751) (0.659, 0.769) (0.677, 0.774) (0.696, 0.785) Moco PT FT RA 0.658 0.703 0.708 0.729 (0.590, 0.724) (0.646, 0.761) (0.656, 0.758) (0.683, 0.772) A comparison via AUC values for supervised vs. self-supervised pretraining for the SIP task is shown in Table 2. AUC values generally increased over longer time intervals. MoCo models had the highest AUC scores for predicting adverse events at all time windows, achieving signi\ufb01cance at 72 and 96 hours. The MoCo PT CL model (pretrained with MoCo, only \ufb01ne-tuning classi\ufb01cation layer) had the highest AUC at 24 hours, whereas the PT FT model (which allowed the entire model to \ufb01ne-tune) achieved highest AUC at 48, 72 and 96 hours. The Scratch method performed worst across all architectures and pretraining methods. Performance improved between Scratch and supervised pretraining, and then again between supervised pretraining and MoCo pretraining. A comparison via AUC values for Supervised vs. Self-Supervised pretraining for the ORP task is shown in Table 3. The time course trend is inverted relative to Table 2, with O2 models becoming less accurate at larger time windows. In this case, Supervised pretraining models performed more similarly to MoCo-pretrained models. The MoCo PT FT model performing had the best AUC at 24 and 48 hours, while the Supervised PT CL model had the best AUC at 72 and 96 hours. Only 24 hour prediction using MoCo was signi\ufb01cant based on the bootstrap tests. 4.4 SIP Comparison to COVID-GMIC COVID-GMIC is a previous neural network image analysis model developed for the purpose of COVID prognosis prediction [2]. COVID-GMIC utilizes supervised pretraining with an architecture explicitly designed for synthesizing information at coarse and \ufb01ne image scales [37] with supervised pretraining on the NIH dataset [17]. Our SIP experiments used the same test set that was used for the COVID-GMIC, so we compare our models directly in Table 4. For this comparison we included a different MoCo pretraining procedure where we used a combination of both the MIMIC-CXR [18] and CheXpert [16] data sets for pretraining. Despite having more data, in our 5-fold cross-validation of the train,val split the same the same model was selected as in Section 4.3. 8 \fA PREPRINT JANUARY 26, 2021 Table 3: ORP AUCs Across Ablations (95% CI Method: Bootstrap) AUC of O2 > 6L requirement Prediction 24 hours 48 hours 72 hours 96 hours Scratch FT (No PT) 0.696 0.670 0.654 0.632 (0.636, 0.754) (0.619, 0.720) (0.611, 0.697) (0.591, 0.674) Supervised PT CL 0.749 0.706 0.712 0.706 (0.698, 0.793) (0.659, 0.750) (0.672, 0.751) (0.671, 0.741) Supervised PT FT 0.727 0.701 0.707 0.694 (0.676, 0.777) (0.655, 0.746) (0.668, 0.746) (0.658, 0.729) Supervised PT FT RA 0.715 0.685 0.695 0.679 (0.655, 0.769) (0.641, 0.731) (0.657, 0.731) (0.644, 0.716) Moco PT CL 0.699 0.683 0.690 0.686 (0.644, 0.750) (0.635, 0.724) (0.654, 0.728) (0.652, 0.722) Moco PT FT 0.765* 0.713 0.702 0.688 (0.719, 0.807) (0.671, 0.753) (0.666, 0.737) (0.654, 0.722) Moco PT FT RA 0.722 0.696 0.698 0.694 (0.671, 0.767) (0.651, 0.739) (0.661, 0.734) (0.658, 0.725) Table 4: SIP AUCs Across Methods (95% CI Method: Bootstrap) AUC of Any Adverse Event Prediction 24 hours 48 hours 72 hours 96 hours DenseNet MoCo (best) 0.686 0.719 0.726 0.742 (0.619, 0.751) (0.659, 0.769) (0.677, 0.774) (0.696, 0.785) COVID-GMIC 0.695 0.716 0.717 0.738 (0.627, 0.754) (0.661, 0.766) (0.661, 0.766) (0.691, 0.781) In this case we did not apply signi\ufb01cance testing since we did not have the raw predictions from this paper. Both the MoCo and COVID-GMIC methods are comparable across time points, with wide overlaps in con\ufb01dence intervals. The widest gap in performance was for predicting adverse events in the next 24 hours, where COVID-GMIC had a 0.02 higher AUC. At longer time intervals the performance of MoCo methods began to improve, showing higher AUC scores than COVID-GMIC on the test set at 48, 72 and 96 hours. 4.5 Multi-Image Task Ablations Table 5 shows ablations for the MIP task. In this case, we compared and tested the difference between the MoCo PT CL ablation and the Transformer model built on top of MoCo representations. We trained each model with the Adam optimizer [29] for 50 epochs with a batch size of 32. We tuned the following hyperparameters using grid search: learning rate, drop image probability (pdrop), projection dimension and the pooling method. We searched over learning rate values of 10{\u22123,\u22122,\u22121}, pdrop values in {0, 0.1, 0.2, 0.5}, projection dimension in {16, 32, 64, 128}. For the pooling method, we used either sum pooling or simply taking the \ufb01nal time step. We set the dropout value for the transformer and the CPE to be 0.5 and weight decay to 10\u22125. Table 5: MIP AUCs Across Ablations (95% CI Method: Bootstrap) AUC of Any Adverse Event Prediction 24 hours 48 hours 72 hours 96 hours MoCo PT CL 0.747 0.755 0.758 0.768 (0.719, 0.776) (0.730, 0.780) (0.734, 0.782) (0.746, 0.790) MoCo PT + Transformer 0.767* 0.778* 0.778* 0.786 (0.740, 0.796) (0.752, 0.802) (0.755, 0.800) (0.764, 0.807) 9 \fA PREPRINT JANUARY 26, 2021 The Transformer method performed better than the single-image MoCo method for predicting adverse events at every time window with statistical signi\ufb01cance at 24, 48, and 72 hours. A more detailed version of Table 5 comparing Single-image and multi-image prediction for other adverse events is shown in TableA1 of Appendix B. The MIP Transformer model was also better at predicting ICU transfers, intubations, and mortalities. The improvements for ICU transfers were signi\ufb01cant at all time windows, while mortalities were signi\ufb01cant at 72 and 96 hours. The performance differences for intubation were not statistically signi\ufb01cant. 4.6 Reader Study We compared the performance of our model with two chest radiologists from NYU Langone Health in a pilot reader study with a sample of 200 image sequences from the test set (note: this was a new, distinct reader study from that used previously [2]). The results of the study for \u201cany\u201d adverse events are shown in Table 6, while Table A2 in Appendix C shows an extended version of the data with breakdowns by adverse event category. We also present the results from averaging the prediction of both radiologists (shown as \u201cRadiologist A + B\u201d). Table 6: MIP AUCs Compared to Radiologists (95% CI Method: Bootstrap) AUC of Any Adverse Event Prediction 24 hours 48 hours 72 hours 96 hours MoCo PT + Transformer 0.785 0.801 0.790 0.790 (0.717, 0.852) (0.739, 0.861) (0.726, 0.855) (0.727, 0.853) Radiologist A 0.786 0.769 0.733 0.739 (0.694, 0.840) (0.709, 0.840) (0.664, 0.804) (0.658, 0.797) Radiologist B 0.789 0.787 0.762 0.751 (0.709, 0.846) (0.716, 0.846) (0.693, 0.826) (0.684, 0.816) Radiologist A + B 0.784 0.787 0.761 0.754 (0.703, 0.774) (0.722, 0.786) (0.687, 0.754) (0.678, 0.745) For this task the radiologists indicated the presence of intubation support devices in the X-ray. We performed an audit of our selection criteria and found that in some cases these patients did not have an intubation event in the anonymized electronic health records despite having the support devices visible on the chest X-ray. In other cases, the adverse event was recorded later. Given that images with support devices were likely in the training set, we allowed radiologists to simply predict positive for any cases where they saw a support device, as this information was available to the model as well. Both radiologists exhibited wide con\ufb01dence intervals as indicated by bootstrap methods. Radiologist AUC values trended better than our model for the task of intubation prediction (Appendix C), while our model trended better for ICU transfer and mortality prediction. On the task of predicting any adverse event, our model performed better for longer term prediction. The largest differences for AUC scores between radiologists and our model occurred when predicting mortalities, where our model had better AUC scores. Radiologists were aware of the primary timeframe of the dataset (i.e., April-June 2020). 5 Discussion 5.1 Performance Among Proposed Models Our study examined whether performance for COVID prognosis prediction could be improved via the use of two techniques: 1) self-supervised pretraining and 2) prediction based on image sequences. Self-supervised pretraining achieved the highest AUC scores on our hold-out test set for all SIP tasks of interest. Our bootstrap statistical tests found signi\ufb01cance at 72 and 96 hours. For ORP tasks, supervised pretraining performed similarly to MoCo pretraining, except for 24 hours. This may indicate that features associated with pathological lung \ufb01ndings are more predictive for oxygen requirements than for adverse events, but this topic needs to be studied further. Interestingly, the predictions for the SIP tasks became more accurate over longer time intervals, whereas the predictions for the ORP tasks became more accurate over shorter time intervals. The SIP results could have improved due to the presence of more positive labels at the later time windows. Label scarcity hindered model performance in all test sets, as indicated by the bootstrap con\ufb01dence intervals in Section 4.4 and Appendix B. However, the inversion of the trend 10 \fA PREPRINT JANUARY 26, 2021 for ORP cannot be explained by label scarcity. It could be that there are image features that become readily apparent for the ORP task close to the actual moment of increased oxygen needs. Our greatest performance gain occurred when when predicting based on image sequences rather than single images. These improvements were statistically signi\ufb01cant via our non-parametric tests. Radiologist sentiment indicated that it is dif\ufb01cult to identify patient conditions from single images, as temporal trajectories are critical for performing analyses. The results of our transformer models support this intuition, with AUC scores improving for all tasks by at least 0.01 and as much as 0.05 when using image sequences rather than individual images. 5.2 Test Set Uncertainty A recent meta-study noted that many prediction models targeting COVID are at a high risk of over\ufb01tting and bias [38]. We addressed this via two methods: 1) pretraining on non-COVID data and 2) explicit sample selection investigation via bootstrap experiments. We observed that the \ufb01nal bootstrap con\ufb01dence intervals in Section 4.4 were wide enough to include broad classes of ablations, corroborating the uncertainties associated with small datasets. The greatest sensitivity was with respect to positive labels, which were infrequent among the test splits for the various tasks. It would be reasonable to expect forward performance anywhere within the range of the bootstrapped con\ufb01dence intervals. This uncertainty is on top of other caveats already well-known in the medical \ufb01eld, namely, uncertainty arising due to changes in the patient population and treatment effects. 5.3 Pilot Clinical Study Limitations Our pilot clinical study found that our multi-image prediction model had comparable performance to human radiologists for most tasks and surpassed human radiologists in predicting mortalities. However, this study has some limitations. We asked radiologists to rate the probability of adverse events over the upcoming time windows. Depending on the hospital and local practices, this may not be a part of the standard radiologist work\ufb02ow. Radiologists are typically asked to summarize imaging \ufb01ndings and to convey nuance with how the information in image features may impact clinical care. It is possible that with further calibration, radiologist performance could be improved for the task of adverse event prediction. We present these results to help readers put model performance in the context of currently possible clinical predictions. 5.4 Alternative Approaches Supervised DenseNet pretraining was previously used [2] to generate higher AUC scores than we report here. The previous work included further optimizations, including a larger hyperparameter sweep and averaging predictions of several models. Some of these optimizations in particular model ensembling were not investigated in our study, but could lead to a closing of the gap in performance between supervised and self-supervised models that we observe in this paper. Since the start of our work, new approaches for self-supervision have been published in the literature. Clustering can be used to relax the approach in this paper that only looks at a single image for matching [15]. Based on our ORP results, another approach that might be bene\ufb01cial would be including labels as a form of weak supervision along side a constrastive loss. Investigating these alternative approaches may provide further performance improvements for the self-supervised methods. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1910.12325v4",
+                    "title": "GrappaNet: Combining Parallel Imaging with Deep Learning for Multi-Coil MRI Reconstruction",
+                    "abstract": "Magnetic Resonance Image (MRI) acquisition is an inherently slow process\nwhich has spurred the development of two different acceleration methods:\nacquiring multiple correlated samples simultaneously (parallel imaging) and\nacquiring fewer samples than necessary for traditional signal processing\nmethods (compressed sensing). Both methods provide complementary approaches to\naccelerating the speed of MRI acquisition. In this paper, we present a novel\nmethod to integrate traditional parallel imaging methods into deep neural\nnetworks that is able to generate high quality reconstructions even for high\nacceleration factors. The proposed method, called GrappaNet, performs\nprogressive reconstruction by first mapping the reconstruction problem to a\nsimpler one that can be solved by a traditional parallel imaging methods using\na neural network, followed by an application of a parallel imaging method, and\nfinally fine-tuning the output with another neural network. The entire network\ncan be trained end-to-end. We present experimental results on the recently\nreleased fastMRI dataset and show that GrappaNet can generate higher quality\nreconstructions than competing methods for both $4\\times$ and $8\\times$\nacceleration.",
+                    "authors": "Anuroop Sriram, Jure Zbontar, Tullie Murrell, C. Lawrence Zitnick, Aaron Defazio, Daniel K. Sodickson",
+                    "published": "2019-10-27",
+                    "updated": "2020-03-30",
+                    "primary_cat": "eess.IV",
+                    "cats": [
+                        "eess.IV",
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Magnetic Resonance Imaging (MRI) is the leading diagnostic modality for a wide range of disorders including musculoskeletal, neurological, and oncological diseases. However, the physics of the MRI data acquisition process make it inherently slower than alternate modalities like CT or X-Ray. As a consequence, increasing the speed of MRI acquisition has been a major ongoing research goal for decades. Parallel Imaging (PI) is one of the most important and successful developments in reducing MRI scan time [5, 16]. The technique requires the use of multiple physical receiver coils to simultaneously record different views of the object being imaged. Parallel imaging is the default option for many scan protocols and it is supported by almost all modern clinical MRI scanners. Another approach to accelerating MR imaging is the use of Compressed Sensing (CS), which can speed up MRI acquisition by acquiring fewer samples than required by traditional signal processing methods. To overcome aliasing artifacts introduced by violating the Shannon-Nyquist sampling theorem, CS methods incorporate additional a priori knowledge about the images. Recently, the use of learned image priors through the use of deep learning have rapidly gained in popularity [6, 19, 22, 25]. These approaches have shown a signi\ufb01cant improvement in image reconstruction quality, particularly for non-parallel MRIs. In this paper, we show that a novel combination of classical parallel imaging techniques with deep neural networks can achieve higher acceleration factors than using either approach alone. Utilizing parallel imaging in deep learning approaches to reconstruction is challenging. The relation between the captured views changes for each scan and is dependent on the con\ufb01guration of the detectors with respect to the object being imaged. To address this challenge we introduce GrappaNet, a new neural network architecture that incorporates parallel imaging. GrappaNet contains a GRAPPA layer that learns a scan-speci\ufb01c reconstruction function to combine the views captured during parallel imaging. To allow the network to fully utilize all the information captured during parallel imaging, the reconstruction is performed jointly across all the complex-valued views captured during the parallel imaging process. Unlike many previous approaches [6], the views are not combined until the \ufb01nal layer to produce the output reconstruction. The model uses a progressive re\ufb01nement approach in both k-space (frequency domain) and image space to both aid in the optimization and to take advantage of the complementary properties of the two spaces. Most previous approaches typically focus on either reconstructing in image space [6] or k-space [8]. We evaluate the performance of our method on the recently released fastMRI [24] dataset. First, we present a brief introduction to parallel MR imaging and review some deep learning methods for parallel MRI arXiv:1910.12325v4  [eess.IV]  30 Mar 2020 \freconstruction in section 2. Next, we provide a description of the GrappaNet model in section 3, followed by a description of our experiments in section 4. Finally, we conclude with a discussion of future work in section 5. 2. Background and Related Work 2.1. Parallel MRI MR scanners image a patient\u2019s anatomy by acquiring measurements in the frequency domain using a measuring instrument called a receiver coil. In the MRI literature, these frequency-domain measurements are called k-space samples, where k refers to the spatial wave number. The image can then be obtained by applying an inverse multidimensional Fourier transform F\u22121 to the measured k-space samples. The underlying image x \u2208CM is related to the measured k-space samples k \u2208CM as k = F(x) + \u03f5, (1) where \u03f5 is the measurement noise. Most modern scanners support parallel imaging: they employ an array of multiple receiver coils that simultaneously obtain k-space samples from the anatomy being imaged. The k-space samples measured by each coil are modulated by their sensitivity to the MR signal arising from different regions. In particular, the k-space sample measured by the i-th coil is ki = F(Six) + \u03f5i, i = 1, 2, . . . , N, (2) where Si is a complex-valued diagonal matrix encoding the position dependent sensitivity map of the i-th coil and N is the number of coils. Different coils are typically sensitive to different but overlapping regions. It is important to note that the coil sensitivities vary per scan since they depend not only on the con\ufb01guration of the coils but also on their interaction with the anatomy being imaged. 2.2. Accelerated MRI The speed of MRI acquisition is limited by the number of k-space samples obtained. This process can be accelerated by obtaining only a subset of the k-space data: ki = MF(Six) + \u03f5i, i = 1, 2, . . . , N, (3) where M is a binary mask operator that selects a subset of the k-space points. The same mask is used for all coils. Applying an inverse Fourier transform naively to this undersampled k-space data results in aliasing artifacts. Parallel MRI can be used to accelerate imaging by exploiting the redundancies in k-space samples measured by different coils to estimate the missing k-space points from the observed points. Various parallel imaging methods have been proposed but they can be divided into two broad classes: a) SENSE-type methods [16] that operate in the image space, and b) GRAPPA-type methods [5] that operate locally in k-space. The latter is relevant to this work. The GRAPPA algorithm estimates the unobserved kspace points as a linear combination of the neighboring observed k-space points from all coils. The same set of weights are used at all spatial locations, which can be seen as a complex-valued convolution in k-space from N channels to N channels, where N is the number of coils. Formally, the unobserved k-space points ku are computed from the observed k-space points k by convolving with GRAPPA weights G: ku = G \u2217k. (4) During acquisition, the central region of k-space (which corresponds to low spatial frequencies) is fully sampled. This region, called the Auto-Calibration Signal or ACS, is used to estimate the GRAPPA weights G. We can simulate under-sampling in the ACS by masking out certain k-space points. Let the simulated observed and unobserved k-space points in the ACS be k\u2032 and ku\u2032 respectively. From equation 4, the convolution of G and k\u2032 should be equal to ku\u2032. Thus, we can estimate G by solving the following optimization problem: \u02c6 G = argminG \u2225ku\u2032 \u2212G \u2217k\u2032\u22252 . (5) The knee images in the fastMRI dataset [24] were acquired using machines that employ 15 receiver coils and can generally support 2\u00d7 acceleration for imaging of the knee using this approach. Higher acceleration factors lead to aliasing artifacts that cannot be removed by standard parallel imaging methods. 2.3. Compressed Sensing for Parallel MRI Reconstruction Compressed Sensing [4] enables reconstruction of images by using fewer k-space measurements than is possible with classical signal processing methods by enforcing suitable priors. Compressed sensing has been combined with parallel imaging to achieve higher acceleration factors than those allowed by parallel imaging alone. Classical compressed sensing methods use sparsity in some transform domain as a prior. Many classical compressed sensing methods operate in the image domain and solve the following optimization problem: \u02c6 x = argminx 1 2 X i \u2225F(Six) \u2212ki\u22252 + \u03bb\u03a8(x), (6) where \u03a8 is a regularization function that enforces a sparsity constraint in some transform domain such as gradients in the image domain. This problem can be solved by iterative gradient descent style methods. \fIn the last few years, there has been rapid development of deep learning based approaches to MRI reconstruction [6, 11, 13, 19, 22, 25]. MRI reconstruction can be viewed as an inverse problem and several previous research papers have proposed neural networks whose design is inspired by the optimization procedure to solve such an inverse problem [6, 11, 13]. One approach in this direction for the multi-coil reconstruction problem is the Variational Network (VN) [6]. The VN model is a deep neural network, each of whose layers implements one gradient update step for the optimization problem in equation 6. The VN uses pre-computed sensitivity maps and achieves excellent reconstructions at low acceleration factors. Computing sensitivity maps becomes more challenging at higher accelerations, which may limit the maximum acceleration this method can achieve. An alternate line of work operating in k-space is the RAKI model [1] which replaces the single convolution operation in GRAPPA with a deep convolutional network that is trained independently for each scan. The RAKI method emphasizes the importance of using a scan speci\ufb01c model for multi-coil reconstruction. This method is complementary to our work and can be integrated into the GrappaNet by replacing the GRAPPA layer with the RAKI network. A comprehensive survey of recent developments in using deep learning for parallel MRI reconstruction can be found in [12]. 3. GrappaNet The GrappaNet is a neural network that takes undersampled, multi-coil k-space data as input and outputs the reconstructed image. Figure 1 shows a diagram of the network architecture that contains three important properties. First, the differentiable GRAPPA layer enables the network to take advantage of the known physical proprieties of parallel imaging. Next, each convolutional network is applied across all complex-valued views jointly, before being combined in the \ufb01nal stage. This enables the network to take advantage of all the information captured during parallel imaging. Several previous approaches [6, 8], performed reconstruction after collapsing to a single view. Finally, image-to-image mappings using U-Nets are performed in both k-space and image space. Convolutions, pooling, and up-sampling result in very different operations in image space and k-space. We demonstrate in Section 4 that using both these complementary spaces provides improved accuracy. The network consists of two convolutional neural networks, with the application of the GRAPPA operator in between them. Denoting the input under-sampled k-space data by k, the network computes the following function: x = h \u25e6f2(G \u2217f1(k)), (7) where f1, and f2 are convolutional networks that map multicoil k-space to multi-coil k-space and h combines the multicoil k-space data to a single image by \ufb01rst applying an inverse fourier transform followed by an root sum-of-squares (RSS) operation (equation 8). The \ufb01rst network, f1 takes the multi-coil k-space data with R-fold under-sampling and maps it to an R\u2032-fold undersampled k-space dataset with the same number of coils. The GRAPPA operator, G, which is separately obtained from the ACS, is then applied to this R\u2032-fold under-sampled dataset to \ufb01ll in the missing k-space data. This allows the network to take advantage of the known physical proprieties of the parallel imaging process. R\u2032 is chosen to be small enough that traditional parallel imaging methods like GRAPPA can reconstruct the image accurately. We use R\u2032 = 2 for our experiments. 3.1. U-Net Both f1 and f2 are composed of multiple U-Nets [18], which are convolutional networks that operate at multiple scales. U-Net models and their variants have successfully been used for many image-to-image mapping tasks including MRI reconstruction [7, 9] and image segmentation [17]. The U-Nets used in this work are based on the U-Net baseline models from [24]. A U-Net is useful for image to image mapping tasks like semantic segmentation because the presence of pooling and up-sampling layers allow it to learn useful feature maps at multiple scales and abstraction levels. This multi-resolution feature representation helps the U-Net predict the higher level details of the output at the lowest level of the decoder and gradually adds \ufb01ner, higher frequency details as the up-sampling layers are applied. The baseline model, described in [24], used such a U-Net model for MRI image reconstruction. However, that model is only able to perform denoising since it is applied after combining the different views using a root-sum-of-squares (RSS) transform (equation 8). This prevents the baseline model from learning how to combine all of the coils and using the phase information. As a result, the reconstructions from this baseline model are too smooth and lose much of the medically relevant high frequency information (see Figure 3). We show in section 4 that simply applying a U-Net to the real and imaginary data from all coils can signi\ufb01cantly improve upon this model. Such a U-Net can potentially learn to combine information from different coils together, which improves performance. Han et al [8] show that a U-Net can also be applied directly to under-sampled k-space data. Their work was motivated by connections between encoder-decoder models and a classical CS algorithm called the annihilating \ufb01lter-based low-rank Hankel matrix approach (ALOHA) [10]. The input to the ALOHA U-Net is zero-\ufb01lled k-space data and the model \ufb01lls in the missing information. In an approach \fConvolutional  Network Zero-filled  K-Space Apply 2X  GRAPPA Convolutional  Network IFT + RSS Reconstruction Estimate  GRAPPA Kernel U-Net in  K-Space Data  Consistency  + IFT U-Net in  Image Space FT + Data  Consistency Convolutional Network Figure 1. Left: The full GrappaNet model which takes under-sampled k-space samples as input and outputs the reconstructed image. Right: Details about each of the convolutional networks, which take multi-coil k-space as input and output multi-coil k-space. Here, FT, IFT & RSS refer to 2D Fourier transform, 2D inverse Fourier transform and root sum-of-squares operations (equation 8) respectively. similar to the fastMRI baseline model [24], Han et al [8] also apply their U-Net after combining all of the coils into a single coil. Taking insight from algorithms like GRAPPA, we posit that it would be bene\ufb01cial to apply convolutions directly to the multi-coil k-space data. We show in section 4 that such a model outperforms the baseline models. The functions f1 and f2 apply the following series of operations to the input k-space data (see \ufb01gure 1): a U-Net in k-space followed by a hard data consistency, inverse 2D Fourier transform to convert to image space, a U-Net in the image space, followed by a 2D Fourier transform and data consistency. Each of the U-Nets map 15 complex-valued channels to 15 complex-valued channels. Here, the hard data consistency operations simply copy all of the observed k-space samples to the correct locations in k-space. This ensures that the model only \ufb01lls in the missing k-space points. The function h combines the reconstructed multi-coil kspace data into a single real-valued image by \ufb01rst applying an inverse 2D Fourier transform to each coil, followed by a root sum-of-squares (RSS) operation. The RSS operation combines all the coils into a single real-valued image: RSS(x1, . . . xN) =  N X n=1 |xn|2 !1/2 , (8) where x1, . . . , xN are the images from the N coils. 3.2. GRAPPA Layer As explained in the previous section convolutional networks in k-space or image space applied to all coils can, to a limited extent, learn to combine all of the coils. However, as described in 2.2, the coil sensitivities can vary from one imaging examination to another. Traditional parallel imaging methods take this into consideration by estimating distinct sensitivity maps or GRAPPA kernels for each scan. This motivates the need to include a scan-speci\ufb01c component within the neural network that can adapt to differences in the sensitivity pro\ufb01le to improve generalization of the reconstruction model. We achieve this adaptation by introducing a new neural network layer that we call the GRAPPA layer. The GRAPPA layer estimates the GRAPPA kernel from the ACS region and then applies a two dimensional convolution with the estimated kernel. Because the application of GRAPPA is differentiable, the entire network can be trained in an end-toend fashion using backpropagation. 4. Experimental Results We ran all our experiments on the multi-coil knee MRIs from the fastMRI dataset [24], which consists of raw k-space data from 1594 knee MRI exams from four different MRI machines. The dataset contains two types of MRI sequences that are commonly used for knee exams in clinical practice: a Proton Density (PD) weighted sequence and a Proton Density weighted sequence with Fat Saturation (PDFS). We used the same train, validation and test splits as in the original dataset. The training data consisted of 973 volumes which contained k-space data of different sizes. During training, we omitted k-space data with a width greater than 372, which is about 7% of the training data. We evaluated various models on all test images. For training our models, we used random masks with 4\u00d7 and 8\u00d7 accelerations, based on code released with the fastMRI dataset1. We experimented with the following models: 1. Classical CS baseline based on Total Variation minimization [24] 2. U-Net baseline model applied to RSS inputs [24] 3. Variational Network model introduced in [6] 4. U-Net applied in k-space to 15 coil input 5. U-Net applied in image space to 15 coil input 6. GrappaNet model 1https://github.com/facebookresearch/fastMRI \fGround Truth Variational Net GrappaNet Image U-Net Baseline U-Net Figure 2. Example reconstructions for 4\u00d7 under-sampling. The top row shows PD images without fat suppression, and the bottom row shows PDFS images with fat suppression. Ground Truth Variational Net GrappaNet Image U-Net Baseline U-Net Figure 3. Example reconstructions for 8\u00d7 under-sampling. The top row shows PD images without fat suppression and the bottom row shows PDFS images with fat suppression. We used the original implementation of the Variational Network2. This code runs the ESPIRiT algorithm [21] to estimate sensitivity maps from the densely sampled ACS region. These maps are used both as input to the network and also to combine the fully sampled coil responses to compute the training targets. For experiments with 8\u00d7 accelerations, the input k-space contains very few ACS lines, which yields poor quality sensitivity maps for the Variational 2https://github.com/VLOGroup/mri-variationalnetwork/ Network. The training targets computed using these poor quality sensitivity maps contain aliasing artifacts that make them unsuitable for training. To mitigate this problem, we always use 30 low frequency lines to compute the training target for 8\u00d7 experiments. The sensitivity maps used as inputs to the network are still computed from the ACS region. We did not change the model architecture or training procedure from the original implementation, except for the use of random masks. \fGround Truth Variational Net GrappaNet Baseline U-Net Image U-Net Figure 4. Example reconstructions for 4\u00d7 under-sampling with the diagnostically important regions zoomed in. Figure 5. Left: Ground truth, Middle: Reconstruction from baseline, Right: Reconstruction from GrappaNet. The arrow points to the meniscus region, which appears \"\ufb01lled in\" with the baseline method. Highly accurate reconstruction of the meniscus is important for radiologists in diagnosing certain pathologies. For the k-space U-Net, the image space U-Net, and the GrappaNet models, we followed the training procedure for the baseline models in [24]. To deal with complex-valued inputs, we simply treated the real and imaginary parts as two distinct channels. Hence, 15-coil complex-valued k-space or image data were treated as 30-channel data. These models were trained using the RMSProp [20] algorithm to minimize a linear combination of Structural Similarity (SSIM) [23] and L1 losses: J(\u02c6 x, x) = \u2212SSIM(\u02c6 x, x) + \u03bb \u2225\u02c6 x \u2212x\u22251 , (9) where \u02c6 x is the reconstruction and x is the ground truth image, after cropping to the central 320 \u00d7 320 region. Lambda was set to 0.001. The models were trained for 20 epochs with a \ufb01xed learning rate of 0.0003. All models were trained on a machine with 8 NVIDIA Volta V100 GPUs using data parallel training for about 3 days. The U-Net models applied either to 15-coil k-space input or 15-coil image input start with 384 channels, which are doubled after each pooling. The GrappaNet model contains a total of 4 U-Nets, each of which starts with 192 channels. All three models have roughly 480M parameters. Experimental results are shown in table 1, which lists three metrics that are computed in the same manner as [24]: normalized mean squared error (NMSE), peak signal to noise ratio (PSNR) and structural similarity (SSIM) [23]. All of the proposed models perform signi\ufb01cantly better than the baselines. The large difference in performance between a U-Net applied to all 15 coils versus the U-Net baseline underscores the importance of letting the neural network \ufb01gure out how to combine the coil images. The GrappaNet performs best according to all metrics. The improved performance of the GrappaNet can be attributed to the inclusion of the GRAPPA layer to implement parallel imaging within the network. Some example reconstructions are shown in \ufb01gures 2 and 3 for 4\u00d7 and 8\u00d7 accelerations, respectively. Figure 4 and 5 show some of the medically relevant regions zoomed in for 4\u00d7 acceleration. The baseline U-Net model is able to remove aliasing artifacts, but this comes at the cost of severe over-smoothing. The reconstruction lacks some of the high frequency detail that is clinically relevant. The reconstructions from the image U-Net model are signi\ufb01cantly better than the baseline, but they are not as sharp as the reconstructions from the GrappaNet model. The Variational Net model makes heavy use of estimated sensitivity maps throughout the network, including in the data consistency terms. It is able to generate good reconstructions with 4\u00d7 acceleration, which retains a suf\ufb01cient number of low frequency lines to estimate sensitivity maps. When the acquisition is accelerated by 8\u00d7, however, the performance degrades signi\ufb01cantly since it is not possible to accurately estimate sensitivity maps for this case. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1711.01567v1",
+                    "title": "Robust Speech Recognition Using Generative Adversarial Networks",
+                    "abstract": "This paper describes a general, scalable, end-to-end framework that uses the\ngenerative adversarial network (GAN) objective to enable robust speech\nrecognition. Encoders trained with the proposed approach enjoy improved\ninvariance by learning to map noisy audio to the same embedding space as that\nof clean audio. Unlike previous methods, the new framework does not rely on\ndomain expertise or simplifying assumptions as are often needed in signal\nprocessing, and directly encourages robustness in a data-driven way. We show\nthe new approach improves simulated far-field speech recognition of vanilla\nsequence-to-sequence models without specialized front-ends or preprocessing.",
+                    "authors": "Anuroop Sriram, Heewoo Jun, Yashesh Gaur, Sanjeev Satheesh",
+                    "published": "2017-11-05",
+                    "updated": "2017-11-05",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Automatic speech recognition (ASR) is becoming increasingly more integral in our day-to-day lives enabling virtual assistants and smart speakers like Siri, Google Now, Cortana, Amazon Echo, Google Home, Apple HomePod, Microsoft Invoke, Baidu Duer and many more. While recent breakthroughs have tremendously improved ASR performance [1, 2] these models still suffer considerable degradation from reasonable variations in reverberations, ambient noise, accents and Lombard re\ufb02exes that humans have little or no issue recognizing. Most of these problems can be mitigated by training the models on a large volume of data that exemplify these effects. However, in the case of non-stationary processes, such as accents, accurate data augmentation is most likely infeasible, and in general, collecting high quality datasets can be expensive and time-consuming. Past robust ASR literature has considered hand-engineered front-ends and data-driven approaches in an attempt to increase the value of relatively parsimonious data with desired effects [3, 4]. While these techniques are quite effective in their respective operating regimes, they do not generalize well to other modalities in practice due to the aforementioned reasons. Namely, it is dif\ufb01cult to model anything beyond reverberation and background noise from the \ufb01rst principles. Existing techniques \u2217equal contribution. do not directly induce invariance for ASR or are not scalable. And, due to the sequential nature of speech, alignments are needed to compare two different utterances of the same text. In this work, we employ the generative adversarial network (GAN) framework [5] to increase the robustness of seq-to-seq models [6] in a scalable, end-to-end fashion. The encoder component is treated as the generator of GAN and is trained to produce indistinguishable embeddings between noisy and clean audio samples. Because no restricting assumptions are made, this new robust training approach can in theory learn to induce robustness without alignment or complicated inference pipeline and even where augmentation is not possible. We also experiment with encoder distance objective to explicitly restrict the embedding space and demonstrate that achieving invariance at the hidden representation level is a promising direction for robust ASR. The rest of the paper is organized as follows. Related work is documented in Section 2. Section 3 de\ufb01nes our notations and details the robust ASR GAN. Section 4 explains the experimental setup. Section 5 shows results on the Wall Street Journal (WSJ) dataset with simulated far-\ufb01eld effects. Finishing thoughts are found in Section 6. 2. RELATED WORK A vast majority of work in robust ASR deals with reverberations and ambient noise; [3] provides an extensive survey in this effort. One of the most effective approaches in this variability is to devise a strong front-end such as the weighted prediction error (WPE) speech dereverberation [7, 8] and train the resulting neural network with realistic augmented data [9, 10]. A shift from more traditional signal processing techniques to more modern, data-driven methods was seen when the denoising autoencoder [11] was employed to induce invariance to reverberations [12]. This is novel in that the autoencoder is explicitly trained to predict the original audio features from a perturbed version convolved with an impulse response. While denoising autoencoder models for enhancing speech have been shown to improve perceptual quality of the produced speech, they have not demonstrated signi\ufb01cant improvement for the task of speech recognition. This is because autoencoders are trained to reconstruct all aspects of the original audio, including many features that are not imarXiv:1711.01567v1  [cs.CL]  5 Nov 2017 \fportant for speech recognition, such as the voice and accent of the speaker, background noises etc. In fact, ASR systems learn to remove such artifacts of the input audio as they can hinder speech recognition performance. [13] proposed multiple rounds of joint denoising and ASR training for each audio sample, but this approach is not scalable for large datasets. A similar approach in spirit is to minimize the distance in the embedding space between clean and noisy audio. The intuition here is that the embedding distance is a measure of semantic similarity [14]. However, the perturbed speech may have a different time duration than the reference audio; dynamic time warping [15] can be used to approximate the alignment and compare sequences of varying lengths, but there is an increased computational overhead. [16] uses the generative adversarial networks (GAN) for domain adaptation to make the simulated images look more realistic to improve the task of robotic hand grasping. GAN [5] is an unsupervised learning framework, where the generator network learns to produce increasingly more realistic data in attempt to fool a competing discriminator. Because equilibrium is reached at a saddle point, it is notoriously hard to train. There have been many improvements to this technique. For example, Wasserstein GAN [17] uses the Earth-Mover distance to mitigate optimization issues. It is also less susceptible to architectural choices. For speech, [18] proposes a GAN based speech enhancement method called SEGAN but without the end goal of speech recognition. SEGAN operates on raw speech samples and hence it is computationally impractical for large scale experiments. 3. ROBUST ASR 3.1. Encoder distance enhancer As explained in Section 2, denoising reconstruction and perceptual enhancement do not signi\ufb01cantly improve ASR. A better approach would be to reconstruct only those aspects of the audio which are important for predicting the text spoken and ignore everything else. We hypothesize that the encoders of well trained ASR systems would learn to retain only this information from the input audio. Based on this idea, we propose a new sequence-to-sequence architecture for robust speech recognition that tries to match the output of the encoder for clean audio and noisy audio. The system works as follows: the same encoder, g, is applied to the clean audio x and the corresponding noisy audio e x to produce hidden states z = g(x) and e z = g(e x). The decoder, h, models the conditional probability p(y|x) = p(y|z) and is used to predict the output text sequence one character at a time. This architecture is described in Figure 1. The entire system is trained end-to-end using a multi-task objective that tries to minimize the cross-entropy loss of predicting y Fig. 1. Architecture of the enhancer models introduced in this paper. The discriminator loss can be L1-distance or WGAN loss. The entire model is trained end-to-end using both the discriminator loss and the cross-entropy loss. We use RIR convolution to simulate far-\ufb01eld audio. It\u2019s also possible to train this model with the same speech recorded in different conditions. from e x and the normalized L1\u2212distance between z and e z: E(x,y)\u223cD \u0014 H(h(e z), y) + \u03bb \u2225z \u2212e z\u22251 \u2225z\u22251 + \u2225e z\u22251 + \u03f5 \u0015 . (1) 3.2. GAN enhancer In our experiments, we found the encoder distance penalty to yield excellent results but it has the disadvantage that the encoder content between clean and noisy audio has to match frame for frame. Instead, employing the GAN framework, we can have a discriminator output a scalar likelihood of the entire speech being clean, and train the encoder to generate embeddings that are indistinguishable by the discriminator. In this paper, Wasserstein GAN (WGAN) [17] is used. Following the notations of WGAN, we parametrize the seqto-seq and discriminator models with \u03b8 and w respectively. The overall architecture depicted in Figure 1 remains the same, but the encoder distance in (1) is now replaced with the dual of Earth-Mover (EM) distance max w\u2208W {Ex [fw(g\u03b8(x))] \u2212Ee x,\u03b5 [fw(g\u03b8(e x + \u03b5)]} . (2) We treat the embedding of the clean input x as real data and the embedding of e x, which can either be augmented from x or drawn from a different modality, as being fake. And so, as GAN training progresses, the encoder g\u03b8 should learn to remove extraneous information to ASR to be able to fool the discriminator. In practice, we found that including a random Gaussian noise \u03b5 to the input prior of the generator helps improve training. Also, weights in the parameter set W should be clipped to ensure the duality of (2) holds up to a constant multiple [17]. The adapted WGAN training procedure is detailed in Algorithm 1. \fData: ncritic, the number of critic per robust ASR updates. c, the clipping parameter. m, the batch size. while \u03b8 has not converged do for t = 1, . . . , ncritic do Sample {(x(i), y(i)) \u223cD}m i=1 a batch of labeled speech data. Sample {e x(i)}m i=1 by augmentation or from a different distribution. Sample {\u03b5(i)}m i=1 a batch of prior noise. g\u03b8 \u2190\u2207\u03b8 \u0002 1 m Pm i=1 H(h\u03b8(g\u03b8(x(i))), y(i)) \u0003 \u03b8 \u2190\u03b8 \u2212Optimizer(\u03b8, g\u03b8) gw \u2190\u2207w \u0002 1 m Pm i=1 fw(g\u03b8(x(i))) \u22121 m Pm i=1 fw(g\u03b8(e x(i) + \u03b5(i))) \u0003 w \u2190w + RMSProp(w, gw) w \u2190clip(w, \u2212c, c) end Sample {(x(i), y(i)) \u223cD}m i=1 a batch of labeled speech data. Sample {e x(i)}m i=1 by augmentation or from a different distribution. Sample {\u03b5(i)}m i=1 a batch of prior noise. g\u03b8 \u2190\u2207\u03b8 \u0002 1 m Pm i=1 H(h\u03b8(g\u03b8(x(i))), y(i)) \u2212\u03bb 1 m Pm i=1 fw(g\u03b8(e x(i) + \u03b5(i))) \u0003 \u03b8 \u2190\u03b8 \u2212Optimizer(\u03b8, g\u03b8) end Algorithm 1: WGAN enhancer training. The seq-to-seq model was trained using the Adam optimizer in our experiments. If e x can be generated from x, data augmentation can also be used to update the seq-to-seq model. 4. EXPERIMENTAL SETUP 4.1. Corpora and Tasks We evaluated the enhancer framework on the Wall Street Journal (WSJ) corpus with simulated far-\ufb01eld effects. The dev93 and eval92 sets were used for hyperparameter selection and evaluation respectively. The reverberant speech is generated with room impulse response (RIR) augmentation as in [19], where each audio is convolved with a randomly chosen RIR signal. The clean and far-\ufb01eld audio durations are kept the same with valid convolution so that the encoder distance enhancer can be applied. We collected 1088 impulse responses, using a linear array of 8 microphones, 120 and 192 of which were held out for development and evaluation. The speaker was placed in a variety of con\ufb01gurations, ranging from 1 to 3 meters distance and 60 to 120 degrees inclination with respect to the array, for 20 different rooms. Mel spectrograms of 20 ms samples with 10 ms stride and 40 bins were used as input features to all of our baseline and enhancer models. 4.2. Network Architecture For the acoustic model, we used the sequence-to-sequence framework with soft attention based on [6]. The architecture of the encoder is described in Table 1. The decoder consisted of a single 256 dimensional GRU layer with a hybrid attention mechanism similar to the models described in [20]. The discriminator network of the WGAN enhancer is described in Table 2. All convolutional layers use leaky ReLU activation [21] with 0.2 slope for the leak, and batch normalization [22]. Bidirectional GRU (dimension = 256, batch norm) Pooling (2x1 striding) Bidirectional GRU (dimension = 256, batch norm) Pooling (2x1 striding) Bidirectional GRU (dimension = 256, batch norm) Pooling (2x1 striding) Bidirectional GRU (dimension = 256, batch norm) Bidirectional GRU (dimension = 256, batch norm) Bidirectional GRU (dimension = 256, batch norm) Table 1. Architecture of the encoder. 7x2 Convolution, 32 \ufb01lters, 5x1 striding 3x3 Convolution, 64 \ufb01lters, 2x1 striding Bidirectional LSTM (dimension = 32) 3x3 Convolution, 64 \ufb01lters, 2x1 striding 3x3 Convolution, 96 \ufb01lters, 1x1 striding Bidirectional LSTM (dimension = 32) Linear projection to per-time step scalar Sigmoid Mean pool of likelihood scores Table 2. Architecture of the critic. (feature)\u00d7(time). 4.3. Training To establish a baseline, in the \ufb01rst experiment, we trained a simple attention based seq-to-seq model. All the seq-to-seq networks in our experiments were trained using the Adam optimizer. We evaluate all models on both clean and far-\ufb01eld test sets. \fModel Near-Field Far-Field CER WER CER WER seq-to-seq 7.43% 21.18% 23.76% 50.84% seq-to-seq + far-\ufb01eld Augmentation 7.69% 21.32% 12.47% 30.59% seq-to-seq + L1-Distance Penalty 7.54% 20.45% 12.00% 29.19% seq-to-seq + GAN Enhancer 7.78% 21.07% 11.26% 28.12% Table 3. Speech recognition performance on the Wall Street Journal Corpus To study the effects of data augmentation, we train a new seq-to-seq model with the same architecture and training procedure as the baseline. However this time, in each epoch, we randomly select 40% of the training utterances and apply the train RIRs to them (in our previous experiments we had observed that 40% augmentation results in the best validation performance). For the enhancer models, \u03bb in Equation 1 was tuned over the dev set by doing a logarithmic sweep in [0.01, 10]. \u03bb = 1 gave the best performance. We use Algorithm 1 to train the WGAN enhancer. The clipping parameter was 0.05 and \u03b5 was random normal with 0.001 standard deviation. We found that having a schedule for ncritic was crucial. Namely, we do not update the encoder parameters with WGAN gradients for the \ufb01rst 3000 steps. Then, we use the normal ncritic = 5. We hypothesize that the initial encoder embedding is of poor quality and encouraging invariance at this stage through the critic gradients signi\ufb01cantly hinders seq-to-seq training. 5. RESULTS We present results in Table 3. All of the evaluations were performed using greedy decoding and no language models. To provide context, our near-\ufb01eld result is comparable to the 18.6% WER of [6] obtained with language model beam decoding with 200 beam size. We can see that a seq-to-seq model trained only on near-\ufb01eld audio data performs extremely poorly on far-\ufb01eld audio. This suggests that it is non-trivial for an ASR model to generalize from homogeneous near-\ufb01eld audio to far-\ufb01eld audio. To overcome this, we train a stronger baseline with simulated far-\ufb01eld audio examples. This model had the same architecture but 40% of the examples that the model was trained on were convolved with a randomly chosen room impulse response during training. We can see from Table 3 that simple data augmentation can signi\ufb01cantly improve performance on far-\ufb01eld audio without compromising the performance on near-\ufb01eld audio, implying that seq-to-seq models have a strong ability to learn from far-\ufb01eld examples. Even with data augmentation, however, there is still a large gap between the WERs on near-\ufb01eld and far-\ufb01eld test sets. The bottom two rows of Table 3 show the performance of the methods introduced in this paper on the same test sets. An L1-distance penalty can lower the test set WER by 1.32% absolute. Using a GAN enhancer can reduce the WER by an additional 1.07%. Overall, the gap between near-\ufb01eld and far\ufb01eld performance decreases by almost 27% compared to the model that only uses data augmentation. An additional bene\ufb01t of our methods is that the L1distance penalty and GAN loss function act as regularizers which reduce generalization error on near \ufb01eld data. The enhancer models have lower WERs even on near-\ufb01eld data compared to the baseline models. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1708.06426v1",
+                    "title": "Cold Fusion: Training Seq2Seq Models Together with Language Models",
+                    "abstract": "Sequence-to-sequence (Seq2Seq) models with attention have excelled at tasks\nwhich involve generating natural language sentences such as machine\ntranslation, image captioning and speech recognition. Performance has further\nbeen improved by leveraging unlabeled data, often in the form of a language\nmodel. In this work, we present the Cold Fusion method, which leverages a\npre-trained language model during training, and show its effectiveness on the\nspeech recognition task. We show that Seq2Seq models with Cold Fusion are able\nto better utilize language information enjoying i) faster convergence and\nbetter generalization, and ii) almost complete transfer to a new domain while\nusing less than 10% of the labeled training data.",
+                    "authors": "Anuroop Sriram, Heewoo Jun, Sanjeev Satheesh, Adam Coates",
+                    "published": "2017-08-21",
+                    "updated": "2017-08-21",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL"
+                    ],
+                    "main_content": "Introduction Sequence-to-sequence (Seq2Seq) (Bahdanau et al., 2015) models have achieved state-of-the-art results on many natural language processing problems including automatic speech recognition (Bahdanau et al., 2016; Chan et al., 2015), neural machine translation (Wu et al., 2016), conversational modeling (Vinyals & Le, 2015) and many more. These models learn to generate a variable-length sequence of tokens (e.g. texts) from a variable-length sequence of input data (e.g. speech or the same texts in another language). With a suf\ufb01ciently large labeled dataset, vanilla Seq2Seq can model sequential mapping well, but it is often augmented with a language model to further improve the \ufb02uency of the generated text. Because language models can be trained from abundantly available unsupervised text corpora which can have as many as one billion tokens (Jozefowicz et al., 2016; Shazeer et al., 2017), leveraging the rich linguistic information of the label domain can considerably improve Seq2Seq\u2019s performance. A standard way to integrate language models is to linearly combine the score of the task1Baidu Research, Sunnyvale, CA, USA. Correspondence to: Anuroop Sriram <sriramanuroop@baidu.com>. speci\ufb01c Seq2Seq model with that of an auxiliary langauge model to guide beam search (Chorowski & Jaitly, 2016; Sutskever et al., 2014a). Gulcehre et al. (2015) proposed an improved algorithm called Deep Fusion that learns to fuse the hidden states of the Seq2Seq decoder and a neural language model with a gating mechanism, after the two models are trained independently. While this approach has been shown to improve performance over the baseline, it has a few limitations. First, because the Seq2Seq model is trained to output complete label sequences without a language model, its decoder learns an implicit language model from the training labels, taking up a signi\ufb01cant portion of the decoder capacity to learn redundant information. Second, the residual language model baked into the Seq2Seq decoder is biased towards the training labels of the parallel corpus. For example, if a Seq2Seq model fully trained on legal documents is later fused with a medical language model, the decoder still has an inherent tendency to follow the linguistic structure found in legal text. Thus, in order to adapt to novel domains, Deep Fusion must \ufb01rst learn to discount the implicit knowledge of the language. In this work, we introduce Cold Fusion to overcome both these limitations. Cold Fusion encourages the Seq2Seq decoder to learn to use the external language model during training. This means that Seq2Seq can naturally leverage potentially limitless unsupervised text data, making it particularly pro\ufb01cient at adapting to a new domain. The latter is especially important in practice as the domain from which the model is trained can be different from the real world use case for which it is deployed. In our experiments, Cold Fusion can almost completely transfer to a new domain for the speech recognition task with 10 times less data. Additionally, the decoder only needs to learn task relevant information, and thus trains faster. The paper is organized as follows: Section 2 outlines the background and related work. Section 3 presents the Cold Fusion method. Section 4 details experiments on the speech recognition task that demonstrate Cold Fusion\u2019s generalization and domain adaptation capabilities. arXiv:1708.06426v1  [cs.CL]  21 Aug 2017 \fCold Fusion: Training Seq2Seq Models Together with Language Models 2. Background and Related work 2.1. Sequence-to-Sequence Models A basic Seq2Seq model comprises an encoder that maps an input sequence x = (x1, . . . , xT ) into an intermediate representation h, and a decoder that in turn generates an output sequence y = (y1, . . . , yK) from h (Sutskever et al., 2014b). The decoder can also attend to a certain part of the encoder states with an attention mechanism. The attention mechanism is called hybrid attention (Chorowski et al., 2015b), if it uses both the content and the previous context to compute the next context. It is soft if it computes the expectation over the encoder states (Bahdanau et al., 2015) as opposed to selecting a slice out of the encoder states. For the automatic speech recognition (ASR) task, the Seq2Seq model is called an acoustic model (AM) and maps a sequence of spectrogram features extracted from a speech signal to characters. 2.2. Inference and Language Model Integration During inference, we aim to compute the most likely sequence \u02c6 y: \u02c6 y = argmax y log p(y|x). (1) Here, p(y|x) is the probability that the task-speci\ufb01c Seq2Seq model assigns to sequence y given input sequence x. The argmax operation is intractable in practice so we use a left-to-right beam search algorithm similar to the one presented in (Sutskever et al., 2014a). We maintain a beam of K partial hypothesis starting with the start symbol \u27e8s\u27e9. At each time-step, the beam is extended by one additional character and only the top K hypotheses are kept. Decoding continues until the stop symbol \u27e8/s\u27e9is emitted, at which point the hypothesis is added to the set of completed hypotheses. A standard way to integrate the language model with the Seq2Seq decoder is to change the inference task to: \u02c6 y = argmax y log p(y|x) + \u03bb log pLM(y), (2) where pLM(y) is the language model probability assigned to the label sequence y. (Chorowski & Jaitly, 2016; Wu et al., 2016) describe several heuristics that can be used to improve this basic algorithm. We refer to all of these methods collectively as Shallow Fusion, since pLM is only used during inference. (Gulcehre et al., 2015) proposed Deep Fusion for machine translation that tightens the connection between the decoder and the language model by combining their states with a parametric gating: gt = \u03c3(v\u22a4sLM t + b) (3a) sDF t = [st; gtsLM t ] (3b) yt = softmax(DNN(sDF t )), (3c) where st, sLM t and sDF t are the states of the task speci\ufb01c Seq2Seq model, language model and the overall deep fusion model. In (3c), DNN can be a deep neural network with any number of layers. [a; b] is the concatenation of vectors a and b. In Deep Fusion, the Seq2Seq model and the language model are \ufb01rst trained independently and later combined as in Equation (3). The parameters v and b are trained on a small amount of data keeping the rest of the model \ufb01xed, and allow the gate to decide how important each of the models are for the current time step. The biggest disadvantage with Deep Fusion is that the taskspeci\ufb01c model is trained independently from the language model. This means that the Seq2Seq decoder needs to learn a language model from the training data labels, which can be rather parsimonious compared to the large text corpora available for language model training. So, the fused output layer of (3) should learn to overcome this bias in order to incorporate the new language information. This also means that a considerable portion of the decoder capacity is wasted. 2.3. Semi-supervised Learning in Seq2Seq Models A few methods have been proposed for leveraging unlabeled text corpora in the target domain, for both better generalization and domain transfer. Sennrich et al. (2015) proposed backtranslation as a way of using unlabeled data for machine translation. Backtranslation improves the BLEU score by increasing the parallel training corpus of the neural machine translation model by automatically translating the unlabeled target domain text. However, this technique does not apply well to other tasks where backtranslation is infeasible or of very low quality (like image captioning or speech recogntion). Ramachandran et al. (2016) proposed warm starting the Seq2Seq model from language models trained on source and target domains separately. Unsupervised pre-training shows improvements in the BLEU scores. (Ramachandran et al., 2016) also show that this improvement is from improved generalization, and not only better optimization. While this is a promising approach, the method is potentially dif\ufb01cult to leverage for the transfer task since training on the parallel corpus could end up effectively erasing the knowledge of the language models. Both back-translation and unsupervised pre-training are simple methods that require no change in the architecture. \fCold Fusion: Training Seq2Seq Models Together with Language Models Model Prediction Ground Truth where\u2019s the sport in that greer snorts and leaps greer hits the dirt hard and rolls Plain Seq2Seq where is the sport and that through snorks and leaps clear its the dirt card and rules Deep Fusion where is the sport and that there is north some beliefs through its the dirt card and rules Cold Fusion where\u2019s the sport in that greer snorts and leaps greer hits the dirt hard and rolls Cold Fusion (Fine-tuned) where\u2019s the sport in that greer snorts and leaps greer hits the dirt hard and rolls Ground Truth jack sniffs the air and speaks in a low voice Plain Seq2Seq jacksonice the air and speech in a logos Deep Fusion jacksonice the air and speech in a logos Cold Fusion jack sniffs the air and speaks in a low voice Cold Fusion (Fine-tuned) jack sniffs the air and speaks in a low voice Ground Truth skipper leads her to the dance \ufb02oor he hesitates looking deeply into her eyes Plain Seq2Seq skip er leadure to the dance \ufb02oor he is it takes looking deeply into her eyes Deep Fusion skip er leadure to the dance \ufb02oor he has it takes looking deeply into her eyes Cold Fusion skipper leads you to the dance \ufb02oor he has a tates looking deeply into her eyes Cold Fusion (Fine-tuned) skipper leads her to the dance \ufb02oor he hesitates looking deeply into her eyes Table 1. Some examples of predictions by the Deep Fusion and Cold Fusion models. 3. Cold Fusion Our proposed Cold Fusion method is largely motivated from the Deep Fusion idea but with some important differences. The biggest difference is that in Cold Fusion, the Seq2Seq model is trained from scratch together with a \ufb01xed pre-trained language model. Because the Seq2Seq model is aware of the language model throughout training, it learns to use the language model for language speci\ufb01c information and capture only the relevant information conducive to mapping from the source to the target sequence. This disentanglement can increase the effective capacity of the model signi\ufb01cantly. This effect is demonstrated empirically in Section 4 where Cold Fusion models perform well even with a very small decoder. We also improve on some of the modeling choices of the fusion mechanism. 1. First, both the Seq2Seq hidden state st and the language model hidden state sLM t can be used as inputs to the gate computation. The task-speci\ufb01c model\u2019s embedding contains information about the encoder states which allows the fused layer to decide its reliance on the language model in case of input uncertainty. For example, when the input speech is noisy or a token unseen by the Seq2Seq model is presented, the fusion mechanism learns to pay more attention to the language model. 2. Second, we employ \ufb01ne-grained (FG) gating mechanism as introduced in (Yang et al., 2016). By using a different gate value for each hidden node of the language model\u2019s state, we allow for greater \ufb02exibility in integrating the language model because the fusion algorithm can choose which aspects of the language model it needs to emphasize more at each time step. 3. Third, we replace the language model\u2019s hidden state with the language model probability. The distribution and dynamics of sLM t can vary considerably across different language models and data. As a concrete example, any fusion mechanism that uses the LM state is not invariant to the permutation of state hidden nodes. This limits the ability to generalize to new LMs. By projecting the token distribution onto a common embedding space, LMs that model novel uses of the language can still be integrated without state discrepancy issues. This also means that we can train with or swap on n-gram LMs during inference. The Cold Fusion layer works as follows: hLM t = DNN(\u2113LM t ) (4a) gt = \u03c3(W[st; hLM t ] + b) (4b) sCF t = [st; gt \u25e6hLM t ] (4c) rCF t = DNN(sCF t ) (4d) \u02c6 P(yt|x, y<t) = softmax(rCF t ) (4e) \u2113LM t is the logit output of the language model, st is the state of the task speci\ufb01c model, and sCF t is the \ufb01nal fused state used to generate the output. Since logits can have arbitrary offsets, the maximum value is subtracted off before feeding into the layer. In (4a), (4d), the DNN can be a deep neural network with any number of layers. In our experiments, we found a single af\ufb01ne layer, with ReLU activation prior to softmax, to be helpful. \fCold Fusion: Training Seq2Seq Models Together with Language Models Figure 1. Cross-entropy loss on the dev set for the baseline model (orange) and the proposed model (purple) as a function of training iteration. Training with a language model speeds up convergence considerably. 4. Experiments 4.1. Setup For our experiments, we tested the Cold Fusion method on the speech recognition task. The results are compared using the character error rate (CER) and word error rate (WER) on the evaluation sets. For all models which were trained on the source domain, the source CER and WER indicate in-domain performance and the target CER and WER indicate out-of-domain performance. We collected two data sets: one based on search queries which served as our source domain, and another based on movie transcripts which served as our target domain. For each dataset, we used Amazon Mechanical Turk to collect audio recordings of speakers reading out the text. We gave identical instructions to all the turkers in order to ensure that the two datasets only differed in the text domain. The source dataset contains 411,000 utterances (about 650 hours of audio), and the target dataset contains 345,000 utterances (about 676 hours of audio). We held out 2048 utterances from each domain for evaluation. The text of the two datasets differ signi\ufb01cantly. Table 2 shows results of training character-based recurrent neural network language models (Mikolov, 2012) on each of the datasets and evaluating on both datasets. Language models very easily over\ufb01t the training distribution, so models trained on one corpus will perform poorly on a different distribution. We see this effect in Table 2 that models optimized for the source domain have worse perplexity on the target distribution. 4.2. Neural Network Architectures The language model described in the \ufb01nal row of Table 2 was trained on about 25 million words. This model conTable 2. Dev set perplexities for character RNN language models trained on different datasets on source and target domain. Note that i) the model trained on source domain does poorly on target domain and vice-versa indicating that the two domains are very different, and ii) the best model on both domains is a larger model trained on a superset of both corpuses. We use the model trained on the full dataset (which contains the source and target datasets along with some additional text) for all of the LM integration experiments. Model Domain Word Perplexity Count Source Target GRU (3 \u00d7 512) Source 5.73M 2.670 4.463 GRU (3 \u00d7 512) Target 5.46M 3.717 2.794 GRU (3 \u00d7 1024) Full 25.16M 2.491 2.325 tains three layers of gated recurrent units (GRU) (Chung et al., 2014) with a hidden state dimension of 1024. The model was trained to minimize the cross-entropy of predicting the next character given the previous characters. We used the Adam optimizer (Kingma & Ba, 2014) with a batch size of 512. The model gets a perplexity of 2.49 on the source data and 2.325 on the target data. For the acoustic models, we used the Seq2Seq architecture with soft attention based on (Bahdanau et al., 2016). The encoder consists of 6 bidirectional LSTM (BLSTM) (Hochreiter & Schmidhuber, 1997) layers each with a dimension of 480. We also use max pooling layers with a stride of 2 along the time dimension after the \ufb01rst two BLSTM layers, and add residual connections (He et al., 2015) for each of the BLSTM layers to help speed up the training process. The decoder consisted of a single layer of 960 dimensional Gated Recurrent Unit (GRU) with a hybrid attention (Chorowski et al., 2015b). The \ufb01nal Cold Fusion mechanism had one dense layer of 256 units followed by ReLU before softmax. 4.3. Training The input sequence consisted of 40 mel-scale \ufb01lter bank features. We expanded the datasets with noise augmentation; a random background noise is added with a 40% probability at a uniform random SNR between 0 and 15 dB. We did not use any other form of regularization. We trained the entire system end-to-end with Adam (Kingma & Ba, 2014) with a batch size of 64. The learning rates were tuned separately for each model using random search. To stabilize training early on, the training examples were sorted by increasing input sequence length in the \ufb01rst epoch (Amodei et al., 2015). During inference, we used beam search with a \ufb01xed beam size of 128 for all of our experiments. \fCold Fusion: Training Seq2Seq Models Together with Language Models Table 3. Speech recognition results for the various models discussed in the paper. Model Train Domain Test on Source Test on Target CER WER CER WER Domain Gap Baseline Attention Model Source 7.54% 14.68% 23.02% 43.52% 100% Baseline Attention Model Target 8.84% 17.61% 0% Baseline + Deep Fusion Source 7.64% 13.92% 22.14% 37.45% 76.57% + sAM in gate Source 7.61% 13.92% 21.07% 37.9% 78.31% + Fine-Grained Gating Source 7.47% 13.61% 20.29% 36.69% 73.64% + ReLU layer Source 7.50% 13.54% 21.18% 38.00% 78.70% Baseline + Cold Fusion + sAM in gate Source 7.25% 13.88% 15.63% 30.71% 50.56% + Fine-Grained Gating Source 6.14% 12.08% 14.79% 30.00% 47.82% + ReLU layer Source 5.82% 11.52% 14.89% 30.15% 48.40% + Probability Projection Source 5.94% 11.87% 13.72% 27.50% 38.17% We also used scheduled sampling (Bengio et al., 2015) with a sampling rate of 0.2 which was kept \ufb01xed throughout training. Scheduled sampling helped reduce the effect of exposure bias due to the difference in the training and inference mechanisms. 4.4. Improved Generalization Leveraging a language model that has a better perplexity on the distribution of interest should directly mean an improved WER for the ASR task. In this section, we compare how the different fusion methods fare in achieving this effect. Swapping the language model is not possible with Deep Fusion because of the state discrepancy issue motivated in Section 3. All fusion models were therefore trained and evaluated with the same language model that achieved a Table 4. Effect of decoder dimension on the model\u2019s performance. The performance of cold fusion models degrades more slowly as the decoder size decreases. This corroborates the fact that the decoder only needs to learn the task not label generation. Its effective task capacity is much larger than without fusion. Model Decoder size Source CER WER Attention 64 16.33% 33.98% 128 11.14% 24.35% 256 8.89% 18.74% 960 7.54% 14.68% Cold Fusion 64 9.47% 17.42% 128 7.96% 15.15% 256 6.71% 13.19% 960 5.82% 11.52% low perplexity on both the source and target domains (See Table 2). This way, we can measure improvements in transfer capability over Deep Fusion due to the training and architectural changes. Table 3 compares the performance of Deep Fusion and Cold Fusion on the source and target held-out sets. Clearly, Cold Fusion consistently outperforms on both metrics on both domains than the baselines. For the task of predicting in-domain, the baseline model gets a word error of 14.68%, while our best model gets a relative improvement of more than 21% over that number. Even compared to the recently proposed Deep Fusion model (Gulcehre et al., 2015), the best Cold Fusion model gets a relative improvement of 15%. We get even bigger improvements in out-of-domain results. The baseline attention model, when trained on the source domain but evaluated on the target domain gets, 43.5% WER. This is signi\ufb01cantly worse than the 17.6% that we can get by training the same model on the target dataset. The goal of domain adaptation is to bridge the gap between these numbers. The \ufb01nal column in Table 3 shows the remaining gap as a fraction of the difference for each model. The Deep Fusion models can only narrow the domain gap to 76.57% while Cold Fusion methods can reduce it to 38.17%. The same table also shows the incremental effects of the three architectural changes we have made to the Cold Fusion method. Note that applying the same changes to the Deep Fusion method does not yield much improvements, indicating the need for cold starting Seq2Seq training with language models. The use of probability projection instead of the language model state in the fusion layer substantially helps with generalization. Intuitively, the character probability space shares the same structure across different language models unlike the hidden state space. \fCold Fusion: Training Seq2Seq Models Together with Language Models Table 5. Results for \ufb01ne tuning the acoustic model (\ufb01nal row from Table 3) on subsets of the target training data. \u2217The \ufb01nal row represents an attention model that was trained on all of the target domain data. Model Target Target Data CER WER Domain Gap Cold Fusion 0% 13.72% 27.50% 38.17% Cold Fusion 0.6% 11.98% 23.13% 21.30% + \ufb01netuning 1.2% 11.62% 22.40% 18.49% 2.4% 10.79% 21.05% 13.28% 4.8% 10.46% 20.46% 11.00% 9.5% 10.11% 19.68% 7.99% Attention\u2217 100% 8.84% 17.61% 0.00 % 4.5. Decoder Ef\ufb01ciency We test whether cold fusion does indeed relieve the decoder of learning a language model. We do so by checking how a decrease in the decoder capacity affected the error rates. As evidenced in Table 4, the performance of the Cold Fusion models degrades gradually as the decoder cell size is decreased whereas the performance of the attention models deteriorates abruptly beyond a point. It is remarkable that the Cold Fusion decoder still outperforms the full attentional decoder with 4\u00d7 fewer number of parameters. Also, we \ufb01nd that training is accelerated by a factor of 3 (see Figure 1). Attention models typically need hundreds of thousands of iterations to converge (Chorowski et al., 2015a). Most of the training time is spent in learning the attention mechanism. One can observe this behavior by plotting the attention context over time and seeing that the diagonal alignment pattern emerges in later iterations. Because the pretrained, \ufb01xed language model infuses the model with lower level language features like the likely spelling of a word, error signals propagate more directly into the attention context. 4.6. Fine-tuning for Domain Adaptation In the presence of limited data from the target distribution, \ufb01ne tuning a model for domain transfer is often a promising approach. We test how much labeled data from the target distribution is required for Cold Fusion models to effectively close the domain adaptation gap. The same language model from Section 4.4 trained on both the source and target domains was used for all \ufb01ne-tuning experiments. The learning rate was restored to its initial value. Then, we \ufb01ne-tuned only the fusion mechanism of the best Cold Fusion model from Table 3 on various amounts of the labeled target dataset. Results are presented in Table 5. With just 0.6% of labeled data, the domain gap decreases from 38.2% to 21.3%. With less than 10% of the data, this gap is down to only 8%. Note that because we keep the Seq2Seq parameters \ufb01xed during the \ufb01ne-tuning stage, all of the improvements from \ufb01netuning come from combining the acoustic and the language model better. It\u2019s possible that we can see bigger gains by \ufb01ne-tuning all the parameters. We do not do this in our experiments because we are only interested in studying the effects of language model fusion in the Seq2Seq decoder. Some examples are presented in Table 1. Recall that all models are trained on the source domain consisting of the read speech of search queries and evaluated on the read speech of movie scripts to measure out-of-domain performance. Because search queries tend to be sentence fragments, we see that the main mode of error for vanilla attention and Deep Fusion is due to weak grammar knowledge. Cold Fusion on the other hand demonstrates a better grasp of grammar and is able to complete sentences. 5."
+                }
+            ],
+            "C. Lawrence Zitnick": [
+                {
+                    "url": "http://arxiv.org/abs/2206.14331v2",
+                    "title": "Spherical Channels for Modeling Atomic Interactions",
+                    "abstract": "Modeling the energy and forces of atomic systems is a fundamental problem in\ncomputational chemistry with the potential to help address many of the world's\nmost pressing problems, including those related to energy scarcity and climate\nchange. These calculations are traditionally performed using Density Functional\nTheory, which is computationally very expensive. Machine learning has the\npotential to dramatically improve the efficiency of these calculations from\ndays or hours to seconds. We propose the Spherical Channel Network (SCN) to\nmodel atomic energies and forces. The SCN is a graph neural network where nodes\nrepresent atoms and edges their neighboring atoms. The atom embeddings are a\nset of spherical functions, called spherical channels, represented using\nspherical harmonics. We demonstrate, that by rotating the embeddings based on\nthe 3D edge orientation, more information may be utilized while maintaining the\nrotational equivariance of the messages. While equivariance is a desirable\nproperty, we find that by relaxing this constraint in both message passing and\naggregation, improved accuracy may be achieved. We demonstrate state-of-the-art\nresults on the large-scale Open Catalyst dataset in both energy and force\nprediction for numerous tasks and metrics.",
+                    "authors": "C. Lawrence Zitnick, Abhishek Das, Adeesh Kolluru, Janice Lan, Muhammed Shuaibi, Anuroop Sriram, Zachary Ulissi, Brandon Wood",
+                    "published": "2022-06-29",
+                    "updated": "2022-10-13",
+                    "primary_cat": "physics.chem-ph",
+                    "cats": [
+                        "physics.chem-ph",
+                        "cs.CE",
+                        "cs.LG",
+                        "physics.comp-ph",
+                        "I.2.6; J.2"
+                    ],
+                    "main_content": "Introduction Modeling the properties of atomic systems is a foundational challenge in computational chemistry and critical to advancing technologies across numerous application domains. Notable applications include drug discovery [32, 39] and the design of new catalysts for renewable energy storage to help in addressing climate change [48, 33]. For catalyst discovery, new materials are currently evaluated using Density Functional Theory (DFT) that can estimate atomic energies and forces but is computationally very expensive; taking hours or days to evaluate a single material. Machine Learning (ML) has the potential to approximate DFT and dramatically speed up these calculations, allowing for high throughput screening of new materials to help address some of the world\u2019s most pressing challenges. Our goal is to approximate DFT calculations using ML. An ML model takes as input a set of atom positions and their atomic numbers. As outputs, a model calculates the structure\u2019s energy (or other properties) and the per-atom forces, i.e., the forces exerted on each atom by the other atoms. A common approach to this problem is to use Graph Neural Networks (GNNs) [19, 47] where each node represents an atom and the set of nearby atoms as edges [37, 17, 24, 35, 38, 45, 31, 13]. A key challenge in network design is balancing the use of model constraints. When modeling atomic systems, a common constraint is SO(3) rotation equivariance [43, 3, 1, 42, 34, 36], i.e., if the atomic system is rotated, the energies should remain constant and the atomic forces should similarly rotate. While this provides strong priors on the model to help in generalization, especially for smaller datasets[7, 32], it can result in limiting the expressiveness of the network due to restrictions on 36th Conference on Neural Information Processing Systems (NeurIPS 2022). arXiv:2206.14331v2  [physics.chem-ph]  13 Oct 2022 \f(a) (b) Figure 1: (a) Illustration of spherical harmonics up to l = 4. Note the m = 0 bases are symmetric about the z-axis (center green). (b) Visualization of 3 spherical channels with l = 8 when varying the number of orders: all, m \u2208[\u22122, 2], m \u2208[\u22121, 1], m = 0. Note how the resolution decreases with fewer m until values are constant for a given z with m = 0. Spherical projections are shown on the left (only half of the channel is visible) and a 2D grid projection (polar and azimuthal) of the same channels is shown on the right. non-linear transformations for equivariant models [3, 42, 34, 36], or limiting interactions to pairs [35, 38, 45], triplets [13] or quadruplets of atoms [28, 14, 15] for invariant models. Alternatively, a non-equivariant model [23] can provide more freedom to the model, but lead to the model needing to learn approximate equivariance through methods such as data augmentation [40]. To draw an analogy with image detection, the use of a CNN [27] provides translation equivariance and removes the need for the network to learn how to detect the same object at different locations. However, most CNNs are not equivariant to scale or rotation [44, 43], but are still effective in learning approximate equivariance through rotation and scale diversity in the training data. What is the analogous balance of constraints for modeling atomic systems? In this paper, we propose a GNN [19, 47] that balances the use of model constraints to aid in generalization while providing the network with the \ufb02exibility to learn accurate representations. We introduce the Spherical Channel Network (SCN) that explicitly models relative orientations of all neighboring atoms; information that is critical to accurately predicting atomic properties. Each node\u2019s embedding is a set of functions de\ufb01ned on the surface of a sphere (S2 \u2192R). The functions are represented using spherical harmonics, similar to approaches that build strictly equivariant models [42, 1, 3, 10]. During message passing, angular information between atoms is conveyed by rotating or steering [5] the embeddings based on each edge\u2019s orientation [41, 28]. We identify an expanded set of spherical harmonic coef\ufb01cients that are invariant to rotation, which can provide rich information while maintaining a message\u2019s rotation equivariance. In addition, we demonstrate that if the equivariance constraint is relaxed, improved performance can be achieved by using additional coef\ufb01cients. We further improve the expressivity of the network by performing a pointwise non-linear function, which is only approximately equivariant, on the embeddings during message aggregation. We demonstrate our Spherical Channel Network on the large-scale Open Catalyst 2020 (OC20) dataset [6], which contains atomic structures useful for numerous applications important to addressing climate change. State-of-the-art results are achieved for atomic force and initial structure to relaxed energy prediction with improvements of 8% \u221211%. In addition, we demonstrate our model is more sample ef\ufb01cient as compared to other state-of-the-art models. 2 Approach Given an atomic structure with n atoms, our goal is to predict the structure\u2019s energy E and the per-atom forces fi for each atom i \u2208n. These values are estimated using a Graph Neural Network (GNN) [19, 47] where each node represents an atom and edges represent nearby atoms. As input, the network is given the distance dij between atoms i and j, and each atom\u2019s atomic number ai. The neighbors Ni for an atom i are determined using a \ufb01xed distance threshold, or by picking a \ufb01xed number of closest atoms. 2 \f(a) (b) Figure 2: (a) Block diagram of message passing function Fe in Equation 2 for source atom s to target atom t. The atomic numbers as and at, distance between the atoms |dst|, and embeddings x(k) s and x(k) t are given as input. (b) Training curves for SCN and GemNet models for force MAEs per epoch evaluated on a 30k subset of the validation ID dataset. Note how the SCN model is signi\ufb01cantly more sample ef\ufb01cient during training. 2.1 Node Embeddings The angular or relative orientation information between atoms is critical to accurately modeling atomic properties such as energies and forces [13, 14]. Inspired by this, our node embedding models the angular information from all neighboring atoms using spherical functions. Each node i\u2019s embedding si is a set of C functions or channels represented on a sphere (S2 \u2192R), whose argument is a 3D unit vector indicating the orientation. That is, sic(\u02c6 r) is the value of channel c for node i for some orientation \u02c6 r \u2208R3. Since the spherical channels contain orientation information over the entire sphere, the network may reason about geometric information for all neighboring atoms and not just atom pairs, triplets, etc. Each spherical channel c may be represented using several different approaches, such as discrete 2D grids sampled over the sphere, or spherical harmonics. As we describe later, we use spherical harmonics due to their property of being SO(3)-equivariant (3D rotation equivariant). Using real spherical harmonics, a function sc(\u02c6 r) de\ufb01ned on the sphere is represented by a set of weighted spherical harmonics Ylm with sc(\u02c6 r) = P l,m xlmcYlm(\u02c6 r) where l and m are the degree and order of the basis functions with m \u2208[\u2212l, l] and l \u2208L. We refer to the spherical harmonic coef\ufb01cients, xlmc, as the coef\ufb01cients of the spherical channel. For every degree l there exists 2l + 1 spherical harmonics (see Figure 1(a) for an illustration of functions for l \u22644), which results in a total of (L + 1)2 basis functions and coef\ufb01cients up to degree L. Therefore for a maximum degree of L with C channels, x has size (L + 1)2 \u00d7 C. The spherical channels are updated by the GNN through message passing for K layers to obtain the \ufb01nal node embeddings S(K). S(0) is initialized from an embedding based on the atom\u2019s atomic number ai for l = 0 coef\ufb01cients and the l \u0338= 0 coef\ufb01cients are set to zero. The nodes\u2019 embeddings are updated by \ufb01rst calculating a set of messages mij for each edge, which are then aggregated at each node. Finally, the energy and forces are estimated from S(K). We describe each of these steps in turn. 2.2 Message Passing Given a target node t and its neighbors s \u2208Nt we want to update the embeddings x(k) t at iteration k \u2208K. The embeddings x(k) t are a set of spherical harmonic coef\ufb01cients indexed by their degree l, order m, and channel c. An important and useful property of spherical harmonics is the function represented by the coef\ufb01cients is steerable [11, 5], i.e., it can be rotated in 3D space using a linear transformation of the coef\ufb01cients. Speci\ufb01cally, for a 3D rotation matrix R there exists a matrix called a Wigner D-matrix Dl of size (2l + 1 \u00d7 2l + 1) that rotates the coef\ufb01cients of degree l by the rotation R. If xilc are all coef\ufb01cients across orders m for node i of degree l and channel c, then for any 3D 3 \frotation R there exists a Dl that for all orientations \u02c6 r: xilc \u00b7 Yl(R\u02c6 r) = (Dlxilc) \u00b7 Yl(\u02c6 r). (1) When calculating the message mst from atom s to atom t, we want to use the information contained in both x(k) s and x(k) t given the context of the edge\u2019s orientation \u02c6 dst, \u02c6 dst = dst/|dst|. We do this by rotating the embeddings by Rst for which Rst \u02c6 dst = [0, 0, 1]\u22a4, i.e., the direction of the edge\u2019s unit vector \u02c6 dst is aligned with the z-axis. Thus the orientation of the atoms with respect to each other is implicit in the rotated embeddings. This simpli\ufb01es the task of the network since it only needs to learn the relationship between atoms that are aligned along the z-axis and not an arbitrary rotation. After calculating the messages using a neural network Fe, they are rotated back to the global coordinate frame: m(k+1) st (x(k) s , x(k) t , dst, as, at) = D\u22121 st Fe(Dstx(k) s , Dstx(k) t , |dst|, as, at), (2) where the matrices Dst and D\u22121 st perform the rotation on the embeddings\u2019 coef\ufb01cients corresponding to the 3D rotation matrices Rst and R\u22121 st respectively. To simplify notation, we assume the coef\ufb01cients in xs across degree l and order m are \ufb02attened, and Dst \u2208R(L+1)2 \u00d7 R(L+1)2 is a block diagonal matrix constructed from the set of Wigner D-matrices across degrees, l \u2208L. D\u22121 st is the inverse of Dst, with D\u22121 st = D\u22a4 st. In addition to the rotated embeddings, Fe is also provided input edge information that is invariant to rotations; |dst| is the magnitude of the distance between atoms s and t, and as and at their respective atomic numbers. Note that the rotation matrix Rst is not unique, since the roll rotation around the vector dst is not speci\ufb01ed and is randomly sampled during training. The implications of this are discussed below in Section 2.2.1. The message function Fe is computed using a neural network as illustrated in Figure 2(a). The atomic numbers as and at are used to look up two independent embeddings of size E = 128, and a set of 1D basis functions are used to represent |dst| using equally spaced Gaussians every 0.02 \u00c5 from 0 to the 8 \u00c5 with \u03c3 = 0.04 followed by a linear layer with E outputs. Their values are added together and passed through a neural network to produce a vector of size H. The rotated embeddings Dstx(k) s and Dstx(k) t are concatenated and passed through a single layer neural network to produce another vector of size H. These two are multiplied together to combine the edge information with the information contained in the rotated embeddings. Two more fully connected layers are performed with each followed by a SiLU non-linear activation function [22]. Finally, a single linear layer is used to expand the output to the original size of the embeddings. 2.2.1 Message Equivariance For the network to be equivariant to rotations, our message function (Equation (2)) needs to be equivariant. In general, this is not the case since the matrix Dst is not unique, i.e., the roll rotation around the vector dst is not speci\ufb01ed, and in practice it is randomly chosen. The roll rotation corresponds to a rotation about the z-axis after rotating the coef\ufb01cients x(k) s and x(k) t by Dst. Due to this, all of the rotated coef\ufb01cients will vary based on the random roll rotation chosen, except the m = 0 coef\ufb01cients that are symmetric about the z-axis; see Appendix D and the m = 0 bases highlighted in green in Figure 1(a). If rotation equivariance is desired for messages, the input and output coef\ufb01cients to Fe can be restricted to only those for which m = 0. Since the other inputs |dst|, as and at to Fe are also invariant to rotations, Fe is invariant to rotations if only m = 0 coef\ufb01cients are used. The resulting messages mst are equivariant to rotations once they are rotated back to the global coordinate frame using D\u22121 st . While equivariance is a desirable property, as we demonstrate later, using m \u2208[\u22121, 1] coef\ufb01cients can help improve performance even though equivariance is not strictly enforced. If we choose to use coef\ufb01cients beyond just m = 0 for Fe for message passing in Equation (2), two approaches may be taken. The \ufb01rst is to simply add the coef\ufb01cients to the inputs and outputs of Fe and assume the neural network will learn a roughly equivariant mapping through the random sampling of the roll rotation. As we demonstrate in Appendix A, the learned functions are indeed roughly equivariant. The second strategy takes a more direct approach to encouraging the network to learn equivariant mappings by taking advantage of the proprieties of the coef\ufb01cients as they\u2019re rotated by \u03c6 about the z-axis. If D\u03c6 rotates a set x of coef\ufb01cients about the z-axis by \u03c6, the m = 0 4 \fcoef\ufb01cients are constant as a function of \u03c6, while the m \u2208{\u22121, 1} coef\ufb01cients are sine and cosine functions of \u03c6 (see Appendix D): (D\u03c6x)(0) = \u03b3, (3) (D\u03c6x)(\u22121) = \u03b1 sin(\u03c6 + \u03b2) = (D(\u03c6\u2212\u03c0 2 )x)(1), (4) (D\u03c6x)(1) = \u03b1 cos(\u03c6 + \u03b2) = (D(\u03c6+ \u03c0 2 )x)(\u22121), (5) for some set of vectors \u03b1, \u03b2 and \u03b3 of size L. x(0) are the m = 0 coef\ufb01cients and similarly for x(\u22121) and x(1) for the m = \u22121 and m = 1 coef\ufb01cients respectively. Similar proprieties hold for m > 1 and m < \u22121. We take advantage of these properties to encourage the output of the message block towards equivariance by computing Fe at multiple rotations \u03c6 about the z-axis: F \u03c6 e = Fe(D\u03c6Dstx(k) s , D\u03c6Dstx(k) t , |dst|, as, at), (6) For m \u2208[\u22121, 1], we compute four samples or \"taps\" at \u03c6 \u2208{0, 1 2\u03c0, \u03c0, 3 2\u03c0}, and combine them based on m using: m(k+1) st = D\u22121 st 1 4 \u0010 F 0 e(0) + F 1 2 \u03c0 e(0) + F \u03c0 e(0) + F 3 2 \u03c0 e(0) \u0011 + D\u22121 st 1 4 \u0010 F 0 e(\u22121) \u2212F 1 2 \u03c0 e(1) \u2212F \u03c0 e(\u22121) + F 3 2 \u03c0 e(1) \u0011 + D\u22121 st 1 4 \u0010 F 0 e(1) + F 1 2 \u03c0 e(\u22121) \u2212F \u03c0 e(1) \u2212F 3 2 \u03c0 e(\u22121) \u0011 (7) where F \u03c6 e(0), F \u03c6 e(\u22121) and F \u03c6 e(1) are the output coef\ufb01cients for m = 0, m = \u22121 and m = 1 respectively. The top line of Equation 7 is simply taking the average, since the m = 0 coef\ufb01cients should be constant regardless of \u03c6. Similarly for m \u2208{\u22121, 1}, the second and third lines average 4 values that should be equal if Equations 4 and 5 hold, e.g., \u03b1 sin(\u03c6 + \u03b2) = \u2212\u03b1 cos(\u03c6 + \u03b2 + \u03c0/2). If higher m are desired, similar calculations can be performed. However, a larger number of taps will be needed, e.g., m \u2208[\u22122, 2] requires 8 taps separated by 1 4\u03c0 radians. The number of coef\ufb01cients used during message passing when calculating Fe may be varied based on the edge properties to reduce the memory required by the network. For instance, atoms that are far away from each other may not need the same resolution of the spherical functions as those close together, so a lower maximum degree L may be used. By training several message passing networks with varying degrees for different ranges of edge distances, memory usage can be reduced while not impacting accuracy. Similarly, the range of m can typically be truncated at [\u22121, 1] or [\u22122, 2] since their observed utility to the network is substantially reduced for values of m greater than 2 (or less than -2). See Figure 1(b) for examples of the spherical channels when the range of m is reduced. Note that while the number of coef\ufb01cients used by Fe may be reduced, the node embeddings maintain their original size based on the maximum L, so they may be able to aggregate information from all neighboring atoms at different distances and angles. 2.3 Message Aggregation At this point in the network, the per edge messages are computed and rotated back to the global coordinate frame. Ideally, to allow for complex interactions between the messages, a non-linear function would be applied after summing all messages directed at a target node. If we did this by naively passing the summed coef\ufb01cients m(k+1) t = P s m(k+1) st through a fully connected neural network, the network may have dif\ufb01culties learning representations that are approximately rotation equivariant (a fundamental property of atom forces). Another option is to place constraints on the non-linearities performed to enforce equivariance [26, 5, 36, 42, 3, 2]. We propose applying an unconstrained non-linear pointwise function on the sphere, i.e., a function that is applied at every orientation without knowledge of the orientation from which the point was sampled [9]. For our function, we use a non-linear neural network Fc (RC \u2192RC) that combines information across channels. In practice, Fc is applied at a discrete number of orientations, which 5 \fFigure 3: Illustration of message aggregation. The summed messages m(k+1) t = P s m(k+1) st and previous embedding x(k) t are converted from a spherical harmonic representation to a spherical grid representation using G. The channels are concatenated and passed through a 3 layer 1 \u00d7 1 CNN. Each layer is followed by a SiLU activation. Finally, the channels are converted back to a spherical harmonic representation using G\u22121 and added to x(k) t to create x(k+1) t . Figure 4: Illustration of different spherical channels for six structures (top). Blue indicates a positive value and red a negative value on the sphere. The spherical channels were sampled from the last layer before the output blocks using a model with 12 layers and L = 6. Note, how some channels have a higher activation for adsorbates (darker colors), certain elements or the bottom or top of the surface. Illustrations of the structures are shown on the bottom with different colors indicating different elements. results in a transformation that we numerically demonstrate (see Appendix A) is approximately equivariant to rotations: x(k+1) t = x(k) t + G\u22121 \u0010 Fc \u0010 G(m(k+1) t ), G(x(k) t ) \u0011\u0011 , (8) where G is a function that converts a spherical function represented by spherical harmonic coef\ufb01cients, to one represented by point samples on a sphere. For G we use a spherical coordinate system (polar and azimuthal) to generate a 2D discrete representation of the spherical functions, see Figure 3. A 1 \u00d7 1 convolutional neural network Fc is applied to each discrete sample containing C channels, and the result is then converted back to the spherical harmonic representation using G\u22121. This is analogous to having a 2D image represented in the frequency domain, converting it to the spatial domain using an inverse discrete Fourier transform, applying a transformation and converting back to the frequency domain. Since the same operation is applied to all orientations on the sphere, a transformation that is approximately equivariant (up to discrete sampling limitations) to rotations may be learned, see Appendix A for a more detailed discussion. A diagram of the network is shown in Figure 3. To provide additional information to the network, x(k) t is also converted to a spherical grid representation, concatenated with G(m(k+1) t ) and provided as input to the neural network to compute the \ufb01nal updates to the coef\ufb01cients x(k+1) t at iteration k +1. The network can learn circular functions on the sphere if varying resolutions of x(k) are provided to the network, similar to Difference of Gaussian \ufb01lters for 2D images [30]. The resolution of x(k) can be reduced by lowering the degree L before transforming by G. In our experiments, if multiple resolutions or bands are used, we use resolutions of degree L and L \u22121. Three layers of 1 \u00d7 1 convolutions on 2C channels (or 4C if two bands are used) are performed with each followed by a 6 \fSiLU non-linear activation function. To avoid aliasing, the spherical grid is sampled at a resolution of 2 \u2217(L + 1). See Figure 4 for several example illustrations of spherical channels for different atomic structures. 2.4 Energy and Forces Estimation We compute the energy E by \ufb01rst estimating a per-atom energy using a pointwise function on the sphere and taking its integral over all possible orientations. The per-atom energy estimates are summed to obtain the overall system\u2019s energy: E = X i Z Fenergy \u0010 s(K) i (\u02c6 r) \u0011 d\u02c6 r, (9) where Fenergy is a three layer fully connected neural network (RC \u2192R1) with SiLU activation functions. Forces may be calculated using two approaches: First, by calculating the gradients of the energy with respect to the atom positions. This approach enforces energy conservation, but due to the need to back-propagate the gradients is computationally much more expensive. The second approach computes the forces in a manner similar to energy prediction, which is computationally more ef\ufb01cient, but does not enforce energy conservation. In this approach, the forces are calculated by estimating a force magnitude |f| = Fforce(s(K) i (\u02c6 r)) in every direction \u02c6 r over the sphere. Integration is performed after multiplying the magnitude by the orientation \u02c6 r to obtain directional vectors: fi = Z \u02c6 rFforce \u0010 s(K) i (\u02c6 r) \u0011 d\u02c6 r, (10) where Fforce is a three layer fully connected neural network (RC \u2192R1). In practice, a discrete approximation of the integral is performed for equations (9) and (10) using a set of 128 evenly distributed points on the sphere (see Appendix C.2). Note that the application of a pointwise function of an arbitrary neural network at a \ufb01nite number of discrete orientations is not invariant (energies) or equivariant (forces) to rotations. However in practice, numerically this operation does closely approximate these properties (see Appendix A.2). 3 Experiments We present results on the Open Catalyst 2020 (OC20) dataset [6] that is released under a Creative Commons Attribution 4.0 License. OC20 contains over 130M training examples for the task of predicting atomic energies and forces for catalysts used in renewable energy storage and other important applications [48]. This dataset is a popular benchmark for the ML community. We begin by comparing results across all tasks on the test set. Next, we show numerous ablation studies comparing model variations on the smaller OC20 2M dataset. Finally, since several papers only report results on the IS2RE task using approaches that directly predict relaxed energies, we also train a model speci\ufb01cally for this task and approach, and compare in Appendix B. 3.1 Implementation Details The implementation of spherical harmonics and their transformations uses the code provided by Euclidean neural networks (e3nn) [16]. For message passing, two resolutions of spherical harmonics are used based on a atom\u2019s nearest neighbors, see Appendix C.1 for exact parameters. Unless otherwise stated C = 128, K = 16, H = 1024, E = 128, and for Fe in message passing only orders m \u2208[\u22121, 1] are used. All forces are estimated directly as an output of the network, unless stated that the energy conserving gradient-based approach was used. During training, the coef\ufb01cients for the force and energy losses are 100 and 2 respectively. Training is performed using the AdamW optimizer [29] with a learning rate of 0.0004. The effective batch size is increased using data parallelism and PyTorch\u2019s Automatic Mixed Precision (AMP). All model code will be open sourced with an MIT license in the Open Catalyst Github repo. Please see the supplementary for details on the PaiNN[36] baseline. 7 \fOC20 2M Validation S2EF IS2RE Samples / Energy MAE Force MAE Force Cos EFwT Energy MAE EwT Model GPU sec. [meV] \u2193 [meV/\u00c5] \u2193 \u2191 [%] \u2191 [meV] \u2193 [%] \u2191 Median SchNet [38] 1400 78.3 0.109 0.00 DimeNet++ [12] 805 65.7 0.217 0.01 SpinConv [41] 406 36.2 0.479 0.13 GemNet-dT [14] 25.8 358 29.5 0.557 0.61 438 GemNet-OC [15] 18.3 286 25.7 0.598 1.06 407 L # layers H # batch SCN No rotation 4 12 512 64 8.5 410 67.7 0.192 0.01 SCN No 1x1 conv 6 12 1024 128 9.1 313 26.2 0.579 0.74 SCN m = 0 6 12 1024 128 10.4 307 26.5 0.588 0.83 SCN m \u2208[\u22121, 1] 6 12 1024 128 8.3 302 24.6 0.601 0.93 SCN m \u2208[\u22122, 2] 6 12 1024 96 6.9 301 23.4 0.612 1.01 SCN m \u2208[\u2212l, l] 4 12 512 64 7.6 297 24.6 0.595 0.92 SCN grad-forces 4 12 512 128 1.2 307 26.2 0.573 0.81 SCN direct-forces 4 12 512 128 12.1 303 25.3 0.592 0.87 SCN 2 12 256 128 12.9 312 27.6 0.568 0.70 SCN 4 12 512 128 12.1 303 25.3 0.592 0.87 SCN 6 12 1024 128 8.3 302 24.6 0.601 0.93 SCN 8 12 1024 64 5.9 300 23.2 0.620 1.15 SCN 6 12 1024 64 7.7 299 24.3 0.605 0.98 SCN 2-band 6 12 1024 64 5.1 292 23.1 0.622 1.18 SCN 4-tap 6 12 1024 64 3.7 296 22.7 0.638 1.27 SCN 4-tap 2-band 6 12 1024 64 3.5 279 22.2 0.643 1.41 371 11.0 SCN m = 0 6 16 1024 96 7.4 300 25.7 0.600 0.96 394 9.6 SCN m = 0 8 16 1024 64 4.8 296 25.3 0.608 1.01 389 9.9 SCN 6 16 1024 96 5.9 287 22.8 0.623 1.22 371 10.5 SCN 8 16 1024 96 3.5 283 22.7 0.627 1.22 364 11.3 SCN 4-tap 6 16 1024 64 2.6 282 22.2 0.648 1.37 378 10.7 SCN 4-tap 2-band 6 16 1024 64 2.3 279 21.9 0.650 1.46 373 11.0 Table 1: Results on the OC20 2M training dataset and ablation studies for SCN model variations. The validation results are averaged across the four OC20 Validation set splits. All SCN models are trained on 16 GPUs for 12 epochs with the learning rate reduced by 0.3 at 5, 7, 9, and 11 epochs, except SCN with L = 8 and 16 layers that used 32 GPUs to obtain a larger batch size. Batch sizes vary based on the number of instances that can be \ufb01t in 32GB RAM. 3.2 OC20 Tasks We compare against numerous models trained on the OC20 2M, All and MD datasets across the Structure to Energy and Forces (S2EF), Initial Structure to Relaxed Structure (IS2RS) and Initial Structure to Relaxed Energy (IS2RE) tasks [6]. OC20 2M, OC20 All and OC20 MD have 2, 133, and 38 million training examples respectively. The 12 and 16 layer SCN models signi\ufb01cantly outperform state-of-the-art GemNet-OC [15] on S2EF force prediction (\u224814% improvement) and IS2RE energy predictions (\u224810% improvement) when trained on OC20 2M, Table 1. Comparable results are found for energy MAE on S2EF with GemNet-OC. We also report throughput ef\ufb01ciency. Note that while the SCN models process fewer samples per second, they are more sample ef\ufb01cient than GemNet (Figure 2(b)). Ef\ufb01ciency improvements have also been noted for fully equivariant models [3]. We compare our largest SCN models trained on the All and the All+MD datasets (133M + 38M examples) across all OC20 tasks in Table 2. For these experiments we use a deep 1-tap model, and a slightly shallower 4-tap model since the use of 4-taps requires more memory. Our model achieves state-of-the-art results for force MAE (\u22489% improvement) and force cosine for the S2EF tasks. The SCN L = 6 model performs similarly to the recently released GemNet-OC model [15] on energy MAE for S2EF. On the challenging relaxation based IS2RE task that requires models to predict both accurate forces and energies, SCN outperforms GemNet-OC by over 7%. Note that we only train a single model to predict both energy and forces, where GemNet-OC-L-F+E + MD is the combination of two models, one trained speci\ufb01cally for forces and the other for energy. 3.3 Ablation studies We explore numerous model variations in Table 1 trained on the OC20 2M dataset and evaluated on the OC20 validation set. The \ufb01rst set of ablation experiments explores reducing the complexity of our model. If messages are not rotated before and after applying Fm in Equation 2, i.e., Ast 8 \fOC20 Test S2EF IS2RS IS2RE Train Energy MAE Force MAE Force Cos EFwT AFbT ADwT Energy MAE Model #Params time meV \u2193 [meV/\u00c5] \u2193 \u2191 [%] \u2191 [%] \u2191 [%] \u2191 meV \u2193 Median \u2013 2258 84.4 0.016 0.01 Train OC20 All SchNet [38, 6] 9.1M 194d 540 54.7 0.302 0.00 14.4 764 PaiNN [36] 20.1M 67d 341 33.1 0.491 0.46 11.7 48.5 471 DimeNet++-L-F+E [12, 6] 10.7M 1600d 480 31.3 0.544 0.00 21.7 51.7 559 SpinConv (direct-forces) [41] 8.5M 275d 336 29.7 0.539 0.45 16.7 53.6 437 GemNet-dT [14] 32M 492d 292 24.2 0.616 1.20 27.6 58.7 400 GemNet-OC [15] 39M 336d 233 20.7 0.666 2.50 35.3 60.3 355 SCN L=8 K=20 271M 645d 244 17.7 0.687 2.59 40.3 67.1 330 Train OC20 All + MD GemNet-OC-L-E [15] 56M 640d 230 21.0 0.665 2.80 GemNet-OC-L-F [15] 216M 765d 241 19.0 0.691 2.97 40.6 60.4 GemNet-OC-L-F+E [15] 348 SCN L=6 K=16 4-tap 2-band 168M 414d 228 17.8 0.696 2.95 43.3 64.9 328 SCN L=8 K=20 271M 1280d 237 17.2 0.698 2.89 43.6 67.5 321 Table 2: Comparison of SCN to existing GNN models on the S2EF, IS2RS and IS2RE tasks when trained on the All or All+MD datasets. Average results across all four test splits are reported. We mark as bold the best performance and close ones, i.e., within 0.5 meV/\u00c5 MAE, which we found to empirically provide a meaningful performance difference. Training time is in GPU days. Median represents the trivial baseline of always predicting the median training energy or force across all the validation atoms. The SCN L = 8 model has K = 20 layers, C = 128 and E = 256, while the SCN L = 6 model has K = 16 layers, C = 128, E = 128 and an energy loss coef\ufb01cient of 4. is an identity matrix, the results are signi\ufb01cantly worse for energies and even more so for forces since forces are highly dependent on angular information. If we replace the non-linear aggregation (Equation 8) with a simple summation (No 1x1 conv) in a model with m \u2208[\u22121, 1], the results also degrade across all metrics. Results improve for m \u2208[\u22121, 1] as compared to only using m = 0 coef\ufb01cients, which demonstrates that if the equivariant to rotation constraint is relaxed, improved results may be achieved. For higher m, m \u2208[\u22122, 2] or m \u2208[\u22124, 4] diminishing returns are noticed with increased computational cost. In some applications, such as molecular dynamics, energy conserving models are needed. We trained an energy conserving model that estimates forces based on the gradients of the energy with respect to the atom positions. Since this approach is more expensive both in memory and computation, we used a smaller model with L = 4, H = 512 and 12 layers (grad-forces). While the accuracy of the model is good, the results are slightly worse and is 10 times slower than a comparable model that directly estimates forces (direct-forces). When scaling the network, increasing L has a more signi\ufb01cant impact on force estimation, while depth improves energy estimation. This may be due to higher L resulting in greater angular resolution, which is important to force estimation. Greater depth, which allows information to travel further, may lead to better energy predictions since energy is function of the entire atomic structure. Sampling multiple z-axis rotations when computing Fe with 4-taps (Equation 7) produces signi\ufb01cantly improved force predictions, while using two bands (L and L\u22121) when aggregating messages improves both energy and force prediction. The use of 2 bands reduces model throughput a small amount, while sampling with 4-taps reduces throughput by over 2\u00d7. However, sample ef\ufb01ciency is further increased with 4-taps, resulting in similar training times for the same accuracy, Figure 2(b). 4 Related work Molecular and atomic property prediction has made signi\ufb01cant recent progress. Models relying on hand-crafted representations like Behler-Parrinello[4] and sGDML[8] have recently been surpassed by learned feature representations using GNNs [3, 12, 14, 36]. Early GNN developments focused only on invariant representations. CGCNN[45] and SchNet[35] make energy and force estimations using only atomic species and pair-wise distance information. DimeNet[12, 13], SphereNet[28], and GemNet[14, 15] extend this to explicitly capture triplet and quadruplet angles, which are scalars. Utilizing invariant representations offers \ufb02exibility in model architecture design since the node or edge features are inherently invariant and model constraints are not needed to maintain equivariance. The 9 \fenumeration of triplets and quadruplets of atoms to calculate functions of relative or dihedral angles requires careful model design to maintain ef\ufb01ciency. More recently, equivariant models, capturing both scalar and equivariant features, have outperformed traditional GNNs on small molecular datasets including MD17[7] and QM9[32]. While not strictly equivariant, our work falls into this later class of GNNs for atomic property predictions. Models that strictly enforce equivariance to the SO(3) (3D rotations) or E(3) group (3D rigid transformations) share similarities to our use of spherical channels [36, 1, 3, 9, 5]. Both represent atoms by higher-order tensors and not just scalar information. For instance, Tensor Field Networks [42], Cormorant [1], NequIP [3], SEGNN [5] and BOTNet [2] use the same spherical harmonic representations as SCNs, while PaiNN [36] is an l = 1 variation. All models place constraints on the operations that can be performed to ensure equivarance, such as restricting non-linear operators to invariant scalar inputs, e.g., pair-wise atom distances. In networks like TFN, geometric information of higher degree l are mixed into the l = 0 features via tensor products on which a gated non-linear function is applied. Similar to our approach, the recently proposed SEGNN uses the spherical harmonic representation to enable steerable features based on edge orientations. SEGNN introduces steerable MLPs that enforce equivariance by restricting the learnable parameters to those that scale the Clebsch-Gordan tensor products followed by gated non-linearities [43]. In our approach, we increase the expressivity of the model by identifying a set of m = 0 coef\ufb01cients across degrees l that are invariant to rotations given an edge\u2019s orientation to which unconstrained functions may be applied. We also demonstrate that improved performance can be achieved by relaxing the requirement for strict message equivariance by including a broader set of coef\ufb01cients beyond m = 0. Furthermore, we project the spherical channels onto a grid and perform a pointwise 1 \u00d7 1 convolution followed by a non-linearity. Although not strictly equivariant, the pointwise operation allows for complex mixing of different degrees of spherical harmonics resulting in rich geometric descriptions. Perhaps as a result of the network\u2019s increased expessivity, we see an increase in accuracy with higher degrees (L = 4, 6, 8) unlike previous approaches that typically use L = 1 or L = 2. 5 Discussion A limitation of SCN is that it is very computationally expensive if energy conservation is enforced through the estimation of forces using energy gradients with respect to atom positions. This may limit its use in chemistry applications such as molecular dynamics. Non-energy conserving models are still practically useful for applications such as structure relaxations and transition state searches. The SCN model scales O(n2) with respect to L, which may limit using L larger than 8. Using a larger set of m coef\ufb01cients during message passing also faces challenges, since the network may \ufb01nd it more dif\ufb01cult to learn approximately equivariant mappings. In Table 1, we saw that increased depth improves results, especially to energy prediction. It remains an open question whether deeper networks will see further improvement. Given SCN\u2019s increased expressivity due to the relaxation of the equivariance constraint and larger model sizes, it is prone to over\ufb01tting on smaller datasets. SCN\u2019s results on datasets such as MD17[7] and QM9[32] are not state-of-the-art. However, in future work it would be interesting to explore pre-training on larger datasets, such as OC20, and see if improved results can be achieved on smaller datasets through \ufb01netuning [25]. While this work is motivated by problems we face in addressing climate change, advances in chemistry may have numerous use cases. Some of these use cases may have unintended consequences, such as the development of the Haber-Bosch process for ammonia fertilizer production that enabled the world to feed a growing population, but over fertilization has recently led to ocean \u201cdead zones\u201d and its production is very carbon intensive. More alarmingly, the knowledge gained from fertilizer production was used to create explosives during wartime [20]. We hope to steer research in this area towards bene\ufb01cial applications by utilizing datasets such as OC20. In conclusion, we propose a GNN that uses spherical harmonics to represent channel embeddings that contain explicit orientation information. While the network is encouraged to learn output mappings that are equivariant to rotations, best performance is achieved by relaxing the equivariant constraints and allowing for more expressive non-linear transformations. We demonstrate stateof-the-art results across numerous tasks on the large-scale OC20 dataset for modeling catalyst for applications addressing climate change. 10"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2010.09435v1",
+                    "title": "An Introduction to Electrocatalyst Design using Machine Learning for Renewable Energy Storage",
+                    "abstract": "Scalable and cost-effective solutions to renewable energy storage are\nessential to addressing the world's rising energy needs while reducing climate\nchange. As we increase our reliance on renewable energy sources such as wind\nand solar, which produce intermittent power, storage is needed to transfer\npower from times of peak generation to peak demand. This may require the\nstorage of power for hours, days, or months. One solution that offers the\npotential of scaling to nation-sized grids is the conversion of renewable\nenergy to other fuels, such as hydrogen or methane. To be widely adopted, this\nprocess requires cost-effective solutions to running electrochemical reactions.\nAn open challenge is finding low-cost electrocatalysts to drive these reactions\nat high rates. Through the use of quantum mechanical simulations (density\nfunctional theory), new catalyst structures can be tested and evaluated.\nUnfortunately, the high computational cost of these simulations limits the\nnumber of structures that may be tested. The use of machine learning may\nprovide a method to efficiently approximate these calculations, leading to new\napproaches in finding effective electrocatalysts. In this paper, we provide an\nintroduction to the challenges in finding suitable electrocatalysts, how\nmachine learning may be applied to the problem, and the use of the Open\nCatalyst Project OC20 dataset for model training.",
+                    "authors": "C. Lawrence Zitnick, Lowik Chanussot, Abhishek Das, Siddharth Goyal, Javier Heras-Domingo, Caleb Ho, Weihua Hu, Thibaut Lavril, Aini Palizhati, Morgane Riviere, Muhammed Shuaibi, Anuroop Sriram, Kevin Tran, Brandon Wood, Junwoong Yoon, Devi Parikh, Zachary Ulissi",
+                    "published": "2020-10-14",
+                    "updated": "2020-10-14",
+                    "primary_cat": "cond-mat.mtrl-sci",
+                    "cats": [
+                        "cond-mat.mtrl-sci",
+                        "cs.CE",
+                        "cs.LG",
+                        "I.2.6; J.2"
+                    ],
+                    "main_content": "Introduction The increased use of renewable energy is essential for addressing the world\u2019s rising energy needs while reducing climate change. Currently, renewable energy such as solar, wind and hydro generates 17% of the US\u2019s and 28% of the world\u2019s electricity [1, 2]. While hydro accounts for 62% of this generation, it is limited in its ability to expand further due to resource availability and environmental concerns. Meanwhile, solar and wind are projected by 2050 to expand signi\ufb01cantly and account for over 70% of renewable energy generation [2]. With this expansion, a unique and potentially limiting challenge arises. Unlike traditional power sources, such as coal, natural gas, and nuclear, power provided by solar and wind is intermittent. The wind does not always blow, and the sun sets. During these periods, wind and solar plants do not generate any power. Notably, energy demand typically peaks in the early mornings (8am) and evenings (6-7pm) well outside the peak generating time (noon) for solar power [3], Figure 1. On the other hand, during peak mid-day solar power generation, the power from other more conventional sources must be reduced to balance supply and demand. Adjustments in output from conventional plants may not always be able to match the variability of solar and wind power generation (e.g. due to the unpredictability of rapid weather \ufb02uctuations), resulting in an excess of supply and wasted energy [4]. Without further advances, it is estimated the overall penetration of solar power is capped at 30% and costs start to rise after 20% penetration [5]. Currently, areas such as California that generate 14% to 19% of their power from solar [6,7] are approaching these levels. The challenges raised by intermittent power can be addressed through several means, including demand response [9,10], improved power transmission [11], and energy storage [12\u201316]. Demand response provides electricity consumers with the ability to shift power demand to time periods with high supply. The transmission of power across large distances can help both transmit power from renewable energy sites that are far away from cities, and smooth \ufb02uctuations in the generation of intermittent power by combining multiple sites. While both of these techniques are part of the solution, ef\ufb01cient storage of renewable energy is essential to unlocking the full potential of intermittent renewable energy. Numerous storage techniques exist, including pump1 arXiv:2010.09435v1  [cond-mat.mtrl-sci]  14 Oct 2020 \fFigure 1. Hourly electricity demand for California on a typical summer day (August 6th, 2020) as reported by California ISO [8]. The demand (black line) peeks around 19:00, the same time that output from solar and wind power (green line) declines. The remaining demand after the use of wind and solar power is shown by the orange line. Note that energy storage is needed if solar and wind are to meet demand without excess energy loss. ing water uphill (Pumped-Storage Hydroelectricity, PSH) [17], batteries [12], and Hydrogen Energy Storage (HES) [18\u201320]. HES combines electricity from renewable sources with water to generate hydrogen through a process called electrolysis. The hydrogen is stored and used when needed to generate electricity through fuel cells. As described in greater detail in Section 2, each of these storage approaches has trade offs and the appropriate choice is based on the application. PSH and batteries both have high ef\ufb01ciency, but do not scale to the energy storage needs of large grids powered by intermittent renewable energy due to land restrictions (PSH) or cost (batteries). Hydrogen energy storage offers a unique combination of scalability, long-term storage and portability. Speci\ufb01cally, HES can store vast quantities of energy cheaply in the form of hydrogen using the existing gas storage infrastructure. However, other costs associated with HES has limited the adoption of this otherwise very promising technology. Hydrogen energy storage\u2019s high costs result from its relatively low ef\ufb01ciency and the capital costs associated with systems for power conversion (electricity to hydrogen and back to electricity) [21, 22]. The ef\ufb01ciency of HES is estimated to be 35% for round-trip AC to AC [22], and unlikely to exceed 50%. Compared to other systems such as PSH, which can achieve 70 \u221280% ef\ufb01ciency, the cost of the electricity used to feed the system is nearly double. Thus the relative overall cost of HES compared to other systems is highly dependent on \ufb02uctuations in electricity pricing. Assuming the cost of electricity from renewable energy sources continues to decline (solar costs have reduced 100-fold since 1976 [23]) and as the economic necessity of carbon neutral energy sources increases, the attractiveness of HES use should only improve. The power conversion systems for HES involve splitting water into its constituent parts (hydrogen and oxygen) using an electrolyzer and converting the hydrogen back to electricity using fuel cells or other methods such as gas-\ufb01red turbines and engines (Figure 2). The electrolyzer and the fuel cell perform inverse reactions, 2 H2O \u2212 \u2212 \u21922 H2 + O2 and 2 H2 + O2 \u2212 \u2212 \u21922 H2O respectively. Both of these systems rely on materials called electrocatalysts to increase the rate and ef\ufb01ciency of their respective electrochemical reactions. Unfortunately, common state-of-the-art electrocatalysts use expensive noble metals such as iridium and platinum [24]. A major challenge is \ufb01nding low-cost, ef\ufb01cient, and durable electrocatalysts for use in these reactions. If such catalysts can be found, earlier adoption of HES systems could be achieved due to signi\ufb01cant cost reductions. Modern design of catalysts uses computational simulation to determine if a material is suitable for further exploration. The simulation models the interaction of a molecule with an catalyst\u2019s surface in a process called adsorption. The adsorbed molecules\u2014i.e., adsorbates\u2014are typically the key chemical species involved in the chemical reactions of interest, such as OH, O2, or H2O. The simulation is performed using Density Functional Theory (DFT) [25, 26]. While the foundations of DFT were \ufb01rst introduced in the 1960s [27, 28], it wasn\u2019t until the 1990s that advances resulted in practical predictions for quantum chemistry [29]. Now DFT is commonly used to compute the interactions of atoms using a process colloquially called \u201crelaxation\u201d. In relaxations, DFT combines the locations of the atomic nuclei with quantum mechanics to predict the energy of the system and the forces acting on each atom. The locations of the atomic nuclei are then updated to minimize the energy, consequently changing the electronic distributions and energies. This iterative process continues until the energy of the system reaches a local minima. By examining the energy of the resulting relaxed system (system with lowest energy), we can gain insight into the overall ef\ufb01ciency of the reaction, i.e., how much energy results from or is needed to drive the reaction. Unfortunately, the DFT computation is extremely expensive and scales O(n3) with the number of electrons in the system, which can range from O(100) to over O(5,000) electrons in practice. The result is computations that take hours or days per relaxation on O(10\u2013100) core CPUs. Beyond DFT, alternative approaches to modeling quantum mechanics include the use of Coupled Cluster Single Double Triple (CCSDT) [30] that improves on the accuracy of DFT but is rarely used in practice due to its even higher computational cost, and force \ufb01elds which are computationally very ef\ufb01cient but fall short in accuracy [31,32]. To enable large-scale exploration of new catalysts, ef\ufb01cient computational approximations to DFT are needed. 2 \fFigure 2. Diagram of the hydrogen (H2) and methane (CH4) energy storage process using renewable energy. The stored hydrogen and methane is used to generate electricity when renewable energy generation is not able to meet demand. Renewable energy (bottom left) may either be fed directly into the grid or used to generate hydrogen by electrolysis. The hydrogen is stored for future electricity generation by fuel cells or used in the methanation process to generate methane. The generation of electricity from stored methane can potentially be carbon neutral by recycling the CO2 for later use in methanation. One particularly promising approach is to use machine learning [33]. Given a training dataset of DFT calculations, machine learning techniques, such as Deep Learning (DL), have shown the ability to learn good approximations for certain problems. These techniques take as input the state of the system, including the atom positions, element types, bond information, etc., and predict system properties such as their energy [34\u201337]. Popular datasets for this task include QM9 [38] containing over 134k small organic molecules, the Material Project containing 70k inorganic compounds [39], and ISO17 containing motion trajectories for 129 molecules from QM9 [35]. Unfortunately, similar datasets relevant to our problem of interest remain limited [40,41]. An ideal dataset containing catalysts for renewable energy storage would have two main properties. First, it would be large scale, containing DFT simulations of relevant catalysts and adsorbates. Second, the dataset would enable the training of models that take as input rough initial starting positions of the system\u2019s atoms from which the relaxed geometries and energies could be computed. The large majority of DFT calculations are spent in the iterative computation of relaxed geometries, so ef\ufb01cient approximations of this process are needed. This paper provides an introduction to how the Open Catalyst Project OC20 [42] dataset addresses this task and how to apply it to training ML models. If ef\ufb01cient approximations of DFT calculations can be performed, it opens the door to large scale exploration of new catalysts. Assuming catalysts are created from up to three of forty potential metals, over 10k combinations exist. Since each set of metals can be combined in various ratios and con\ufb01gurations, the total number of possibilities increases ad in\ufb01nitum. As discussed in Section 3, determining whether a potential catalyst may be effective requires simulating its interaction with the set of potential adsorbates at all potential surfaces and binding sites. This results in O(1,000) potential simulations per catalyst. The result is a search space containing millions of possibilities requiring billions of simulations to explore! While this may seem overwhelming, with ef\ufb01cient DFT approximations it may be possible to compute all of them through brute force. The most promising candidates may then be veri\ufb01ed through traditional DFT calculations before further exploration into their real-world feasibility. While the discussion so far has focused on HES, ef\ufb01cient DFT approximations could be applied to numerous highimpact problems. Two notable examples include methanation and the production of ammonia fertilizer. Methanation refers to the reaction of carbon dioxide and hydrogen (generated as we described above using renewable energy) to form methane and water. Instead of storing renewable energy in the form of hydrogen this would allow storing it as methane, while having the bene\ufb01cial side effect of reducing carbon dioxide (Figure 2). The potential end result could be a high density, carbon neutral fuel that \ufb01ts into the current energy infrastructure [43]. More complex hydrocarbon fuels beyond methane, such as ethanol, may also be created through the family of Fischer-Tropsch chemical reactions. The production of ammonia, a key ingredient in agricultural fertilizer, combines hydrogen and nitrogen to create ammonia. Currently this energy intensive process, called the Haber-Bosch process, accounts for approximately 1% of the world\u2019s CO2 emissions [44]. This could be greatly 3 \fUS Power Grid 2018 Source Capacity (GW) Generation (TWh) % Natural Gas 537 1,469 35.2% Coal 264 1,146 27.5% Nuclear 104 807 19.3% Hydroelectric 80 293 7.0% Wind 95 273 6.5% Solar 32 64 1.5% Other 84 122 3.0% Total 1,196 4,174 \u2013 Table 1. Power capacity and generation of the US power grid by source in 2018 [45]. reduced by using renewable energy to produce the input hydrogen and by \ufb01nding a more ef\ufb01cient reaction to HaberBosch. 1.1. Overview We hope this paper serves as a gentle introduction for the machine learning community to opportunities presented by the problem of renewable energy storage, and the use of ef\ufb01cient ML models for atomic simulations. For researchers only interested in using the OC20 dataset for model training, Sections 2 and 3 may be skipped, and you may proceed directly to Section 4. We begin by providing a brief overview of renewable energy storage technologies (Sec. 2) followed by an introduction to the design of catalysts using DFT (Sec. 3). We explore applications for deep learning starting in Section 4. This includes the OC20 dataset needed for training the relevant tasks (Sec. 4.1) and potential directions for ML models to addressing these tasks (Sec. 4.2). Finally, future directions are explored in Section 5. 2. Renewable Energy Storage In this section, we provide a brief overview of the two most popular renewable energy storage technologies\u2014 pumped-storage hydroelectricity and batteries\u2014as well as the technologies relevant to this paper: hydrogen energy storage and methane synthesis. A summary of these technologies is provided in Table 2. For a more thorough discussion of the numerous technologies available please see [12\u201315]. Due to the readily available information provided by the US Energy Information Administration, we use the US as an example for our discussion below. However, results will vary by country based on geography, natural resources available, etc. During this discussion we use the terms gigawatts (GW) and terawatt hours (TWh) (1 TW = 1,000 GW). A GW is a unit of measurement for electric capacity, i.e., how much electricity can \ufb02ow at an instance in time. A TWh is a unit of electric energy, i.e., how much electricity was generated in one hour. A GW is equal to one billion watts and a TWh is equal to one trillion watt hours. For instance, a 50 GW plant operating for 1,000 hours at peak capacity would generate 50 TWh of energy. For reference, the largest nuclear power plant (Palo Verde) in the US has a capacity of 4 GW, while the average natural gas plant has 0.5 GW of capacity. In 2018, the US had a maximum capacity of 1,196 GW and generated 4,174 TWh of energy [45]. The percentage of actual energy produced compared to the maximum capacity (known as the capacity factor) is on average 40%. A summary of US power generation by source is provided in Table 1. 2.1. Estimating Needed Storage Capacity Estimating the total stored energy and capacity needed to move to 100% renewable sources is dif\ufb01cult and depends on numerous factors [16, 46, 47]. For illustration, we consider a case where the US experiences a week of cloudy weather in the southwest and low winds in the plains. Let\u2019s assume that during this period 70% of the US energy would need to come from storage. In this case, 56 TWh of energy storage would be needed (4,174 TWh / year x 0.02 years x 70%) with a capacity of 420 GW (1.25 peak to mean ratio x 56 TWh / 168 hours per week). However, this assumes we\u2019re only replacing energy that is currently being supplied through electricity. When including other energy sources such as gasoline powered engines and heating, the US actually uses \u224830, 000 TWh in total energy [48]. Since electric motors are more ef\ufb01cient than internal combustion engines, we could potentially drop this amount by a third to 20,000 TWh. Therefore, to meet 70% of our total energy needs over 7 days with storage, we would need roughly 250 TWh of storage and 2,000 GW of capacity. Note, these numbers should only be used as very rough guides, but provide an approximate \u201corder of magnitude\u201d estimate (Table 2) for our discussion below. 2.2. Pumped-Storage Hydroelectricity Pumped-Storage Hydroelectric (PSH) refers to the conceptually simple technique of using renewable energy to pump water uphill. When additional energy is needed, the water \ufb02ows back downhill through a hydroelectric plant to generate electricity. This mature technology has resulted in over 40 plants being built in the US, with most of their construction happening in the 1970s. No new plants have come online since 2012. An advantage of PSH is their ef\ufb01ciency, which ranges between 70% to 80% for round trip AC to AC. Similar to hydroelectric plants, they are also highly responsive in that they can respond to power \ufb02uctuations in seconds. They are typically used for energy storage of 6 to 20 hours to help in system load balancing. For instance, a nuclear power plant 4 \fEnergy Storage Techniques Source Capacity (GW) Generation (TWh) AC to AC Ef\ufb01ciency Scale to Capacity? Scale to Storage? Pumped-Storage Hydroelectricity 23 24 70% \u221280% No No Batteries <1 <1 60% \u221295% Maybe No Hydrogen Energy Storage 0 0 35% Maybe Yes Methane Synthesis 0 0 25% Maybe Yes Table 2. Comparison of different energy storage techniques, including the installed US capacity and generation, and their round trip AC to AC ef\ufb01ciency. See text for detailed description. HES and methane synthesis are the only two techniques that could scale to meet both capacity and storage needs. can be kept at peak operating ef\ufb01ciency over 24 hours by diverting power to PSH in the night time and using hydroelectric power to meet peak demand during the daytime. In 2018, the US PSH capacity was 23 GW [45]. On average, the PSH plants operated between 8% and 16% of full capacity depending on the time of year (more in the summer when natural water \ufb02ow into reservoirs is reduced). Assuming an average of 12%, the total energy produced by PSH is 24 TWh, or 0.6% percent of the power (4,174 TWh) generated by the US in 2018. The primary limitation of PSH is its scalability. There are not enough places to store water to meet the national energy storage needs. Current yearly generation using hydroelectric is 293 TWh with a capacity of 80 GW [45], or only 7% of the national total (Table 1). To meet the 2000 GW of capacity needed during a week of cloudy still weather, we would need to build 25 times more hydroelectric dams. This would be quite a feat, especially when all the easy to build areas are already taken. A study initiated by the US Department of Energy concluded that only 65 GW of undeveloped hydroelectric potential from streams exists in the US, which wouldn\u2019t even double current capacity [49]. In another US study, it was found that if every non-hydroelectric dam was converted to hydroelectric an additional 12 GW (15% of current capacity) of capacity could be created [50]. We\u2019re also only considering capacity; this doesn\u2019t include the infrastructure needed to pump and store the water necessary to generate 250 TWh of energy. By one estimate, the needed reservoirs\u2019 area would be equivalent to the size of Lake Erie [46]! Even though PSH may not grow to the scale needed for nation-sized grids, PSH remains the best option for areas that do have suf\ufb01cient water storage capacity. 2.3. Batteries When considering electricity storage technologies, batteries quickly come to mind. Battery types can vary signi\ufb01cantly in cost, cycle life (how many times they can be discharged and recharged), energy density, and ef\ufb01ciency. Similar to PSH, batteries have high ef\ufb01ciency (80% to 95%), which would seem to make them promising candidates for large scale energy storage. Of the numerous types that exist, the most popular include lead-acid, sodium-sulfur (NaS) and lithium-ion (Li-ion) batteries [12,51]. Lead-acid batteries are relatively inexpensive and have high ef\ufb01ciency, but have decreased in popularity due to their lower power density and limited recharge cycles (< 500). NaS is a promising solution for large scale systems with a greater cycle life (> 1, 500) and low cost, but needs to operate at high temperatures (350 \u00b0C). Li-ion batteries offer a high cycle life (3,500) [52] and high ef\ufb01ciency (95%). Due to the market for electric cars, the price of Li-ion batteries has decreased signi\ufb01cantly, making them the lowest in per cycle cost. Currently, nearly all of the installed battery storage in the US uses Li-ion batteries for a combined capacity of 0.77 GW [53]. Would Li-ion batteries scale to the size of the US grid [47, 54]? Recently Tesla installed a 100 MW (129 MWh) battery in Australia at a cost of $66 million. Doing the simple math, the cost of a system with 2,000 GW of capacity would be $1.23 trillion (other estimates put the cost at $2.5 trillion [54]). While this expense may be daunting, it could potentially be feasible to meet capacity demands. However, it becomes prohibitive if we consider the cost of storage. For 250 TWh of power (7 days worth of power in our cloudy low wind scenario), the costs rise to $128 trillion! Using another estimate, the US DOE set a goal of $100 per kWh for lithium-ion battery cells [55]. Ignoring the other costs associated with batteries, this would still cost $25 trillion. While the cost of Li-ion batteries have dropped signi\ufb01cantly (in the last 10 years prices have dropped by 10x), it is unlikely their cost will drop by another order of magnitude over the next several decades [56]. Even if cost was not an issue, there might not be enough lithium on the planet to build a battery of this magnitude [47]. Finally, the cost of other rare metals used in Li-ion batteries has already signi\ufb01cantly increased due to high demand [57]. As we saw in the discussion above, batteries scale better as a function of capacity than energy storage. This makes them ideal for scenarios where energy only needs to be stored for short periods. For instance, power from solar can be stored during the middle of the day to be used in the evening. Similar to PSH, this is useful in keeping more conventional plants running steadily at peak operating ef\ufb015 \fciency. Batteries also have high energy density\u2014i.e., they take up relatively little volume\u2014which has led to their recent popularity in electric vehicles. Unfortunately, batteries also have low speci\ufb01c energy\u2014i.e., they require a large amount of mass per energy. This has prevented their use in planes, where excess mass reduces the energy ef\ufb01ciency of \ufb02ight. 2.4. Hydrogen Energy Storage Hydrogen Energy Storage (HES) refers to the process where renewable energy is stored in the form of hydrogen. Hydrogen is generated using electrolyzers powered by renewable energy to split water into hydrogen and oxygen. The hydrogen is stored for later use in fuel cells or gas-\ufb01red turbines to generate electricity. While hydrogen has historically been viewed as one of the promising technologies for moving beyond fossil fuels [58, 59], it is still in the early stages of adoption. When considered as an option for large scale energy storage, hydrogen offers signi\ufb01cant potential. Storing large scale reserves of hydrogen is feasible. Currently, there exists 4.3 trillion cubic feet of underground natural gas storage in the US using depleted gas reservoirs, salt caverns and aquifers [60]. Hydrogen could be stored in the same manner in place of the natural gas. Assuming hydrogen has 0.08 kWh of energy per cubic foot, 4 trillion cubic feet of stored hydrogen would contain 344 TWh of energy. Assuming a fuel cell ef\ufb01ciency of 60%, this corresponds to 206 TWh of stored energy; an amount very close to our target of 250 TWh of energy needed to run the US when solar and wind generation lags for a week! Moreover, if the infrastructure used to store natural gas was converted to hydrogen storage, the capital costs would be negligible. Hydrogen is portable. It can be compressed and transported in pipes similar to natural gas for local electricity generation or transportation uses. For transport, hydrogen has more than twice the speci\ufb01c energy (kWh per kg) of gasoline, but 1/7th the energy density (at 700 bar of pressure). In other words: hydrogen storage is less ef\ufb01cient than fossil fuels in terms of energy stored per volume/pressure, but hydrogen is more ef\ufb01cient in terms of mass. This means that HES could be suitable for aircrafts. Hydrogen automobiles already have shown ranges over 300 miles (500 km). The dif\ufb01culty with hydrogen is not storage. It is in its generation and conversion back to electricity. One significant limiting factor is the 35% ef\ufb01ciency for round trip AC to AC of HES, which is much lower than batteries and PSH. If HES is to be adopted, it needs to be less expensive than energy generated through fossil fuels. At 35% ef\ufb01ciency, the price of renewable energy feeding HES needs to be 2.85x less expensive than energy generated through other means. Currently, the 2023 cost projection for natural gas is $40\u2013$42 per MWh [61], which means renewable sources would need to cost $14 per MWh. 2023 projections for wind and solar fall short of this goal with prices of $43 and $48 per MWh, respectively [61]. However, the cost of renewable energy has been dropping at a high rate. For instance, the price of solar electricity has dropped from $280 per MWh in 2010 to $60 in 2017. The goal is to drop this to $30 per MWh by 2030 [62]. In considering HES systems, we need to consider their capital costs and not just the electricity feeding them. Using a fuel cell that is 60% ef\ufb01cient, a kg of hydrogen will produce 20 kWh of power. Thus to generate a MWh of power we need 50 kg of hydrogen. The electrolyzer capital costs associated with producing a kg of hydrogen was estimated to be $1.33 in 2014 and projected to fall to $0.53 in 2025 [63]. This corresponds to an additional $26.5 per MWh in additional costs using the 2025 projections. A 250 kW fuel cell\u2019s capital cost is approximately $300,000 [64]. With a 5-year life-cycle, this corresponds to a cost of $27.3 per MWh, similar to the estimates of the electrolyzer. In summary, the total approximate costs for a HES system is $113 per MWh ($60 electricity, $26.5 electrolyzer and $27.3 fuel cell). For comparison, an estimate from 2009 put the total cost at $240-280 per MWh [21]. While this is still 2x to 3x the cost of electricity from natural gas, a combination of reductions in renewable energy costs and capital costs could lead to HES being price competitive. Capital costs will begin to dominate as electricity prices decline. These costs are heavily dependent on the electrocatalysts used for both the electrolyzers and fuel cells. For instance, popular Polymer Electrolyte Membrane (PEM) fuel cell\u2019s high costs are due in part to their use of platinum as their electrocatalysts [63,65\u201367]. A study of automotive fuel cell experts showed that 76% believed the cost of platinum was the primary barrier to reducing costs [68]. The percentage of capital costs for material platinum alone is estimated to be 8% for commercial scale electrolyzers [67] and 11 \u221212% for transportation fuel cells [66]. When taking into consideration the entire cost of the electrocatalyst, which includes support structures that are needed to increase the power density of platinum, the percentage of capital costs related to platinum for fuel cells jumps to over 40% [66]. Does there exist enough platinum to generate the needed hydrogen? We would need to generate approximately 35 TWh of power per day, assuming 70% of the daily power would be generated through storage in our cloudy low wind scenario. This is equivalent to 1.75 million metric tons of hydrogen (assuming the use of a 60% ef\ufb01ciency fuel cell). A 1 MW electrolyzer plant could produce 400 kg per day [67], resulting in the need for 4.4 million plants. Each plant uses 420 grams of platinum, for a total of roughly 2,000 metric tons of platinum. For comparison, the known platinum reserves in the world are approximately 70,000 metric tons [69]. These amounts are pushing the boundaries 6 \fof feasibility, especially when considering the world\u2019s energy needs and not just the US\u2019s. As we explore later in this paper, discovering new electrocatalysts that don\u2019t use rare expensive metals could lead to a signi\ufb01cant reduction in costs, and to the potentially wide-spread adoption of HES systems. Already, research has shown that a 90% reduction in the amount of platinum may be feasible [70,71]. Other methods for hydrogen generation: Currently, the most economical process for generating hydrogen is called Steam Methane Reforming (SMR), which involves using high-temperature steam to produce hydrogen from methane. Unfortunately, a byproduct of SMR is the production of CO2 (9 kg of CO2 is produced for every kg of H2 [72,73]), partially defeating our goal of using hydrogen in the \ufb01rst place. Currently, 95% of hydrogen generation in the US uses SMR [74, 75]. Due to the low price of natural gas, the price per kg of hydrogen from SMR is currently below $1.50, making it competitive to gasoline [75] (1 kg of H2 has roughly the same energy as 1 gallon of gas). Currently, 10 million metric tons of hydrogen is produced annually in the US, with 70% of it used for petroleum re\ufb01ning and 20% in fertilizer production (ammonia) [76]. 2.5. Methane synthesis Methane synthesis is similar to HES except energy is stored in the form of methane instead of hydrogen. Methane is the primary ingredient of natural gas (90%). Methane offers two main advantages: nearly 4x higher energy density than hydrogen (by volume), and it can be used in existing natural gas infrastructure for power generation and transport. Higher energy density offers advantages in storage, especially for transportation uses. Methane synthesis uses a process called methanation (also known as the Sabatier reaction [77]), to combine H2 and CO2 to form methane CH4. The input hydrogen is generated using the same electrolyzers as used in HES. The CO2 would ideally be captured from CO2 emissions from other sources, such as natural gas power plants. While capturing CO2 directly from the air is feasible, it adds significant cost due to the low concentration of CO2 in the air. The methanation process results in an additional 18 \u221225% energy loss [22], resulting in AC to AC ef\ufb01ciency of approximately 25% [78]. While methanation offers a potentially carbon neutral solution to energy, it is unlikely to be adopted in the near term due to low natural gas prices. Estimated costs are approximately double that of HES [78]. Longer-term, methanation may become economically feasible if natural gas prices increase, renewable energy and hydrogen prices decrease, or carbon taxes are levied [79]. Meanwhile, methanation offers interesting challenges for research due to the more complex chemical reaction involved. Beyond methane, other even higher density and more desirable fuels may be generated, such as ethanol. Unlike methane which is a powerful greenhouse gas that is prone to leaks, ethanol is a liquid and easier to safely store. However, fuels such as ethanol have the same near-term adoption problems as methane while being even more complex to generate. Returning to our earlier discussion on the current use of expensive, rare metals like platinum as catalysts, we next discuss the design of electrocatalysts. 3. Design of Electrocatalysts Inexpensive and effective electrocatalysts are essential for the wide-scale adoption of renewable energy storage technologies such as HES and methane synthesis. In this section, we dive into what an electrocatalyst is and their desired properties. This section is meant as an introduction to electrocatalysis for the non-expert that may want to experiment with building machine learning models to predict various molecular proprieties. For those interested in a more detailed exposition, please refer to [80]. 3.1. Electrocatalyst Properties An electrocatalyst is a catalyst used to increase the reaction rate and ef\ufb01ciency of an electrochemical reaction (a chemical reaction that generates electricity). As is true for all catalysts, an electrocatalyst is not consumed during the reaction. In the design of an catalyst, we must consider several properties to determine whether it will be useful in a practical scenario. These include: \u2022 Ef\ufb01ciency What fraction of energy is lost when driving the reaction? \u2022 Reaction Rate How fast does the reaction occur? Typically measured in moles per second. \u2022 Stability How durable is the catalyst, i.e., how long will it last? \u2022 Selectivity Does it generate only the desired products, or other byproducts? \u2022 Cost How expensive is it to manufacture the catalyst? In this paper, we will primarily discuss the ef\ufb01ciency and reaction rate of an catalyst. The intent of focusing on these characteristics is to screen cheaper catalysts by their ef\ufb01ciency and reaction rates. However, the other factors are still very important. The stability of an catalyst determines whether it will degrade during prolonged use in a reaction. If an catalyst breaks down, its costs will signi\ufb01cantly increase due to the need to replace it. Selectivity refers to the percentage of the desired product produced by the reaction. In many reactions, unwanted byproducts may be produced. While we don\u2019t discuss selectivity speci\ufb01cally, similar tools to those used to determine a reaction\u2019s ef\ufb01ciency 7 \fFigure 3. Illustration of a PEM fuel cell. The anode (red) splits hydrogen into protons H+ and electrons e\u2013. The cathode (blue) combines the protons and electrons with oxygen to produce water. The membrane (yellow) only allows protons to pass through it, which forces the electrons to travel through the electric circuit, creating an electric potential. and reaction rates can also be used to calculate its selectivity. Finally, the manufacturing cost of the catalyst must be taken into account. This is driven by two primary factors: the cost of the materials used in its construction, and the complexity of the structure. Ideally, an catalyst should be constructed from abundantly available inexpensive elements, and not platinum, iridium, etc. 3.2. Fuel Cells To ground our discussion, we begin by describing fuel cells, which are a speci\ufb01c use case for catalysts. While we use this as an illustrative example, other electrochemical reactions behave in a similar manner, such as electrolyzers. A fuel cell is a conceptually simple device that converts hydrogen and oxygen into water and electricity. A fuel cell contains three main parts; the anode, cathode, and membrane (Figure 3). In the popular Proton-Exchange Membrane (PEM) fuel cell, the anode breaks the hydrogen into protons H+ and electrons e\u2013: H2 \u2212 \u2212 \u21922 H+ + 2 e\u2212 The cathode combines oxygen O2 with the protons and electrons from the anode reaction to create water H2O: 1 2 O2 + 2 H+ + 2 e\u2212\u2212 \u2212 \u2192H2O Taken together, the overall chemical reaction is: H2 + 1 2 O2 \u2212 \u2212 \u2192H2O How does a chemical reaction that combines hydrogen and oxygen to create water also generate electricity? The secret is in the membrane. The membrane only allows protons to pass through it, which forces the electrons to take an alternative path through the electrical circuit generating electricity! The amount of electricity generated is dependent on how fast the reactions on the anode and cathode occur. As we discuss next, the rate of these reactions is highly dependent on the type of catalysts used. 3.3. Electrocatalysis A catalyst is a substance used to increase the rate of a chemical reaction that is not consumed in the process. A chemical reaction pathway typically has several intermediate steps. The catalyst works by providing an alternative reaction pathway that reduces the amount of energy, voltage, or pressure needed for the reaction to occur. There are three main types of catalysts: homogeneous catalysts that are in the same phase as the reactant (typically gas-gas or liquid-liquid), heterogeneous catalyst that are in a different phase (typically gas-solid), and enzymes that are biological catalysts. We focus on heterogeneous catalysts, since they allow for the easy separation of the catalyst (solid) from the reactants (gas) in practical applications. An electrocatalyst is a catalyst used during a electrochemical reaction. In our fuel cell example, both the anode and cathode would use a catalyst to increase their reaction rates. To understand how this is done, let us consider the Oxygen Reduction Reaction (ORR) that occurs at the fuel cell\u2019s cathode, which involves the splitting of O2 to create water [81\u201383]: 1 2 O2 + 2 H+ + 2 e\u2212\u2212 \u2212 \u2192H2O (1) Most reactions proceed along a pathway containing a series of smaller steps. For any given reaction there may even be multiple possible pathways, e.g., Figure 18 for syngas. One potential reaction pathway (the dissociative pathway [81]) for (1) involves three intermediate steps1: 1 2 O2 + \u2217\u2212 \u2212 \u2192*O (2) *O + H+ + e\u2212\u2212 \u2212 \u2192*OH (3) *OH + H+ + e\u2212\u2212 \u2212 \u2192H2O + \u2217 (4) where * indicates a bonding site on the catalyst. In (2), oxygen gas O2 comes in contact with the catalyst\u2019s surface which splits apart the two oxygen atoms. While still on the surface of the catalyst, an oxygen atom O* combines with a proton H+ and electron e\u2013 to form OH in (3). Finally in 1The higher complexity associative pathway containing \ufb01ve steps is more likely to be important for platinum catalysts [83] 8 \fFigure 4. Illustration of the potential energy between two atoms as the distance d between them varies. When the atoms are far apart (right), the weaker van der Waals forces attract atoms towards each other. As the atoms move closer together (left), the stronger Coulombic forces repel the atoms. The minimum is the stable state where these two forces cancel out. The depth of the minimum is dependent on whether a bond is formed (chemisorption) or not (physisorption) by the atoms. (4), another proton and electron combine with the OH to form a water molecule. The water molecule has a weaker bond with the catalyst, allowing it to \ufb02oat away. Why is the catalyst needed? Without the presence of a catalyst, breaking the bond between the pair of oxygen atoms and keeping them separated so that they can bond with the protons and electrons would require a signi\ufb01cant amount of energy (heat). When it comes in contact with the catalyst, the bond between the oxygen atoms is weakened as they bond with the catalyst. This allows the rate of the reaction to increase with less energy. To understand this process in more detail, we need to de\ufb01ne a few terms. The amount of usable energy for a closed system with constant temperature and pressure is referred to as the Gibbs free energy, or Gibbs energy and is de\ufb01ned as: G = H \u2212TS (5) where H is the enthalpy (energy contained in the bonds between atoms) of the system, T is the temperature (measured in Kelvin) and S is the entropy. The entropy of a system increases when molecules break their bonds and decreases when they form new ones. The computation of H involves the potential energy between atoms, which is illustrated for two atoms as a function of distance in Figure 4. When the two atoms are far apart, they are weakly attracted to each other. As they move closer together this attraction is maximized resulting in a minimum potential energy. If the atoms move even closer together, they begin to repel and the poFigure 5. Illustration of a reaction\u2019s free and activation energy. The primary role of a cataylst is to minimize the activation energy of a reaction. The sign of the free energy indicates whether the reaction is spontaneous (negative) or not (positive). Plot values obtained from CatApp [84] for Pt(111) surface. tential energy increases. If a bond is formed between two atoms (chemisorption), their attraction is stronger and the minimum potential energy is lower than if a bond is not formed (physisorption). The kinetic energy measures the energy due to the motion of the system, i.e., how fast are the atoms moving or simply the heat of the system. An important property in determining whether a reaction is spontaneous (whether it will occur without energy input) is its change in Gibbs energy from the initial state to the \ufb01nal state. The change in Gibbs energy is referred to as the \u201cfree energy\u201d for a reaction, and is de\ufb01ned as: \u2206G = \u2206H \u2212T\u2206S (6) If the difference in Gibbs energy is negative the system is said to be exergonic, i.e., it will occur spontaneously (similar to a cart being pulled down a hill by gravity). Otherwise the reaction is endergonic with the Gibbs energy increasing. By its de\ufb01nition, an endergonic reaction requires energy to be added to the system for it to occur (the cart needs to be pushed up the hill). For scenarios in which the temperature is relatively low, the reaction\u2019s free energy will be dominated by the change in enthalpy (\u2206H term). A negative \u2206H means heat is released by the reaction (exothermic reaction), where as a reaction that absorbs heat has a positive \u2206H (endothermic reaction). At \ufb01rst glance, it may be tempting to assume that all exergonic or exothermic reactions will spontaneously occur at a high rate. After all, any system will want to reduce its potential energy much like a cart let go on a hill. However, this isn\u2019t always true. In many cases, a set of atoms will be bonded together, and be quite satis\ufb01ed in their current state, such as the O2 gas molecule in reaction (2). In terms of our analogy: The cart could have a wedge under one 9 \fFigure 6. Illustration of Gibbs free energy diagram for platinum (111) and nickel (111) catalysts for the dissociative pathway of reaction (1). Activation energies are approximated from equivalent thermal reactions. of its wheel to prevent it from rolling, even though it may \u201cwant to\u201d roll downhill. In order for the oxygen atoms to bond with the catalyst, they \ufb01rst need to break their oxygenoxygen bonds. Breaking this bond requires kinetic energy, and the amount of energy needed is referred to as the activation energy for the reaction, Figure 5. In our cart analogy: This is how much effort it takes to remove the wedge from under the cart\u2019s wheel. As an extreme example, the decay of diamonds to graphite is a exergonic reaction, but due to its very high activation energy this process takes millions of years to occur! We may think of each state in the reaction as a local minimum in the Gibbs energy with the activation energy being the height of the hills between the valleys. The primary goal of a catalyst is to reduce the activation energy, or the height of the hills between the valleys. As shown in Figure 5, the free energy between O2 and 2 O* in (2) is -1.5eV with a platinum catalyst (eV is a unit of energy measurement that is commonly used for chemical reactions). Even though the reaction is clearly exothermic, the activation energy of reaction (2) is still 0.62 eV to break the O2 bond. Therefore, energy (heat) needs to be added to the system in order for the reaction to occur, even though the \ufb01nal potential energy is lower than the initial energy. Note that in a real system, the activation energy does not act as an on/off switch. Instead, the rate of reaction gradually increases as the activation energy is approached. The cathode\u2019s oxygen reduction reaction (1) contains three intermediate steps. To get a complete picture of the overall reaction, we can string together Gibbs free energy plots for the three intermediate reactions to obtain Figure 6 (two steps are repeated twice to create two water molecules). The \ufb01gure shows the reaction in the presence of two different catalysts, platinum and nickel. The overall reaction is exothermic using either catalyst. However, platinum has a much faster rate of reaction. Why is this? Note that the reaction\u2019s overall free energy is independent Figure 7. Plot of reaction activities or rates (vertical axis) vs. the oxygen adsorption energy \u2206EO for different metals. Lines indicate the \u201cvolcano plot\u201d using Sabatier analysis to predict the reaction rates. Note that expensive metals such as platinum (Pt) and palladium (Pd) are near the ideal peak, where as the more common metals iron (Fe) and nickel (Ni) have signi\ufb01cantly lower rates. Reprinted with permission from [81]. Copyright 2004 American Chemical Society. of the catalyst used. The choice of catalyst only impacts the free energy and activation energies of the intermediate reactions. Using platinum, most of the intermediate reactions have lower activation energies. Using nickel, the oxygen atoms bind very strongly with the catalyst, resulting in a signi\ufb01cantly lower free energy. In fact, no activation energy is needed to split O2 in the \ufb01rst reaction. However due to the strong oxygen binding, the second two reactions have high activation energies, which slows the overall reaction rate. Figure 7 shows the reaction rates of various metals as a function of the binding energy between oxygen and the catalyst. Note the reaction rate is like Goldilocks. If the binding energy is too low (strong bond) the oxygen has a hard time reacting and separating from the catalyst (nickel Ni, iron Fe). On the other hand, if the binding energy is too high (weak bond), the catalyst (silver Ag or gold Au) has a hard time attracting and holding onto oxygen atoms so that the desired reactions can take place. Platinum conveniently sits near the peak, with the \u201cjust right\u201d bonding energy. In the discussion above, we\u2019ve considered the fuel cell system as if it was short-circuited, i.e., no resistance was placed on the electronic current. In practice, the circuit will have resistance, e.g., to turn on a light bulb. What effect does this have on the Gibbs free energy? Resistance in the circuit between the electrodes creates potential energy that can also be measured in volts. To \ufb01nd the resulting free energy given different electrode potentials, we can just subtract the potential energy per electron for each step in our reactions. As shown in Figure 8, the equilibrium potential 10 \fFigure 8. Illustration of Gibbs free energies for the dissociative pathway of the ORR reaction (1) with adjustments based on different electric potentials U. These include zero potential (U = 0V ), the highest potential where all steps are exothermic (U = 0.78V ), and the equilibrium potential (U = 1.23V ). Adapted with permission from [81]. Copyright 2004 American Chemical Society. (when the overall free energy is zero) is 1.23 eV, and the highest potential in which all reactions are exothermic is 0.78 eV. Thus the free energy of an electrochemical reaction not only helps determine its spontaneity, but also the electric potential that may be generated. 3.3.1 Reaction rate The rate of a chemical reaction typically follows the form: r = k[A][B] (7) where r is the rate of the reaction, k is the \u201crate constant\u201d, [A] is the concentration of one reactant, and [B] is the concentration of the other reactant. Thus the reaction rate is dependent on two types of quantities: the concentrations of the reactants (the inputs to the reaction) and the reaction\u2019s rate constant. Intuitively if the concentration of the reactants is higher, the reaction rate will increase. The reaction rate constant k, for which the reaction rate varies linearly, is computed using the Arrhenius equation that contains several terms: k = Ae\u2212Ea/RT (8) where the constant A encodes how frequently the reactant molecules collide and whether they have the correct orientation to react. The second term e\u2212Ea/RT measures the probability of a collision having enough energy (velocity of the molecules) for a reaction to occur. It is in this second term that the activation energy Ea is needed. Ea is divided by the temperature T and the gas constant R. Thus, the higher the activation energy, the higher the temperature needed to drive the reaction. If an catalyst lowers the activation energy, the reaction rate will increase for the same temperature, increasing the reaction rate. 3.3.2 Estimating reaction rates in practice There are many factors that impact the overall rate, such as temperature, pressure, coverage (percentage of binding sites occupied), surface area of the catalyst, and the rate constant. However, there are some methods for changing the rate constant that extend beyond the relatively simple Arrhenius equation, such as changing the electrolytes used in the electrochemical cell or using promoters [85,86] (additional non-catalytic substances used to increase the reaction rate). Given all these factors it can be very dif\ufb01cult to directly compute the free and activation energies needed to compute the rate constant and reaction rate. However, it has been shown experimentally that the variation in the reaction rate can be approximated using only a small number of factors. In most reactions there is a single reaction step that limits the rate of the reaction\u2014i.e., a bottleneck reaction step. Thus only this intermediate reaction needs to be considered for determining the overall reaction rate. An intermediate reaction can be described using the change in Gibbs free energy (free energy) and the activation energy. The Gibbs free energies are computed in part (during the computation of \u2206H) based on the adsorption energy of the adsorbate on the catalyst, i.e., the change in energy of the system before and after the adsorbate bonds with the catalyst. Perhaps surprisingly, it has been shown that adsorption energies and activation energies are related by a linear function across a wide range of potential catalysts [87,88]. For instance, consider reaction (2) in our oxygen reduction reaction. If a catalyst bonds more strongly to the individual oxygen atoms, it is also more likely to be able to pull apart the oxygen atoms in O2. The linear relation between energies results in the the ability to reasonably approximate the overall reaction rate using a linear function to predict activation energies from the adsorption energies of the intermediate adsorbates. Depending on the catalyst, the rate limiting reaction will change. Using a technique called Sabatier Analysis [89], the various linear relations for each rate limiting reaction can be combined by taking their minimum. The end result are plots, sometimes referred to as \u201cvolcano plots\u201d, like those shown in Figures 7 and 9 from [81] and [83] respectively. In Figure 7, the two lines signify two different rate limiting reactions: combining O and H+ (left) and separating O2 (right). The point indicates the transition between the two, and the optimal adsorption energy (\u223c1.8eV) in reaction (2). A similar plot is shown in 9 for a different reaction pathway (associative vs. dissociative) for reaction (1), but in two dimensions using the energies for two interme11 \fFigure 9. Plot showing reaction activity (red higher, blue lower) for the ORR reaction (1) using the associative pathway. Plotted as a function of the binding free energy of *OH and *OOH. Note that most known catalysts lie along a line due to the high correlation in binding or adsorption energies between OH and OOH. Finding a catalyst closer to the red peak remains an open research problem. Reprinted with permission from [83]. Copyright 2018 American Chemical Society. diate adsorbates (OH and OOH). Using these approximations, the reaction rates of new catalysts can be optimized by searching for the appropriate adsorption energies per adsorbate. It is interesting to note that for a wide range of different catalysts, they all lie roughly on a line in Figure 9. Why is this? The reason is the adsorption energies of OOH (vertical axis) and OH (horizontal axis) to a catalyst are highly correlated, since OH and OOH bond through their O atoms and not their H atoms [83]. Essentially, the catalysts have a very dif\ufb01cult time distinguishing between these two molecules. This places limits on the types of catalyst that are likely to be found, since it is highly unlikely that a new catalyst will deviate from this trend. In order to differentiate between OH and OOH, research into more complex structures is needed [83]. Up until this point, we\u2019ve assumed that the Gibbs, adsorption, and activation energies, which are important for determining a reaction\u2019s spontaneity and rate, are known. How are these values computed in practice? Determining these values experimentally can be dif\ufb01cult, since most chemical reactions are comprised of multiple potentially unknown steps. Instead, these values are computed through numerical approximation from \ufb01rst principles. A popular technique for this is called Density Functional Theory (DFT) [25,26]. As we explore next, DFT can produce accurate results for problems like those discussed above, but the computational cost of DFT is high. Which raises the central question of this paper: Can we train deep learning models to ef\ufb01ciently approximate DFT and enable the large-scale exploration of new catalysts? 3.4. Density Functional Theory Density Functional Theory (DFT) is a popular quantum chemistry method for computing the electronic properties of an atomic structure including its energy and per-atom forces [26]. Here, a structure refers to a set of atoms and their 3D positions, which are input and assumed to be \ufb01xed. In our scenario, the structure includes the atoms of the adsorbate and the catalyst. Adsorbates are molecules that participate in the chemical reaction of interest, e.g., O, HO and H2O in the oxygen reduction reaction (1). Using the energies computed at multiple structures, we can determine the reaction\u2019s free, adsorption, and activation energies. For instance, we can determine the adsorption energy by taking the energy difference between a structure where the adsorbate is in contact with the catalyst and a structure where the adsorbate is far away from the catalyst. A reaction\u2019s free energy can be computed by taking the difference in energy between the \ufb01nal molecule and the initial molecule along with some additional corrections. Finally, if the transition state (the state corresponding to the peak in energy between two reaction steps) of a system is known, the activation energy can be computed by \ufb01nding the energy difference between the reaction\u2019s initial molecule structure and the transition structure. DFT takes as input the positions of the structure\u2019s atomic nuclei and computes its energy using the electron density [25]. The energy of a structure, which is known as its electronic energy, refers to the potential energy, kinetic energy and other energies related to the interactions of the electrons [90]. While we assume in our experiments that the atom nuclei are stationary (no kinetic energy), the electrons move and have both kinetic and potential energy. The energy is found using an approach that iteratively estimates the electron density of the structure until convergence. The electron density is represented using a set of spatially dependent functions, which may be either spatially localized for isolated molecules, or periodic functions for extended surfaces. This process is made possible using the KohnSham equations [28]. Note the term density functional theory comes from the use of functionals of the electron density, i.e., functions of the electron densities that are represented as functions. 3.4.1 Relaxations When estimating the energy of an adsorbate interacting with a catalyst, the positions of the atoms\u2019 nuclei are typically 12 \fonly roughly known. If we used DFT to calculate energies using these rough estimates, the predicted energies could be signi\ufb01cantly overestimated, since real-world systems seek the state of lowest energy. Instead, the energy should be computed using the \u201crelaxed\u201d structure of the system. That is, the structure in which the atomic positions have been optimized to \ufb01nd the con\ufb01guration with the lowest energy. The relaxation process is done iteratively using second order optimization methods. The initial atom positions are determined through heuristics based on the expected bonds. Assuming the nuclei positions are \ufb01xed, DFT is used to compute the electron densities and energy. Using derivatives of the structure\u2019s energy, the forces on the nuclei are found and their positions updated. This process is repeated until the nuclei positions converge and the minimum energy is found. Typical relaxations require 50\u2013400 iterations to converge. By making thermodynamic corrections (such as changes, which we don\u2019t describe, from the TS term in Eqn.(5)) to the resulting energy, the structure\u2019s Gibbs free energy can also be computed. For a more detailed explanation for how the energies and electron densities are computed, please refer to [25]. Unfortunately, there exist limitations to the use of DFT. Most importantly, it is computationally expensive and scales O(n3) with respect to the number of electrons in the structure. In practice, this results in DFT relaxations taking hours\u2013weeks per simulation using O(10\u2013100) core CPUs on structures containing O(10\u2013100) atoms. As a result, complete exploration of catalysts using DFT is infeasible. DFT relaxations also fail more often when the structures become large (number of atoms) and complex. It is not uncommon for relaxations to not converge, especially if the initial structure is not set properly. In practice, failure rates vary from 20% to 50% for the types of structures we consider in this paper. DFT would fail completely for large and more complex catalysts such as enzymes. The hope is that machine learned approximations to DFT can be both faster and potentially more stable enabling the exploration of more and larger structures. 3.4.2 Initial Structure Before a relaxation is run, an initial structure needs to be de\ufb01ned. In this paper, we consider structures containing three attributes: a catalyst, an adsorbate, and a binding site. Other attributes, such as temperature and pressure, will be held constant. The catalyst is assumed to have a crystalline structure, i.e., a structure containing a repeating pattern called a \u201cslab\u201d. To make the DFT computations feasible, the catalyst is modeled using a single \u201cslab\u201d that is tiled to create the catalyst\u2019s 2D surface. A typical slab is 6 atoms deep and 4-6 atoms across. The adsorbate is selected from the set of molecules in the reactions of interest and their intermediFigure 10. Illustration of three binding sites for OH on a metallic surface. The oxygen atom is shown bonding with one (top), two (middle) and three (bottom) metal atoms. ates. The binding site refers to selecting the atoms that bond between the adsorbate and the catalyst. To improve the likelihood of convergence for DFT we want to select initial positions that are as close as possible to the potential binding sites. A 2D example of bonding sites is shown in Figure 10 between one, two and three atoms on the catalyst\u2019s surface and an adsorbate with two atoms. If the adsorbate has multiple atoms, the atom of the adsorbate that bonds must also be selected. We discuss the speci\ufb01c set of catalysts, adsorbates and binding sites used for relaxations in Section 4.1. 3.5. Classes of Materials for Electrocatalysts There exists an enormous space of potential materials for electrocatalysts [91]. It would be overwhelming to catalog all the materials in one review. Instead, we provide some perspective on the space of materials to be considered. It is important to note that any materials dataset will necessarily contain limitations and bias. Of the different types of materials, metal surfaces have received the greatest attention due to their reaction attributes, durability, and manufacture-ability. However, other materials are also possible, including (among many many others): surfaces with non-metallic elements, such as oxides (materials containing oxygen) [92] or nitrides (materials containing nitrogen) [93]; metal-doped 2D materials such as graphene [94] or boronitrides [95]; Perovskites (metallic oxide minerals) [96] that are more earth abundant and potentially less expensive; and other metal-free catalysts [97]. While these other material types show promise, we focus on metal surfaces in this paper given their potential for nearer term practical use and the clear applicability of DFT relaxations to their study. However, we hope the 13 \fFigure 11. Illustration of different metallic catalyst bulk structures: pure metals, multi-metallic (1-3 metals), intermetallic (ordered multimetallic), overlayer (thin layers of metal applied to the surface), and high-entropy (more than 3 metals). Figure 12. Illustration of different surface structures (facets) using three different cutting orientations (100), (110) and (111). The three values (0 or 1) indicate the Miller indices for the cutting plane. techniques developed for metal surfaces will also be useful for other material types. Figure 11 illustrates several different types of metal structures. These include pure metals, multi-metallics (multiple unordered metal alloys), intermetallics (ordered multimetallics), single-atom catalysts in host materials [98], and overlayers (surface layers of one or more metals). Alloys that contain more than three different metals are referred to as \u201chigh-entropy alloys\u201d. Historically, the most thoroughly studied are the simplest structures containing pure metals. In contrast, the space of multi-metallics, intermetallics and overlays is signi\ufb01cantly less well explored given the vastness of the space. For instance, choosing up to three metals from the roughly 40 possible, results in almost 10,000 combinations, and this does not even include all the possible combinations of different ratios and patterns! Given the same underlying bulk structure, there are multiple different surfaces that may be enumerated based on which orientation you slice the material. As shown in Figure 12, the orientation of a crystalline surface or facet is typically indicated using coordinates on a unit cube. For instance, a diagonal facet is speci\ufb01ed as (111) and results in the surface having a hexagonal pattern, whereas a (100) surface that cuts along a single dimension has a rectangular grid. In addition to \ufb02at surface types, surfaces may also contain steps, defects or other complications [99]. All of these variables impact the number of different bonding sites available and how many atoms may bond to the catalyst at one time (Figure 10), which in turn impacts the overall reaction rate. 3.6. Other considerations While we covered many of the basics for modeling catalysts, numerous improvements could be made to further improve the realism of DFT calculations. While we do not cover these items in detail, we want to make the reader aware of these topics. For those interested in a more indepth discussion, we recommend [80]. Coverage: The coverage of a catalyst refers to the percentage of surface sites that are occupied by adsorbates. As the coverage increases, the adsorbates move closer together and impact the system\u2019s energy. For instance, the adsorption energy of the \ufb01rst oxygen placed on a Pt(111) surface is smaller than the adsorption energy of the 2nd, third, or fourth oxygen placed on the surface [100]. Attractive and repulsive interactions between the adsorbates can also create ordered patterns of adsorbates on the surface [101]. Poisoning: Other species in the reacting mixture (even at trace levels) may poison the catalyst surface through a competitive adsorption process. Other adsorbates that do not participate directly in the reaction may adsorb strongly such that the coverage of the desired molecule and accompanying reaction rates are signi\ufb01cantly reduced. Worse, they may bind so strongly that they never leave, thereby rendering the catalyst inert. These molecules occupy the potential bonding sites of the desired adsorbates, and essentially \u201cpoison\u201d the catalyst. A common example is the poisoning of noble metal catalysts by lead, or poisoning of platinum catalysts by trace carbon monoxide. Electrolytes: Most electrochemical reactions occur in the presence of an aqueous or solid electrolyte to conduct charged particles to the surface, e.g. they are submerged in water/electrolyte solution to increase the conductivity of the system and therefore improve reactivity. The precise solvent (water or otherwise) can affect adsorption energies, 14 \fmost notably for adsorbates that can form hydrogen bonds with the water layer. Further, electrolytes such as KOH or NaCl can be added to the water to further improve reaction performance. They do this by interacting with the adsorbates and catalyst to stabilize the transition states of particular reaction intermediates, thereby reducing the activation energy and accelerating the reaction rate. Sometimes these effects are modeled by making simple corrections to the energies found without electrolytes, but ideally the electrolytes would be modeled directly. There are major scienti\ufb01c challenges remaining in our understanding of these processes. Catalyst Dynamics: In the real world, the atoms in the system are constantly moving. The more heat that is applied to a system, the faster the atoms move. Molecular dynamics models the movement of these atoms by integrating the standard equations of motion (F = ma). Since the atoms may be fast moving and visiting a wider range of potential surfaces and sites, modeling these systems can be more challenging than the relaxations (with zero kinetic energy) we describe in this paper. The real catalyst may re-arrange under reaction conditions or form an ensemble of different states that all contribute to the \ufb01nal activity. Modeling these processes requires molecular dynamics simulations, but accessing experimentally-relevant timescales (nanoseconds) requires approximately 106 force calls at the atomic (femtosecond) timescale and are thus vastly more expensive than standard DFT relaxations. Approaches to exploring this problem use both standard DFT [102] and learned models [103\u2013106]. 4. ML for Electrocatalysis In this section, we begin by providing a short overview of the problem, followed by the dataset, tasks, and approaches for addressing it. For those interested in a more detailed discussion of the problem, please refer to Sections 2 and 3. As intermittent renewable energy sources (solar and wind) become more prevalent, the storage of their energy from times of peak generation to peak demand will increase in importance. The use of renewable energy to generate other fuels that are easily stored provides a potential solution to this problem that can scale to nation-sized grids. For instance, renewable energy can be used to split water into hydrogen and oxygen gas. The stored hydrogen is then used to generate electricity at a later time using fuel cells. Currently, a major limiting factor is \ufb01nding ef\ufb01cient, durable and cheap catalysts for driving the electrochemical reactions involved in this process. A catalyst increases the rate of a chemical reaction while not being consumed in the process. During a chemical reaction the molecules involved in the reaction, called adsorbates, interact with the catalyst\u2019s surface. The change in energy when the adsorbate comes in contact with the catalyst, called the adsorption energy, is the key factor is determining the overall reaction\u2019s rate (Section 3.3.2). The adsorption energy must be neither too high (the catalyst holds on to the adsorbates too strongly) nor too low (the adsorbates \ufb02oat away from the catalyst). If the reaction\u2019s rate is increased, the overall ef\ufb01ciency of the system increases and costs decrease. The challenge is \ufb01nding catalysts that not only increase the reaction\u2019s rate, but are also cheap to produce, e.g., are not made of expensive metals such as platinum. The adsorption energy is computed by \ufb01rst \ufb01nding the positions of the atoms when the adsorbate comes in contact with the catalyst\u2019s surface. In the real-world, the atoms seek a con\ufb01guration that reduces the overall energy of the system. The set of 3D atom positions at a local energy minimum is referred to as a \u201crelaxed structure\u201d and the corresponding system energy as the \u201crelaxed energy\u201d. How is a relaxed structure estimated computationally? An iterative process is used that \ufb01rst roughly places the adsorbate near the catalyst to create an initial structure. For any given pair of adsorbate and catalyst, numerous initial structures may exist based on the binding site chosen, Figure 10. The selection of an adsorbate\u2019s binding site and its initial positioning relative to it is heuristically determined. Next, the per-atom forces are computed, from which atom positions are updated using BFGS [107] or conjugate gradient techniques. The forces are re-estimated and atom positions updated until the forces converge to zero at which point a local energy minimum has been found. Typically, hundreds of iterations are necessary to converge to a relaxed structure. While this process seems simple, the computation of the atom forces is highly complex and computationally very expensive. Atom forces and a structure\u2019s energy are computed using Density Functional Theory (DFT) [25, 26], which is computationally expensive and can take hours to calculate for a single structure. How can we scale the simulation of new catalysts from hundreds or thousands to billions? A promising alternative to expensive DFT calculations is to train ef\ufb01cient machine learning models to approximate DFT. The model takes as input the set of atoms, including their atomic numbers and 3D positions, and computes the per-atom forces. Similar to DFT, the force estimates may be used to \ufb01nd relaxed structures. To eliminate the need for DFT entirely, models may be trained to also estimate a structure\u2019s energy. Machine learning models that employ deep learning require large datasets for training. We begin by describing the Open Catalyst Dataset (OC20) created for this purpose; followed by deep learning approaches that have been proposed for estimating DFT and metrics for evaluating the models. We strive to release all data and code in accordance with FAIR (Findable, Accessible, Interoperable, Reusable) principles [108]. 15 \fFigure 13. Examples of cataylsts (larger atoms) and adsorbates (smaller atoms) in the OC20 dataset. Spheres represent the atoms with the color and size indicating the atomic number (element type) and the atomic radii, respectively. For example, the small white, red, grey, and blue spheres are hydrogen, oxygen, carbon and nitrogen, and the larger spheres are mostly metals. Figure 14. Example initial and relaxed structures from the OC20 dataset. Note that the changes from initial to relaxed structures can appear to be quite small, especially for adsorbates containing only a few atoms. However, the changes in energy can be signi\ufb01cant. 4.1. Open Catalyst Dataset (OC20) An ideal dataset would enable the exploration of new techniques for estimating DFT calculations while also helping the catalysis community explore new materials and gain a deeper understanding of the physics underlying catalysts\u2014overall accelerating progress towards discovering better alternatives to commonly used catalysts. This requires both a large (for learning) and diverse (for generalization and exploring new materials) dataset. For the dataset, priority is given to catalysts with relatively near-term practical potential, and adsorbates relevant to the reactions used in renewable energy storage. Speci\ufb01cally, we explore metallic surfaces, which are currently the most popular catalysts in use due to their durability, ease of manufacture, and ef\ufb01ciency. Our OC20 dataset [42] contains over a million relaxations of randomly selected catalysts and adsorbates from a set of potential candidates. On average each relaxation contains approximately 200 structures in its trajectory from the initial structure to the relaxed structure, which results in over 130+ million structures for training with the rest used for validation and testing. Example adsorbate and catalyst pairs are shown in Figure13, and example initial and relaxed structures in Figure 14. All calculations are performed using VASP (Vienna Ab initio Simulation Package) [109\u2013112], which is a popular package for performing the types of DFT calculations used in this paper. 16 \fFigure 15. Illustration of the relationship between tasks in computing a relaxed structure or relaxed energy from an initial structure. (a) the traditional approach uses DFT along with a linear optimizer, such as BFGS or conjugate gradient, to iteratively update the atom positions until the relaxed structure and energy are found. (b) using ML models trained to predict the energy and forces of a structure, S2EF can be used as a direct replacement for DFT. (c) the relaxed structure could potentially be directly regressed from the initial structure and S2EF used to \ufb01nd the energy. Finally, if only the relaxed energy is desired, (d) could compute the relaxed energy directly from the initial state. A detailed description of the OC20 dataset and its various design choices are provided in [42]. In this paper, we cover the necessary details to build and evaluate models with the dataset. We begin by describing the various tasks, followed by the inputs and data splits. 4.1.1 Tasks and Evaluation Metrics We identify three tasks based on their practical applicability to cataylst design. Since structure relaxations are the primary tool for determining a catalyst\u2019s utility, we focus on modeling this process and its outputs. As shown in Table 4, task outputs include per atom forces and positions, and the energy of a structure. The relationship of each of our tasks is shown in Figure 15. Per task evaluations may be performed on the validation set using the supplied code and ground truth data. Test set evaluations may be performed by uploading results to the Open Catalyst Project leaderboard. Our three tasks are as follows: S2EF Structure to Energy and Forces task is given a structure as input, predict the per-atom forces and structure\u2019s energy as output. This task mirrors the computationally expensive DFT calculation. An ef\ufb01cient approximation to DFT would be bene\ufb01cial to numerous tasks, including relaxations. Evaluation is performed using the Mean Average Error (MAE) between the computed forces and energies and those computed using DFT. In addition, a third evaluation metric determines the percentage of predictions with Energy and Forces within a Threshold (EFwT) of the ground truth. By using tight thresholds, the EFwT metric provides an estimate for how often the predictions would be practically useful. IS2RS Initial Structure to Relaxed Structure task estimates the relaxed structure given the initial structure as input. This task may be performed iteratively using S2EF calculations, similar to how traditional relaxations are computed, Figure 15(b). We refer to these calculations as ML relaxations, since ML models are used to compute the relaxations. An alternative is to compute the \ufb01nal 3D positions of the atoms directly from their initial structure, Figure 15(c). Evaluations are performed using two techniques. The \ufb01rst measures the 3D positional differences between the ML computed relaxed structure and the structure computed using a traditional DFT relaxation. Since small changes in position can lead to signi\ufb01cant changes in forces and energies, distance metrics by themselves are insuf\ufb01cient for determining whether a true relaxed structure is found. The second metric uses a single DFT calculation to determine the forces on the proposed relaxed structure. If a true local minimum was found, the forces as computed by DFT should be close to zero. The \ufb01nal metrics compute the percentage of ML relaxed structures with maximum force magnitudes below a threshold. Since DFT calculations are computationally expensive, IS2RS force metrics reported on the Open Catalyst Project test server are only evaluated on a subset of the test splits. The 3D distance metric is computed using the entire test set. IS2RE Initial Structure to Relaxed Energy task computes the relaxed structure\u2019s energy from the initial structure. The relaxed energy is used to compute the adsorption energy, i.e., the change in energy when the adsorbate comes in contact with the catalyst. As discussed in Section 3.3.2, the adsorption energy is informative in predicting the activation energies and ultimately the rate of the reaction in the presence of the catalyst. While we focus on the system\u2019s energy, other system properties may also be estimated. For instance, the popular QM9 dataset [38] containing small organic molecules, estimates 13 energetic, electronic, and thermodynamic properties. Similar to IS2RS, IS2RE may be accomplished using two approaches: The \ufb01rst iteratively estimates the relaxed structure and its corresponding energy using S2EF, Figure 15(b). The second, directly computes the relaxed energy from the initial structure, Figure 15(d). Using the second 17 \fFigure 16. 2D Illustration of a slab that represents a catalyst\u2019s surface and an adsorbate. The slab is tiled in the X and Y directions to create the surface (neighboring cells shown as atoms with dashed outlines). Only the cells to the left ([\u22121, 0, 0]) and right ([1, 0, 0]) are shown. The adsorbate is also assumed to be tiled with the slab (white, red, and grey atoms). Only the top 2 layers of the slab are allowed to move during a relaxation (dark blue), and the others are \ufb01xed (light blue). Neighboring atoms (black arrows) can be from the same cell or neighboring cells (t and s\u2032). All atoms within a radius (dotted circle) are assumed to be neighbors. approach, IS2RE does not require the estimation of the relaxed structure. Evaluation is performed using MAE between the estimated relaxed energies and those found using DFT relaxations, as well as, the percentage of energy estimates within a tight threshold of the ground truth. 4.1.2 Inputs The inputs to the OC20 tasks include the atoms, catalyst and adsorbate information. We discuss each in turn. Atoms A structure contains catalyst and adsorbate atom information at a speci\ufb01c instance in time. This includes the atoms\u2019 3D positions, along with various per-atom and perpair (of atoms) proprieties, as shown in Table 4. Per-atom information includes the atom\u2019s group and period number, electronegativity, covalent radius, etc. The per-pair properties include the distance between the atoms and their predicted bond type. Catalyst Along with the atom information described above, additional information is provided for the catalyst. Each catalyst contains 1 to 3 types of elements, with 5%, 65%, and 30% of the OC20 catalysts being unary, binary, and ternary materials respectively. Stable catalyst materials were randomly selected from those contained in the Materials Project [39]. A catalyst is modeled as an in\ufb01nite 3D surface with a repeating 2D crystalline structure in the x and y dimensions. The adsorbate sits on top of the catalyst in the z dimension with the catalyst extending in\ufb01nitely below the adsorbate. As input, only a single crystalline structure, known as a slab, is given that can be replicated or tiled in cells to create the full 3D surface. A 2D illustration of this process is shown in Figure 16. When computing atom interactions, the set of neighboring atoms needs to be determined. Since the slab is repeated in each cell, the atoms on the far right of a slab (atom t in Fig. 16) may be neighbors to those on the far right of the next cell over (atom s\u2032 and s in Fig. 16), and similarly for all other x and y directions. The atom positions can be transformed from one cell to the next using a 3 \u00d7 3 matrix M: x\u2217= Mc + x, (9) where c is the 3D cell indicator vector, e.g., [1, 0, 0]T or [\u22121, 1, 0]T, x is the atom\u2019s position in the slab, and x\u2217is the position of the atom in the indicated cell. Instead of tiling the slab in the z direction, it is assumed that only atoms near the surface of the catalyst will move. Each atom in the structure is given a variable that indicates whether it is assumed to be \ufb01xed or is free to move. All atoms in the adsorbate are free, and typically the top two layers of atoms in the catalyst are free and the rest \ufb01xed (dark and light blue atoms in Fig. 16). Note that DFT computes the forces assuming that the slab is not tiled in the z direction, so the forces of the bottom atom layers in the catalyst may be very large as they try to expand. Only free atoms are used when evaluating computed positions and forces. The forces of the \ufb01xed atoms may be used for training if desired. 18 \fReaction Name Purpose 2 H+ + 2 e\u2013 \u2212 \u2212 \u2192H2 Hydrogen Evolution Reaction (HER) Conversion of renewable energy to H2/O2 2 H2O \u2212 \u2212 \u2192O2 + 4 H+ + 4 e\u2013 Oxygen Evolution Reaction (OER) Conversion of renewable energy to H2/O2 H2 \u2212 \u2212 \u21922 H+ + 2 e\u2013 Hydrogen Oxidation Reaction (HOR) Conversion of hydrogen to electricity 4 e\u2013 + O2 + 4 H+ \u2212 \u2212 \u21922 H2O Oxygen Reduction Reaction (ORR) Conversion of hydrogen to electricity CO2 + 4 H2 + \u2212 \u2212 \u2192CH4 + 2 H2O Methanation Production of methane (main component of natural gas) and CO2 reduction N2 + 3 H2 \u2212 \u2212 \u21922 NH3 Haber-Bosch Production of ammonia for fertilizer N2 + 6 H+ + 6 e\u2013 \u2212 \u2212 \u21922 NH3 Nitrogen Reduction Reaction (NRR) Electrochemical production of ammonia for fertilizer from renewable energy 2 NH3 \u2212 \u2212 \u2192N2 + 3 H2 Ammonia Decomposition Ammonia as a hydrogen carrier CO2 + ne\u2013 + nH+ \u2212 \u2212 \u2192products CO2 Reduction Reaction (CO2RR) Renewable chemical building blocks from CO2 CO + H2 \u2212 \u2212 \u2192products Fischer-Tropsch Production hydrocarbons from renewable H2/CO CO + H2O \u2212 \u2212 \u2192CO2 + H2 Water Gas Shift (WGS) H2 production from CO2 C2H6 \u2212 \u2212 \u2192C2H4 + H2 Non-Oxidative Ethane Dehydrogenation Upgrading of waste ethane Table 3. Subset of the many reactions that are relevant to the OC20 dataset. Figure 17. The adsorbates used in the OC20 dataset [42] contain oxygen (red), hydrogen (white), one carbon (grey), two carbon (gray) and nitrogen (blue) molecules useful for renewable energy applications. Adsorbate The adsorbates are randomly chosen from the 82 molecules shown in Figure 17. The set of adsorbates was selected based on the molecules, called intermediates, that are involved in the reactions of interest, such as those in Table 3 for renewable energy storage and sustainable agriculture. The adsorbates are clustered into four groups: small molecules containing a few atoms, small molecules with at least one carbon (C1), larger molecules with at least two carbons (C2), and molecules containing nitrogen. The study of these adsorbates has signi\ufb01cant practical value. For instance, analysis of C1/C2 intermediates will allow new insights into a catalyst\u2019s selectivity (do they produce any undesired byproducts) for CO2 conversion to fuels. Note there is a larger number of high-complexity (i.e., larger) adsorbates. By sampling from the adsorbates uniformly, we implicitly perform more calculations on the larger adsorbates for which there is less prior data. This helps the catalysis research \ufb01eld by generating more previously unexplored data, and provides a more diverse training dataset for the ML community. 4.1.3 Train, Validation, and Test Splits The OC20 dataset contains over 1.3 million relaxations that required over 70 million hours of compute [42]. The data is split into train, validation and test sets. Each relaxation contains the 3D atom positions, atom forces, and energies for the initial, intermediate, and \ufb01nal relaxed structures. The training dataset contains over 650,000 relaxations with over 130 million structures. A model\u2019s ability to generalize may be evaluated using 19 \fInputs Per atom atom positions 3D (x, y, z) positions group number column in the periodic table (1-18) period number row in the periodic table (1-9) electronegativity tendency to attract electrons (0.5-4.0) covalent radius [113] size of an atom part of one covalent bond (25-250 pm) valence electrons # of outer shell electrons (1-12) \ufb01rst ionization energy energy needed to remove outermost electron (1.3-3.3 eV) electron af\ufb01nity energy released when electron attaches to a neutral atom to form a negative ion (\u22123-3.7 eV) block s, p, d, f atomic volume 1.5-4.3 cm3/mol Per pair bond type single, double, etc. (one-hot) distance 3D (x, y, z) delta positions Outputs Per atom atom positions 3D (x, y, z) positions atom forces 3D (fx, fy, fz) values Per system energy E energy Table 4. Summary of model inputs and outputs. different validation and test splits. Generalization to new catalysts and adsorbates is one of high practical impact, since this would enable the application of models to new catalyst materials and chemical reactions. We propose creating validation and test splits speci\ufb01cally for these scenarios. We create four subsplits for validation and testing based on whether the catalyst material or adsorbate is present in the training dataset. A catalyst material is said to be in the training dataset if a catalyst containing the same elements is contained in the training dataset, even if the elements are in different arrangements or proportions. The four subsplits are labeled In Domain (same distribution as training dataset), Out of Domain Adsorbate (OOD Adsorbate), OOD Catalyst and ODD Both (neither the adsorbate or catalyst material is in training). Each subsplit contains approximately 70,000 to 90,000 relaxations. For the S2EF task we randomly select a one million structure subset from the relaxations in each subsplit. A random subset of 25,000 relaxations per split is sampled for the IS2RS and IS2RE tasks. The subsplits of our validation set are exclusive to the subsplits in the test set, e.g., the adsorbates in the validation OOD Adsorbate subsplit are unique from the adsorbates in the test OOD Adsorbate subsplit and similarly for OOD Catalyst and OOD Both. The train and validation datasets are provided to the public. The inputs to the test dataset are also provided. However, the test outputs are not made public to ensure over\ufb01tting to the test set does not occur. To evaluate on the test dataset, results can be uploaded to the Open Catalyst Project leaderboard. 4.2. Models Our goal is to help enable the community to develop models that can predict a structure\u2019s energy and per atom forces (S2EF), relaxed structure (IS2RS) and relaxed energy (IS2RE). High practical impact could be realized if ef\ufb01cient solutions to these problems can be found. Currently, several approaches have been proposed for addressing our three tasks. The results of several state-of-the-art approaches on OC20 can be found in [42] or on the public leaderboard. Numerous deep learning approaches have been proposed for predicting the properties of a structure [34\u201337, 114\u2013 118]. One challenging problem is how to model the interactions of the atoms. Since the atoms do not lie on a regular grid, such as pixels in an image, standard approaches such a convolutional neural networks [119] cannot be applied. Instead, Graph Neural Networks (GNNs) [120\u2013122] are commonly used [34\u201337,103,116,123]. A graph is represented by a set of vertices and edges where each vertex represents an atom and the edges represent the interactions between the atoms. Computation on the graph is done by updating each vertex\u2019s (atom\u2019s) x\u2019s hidden representation hx. The updates iteratively pass messages between vertices connected by the edges: ht+1 x = g(ht x\u222aN(x), vx\u222aN(x), ex\u2194N(x)) (10) N(x) is the set of neighboring atoms of x. The initial states v of the vertices are based on the atom features, such as element type or atomic numbers, and the edge features e are computed from the distances between atoms and potentially other bond features. vx\u222aN(x) refers to the initial states of atom x and its neighbors N(x), and ex\u2194N(x) refers to features of edges with x as an endpoint. The structure property y of interest, such as the energy, is typically estimated locally at each vertex and averaged or summed together for the \ufb01nal result: y = X x g\u2032(hx) (11) Neural networks are trained for functions g and g\u2032 using a regression loss function. An overview of early message passing techniques and how they relate can be found in [34]. To the best of our knowledge, the \ufb01rst to apply a variant of GNNs to materials was SchNet [35]. Improvements to these models include new methods for adding edges, such as using Voronoi tessellation or Knearest neighbors [36, 37], and increasing the amount of 20 \finformation used in message passing [36]. The use of continuous instead of discrete \ufb01lters [34, 123] for pairwise interactions was proposed by [35, 103] that enabled their use in molecular dynamics. An exploration of the tradeoffs between equivariant and invariant pairwise information is provided by [124]. As noted by [125], networks should be rotationally invariant and they propose explicitly enforcing that features are covariant to rotations. In [126, 127], global attention layers are used to model higher-order graph properties, while [117] use directional message passing to model angular information between triplets of atoms. The limits of modeling higher-order terms by only looking at 3 or 4 body correlations is explored in [128]. A promising approach to achieving higher accuracy while improving ef\ufb01ciency is the incorporation of more domain speci\ufb01c features, such as those obtained from mean-\ufb01eld electronic structure methods [116], or atomic orbital interaction features for crystal structures [129]. The per-atom forces for the S2EF task have been computed using two approaches. The \ufb01rst is to estimate directly the forces fx \u2208R3 for atom x in the same manner as the system properties as described above [104]: fx = g(hx) (12) However, estimating each atom\u2019s forces in isolation may lead to inconsistencies between energies and forces, i.e., forces along a closed path may not sum to zero [106]. The second approach is to estimate the forces directly from the energy by taking the partial derivative of the energy E relative to the atom\u2019s positions px [35,103]: fx = \u2212\u2202E \u2202px (v, e) (13) A loss based on the computed forces may also be added during training [103]. This approach makes the assumption that the network is differentiable with respect to the atom positions. For instance, the pairwise features that encode the distances between atoms should not use a discrete (bucket) representation [123] that results in discontinuities in the derivatives. Instead continuous distance \ufb01lters are a better choice [35,103]. Research in predicting the relaxed structure from the initial structure (IS2RS) through the use of ML models is still an emerging area of research. Please see the OC20 public leaderboard to view evaluations on the latest models. If relaxed structures are estimated through an ML relaxation process (Figure 15(b)), the forces estimated (S2EF) at each iteration need to be of high accuracy. While the accuracy of energy and force predictions are improving, as shown in [42], less than 1% of S2EF structure calculations performed by baseline models (SchNet [103], CGCNN [37], DimeNet [117]) are within accuracy thresholds used to estimate practical utility. Not surprisingly, when these force predictions are used in the IS2RS task, less than 1% of relaxed structures are similarly found to be practically useful. While signi\ufb01cant room for improvement exists, it is encouraging that lower force prediction errors does lead to improvements in the prediction of relaxed structures [42]. Furthermore, while ML relaxations may not at \ufb01rst accurately estimate relaxed structures, they may still be useful in speeding up DFT relaxations. For instance, the output of an ML relaxation may be used as the initial structure for a DFT relaxation, for which signi\ufb01cantly fewer iterations (DFT calculations) may be needed to converge. Another approach to the IS2RS task is to regress the atom positions directly from an initial structure. While this may still be worth testing, it is unlikely to be effective. In typical regression problems, a neural network will regress to the mean. For the problem we are addressing, the mean of several local minima is unlikely to be a minimum itself (the mean of an atom sitting on either side of another atom will place both atoms in the same location). For the IS2RE task the same approach as S2EF is typically used, except the initial structure is given as input. However, the accuracy of the estimated relaxed energy has been shown to be signi\ufb01cantly worse if the initial structure is given as input rather than the relaxed structure [34]. Thus, one hypothesis is that the estimation of the relaxed structure is necessary to determine the relaxed energy accurately. 4.3. Related Datasets Datasets for catalysis are growing in popularity, especially as the use of machine learning models has increased. These datasets range in size from a few thousand to 90,000 reaction and adsorption energies. An online collection of over 50 datasets can be found on Catalysis Hub [130], containing approximately 109,000 computed energies. Included in this collection is a dataset [131] that computed 90,000 relaxations to \ufb01nd adsorption and reaction energies for 2,035 binary alloy surfaces with 12 adsorbates. Datasets for intermetallic surfaces with small molecule adsorbates include those released by [40] and [41] that contain 40,000 and 47,000 energies respectively using approximately 2,000 catalyst surfaces. Datasets containing less than a few thousand calculations include those for (111) transition metal surfaces [132] and high-entropy alloys [133], while others focus on speci\ufb01c reactions such as the CO2 reduction reaction [134]. If considering bulk material properties (not surfaces or adsorbates), [135] is a benchmark across 13 diverse material properties with over 100,000 samples, OQMD [136] contains information on over 600,000 materials, and AFLOW [137] studies millions of materials. Information covering a broad set of materials and molecules can be found in the Materials Project [39]. The catalyst materials for OC20 are sampled from those in the Materials Project. 21 \fFigure 18. Diagram of the over 2,000 possible reaction pathways from syngas (CO+H2) to CO2, water, methane, methanol, acetaldehyde and ethanol. Determining a catalyst\u2019s impact across numerous reaction pathways remains an open question. Figure adapted from [138]. 5. Discussion Training ef\ufb01cient and accurate models for approximating DFT is the \ufb01rst step in using machine learning to aid in the design of new electrocatalysts. If this is accomplished, numerous potential areas for future research exist. Currently, the understanding of how larger molecules interact with an catalyst is limited. The production of fuels with higher energy densities depends on modeling these larger (C2 and C3) molecules [139]. In these molecules, it is likely that multiple atoms (the carbons) bond with the surface of the catalyst, creating a much larger and complex search space when computing potential energies. Ef\ufb01cient and accurate approximations could lead to to new insights for these reactions. For many reactions, numerous reaction pathways exists. The large set of possible reaction pathways for the relatively simple reaction of syngas (CO+H2) is shown in Figure 18 [138]. The pathways realized in practice, which impacts the level of selectivity (percentage of desirable products), are dependent on the choice of catalyst. For example, a catalyst might increase a pathway\u2019s rate that produces methane. However, it may also result in increasing in the rate of another pathway that produces the unwanted byproduct CO2. This decrease in the desired selectivity of the catalyst reduces its practical usefulness. The design of new catalysts that optimize their impact on numerous reaction pathways is an open and challenging problem. Developing more realistic simulations is of high importance to the catalyst community. These simulations typically require modeling a larger number of atoms than is currently feasible with DFT. The types of interactions that could be modeled include: the interactions of adsorbates with each other, the use of electrolytes [85] to increase electron transfer, promoters [86] to increase the reaction rate, and the poisoning of catalysts [140]. It remains an open question whether ML approximations of DFT can scale to accurately handle larger systems. In summary, ef\ufb01cient and accurate approximations to DFT could lead to the discovery of new catalysts for use in the cost-effective storage of renewable energy and reducing climate change. We explored different methods for energy storage and discussed their scalibility to nation-sized grids. We introduced the basic concepts behind the design of catalysts and how DFT is utilized in this process. Finally, we described how the OC20 dataset may be used to train ML models for approximating DFT. We are hopeful that research at the intersection of AI and catalysis offers both interesting scienti\ufb01c challenges and the opportunity for long-term practical impact on pressing global problems."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1608.08716v1",
+                    "title": "Measuring Machine Intelligence Through Visual Question Answering",
+                    "abstract": "As machines have become more intelligent, there has been a renewed interest\nin methods for measuring their intelligence. A common approach is to propose\ntasks for which a human excels, but one which machines find difficult. However,\nan ideal task should also be easy to evaluate and not be easily gameable. We\nbegin with a case study exploring the recently popular task of image captioning\nand its limitations as a task for measuring machine intelligence. An\nalternative and more promising task is Visual Question Answering that tests a\nmachine's ability to reason about language and vision. We describe a dataset\nunprecedented in size created for the task that contains over 760,000 human\ngenerated questions about images. Using around 10 million human generated\nanswers, machines may be easily evaluated.",
+                    "authors": "C. Lawrence Zitnick, Aishwarya Agrawal, Stanislaw Antol, Margaret Mitchell, Dhruv Batra, Devi Parikh",
+                    "published": "2016-08-31",
+                    "updated": "2016-08-31",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.CL",
+                        "cs.CV",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Humans have an amazing ability to both understand and reason about our world through a variety of senses or modalities. A sentence such as \u201cMary quickly ran away from the growling bear.\u201d, conjures both vivid visual and auditory interpretations. We picture Mary running in the opposite direction of a ferocious bear with the sound of the bear being enough to frighten anyone. While interpreting a sentence such as this is effortless to a human, designing intelligent machines with the same deep understanding is anything but. How would a machine know Mary is frightened? What is likely to happen to Mary if she doesn\u2019t run? Even simple implications of the sentence, such as \u201cMary is likely outside\u201d may be nontrivial to deduce. How can we determine if a machine has achieved the same deep understanding of our world as a human? In our example sentence above, a human\u2019s understanding is rooted in multiple modalities. They can visualize a scene depictFigure 1. Example image captions written for an image sorted by caption length. ing Mary running, they can imagine the sound of the bear, and even how the bear\u2019s fur might feel when touched. Conversely, if shown a picture or even an auditory recording of a woman running from a bear, a human may similarly describe the scene. Perhaps machine intelligence could be tested in a similar manner? Can a machine use natural language to describe a picture similar to a human? Similarly, could a machine generate a scene given a written description? In fact these tasks have been a goal of arti\ufb01cial intelligence research since its inception. Marvin Minsky famously stated in 1966 [8] to one of his students,\u201cConnect a television camera to a computer and get the machine to describe what it sees.\u201d At the time, and even today, the full complexities of this task are still being discovered. 2. Image Captioning Are tasks such as image captioning [3, 22, 27, 16, 19, 15, 6, 11, 25, 21, 20, 32] promising candidates for testing arti\ufb01cial intelligence? These tasks have advantages, such as being easy to describe and being capable of capturing the imagination of the public [26]. Unfortunately, tasks such as image captioning have proven problematic as actual tests of intelligence. Most notably, the evaluation of image captions may be as dif\ufb01cult as the image captioning task itself [12, 30, 19, 22, 27]. It has been observed that captions judged as \u201cgood\u201d by human observers may actually contain 1 arXiv:1608.08716v1  [cs.AI]  31 Aug 2016 \fA giraffe standing in the grass  next to a tree. Figure 2. (left) An example image caption generated from [15]. (right) A set of semantically similar images in the MS COCO training dataset for which the same caption could apply. signi\ufb01cant variance even though they describe the same image [30]. For instance see Figure 1. Many people would judge the longer more detailed captions as better. However, the details described by the captions varies signi\ufb01cantly, e.g. \u201ctwo hands\u201d, \u201cwhite t-shirt\u201d, \u201cblack curly hair\u201d, \u201clabel\u201d, etc. How can we evaluate a caption if there is no consensus on what should be contained in a \u201cgood\u201d caption? However, for shorter less detailed captions that are commonly written by humans a rough consensus is achieved \u201cA man holding a beer bottle.\u201d This leads to the somewhat counterintuitive conclusion that captions humans like aren\u2019t necessarily \u201chuman-like\u201d. The task of image captioning also suffers from another less obvious drawback. In many cases it might be too easy! Consider an example success from a recent paper on image captioning [15], Figure 2. Upon \ufb01rst inspection this caption appears to have been generated from a deep understanding of the image. For instance, in Figure 2 the machine must have detected a giraffe, grass and tree. It understood that the giraffe was standing, and the thing it was standing on was grass. It knows the tree and giraffe are \u201cnext to\u201d each other, etc. Is this interpretation of the machine\u2019s depth of understanding correct? When judging the results of an AI system, it is not only important to analyze its output, but the data used for its training. The results in Figure 2 were obtained by training on the Microsoft Common Objects in Context (MS COCO) dataset [23]. This dataset contains \ufb01ve independent captions written by humans for over 120,000 images [5]. If we examine the image in Figure 2 and the images in the training dataset we can make an interesting observation. For many testing images, there exists a signi\ufb01cant number of semantically similar training images, Figure 2(right). If two images share enough semantic similarity, it is possible a single caption could describe them both. This observation leads to a surprisingly simple algorithm for generating captions [9]. Given a test image, collect a set of captions from images that are visually similar. From this set, select the caption with highest consensus [30], i.e. the caption most similar to the other captions in the set. In many cases the consensus caption is indeed a good caption. When judged by humans, 21.6% of these borrowed captions are judged to be equal to or better than those written by humans for the image speci\ufb01cally. Despite its simplicity, this approach is competitive with more advance approaches using recurrent neural networks [6, 11, 25, 21, 20, 32] and other language models [15] which can achieve 27.3% when compared to human captions. Even methods using recurrent neural networks commonly produce captions that are identical to training captions even though they\u2019re not explicitly trained to do so. If captions are \u201cgenerated\u201d by borrowing them from other images, these algorithms are clearly not demonstrating a deep understanding of language, semantics and their visual interpretation. The odds of two humans repeating a sentence is quite rare. One could make the case that the fault is not with the algorithms but in the data used for training. That is, the dataset contains too many semantically similar images. However, even in randomly sampled images from the web, a photographer bias is found. Humans capture similar images to each other. Many of our tastes or preferences are universal. 3. Visual Question Answering As we demonstrated using the task of image captioning, determining a multimodal task for measuring a machine\u2019s intelligence is challenging. The task must be easy to evaluate, yet hard to solve. That is, it\u2019s evaluation shouldn\u2019t be as hard as the task itself, and it must not be solvable using \u201cshortcuts\u201d or \u201ccheats\u201d. To solve these two problems we propose the task of Visual Question Answering (VQA) [1, 18, 24, 29, 4, 17]. The task of VQA requires a machine to answer a natural language question about an image as shown in Figure 3. Unlike the captioning task, evaluating answers to questions is relatively easy. The simplest approach is to pose the questions with multiple choice answers, much like standardized tests administered to students. Since computers don\u2019t get tired of reading through long lists of answers, we can even increase the length of the answer list. Another more challenging option is to leave the answers open-ended. Since most answers are single words such as \u201cyes\u201d, \u201cblue\u201d, or \u201ctwo\u201d evaluating their correctness is straightforward. Is the visual question answering task challenging? The task is inherently multimodal, since it requires knowledge of language and vision. Its complexity is further increased by the fact that many questions require commonsense knowledge to answer. For instance, if you ask \u201cDoes the man have \u201c20/20\u201d vision?\u201d, you need the commonsense knowledge that having 20/20 vision implies you don\u2019t wear glasses. Going one step further, one might be concerned 2 \fDoes it appear to be rainy? Does this person have 20/20 vision? Is this location good for a tan?  What flag is being displayed?  How many slices of pizza are there? Is this a vegetarian pizza? What color are her eyes? What is the mustache made of? Figure 3. Example images and questions in the Visual Question Answering dataset (http://visualqa.org). that commonsense knowledge is all that\u2019s needed to answer the questions. For example if the question was \u201cWhat color is the sheep?\u201d, our commonsense would tell us the answer is \u201cwhite\u201d. We may test the suf\ufb01ciency of commonsense knowledge by asking subjects to answer questions without seeing the accompanying image. In this case, humans subjects did indeed perform poorly (33% correct), indicating that commonsense may be necessary but not suf\ufb01cient. Similarly, we may ask subjects to answer the question given only a caption describing the image. In this case the humans performed better (57% correct), but still not as accurately as those able to view the image (78% correct). This helps indicate the VQA task requires more detailed information about an image than is typically provided in an image caption. How do you gather diverse and interesting questions for 100,000\u2019s of images? Amazon\u2019s Mechanical Turk provides a powerful platform for crowdsourcing tasks, but the design and prompts of the experiments must be careful chosen. For instance, we ran trial experiments prompting the subjects to write questions that would be dif\ufb01cult for a \u201ctoddler\u201d, \u201calien\u201d, or \u201csmart robot\u201d to answer. Upon examination, we determined that questions written for a smart robot were most interesting given their increased diversity and dif\ufb01culty. In comparison, the questions stumping a toddler were a bit too easy. We also gathered three questions per image and ensured diversity by displaying the previously written questions and stating \u201cWrite a different question from those above that would stump a smart robot.\u201d In total over 760,000 questions were gathered 1. The diversity of questions supplied by the subjects on Amazon\u2019s Mechanical Turk is impressive. In Figure 4, we show the distribution of words that begin the questions. 1http://visualqa.org Figure 4. Distribution of questions by their \ufb01rst four words. The ordering of the words starts towards the center and radiates outwards. The arc length is proportional to the number of questions containing the word. White areas indicate words with contributions too small to show. The majority of questions begin with \u201cWhat\u201d and \u201cIs\u201d, but other questions include \u201cHow\u201d, \u201cAre\u201d, \u201cDoes\u201d, etc. Clearly no one type of question dominates. The answers to these questions have a varying diversity depending on the type of question. Since the answers may be ambiguous, e.g. \u201cWhat is the person looking at?\u201d we collected ten answers per question. As shown in Figure 5, many question types are simply answered \u201cyes\u201d or \u201cno\u201d. Other question types such as those that start with \u201cWhat is\u201d have a greater variety of answers. An interesting comparison is to examine the distribution of answers when subjects were asked to answer the questions with and without looking at the image. As shown in Figure 5 (bottom), there is a strong bias to many questions when subjects do not see the image. For instance \u201cWhat color\u201d questions invoke \u201cred\u201d as an answer, or for questions that are answered by \u201cyes\u201d or \u201cno\u201d, \u201cyes\u201d is highly favored. Finally it is important to measure the dif\ufb01culty of the questions. Some questions such as \u201cWhat color is the ball?\u201d or \u201cHow many people are in the room?\u201d may seem quite simple. In contrast, other questions such as \u201cDoes this person expect company?\u201d or \u201cWhat government document is needed to partake in this activity?\u201d may require quite advanced reasoning to answer. Unfortunately, the dif\ufb01cultly of a question is in many cases ambiguous. The question\u2019s dif\ufb01cultly is as much dependent on the person or machine answering the question as the question itself. Each person or machine has different competencies. 3 \fAnswers with Images Answers without Images Figure 5. Distribution of answers per question type when subjects provide answers when given the image (top) and when not given the image (bottom). In an attempt to gain insight into how challenging each question is to answer, we asked human subjects to guess how old a person would need to be to answer the question. It is unlikely most human subjects have adequate knowledge of human learning development to answer the question correctly. However, this does provide an effective proxy for question dif\ufb01culty. That is, questions judged to be answerable by a 3-4 year old are easier than those judged answerable by a teenager. Note, we make no claims that questions judged answerable by a 3-4 year old will actually be answered correctly by toddlers. This would require additional experiments performed by the appropriate age groups. Since the task is ambiguous, we collected ten respondences for each question. In Figure 6 we show several questions for which a majority of subjects picked the speci\ufb01ed age range. Surprisingly the perceived age needed to answer the questions is fairly well distributed across the different age ranges. As expected the questions that were judged answerable by an adult (18+) generally need specialized knowledge, where those answerable by a toddler (3-4) are more generic. 4 \f3-4 (15.3%) 5-8 (39.7%) 9-12 (28.4%) 13-17 (11.2%) 18+ (5.5%) Is that a bird in the sky?  How many pizzas are shown?  Where was this picture taken? Is he likely to get mugged if he walked  down a dark alleyway like this? What type of architecture is this?  What color is the shoe? What are the sheep eating? What ceremony does the cake  commemorate? Is this a vegetarian meal? Is this a Flemish bricklaying  pattern? How many zebras are there? What color is his hair? Are these boats too tall to fit  under the bridge? What type of beverage is in the glass? How many calories are in this  pizza? Is there food on the table? What sport is being played? What is the name of the white  shape under the batter? Can you name the performer in the  purple costume? What government document is  needed to partake in this activity? Is this man wearing shoes? Name one ingredient in the skillet. Is this at the stadium? Besides these humans, what other  animals eat here? What is the make and model of  this vehicle? Figure 6. Example questions judged to be answerable by different age groups. The percentage of questions falling into each age group is shown in parentheses. 4. Abstract Scenes The visual question answering task requires a variety of skills. The machine must be able to understand the image, interpret the question and reason about the answer. For many researchers exploring AI, they may not be interested in exploring the low-level tasks involved with perception and computer vision. Many of the questions may even be impossible to solve given the current capabilities of state-ofthe-art computer vision algorithms. For instance the question \u201cHow many cellphones are in the image?\u201d may not be answerable if the computer vision algorithms cannot accurately detect cellphones. In fact, even for state-of-the-art algorithms many objects are dif\ufb01cult to detect, especially small objects [23]. Are the kids in the room the grandchildren of  the adults? What is on the bookshelf? Is this person expecting company? What is just under the tree? Do you think the boy on the ground has  broken legs? Why is the boy on the right freaking out? How many glasses are on the table? What is the woman reaching for? Figure 7. Example abstract scenes and their questions in the Visual Question Answering dataset (http://visualqa.org). To enable multiple avenues for researching VQA, we introduce abstract scenes into the dataset [2, 34, 35, 36]. Abstract scenes or cartoon images are created from sets of clip art, Figure 7. The scenes are created by human subjects using a graphical user interface that allows them to arrange a wide variety objects. For clip art depicting humans, their poses and expression may also be changed. Using the interface a wide variety of scenes can be created including ordinary scenes, scary scenes, or funny scenes. Since the type of clip art and it\u2019s properties are exactly known, the problem of recognizing objects and their attributes is greatly simpli\ufb01ed. This provides researchers an opportunity to more directly study the problems of question understanding and answering. Once computer vision algorithms \u201ccatch up\u201d, perhaps some of the techniques developed for abstract scenes can be applied to real images. The abstract scenes may be useful for a variety of other tasks as well, such as learning common sense knowledge [35, 2, 7, 10, 31]. 5. Discussion While visual question answering appears to be a promising approach to measuring machine intelligence for multimodal tasks, it may prove to have unforseen shortcomings. We\u2019ve explored several baseline algorithms that perform poorly when compared to human performance. As the dataset is explored, it is possible that solutions may be found that don\u2019t require \u201ctrue AI\u201d. However, using proper analysis we hope to continuously update the dataset to re\ufb02ect the current progress of the \ufb01eld. As certain question or image types become too easy to answer we can add new questions and images. Other modalities may also be explored such as audio and text-based stories [13, 14, 33, 28]. In conclusion, we believe designing a multimodal challenge is essential for accelerating and measuring the progress of AI. Visual question answering offers one approach for designing such challenges that allows for easy evaluation while maintaining the dif\ufb01cultly of the task. As the \ufb01eld progresses our tasks and challenges should be continuously reevaluated to ensure they are of appropriate dif\ufb01cultly given the state of research. Importantly, these tasks should be designed to push the frontiers of AI research, and help ensure their solutions lead us towards systems that are truly \u201cAI complete\u201d. 5"
+                }
+            ],
+            "Devi Parikh": [
+                {
+                    "url": "http://arxiv.org/abs/2005.07328v2",
+                    "title": "Exploring Crowd Co-creation Scenarios for Sketches",
+                    "abstract": "As a first step towards studying the ability of human crowds and machines to\neffectively co-create, we explore several human-only collaborative co-creation\nscenarios. The goal in each scenario is to create a digital sketch using a\nsimple web interface. We find that settings in which multiple humans\niteratively add strokes and vote on the best additions result in the sketches\nwith highest perceived creativity (value + novelty). Lack of collaboration\nleads to a higher variance in quality and lower novelty or surprise.\nCollaboration without voting leads to high novelty but low quality.",
+                    "authors": "Devi Parikh, C. Lawrence Zitnick",
+                    "published": "2020-05-15",
+                    "updated": "2020-05-22",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.HC"
+                    ],
+                    "main_content": "Introduction How can one best collaborate with humans in a creative process? Insights towards this can inform what roles machines can (or should not) play when co-creating with humans. Speci\ufb01cally, we consider a scenario where agents take turns collaboratively drawing a sketch on a simple web interface (Figure 1). During each iteration, multiple agents propose strokes to add to the sketch. Agents then vote on the proposals, and the preferred set of strokes is added to the sketch. This process is repeated for a \ufb01xed number of iterations to create a \ufb01nal sketch. The roles of creating stroke proposals and voting could each be ful\ufb01lled by either humans (H) or machines (M). Borrowing terminology from Generative Adversarial Networks (Goodfellow et al. 2014), we can call the former role a generator (G), and the latter a discriminator (D). This allows for 4 {H,M} \u00d7 {G,D} co-creation scenarios. Further, different individuals could play the role of generators/discriminators across iterations, leading to crowd co-creation. In this work, as a step towards human-machine cocreation, we study various human-human crowd co-creation scenarios. In the \ufb01rst, Individual, a single human creates the entire sketch (no discriminator D, and no crowd). Second, in Collaborative the sketch is generated by multiple human agents (crowd) iteratively taking turns adding strokes. That is, all the agents act as generators G and there is no voting or discriminator D. The third, Collaborative + voting, is where multiple human agents (generators) propose new strokes at each iteration. Another set of human agents (discriminators) vote on which set of strokes to add to the sketch. Finally, we explore Individual with collaborative Figure 1: As a \ufb01rst step towards human-machine cocreation, we explore human-human collaboration for creating digital sketches on a simple web interface shown above. Video: https://youtu.be/9fikuKPYPd0 prompts, for which the crowd is involved indirectly. A single human creates the entire sketch, but by following text prompts that describe the evolution of a sketch that was created in the Collaborative scenario. We evaluate the qualitative difference between the sketches produced via these four scenarios. We \ufb01nd that the collaborative setting with a voting mechanism (Collaborative + voting) leads to sketches that are rated by human subjects as most creative (and are preferred along a variety of other dimensions). The lack of either one of these components results in less creative sketches: Individual sketches have decent quality (value) but low novelty, while Collaborative sketches have high novelty but low value. Individual with collaborative prompts results in high novelty but even worse quality. Overall, among these four scenarios, Collaborative + voting best hits the sweet spot for creativity: value + novelty (Boden 1992). Related Work In the context of crowd-based sketching, (Tuite and Smith 2012) analyze user actions and large-scale behavior patterns in 50k sketches from Sketch-a-bit, a collaborative mobile drawing application. Different from our incremental contriarXiv:2005.07328v2  [cs.AI]  22 May 2020 \fFigure 2: Few iterations of a sketch being created in the Collaborative + voting scenario. Once a parent sketch gets \ufb01ve children, it gets selected as the next iteration of the sketch (black outline), and the \ufb01ve children become the parents for the next iteration. Temporal visualization: https://youtu.be/JQmGALAhhMU. Examples with all iterations: Figures 7 and 8. bution + voting mechanisms, (Yu and Nickerson 2011) and (Gingold et al. 2012) explore combination and averaging of sketches respectively as collaboration strategies. Several AI systems have been trained to recognize sketches (e.g., models trained on The Quick, Draw! Dataset1). These may form useful building blocks for the next stages of our work. However, as seen in Figure 3, our sketches tend to be complex scenes and often abstract as opposed to concrete individual objects, which has been the focus of most existing work in automatic sketch recognition. There is also work on generating images based on sketches (Chen and Hays 2018). (Davis et al. 2016) employ a cognitive science framework called participatory sense-making to study co-creation in sketches. Central to their study is the back and forth interaction (dialog) between the human and machine as they take turns. Our work is focussed on a crowd setting where no two agents interact again in the future. (Karimi et al. 2020) study human-AI co-creativity in the context of humans sketching for a particular design goal. Our work falls in the category of \u201ccasual creators\u201d (Compton and Mateas 2015) \u2013 systems that support exploratory as opposed to goal-driven creativity. Sketching Interface Human agents create sketches using the JavaScript based interface shown in Figure 1. Strokes can be varied across four thicknesses and ten colors and have a paint-like texture. The number and length of the strokes an agent may draw is limited during each iteration. Feedback on how close they are to the cutoff is provided in real-time by the stroke limit bar. Thicker strokes count more towards the limit. Strokes drawn by the agent during the current iteration may be undone. See https://youtu.be/9fikuKPYPd0 for a video of the interface. Our interface is publicly available. Co-creation Scenarios We explore four scenarios for collaborative human-human sketch co-creation. In every scenario, the sketch starts with a blank canvas. During each iteration, a limited number of strokes may be added. The limit roughly corresponds to \ufb01ve 1https://github.com/googlecreativelab/quickdraw-dataset medium-thickness strokes spanning the width of the canvas. 30 iterations are used to create each sketch. Unless stated otherwise, we collected 20 sketches for each scenario. All our studies were conducted on Amazon Mechanical Turk. Subjects can not submit their work till they have contributed the required amount of strokes to the canvas. Individual. The entire sketch is created by a single individual. That is, a single human agent adds all 30 iterations of strokes to \u201cCreate a beautiful, detailed, coherent painting!\u201d. Collaborative. A different human agent contributes strokes for each iteration of the sketch. That is, 30 unique individuals contribute to a sketch. The \ufb01rst subject sees a blank canvas and adds strokes. Every subsequent subject is shown the partial sketch and asked to add to it. They cannot undo strokes from earlier contributors. The prompt is \u201cLet\u2019s collectively create a beautiful, detailed, coherent painting!\u201d. Subjects are given the additional instruction to consider the kind of painting being created and the stage of the painting when deciding upon which strokes to draw. Collaborative + voting. Each subject contributes strokes to a sketch of their choosing from a set of \ufb01ve starting sketch variations. We refer to the chosen starting sketch as a parent, and the sketch created by a subject as the chosen sketch\u2019s child. During each iteration, sketches are gathered until a parent is selected \ufb01ve times. Its children then replace the current \ufb01ve parents and the process is repeated. Children of parents selected less than \ufb01ve times are discarded. See Figure 2, Figures 7 and 8 and https://youtu.be/ JQmGALAhhMU for more examples. This voting strategy allows for the most promising versions of a sketch to go forward. This scenario is robust to the strokes added by any one individual. Of course, it is also signi\ufb01cantly more \u201cexpensive\u201d. In the best case scenario where a single parent gets all 5 children and none of the other parents get a child, it takes 5 times the amount of strokes to create a sketch compared to Collaborative. In the worst case, all 5 parents get 4 children each before a parent gets a \ufb01fth child. This would result in 21 times the number of strokes. In practice we found this factor to be about 12.5 times. Given the increased cost, we reduced the number of iterations in this scenario to 20 (as opposed to 30). On average, 250 unique individuals contribute to a single sketch. Individual with collaborative prompts. A single indi\fFigure 3: Example sketches from four co-creation scenarios along with differences identi\ufb01ed by human subjects between sketches from pairs of scenarios. Collaborative + voting involves \u223c12.5 times the individuals, and so was run for 20 instead of 30 iterations. For comparison, Collaborative sketches are also shown at 20 iterations. More sketches from the four scenarios can be seen in Figures 9, 10, 11, and 12 respectively. Figure 4: Example prompts used in the Individual with collaborative prompts scenario. Figure 5: Evolution of example sketches in the Collaborative scenario. Left: Focus of the sketch shifts from the house to the cat in the rain outside the house. Right: Faced with seemingly incoherent strokes, subjects emphasize structure they see in it so subsequent subjects can add to it. More examples of sketches evolving are in Figures 13 and 14. vidual creates an entire sketch using instructive text prompts provided at each iteration. The individual is instructed to follow the prompts when drawing. The text prompts are generated by asking another individual to describe what changed in a sketch from one iteration to the next in the Collaborative scenario. All text prompts for a sketch are written by a single individual. This is an interesting hybrid of having a single creator, but being guided through prompts that describe the evolution of a sketch as created by 30 unique individuals. We collected three sets of text descriptions for each of the 20 Collaborative sketches. This resulted in a total of 60 Individual with collaborative prompts sketches. In our evaluation, we consider 20 sketches (randomly picking 1 out of the set of 3). See Figure 4 for example prompts. Evaluation Example sketches from these scenarios are shown in Figure 3 as well as in Figures 9, 10, 11, 12. Before we discuss properties of the \ufb01nal sketch, it is worth considering the evolution of a sketch as it is being created. Collaborative sketches evolve in several interesting ways: what seems like the main subject of a sketch changes in a few iterations (Figure 5, left), given seemingly incoherent strokes, subsequent subjects try and emphasize regions that could lead to meaningful structures in the sketch for future subjects to build on (Figure 5, right), and subjects use the color white or other strategies to try and cover parts of the sketch they think are contributing negatively to it. More examples of sketches evolving across iterations can be found in Figures 13 and 14. To assess the qualitative differences between sketches produced from the 4 scenarios, we created a collage of 20 sketches from each scenario (at 20 iterations for Collaborative and Collaborative + voting, 30 for the rest). We showed pairs of collages to subjects on Amazon Mechanical Turk and asked them to describe differences that stood out. Snippets from subjects\u2019 responses are shown in Figure 3. For a quantitative evaluation, we showed subjects pairs of sketches from two different scenarios (i,j). Each subject picked which sketch they prefer along 12 axes. Every pair was evaluated by 5 subjects resulting in 144,000 assessments: 20 (sketches from scenario i) \u00d7 20 (sketches from scenario j) \u00d7 6 (pairs of scenarios) \u00d7 12 (axes) \u00d7 5 (subjects per sketch-pair). The 12 axes were: which painting (1) seems more strange / unusual / different than typical paintings? (2) is a better painting? (3) do you like looking at more? (4) is more creative? (5) is more interesting? (6) is more original? (7) took more skill? (8) is made by an artist more likely to be an adult? (9) would you pay more for? (10) are you more likely to put up in your home? (11) is more likely to be in an art museum? (12) would you be more proud to have made yourself? Some of these axes (e.g., originality, novelty, skill) are from (van der Velde et al. 2015). The % of times each scenario was picked over a competing scenario is shown in Figure 6. For 11 of 12 axes, including creativity, Collaborative + voting is preferred. Collaborative + voting scores well for both novelty (unusual) \fFigure 6: Collaborative + voting sketches are consistently preferred by human subjects over sketches from other scenarios across a variety of dimensions, and notably are rated as most creative. Notice the high variance in Individual sketches. and quality (better, look), which we hypothesize increases its perceived creativity. Individual is rated well for quality but scores poorly on novelty. Across 11 axes, Individual has high variance due to differences in skill + motivation of individuals creating the sketches. Collaborative scores well on novelty, but worse on quality. Individual with collaborative prompts does poorly across all axes except for unusual, which is visually apparent in Figure 12. Of all scenarios, Collaborative + voting falls in the sweet spot for maximizing creativity (value + novelty). Discussion In what way may a machine best contribute to the collaborative creation of a sketch? It is often the case that humans may not be good at generating strokes, but can tell if a sketch looks good or not. This may suggest using machines to generate candidate strokes and having humans vote on which versions should proceed next. The machine may also contribute in a manner similar to the humans in our fourth scenario, i.e., the machine could generate textual prompts as a human draws a sketch. The prompter can have different \u201cpersonalities\u201d based on whether it is trained on sketches generated from Individual (coherent), Collaborative (rich but chaotic) or Collaborative + voting (rich with subtle details and coherent). Humans and machine can generate strokes as a team, either in co-painting scenarios as in (Cabannes et al. 2019), or where the machine provides some visual guidance as in (Lee, Zitnick, and Cohen 2011) or via suggestions for where to draw, what colors to use, etc. as explored in (Oh et al. 2018). We can also train a machine to be a discriminator: given a few different stokes from a human, select which stroke should be added to the sketch next. All our sketches started with a blank canvas. We could instead start sketches with a prompt (subject of the sketch, adjective describing a desired property of the sketch, a picture to be used as inspiration for the sketch, etc.), and have this prompt persist across iterations (or not). It is interesting to consider ideas of ownership in the context of crowd co-creation. While no one individual may feel a complete sense of ownership of the \ufb01nal piece, crowd collaboration may lead to a sense of community and the satisfaction of contributing to a common cause. Finally, while our motivation was human-machine co-creation, studying human-human collaboration in general is, obviously, important and interesting in and of itself. Collaborative creative endeavors may be a fertile ground for such explorations."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2003.01274v1",
+                    "title": "Predicting A Creator's Preferences In, and From, Interactive Generative Art",
+                    "abstract": "As a lay user creates an art piece using an interactive generative art tool,\nwhat, if anything, do the choices they make tell us about them and their\npreferences? These preferences could be in the specific generative art form\n(e.g., color palettes, density of the piece, thickness or curvatures of any\nlines in the piece); predicting them could lead to a smarter interactive tool.\nOr they could be preferences in other walks of life (e.g., music, fashion,\nfood, interior design, paintings) or attributes of the person (e.g.,\npersonality type, gender, artistic inclinations); predicting them could lead to\nimproved personalized recommendations for products or experiences.\n  To study this research question, we collect preferences from 311 subjects,\nboth in a specific generative art form and in other walks of life. We analyze\nthe preferences and train machine learning models to predict a subset of\npreferences from the remaining. We find that preferences in the generative art\nform we studied cannot predict preferences in other walks of life better than\nchance (and vice versa). However, preferences within the generative art form\nare reliably predictive of each other.",
+                    "authors": "Devi Parikh",
+                    "published": "2020-03-03",
+                    "updated": "2020-03-03",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.HC",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Generative art is art that has at least some of its features determined by a non-human autonomous system.1 The autonomous system is typically a computer, and it frequently relies on randomization to determine the art features. Among the human-de\ufb01ned features, some might be \ufb01xed by a generative artist, and other parameters might be open to manipulation. Different settings of these open parameters, combined with the machine-de\ufb01ned features, leads to different instances of the generative art form. Generative artists often make interactive tools available so a lay person can set values of these open parameters and create their own generative art piece. We refer to these tools as interactive generative art tools, and they are the subject of this study. We ask the research question: what does the choices a lay person makes while creating art using an interactive generative art tool tell us about them \u2013 about their personality or preferences in food, fashion, interior design, etc., as well as about their preferences in the speci\ufb01c generative art form? 1Generative Art (Wikipedia): https://en.wikipedia. org/wiki/Generative_art Effectively predicting their preferences in the speci\ufb01c generative art form can lead to a smarter interactive generative art tool. It can help the user create an art piece they like faster by encouraging them to explore a certain part of the parameter space. It can prevent the user from losing interest by discouraging them to explore a different part of the parameter space. Predicting their preferences in other aspects of life can position interactive art generation as a generic personality assessment tool for products and experiences recommendations. Finally, predicting their preferences in generative art from other other known preferences in life can lead to improved art recommendation. We conducted a survey where 311 subjects consented to participate and self-reported their preferences along various parameters in a generative art form, as well as in various walks of life such as food, chocolate, alcoholic beverages, music, interior design, fashion, paintings and their other traits such as gender, personality type, exposure to design principles, artistic inclination, and introspectiveness. We analyze these preferences to identify statistical correlations between pairs of preferences and report some interesting \ufb01ndings. We train machine learning models to predict subsets of these preferences from other preferences. In addition to some black box models, we include models that reveal interpretable relationships between various preferences. Quantitatively, we \ufb01nd that for the speci\ufb01c generative art and machine learning models we experimented with, user\u2019s preferences in other aspects of life cannot be reliably predicted from their preferences in the generative art form (and vice versa). However, their preferences in the generative art form can be predicted with statistical signi\ufb01cance from other preferences in the generative art form. This is a promising result towards demonstrating the feasibility of smarter interactive generative art tools. Related Work Preferences in art and personality. Connections between perception of and preferences in art, and personality traits or preferences in other domains has been studied in several works. We describe those next. However, to the best of our knowledge, this has not been studied in the context of generative art speci\ufb01cally or even digital art in general. As the landscape of art changes with the incorporation of digital tools and more recently AI, it is valuable to consider how arXiv:2003.01274v1  [cs.AI]  3 Mar 2020 \fFigure 1: Example instances of the Strokes generative art form that we study in this work. Figure 2: Interactive generative art tool for creating Strokes. Video: https://youtu.be/YzfzjK8NNMg. these tools can be made smarter to better assist a human in their creative process. The ability to predict a user\u2019s preferences is central to such smart tools. ( \u00a8 Oz, Ozpolat, and Takesen 2015) study the correlations between \ufb01ve personality traits and art preferences among 24 visuals from Renaissance, Cubism, Abstract Art, Traditional Art, Impressionism and Surrealism. They found that extroversion and openness to experience was correlated with a preference for Surrealist works, while tender-mindedness was correlated with Impressionist works. (ChamorroPremuzic et al. 2008) performed a similar study across over 90,000 individuals in UK. (Chamorro-Premuzic et al. 2010) \ufb01nd that personality traits correlate better with art categories when these categories are de\ufb01ned based on emotional valence and complexity as assessed by a collection of observers, than categories de\ufb01ned by researchers or historical art taxonomies. (Ercegovac, Dobrota, and Kuevi 2015) also study the relationship between personality traits and art preferences, but for both visual art and music. They also investigate correlations between music and visual art preferences. (Gridley 2013) study not only the correlations of visual art preferences with personality traits, but also with styles of thinking (concrete versus abstract thinking, sequential versus random thinking, ego functions of intuition and sensation). They also \ufb01nd evidence for cross-modal relations in aesthetic preferences across food, music, and visual stimuli. (Lyssenko, Redies, and Hayn-Leichsenring 2016) study how participants describe abstract artwork and the relationship of these descriptions to various image properties. They also investigate the correlation between personality traits and preferences. They found neuroticism to correlate with a preference for objectively more complex images. (Bhattacharjya 2016) discusses formal models of preferences in the context of computational creativity to embrace the subjective nature of an evaluator judging creative value. (Cook and Colton 2015) design a software system capable of having preferences \u2013 to make and justify subjective decisions beyond using random chance or a pre-de\ufb01ned external heuristic. Authors argue that having such preferences adds to the perception of the software being creative. Casual creators. The interactive generative art tools we study fall in the category of \u201cCasual Creators\u201d. This is in contrast with tools that are designed to assist task-oriented professionals or amateurs in their creative process towards solving a speci\ufb01c problem or designing for a speci\ufb01c task. Casual creators on the other hand are autotelic creativity tools that cater to enjoyable explorative creativity over task completion. (Compton and Mateas 2015), who coined the term, states \u201cA Casual Creator is an interactive system that \fFigure 3: Con\ufb01gurations of Strokes we studied and % times each alternative (b-i) was preferred by subjects over the default (a). encourages the fast, con\ufb01dent, and pleasurable exploration of a possibility space, resulting in the creation or discovery of surprising new artifacts that bring feelings of pride, ownership, and creativity to the users that make them.\u201d They stress the interactive aspect of casual creators where the user is the driver, and the creating process as being core to the experience. A casual creator is an effective tool if it helps users \ufb01nd desirable artifacts without getting stuck in a local minima or being lost in a vast space of bad artifacts. Our work addresses exactly this. The various parameters in the interactive generative art tool de\ufb01ne the space of artifacts, of possibilities, that a user can explore. Our work predicts the user\u2019s preferences. A computational model that uses these predictions can in\ufb02uence the path the user takes in this space; it affects the probability that a user will encounter a certain artifact in their creating process. By effectively predicting their preferences, we increase the chances that users will \ufb01nd a desirable artifact when using a casual creator. Individual user preferences. Existing work, e.g., (Zsolnai-Feh\u00b4 er, Wonka, and Wimmer 2018), models individual preferences of a user as they explore a parametrized design space. Our work learns correlations between preferences across parameters from a population of users. The two directions have a common goal \u2013 helping a user \ufb01nd designs they like \u2013 but are complementary. GANs. Generative Adversarial Networks (Goodfellow et al. 2014) have gained tremendous popularity as tools for AIpowered generative visual art. In this work we focus on a more geometric, abstract, \ufb02at, generative art form that gives the user direct control over speci\ufb01c features of the art. Interactive Generative Art: Strokes We start by de\ufb01ning the generative art form \u2013 Strokes \u2013 around which we design our study. Example art pieces of the Strokes form are shown in Figure 1. We chose Strokes because it is an abstract form, allowing us to focus our study on visual preferences rather than semantic associations. A Strokes piece is a series of overlaid shapes. A shape is started by connecting two random points on a square canvas via a curve of a certain thickness (T). The curve may be a straight line, or a quadratic Bezier curve using a third point as a control point. This control point is chosen to be the midpoint between the two end points perturbed by random noise. The random noise lies uniformly randomly in the range R, which is 10% to 20% the width of the canvas. This noise is either added or subtracted to the x and y coordinates of the mid-point independently. Each of these 4 possibilities has a probability \u03bd of 0.25. Having placed the \ufb01rst curve, the end point of the curve is connected to another random point on the canvas via a curve (again, straight line or quadratic Bezier curve with a noisy control point). This process is repeated. After each curve is drawn, the shape either continues (with probability P = 0.5) or the shape ends and a new shape begins. When the shape ends, the canvas enclosed by the curves and a straight line connecting the start point of the \ufb01rst curve in the shape and end point of the last curve in the shape is colored by a random color from a palette. The color of the curves themselves is a pre-de\ufb01ned background color in the palette. When a total of N curves have been drawn, that shape and the piece overall are complete. Any pixel covered by more than one shape is colored by the most recent color. The generative artist designed this generative process, chose the colors in each palette, the probability P with which a new shape starts after each curve, the range R that determines the amount of noise added to a mid point, and the probability \u03bd of adding noise to each of the 4 directions to form the control point of the quadratic Bezier curve (when applicable). The machine picks the random end points and noise added to the mid point to form the control point of the quadratic Bezier curve (when applicable). The color palette, number of curves N in the piece, thickness of curves T and whether the curves should be a straight line or a quadratic Bezier curve are free parameters. These parameters are provided as options on an interactive tool as seen in Figure 2. The random seed is kept \ufb01xed when a user is changing parameters on the interface so that the only in\ufb02uence changing the piece is input from the user. The user can click on the \u201cGenerate\u201d button to change the random seed that determines the machine\u2019s in\ufb02uence. A video demonstrating the interface is provided here https://youtu.be/YzfzjK8NNMg. As options, the tool provides 6 color palettes, 11 densities which determine the number of curves (N = 2density), 15 line thicknesses, and a binary option of curved or straight lines. In our study, which serves as a proof of concept, \f(a) Cubism (b) Renaissance (c) Modern (d) Traditional (e) Bohemian Chic (f) Business casual Figure 4: Example images shown to subjects when asking for their preferences in paintings: Cubism vs. Renaissance, interior design: Modern vs. Traditional, and fashion: Bohemian Chic vs. Business Casual. Other than showing them as examples to remind subjects of what these categories mean, these images were not used in this work in any way. we restrict the number of palettes to 3, densities to 3, line thicknesses to 4, and retain the straight vs. curved lines option.2 Speci\ufb01cally, we start with a \u201cdefault\u201d con\ufb01guration for each of these options (Figure 3a), and generate 8 versions of the piece by changing one property at a time (Figure 3b-i): switching to one of the other two palettes (gray in Figure 3c and bright in Figure 3g), changing the density to one of the other two values (sparser in Figure 3d and sparsest in Figure 3e), changing the line thickness to one of the other three values (thin in Figure 3i, thicker in Figure 3b, and thickest in Figure 3h), and switching to straight lines (Figure 3f). Collecting Preferences To answer our research question, we collect two classes of preferences from subjects. One is in the context of the Strokes interactive generative art described above. The other is about other walks of life. Each preference is posed as a two-way forced choice question. For Strokes, we generate 8 pairs of comparisons: default in Figure 3a vs. each of the 8 edited versions in Figure 3b-i. Both pieces in each pair are generated with the same random seed so that there is only one cause of variation between the two pieces, but different seeds are used across pairs to ensure that the preferences we collect are generic across seeds. We randomly order the default and the edited version within a pair. The 8 pairs are also randomly ordered. We generate a total of 3 sets of these comparisons with different random seeds. This gives us 24 two-way forced choice pairs in the context of interactive generative art. For each pair, subjects were asked \u201cWhich visual pattern appeals to you more?\u201d For other walks of life, we ask subjects for 12 preferences. \u2022 Do you re\ufb02ect on a regular basis (e.g., write in a journal)? Yes/No \u2022 Which do you prefer? Milk vs. dark chocolate. \u2022 Which do you prefer? Wine vs. beer. \u2022 Which do you prefer? Country vs. rock music. \u2022 Do you have any exposure to design principles? Yes/No. \u2022 Are you artistically inclined? Yes/No \u2022 Which gender do you associate with more? Male/female. \u2022 What personality type do you associate with more? Introvert/extrovert. \u2022 Which do you prefer? Sweet vs. savory food. \u2022 Which of these styles of painting appeals to you more? Cubism vs. Renaissance (Figures 4a, 4b were shown). 2Conceptually, our approach is not restricted to a few parameters, because it learns correlations in preferences of\ufb02ine across a population of users, rather than online for an individual user. \u2022 If you could setup your home however you liked, which of these styles would you go with? Modern vs. traditional (Figures 4c, 4d were shown as examples). \u2022 Irrespective of your gender, which of these fashion styles do you relate to more? Bohemian Chic vs. business casual (Figures 4e, 4f were shown as examples). These 12 combined with the 24 generative art preferences described earlier gives us a total of 36 two-way forced choice questions. We collected these 36 preferences from 311 subjects on Amazon Mechanical Turk. Analyzing Preferences We begin by analyzing the preferences. We \ufb01rst ask how consistent subjects are in their own preferences in generative art. Recall that each subject was shown all 8 generative art comparisons for 3 different seeds. If preferences change signi\ufb01cantly across random seeds, there is limited scope to predict them. But if the choices are consistent across seeds, there may be scope to predict them automatically. Are people consistent? For the 3 sets of responses for each of the 311 subjects and each of the 8 comparisons (i.e., across a total of 2488 sets of 3 responses), we \ufb01nd that on average 2.7 responses are the same. 2.7 out of 30 is 90%. Random guessing would result in 75% of the responses being the same. Another way of looking at inter-human agreement is to check how accurately a person\u2019s response to the 3rd random seed can be predicted if one assumes that their response will be the same as their response to the other 2 random seeds. We \ufb01nd this accuracy to be 81%. Recall that each pair contains the default con\ufb01guration (Figure 3a) and one of the eight alternative versions (Figure 3b-i). We \ufb01nd that across subjects, the default con\ufb01guration is preferred 62% of the time. This is not surprising because the default con\ufb01guration has been set by the generative artist as a generically good con\ufb01guration. We use this prior preference for the default con\ufb01guration to break ties. That is, in cases where the response to the \ufb01rst two random seeds was different (one picking the default and the other picking the alternative version), we assume that for the 3rd seed the subject picked the default version. Note that always assuming prior \u2013 that is, always predicting that the user prefers the default options \u2013 would result in a prediction accuracy of 62% \u2013 signi\ufb01cantly lower than 81% reported above. This suggests that people prefer alternative con\ufb01gurations frequently and that they are consistent in this preference \facross random seeds. That is, there is scope for predicting the personal preferences of a user automatically. What do people like? Which con\ufb01gurations do subjects tend to like better? Figure 3 shows the % of times each alternative is preferred over the default. As discussed above, the default is most preferred (62% of the time), but there is a heavy tail suggesting that the other alternatives are preferred frequently. We \ufb01nd that on average across subjects, 3 out of the 8 alternatives are preferred over the default. As for preferences in other walks of life, Table 1 shows the % of times the various options were preferred. Most preferences are not signi\ufb01cantly skewed in one direction over the other, and may have useful information to predict other preferences. Table 1: Subjects\u2019 preferences in various walks of life. What do subjects like? Milk over dark chocolate 60% Sweet over savory food 34% Wine over beer 53% Country over rock music 25% Cubism over Renaissance 43% Modern over traditional interior 34% Bohemian over business casual 43% What are subjects like? Male 39% Re\ufb02ect regularly 40% Exposure to design principles 26% Artistically inclined 48% Introvert 74% Which preferences are related? Next, we analyze which preferences are most related to each other. For instance, if we wanted to know whether a person will like the gray palette over the default in Strokes, is it best to use information about whether they liked straight over curved lines in Strokes (perhaps those who prefer straight lines prefer the gray palette)? Or whether they like Bohemian Chic fashion over Business Casual? That is, for a target preference of interest, which other preference is the best source such than if the source preference is directly used to predict the target preference, prediction accuracy is the highest across all potential source preferences. To evaluate this, for each of the 36 target preferences, we search over the remaining 35 preferences as a potential source preference and compute the prediction accuracy. We report the best source for some target preferences in Table 2. We can compare this prediction accuracy (Acc) to the prior accuracy (Prior) \u2013 the accuracy we would get if for every target preference we predicted whichever choice is the most common. In addition to reporting the best source for some target preferences, in Table 2 we also report some source preferences that are not necessarily the best, but are still more accurate than this baseline and we thought showcase interesting correlations in our data. Table 2: Analysis of which (source) preferences are good indicators to predict other (target) preferences. Target Source Acc Prior Best source to predict a target preference Re\ufb02ect regularly Design exposure 61% 59% Wine over beer Female 65% 53% Artistic Design exposure 66% 51% Male Beer over wine 65% 60% Grey palette Straight lines 61% 59% Sparsest pattern Sparser pattern 73% 61% Sparser pattern Sparsest pattern 73% 60% Thicker lines Thickest lines 77% 59% Thickest lines Thicker lines 77% 66% Interesting source to predict a target preference Artistic Bohemian>business 56% 51% Thicker lines Bright palette 61% 58% Sparser pattern Thin lines 62% 60% Artistic Sparser pattern 53% 51% Not artistic Milk over dark choc. 53% 51% We see that gender is the best predictor for whether someone prefers wine or beer. Whether someone prefers the gray palette over the default is the best predictor for whether they prefer straight lines over the curved lines in Strokes. Whether someone prefers the bright palette is a good indicator of whether they prefer thicker lines in Strokes. Unsurprisingly, whether someone prefers the sparser pattern is best predicted by whether they preferred the sparsest pattern. Overall, we see promising indications that there are correlations between the different preferences, and there is potential to predict a subset of preferences from other preferences. To build these predictors, we train a variety of machine learning models, which we describe next. Predicting Preferences In order to predict a subset of preferences from other preferences, we train machine learning models. Our setup is the following: We create three groups of preferences \u2013 A: The 8 art (Strokes) preferences (recall that we have 3 responses from each subject for these 8 preferences), L: the 12 preferences from other walks of life, and U: all 20 preferences. Each group can be used as features F to predict each of the preferences in the other group (target T ). A group can also be used as features to predict a preference from the same group by excluding that target preference from the input features. This results in a total of 9 machine learning problems (F, T ) \u2208{A, L, U}\u00d7{A, L, U} where \u00d7 denotes the cartesian product. These include (as non-exhaustive examples): \u2022 Predicting art preferences from other life preferences (i.e., features: A, target: each preference from L, metric: average accuracy across all preferences in L). \u2022 Predicting other life preferences from art preferences (i.e., features: L, target: each preference from A, metric: average accuracy across all preferences in A). \u2022 Predicting an art preference from other art preferences. \fFigure 5: Accuracies across all 9 settings corresponding to choices of source preferences (features F) and target preferences (prediction targets T ) using a linear SVM. We train and test our machine learning models via leaveone-out cross validation. We train on preferences from 310 of 311 subjects, and test on the remaining subject. This is done 311 times (rotating which subject we test on). We report average accuracies across the 311 tests. Recall that we collected preferences from each subject for the 8 Strokes preferences across 3 seeds. We ensure that all 3 preferences from a subject are either in test or train, and not split across. To normalize for the fact that different preferences have a different prior, we report class normalized accuracies. Class normalized accuracy is the average of the accuracies of predicting a subjects preference in both directions. So if we predict that a subject always prefers the default Strokes piece (say), we will have 100% accuracy for class \u2018default\u2019 but will have missed all instances when the subject preferred the alternative, leading to 0% accuracy for class \u2018alternative\u2019. This results in a normalized class accuracy of 50% (average of 0% and 100%). That is, class normalized accuracy of prior is 50% and is the baseline we compare our models to. We experiment with the following approaches: nearest neighbor, logistic regression, linear Support Vector Machines (SVMs), polynomial SVMs, Radial Basis Function (RBF) SVMs, neural networks, and decision trees, as well as a matrix completion approach. Hyper parameters of each of these models are picked via cross validation. The linear SVM performs the best. In Figure 5 we show its accuracy on all 9 con\ufb01gurations of the machine learning setup described earlier. Recall that performance of a prior baseline under our normalized accuracy metric is 50%. We see that our models are unable to predict interactive generative art (Strokes) preferences based on other life preferences and vice versa. However Strokes preferences predict other Strokes preferences well. Knowing other preferences in life further (incrementally) helps predict Strokes preferences. We provide further analysis of our machine learning models for these two settings: using interactive generative art (Strokes) preferences to predict other Strokes preferences, and using all preferences (Strokes + other preferences in life) to predict a held out Strokes preference. Figure 6: Comparison of machine learning models. Figure 7: Predictability of preferences in interactive generative art (Strokes) from all other preferences of a user. While linear SVMs performed the best, nearest neighbor and matrix completion accuracies are informative points of comparison. The nearest neighbor approach assumes that a test subject\u2019s preference is the same as the training subject who has the most number of other preferences is common with the test subject. Matrix completion approaches are commonly used in recommendation systems to predict user preferences. In Figure 6 we show a comparison of these classi\ufb01ers along with the prior baseline. As an additional point of reference, if we use the best single source preference to predict each target preference (similar to the analysis from Table 2), we get a prediction (class normalized) accuracy of 60%. This is lower than the accuracy of the linear classi\ufb01er, demonstrating the bene\ufb01t of using information from multiple preferences to make the prediction. The linear SVM, matrix completion and baseline prior approaches are deterministic. The nearest neighbor approach often \ufb01nds multiple nearest neighbors (all equidistant from the test instance). We break ties randomly. To assess statistical signi\ufb01cance, Figure 6 shows variation due to this stochasticity via a 95% con\ufb01dence interval. As another measure of statistical signi\ufb01cance, particularly for the deterministic approaches, we use 1,000 bootstrap samples and \ufb01nd the 95% con\ufb01dence interval to consistently be \u00b1\u223c0.1%. \fFigure 8: Decision trees to interpret the multi-dimensional correlations between different preferences in our data. Figure 9: Accuracies at predicting Strokes preferences before (left) and after (right) removing Strokes preferences that are linearly related (e.g., thicker vs. thickest lines). Interpreting Predictions We now analyze the machine learning models to get a better insight into the patterns in the data. We start by analyzing which of the Strokes preferences are easiest to predict i.e., are most related to all other preferences of a user (within Strokes and in other walks of life). See Figure 7. We see that it is easiest to predict whether someone will like the thickest setting of the line width or not. On the other hand, it is hardest to predict whether someone will like the bright color palette or not (we cannot guess any better than the prior). Next, we seek interpretable \u201crules\u201d that explain how different preferences are related to each other. For this we turn to decision trees as interpretable machine learning models. We train a decision tree on all preferences except a target Strokes preference that we wish to predict. To aid in interpretability, we restrict the depth of the decision tree to be 3 levels or less. Figure 8 show several instances of rules that we recover for different target preferences. We see that if someone prefers the gray palette, and if they prefer the sparser pattern, they are likely to prefer straight lines. Otherwise (either they prefer the default palette over the gray one or over the sparser pattern), they are likely to prefer the curved lines over the straight lines. If someone prefers straight lines and country music, they are likely to prefer the default palette over the gray palette. Note that these patterns give interesting insights into the data we collected and inspire thought about how one might predict a user\u2019s preferences in interactive generative art. They are are not meant to encourage generalization or stereotypes. One thing that stands out in Figure 8 is that whether someone will prefer thicker lines can be predicted by checking if they like the thickest or thin lines. Same for sparse vs. dense patterns. These preferences along a linear ordering are obviously (and hence somewhat uninterestingly) related (if someone does not like sparse patterns, it is unlikely they will like even sparser patterns). To verify that the predictive power of our machine learning models is not relying primarily on these uninteresting correlations, we reduce our set of Strokes preferences down to 5. We remove the sparsest (Figure 3e), thickest lines (Figure 3h) and thin lines (Figure 3i) alternatives because those were the least preferred alternatives along the thickness and density parameters. The remaining 5 alternatives are all or\fFigure 10: Accuracy with increasing training data. thogonal to each other (with no linear ordering). We retrain our machine learning models. Their performance is shown in Figure 9. We see that while model accuracies go down a little after removing these linear dependencies (e.g., accuracy of a linear SVM at predicting Strokes preferences from other Strokes preferences goes from 64% to 62%), it continues to be signi\ufb01cantly better than the prior baseline (50%). This suggests that we can indeed predict meaningful (non-obvious) dependences between a user\u2019s preferences when interactively creating generative art (Strokes). More data? Finally, we ask the question: would more training data improve the performance of these models? Figure 10 shows the performance of our models at predicting the Strokes preferences from all (Strokes + other walks of life) but the target preference, with increasing amount of data. For a \ufb01xed amount (x-axis), we show accuracies when training on 10 random subsets of data of that size. We \ufb01nd that performance of the linear model has saturated. Performance of a nearest neighbor model, as expected, continues to increase with more training data. The more subjects we collect preferences from, the more likely it is that a subject in our training set will closely mimic a new test subject."
+                }
+            ],
+            "Dhruv Batra": [
+                {
+                    "url": "http://arxiv.org/abs/2011.01975v1",
+                    "title": "Rearrangement: A Challenge for Embodied AI",
+                    "abstract": "We describe a framework for research and evaluation in Embodied AI. Our\nproposal is based on a canonical task: Rearrangement. A standard task can focus\nthe development of new techniques and serve as a source of trained models that\ncan be transferred to other settings. In the rearrangement task, the goal is to\nbring a given physical environment into a specified state. The goal state can\nbe specified by object poses, by images, by a description in language, or by\nletting the agent experience the environment in the goal state. We characterize\nrearrangement scenarios along different axes and describe metrics for\nbenchmarking rearrangement performance. To facilitate research and exploration,\nwe present experimental testbeds of rearrangement scenarios in four different\nsimulation environments. We anticipate that other datasets will be released and\nnew simulation platforms will be built to support training of rearrangement\nagents and their deployment on physical systems.",
+                    "authors": "Dhruv Batra, Angel X. Chang, Sonia Chernova, Andrew J. Davison, Jia Deng, Vladlen Koltun, Sergey Levine, Jitendra Malik, Igor Mordatch, Roozbeh Mottaghi, Manolis Savva, Hao Su",
+                    "published": "2020-11-03",
+                    "updated": "2020-11-03",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.CV",
+                        "cs.LG",
+                        "cs.RO"
+                    ],
+                    "main_content": "Introduction Embodied AI is the study and development of intelligent systems with a physical or virtual embodiment. Over the past few years, signi\ufb01cant advances have been made in developing intelligent agents that can navigate in previously unseen environments. These advances have been accelerated by dedicated software platforms [44, 61, 75, 62] and clear experimental protocols [1, 4]. Navigation research is The authors are listed in alphabetical order. thriving in part due to healthy infrastructure and experimental methodology. An exciting frontier for Embodied AI research concerns interaction and contact between the agent and the environment: tasks that call on the agent to actively engage with and modify the environment in order to accomplish its goals. A number of software platforms support such interaction scenarios [37, 44, 62, 76, 28]. These software platforms simulate realistic onboard perception and in some cases the physical dynamics of the agent, environment, and their interaction. One missing ingredient is a clear task de\ufb01nition that can span different software platforms and catalyze coordinated accumulation of knowledge and ability across research groups. Clear task de\ufb01nitions and evaluation metrics are essential in the common task framework, which is substantially responsible for progress in computer vision and natural language processing [22]. In computer vision, standard tasks such as image classi\ufb01cation and object detection have facilitated the development of foundational techniques and architectures that have enriched the whole \ufb01eld [23, 59]. Language modeling and machine translation have served a similar role in natural language processing [69, 70, 2, 73]. In both \ufb01elds, these standard tasks focus the development and validation of new representations and algorithms, and serve as a source of trained models that can be transferred to other tasks, as with convolutional backbones pretrained for image classi\ufb01cation [35], object detectors [58, 34], and transformers pretrained on natural language [20, 57, 77, 11]. 1 arXiv:2011.01975v1  [cs.AI]  3 Nov 2020 \fGeometric Image Experience Lang\u0011age \"Move toaste\t  e\u000bt to \fo\ree \u000ea\u000fe\t\u0010\" Goal state Current state Task Specification Figure 1. Object rearrangement example. The goal and the current state of the scene are shown in the left and right images, respectively. The agent is required to move objects (e.g., chair) or change their state (e.g., close the fridge) to recover the goal con\ufb01guration. The rightmost panel shows different ways of specifying the rearrangement task. In this report, we develop a task de\ufb01nition that can likewise align and accelerate research in Embodied AI. The task is rearrangement: Given a physical environment, bring it into a speci\ufb01ed goal state. Figure 1 provides an example. We propose rearrangement as a canonical task for Embodied AI because it naturally uni\ufb01es instances that are of clear practical interest: setting the table, cleaning the bedroom, loading the dishwasher, picking and placing orders in a ful\ufb01llment center, rearranging the furniture, and many more. Rearrangement scenarios can be de\ufb01ned with stationary manipulators that operate locally or with mobile systems that traverse complex scenes such as houses and apartments. Many experimental settings that have been explored in robotics can be viewed as instances of rearrangement, as well as many compelling settings that are beyond the reach of present-day systems (Figure 2). We focus on rearrangement of rigid and articulated piecewise-rigid objects. This includes a broad range of interesting and challenging scenarios, as illustrated in Figure 2. In order to focus research on settings that we consider challenging but tractable in the near future, we deliberately exclude within-object transformations, such as melting (spot welding, soldering), destruction (cutting, drilling, sanding), and non-rigid dynamics (liquids, \ufb01lms, cloths, or ropes). Thus boiling water, chopping onions, stomping grapes, pouring coffee, making a smoothie, folding towels, ironing clothes, or making a bed are considered beyond the scope of the presented framework. We expect these scenarios to be incorporated as technology matures in the future. Rearrangement calls upon a variety of abilities in an embodied intelligent system. Successful rearrangement can require recognizing the state of objects in the environment (is the cabinet open or closed?), inferring the differences between the current and the goal state, manipulating objects in cluttered environments (e.g., an object cannot be moved if it is blocked by another object), estimating forces required to move objects and predicting the effect of those forces, planning a sequence of actions, navigating through complex environments while maintaining a persistent internal representation, and learning the pre-conditions and postconditions of each action. Rearrangement is thus a comprehensive framework for research and evaluation in Embodied AI that subsumes component skills such as navigation and manipulation and integrates them into a broader roadmap towards embodied intelligence. The goal state of rearrangement can be speci\ufb01ed in different forms: geometric con\ufb01guration, images of the desired state, natural language, a formal predicate-based speci\ufb01cation, or an embodied speci\ufb01cation that lets the system examine the environment in the goal state before being instantiated in the initial con\ufb01guration. We de\ufb01ne an evaluation protocol that can accommodate all of these speci\ufb01cations by evaluating predicates that score objects and object sets in terms of their compliance to the goal. A rearrangement episode receives a score between 0 and 1, enabling clear ranking of competing approaches. We provide a number of guidelines in order to maximize the practical utility of this research program and assist subsequent deployment on physical systems. First, we emphasize the importance of acting based on perceptual input: sensing the environment with realistic sensors, rather than planning in an idealized map, con\ufb01guration space, or logical abstraction. Embodied AI puts noisy and incomplete onboard perception in the control loop: this makes the development of robust systems challenging but is necessary for successful adoption in the real world. Our second key recommendation is to work either in the physical world or in physical dynamics simulation. Objects should be moved by forces, most commonly applied via contact. Lastly, we advocate prioritizing strong generalization: evaluating rearrangement agents with objects and environments that were never encountered during training, with no access 2 \fFigure 2. Examples of rearrangement tasks. Top row: examples of experimental settings in vision, robotics, and arti\ufb01cial intelligence that can be viewed as rearrangement tasks. From left to right: the MIT copy demo is a seminal demonstration of robotic rearrangement; the Amazon Robotics Challenge involves moving items from a shelf into a tote (image from Hernandez et al. [36]); RoboCup@Home is a longrunning robotics competition involving household tasks (image from Stuckler et al. [68]). Bottom row: people perform many rearrangement tasks that remain challenging for current robotic systems. From left to right: construction toys are a popular activity for children (image from toyhalloffame.org); setting a dining table, cleaning up a bedroom, stocking grocery shelves are all rearrangements performed by people. to privileged information such as 3D models, ground-truth object poses, or perfect localization. Agents should be able to perform rearrangement tasks in previously unseen environments, the content and layout of which are accessibly solely through realistic onboard sensing. To assist near-term research and development, we release suites of rearrangement scenarios in four simulation environments: THOR [44], RLBench [37], SAPIEN [76], and Habitat [62]. We anticipate that other scenario datasets will be released and new simulation platforms will be built to support this line of work. We hope that the task speci\ufb01cation, evaluation protocol, and broader discussion in this report will support healthy development of Embodied AI and the creation of intelligent systems that perceive, act, and accomplish increasingly long-term goals in complex physical environments. 2. Background Rearrangement can be approached in a modular way, for example through a decomposition into submodules such as perception and planning. Here perception converts sensory input to a representation of the world, from which planning produces actions. Planning has been the subject of a long line of work in Arti\ufb01cial Intelligence [60]. Going back to the early days of AI, Newell and Simon [65] proposed GPS, the General Problem Solver, which could be related to even more classical work, the means-ends analysis of Aristotle. More concretely, the Shakey robot project [46] at SRI in the late 1960s led to the STRIPS [25] formulation of a plan as applying operators in sequence with each operator having its precondition, add and delete lists expressed in a logical formalism. The representation language used by STRIPS for specifying planning problems has been replaced by PDDL (Problem Domain Description Language), introduced by Ghallab et al. [31], and there is a \ufb02ourishing community with an annual conference and challenge competitions. A number of books and surveys cover this body of work [30, 32, 40]. This style of planning is sometimes called task planning to distinguish it from motion planning, which has been the subject of extensive study in robotics. The focus in motion planning is much more geometric \u2013 planning paths for moving objects while avoiding obstacles in con\ufb01guration spaces. LaValle [47] is a good entry into this literature. The combination of task planning and motion planning has also been explored, for example by Kaelbling and Lozano-P\u00b4 erez [39, 29]. In the task and motion planning literature, there has been a rich body of work on \u2018rearrangement planning\u2019 [6, 16, 18, 21, 41, 42, 45, 63, 64, 66], the problem of searching for a sequence of actions that transform an initial con\ufb01guration of objects into some goal con\ufb01guration. A variety of problem formulations exist, with different kinds of permitted transformations, such as pushing versus grasping. Such rearrangement planners usually do not directly address the problem of perception, in the sense that they do not deal 3 \fwith raw sensory input such as pixels; instead, they typically assume that the shape and pose of objects are already provided, possibly partially and with uncertainty. Our point of departure from the classic task and motion planning literature is our emphasis on standardized end-toend evaluation. End-to-end evaluation measures the capability of the full system from raw sensory input to actuation. Such evaluation is agnostic to the choice of approach and internal decomposition, putting more focus on real-world perception and action, as with physical sensors and actuators on robots and in physical simulation. We believe that standardized end-to-end evaluation is synergistic with research on task and motion planning. Many of the existing task and motion planning methods are not easily comparable to each other due to different input/output assumptions. Standardized end-to-end evaluation can shed light on the role of planning techniques in the context of a full system that must handle real perception and actuation. It also allows comparing approaches that differ in design philosophy, such as modular systems based on classic planning techniques and monolithic neural networks that directly map sensory input to actuation commands. Our work is of course not the \ufb01rst to present standardized evaluation protocols or benchmarks for Embodied AI problems. A number of efforts in robotics, computer vision, and machine learning have proposed various standardized evaluation methods, including protocols for evaluating robotic grasping [52, 72, 49], navigation [1, 4], manipulation tasks focusing on standardized real-world object sets [13], and competition-style setups, such as RoboCup [43, 67] and the International Planning Competition [53], as well as simulated benchmarks focusing on speci\ufb01c algorithmic approaches such as reinforcement learning [5, 9, 24, 79, 26]. Our proposal addresses a need that we believe is not met by these prior works. Our aim is to develop a framework and protocol for evaluating Embodied AI systems that is general enough to cover a wide range of different capabilities (as compared, for example, to more narrow evaluations focusing on speci\ufb01c tasks, such as grasping) and can compare a large variety of algorithmic approaches on the same footing. We believe that such a framework for evaluation of Embodied AI systems will empower researchers to pursue general and ambitious goals in terms of capabilities, without constraining to a speci\ufb01c and narrow range of approaches. 3. Rearrangement In this section, we provide a concrete and general but application-grounded de\ufb01nition for rearrangement. This is followed by recommendations on agent embodiment and sensor speci\ufb01cation. Evaluation criteria are discussed in Sec. 4. Notation. For a set S, we will use 2S to denote the power set (the set of all possible subsets) of S. We specify Rearrangement using the language and notation of Partially Observable Markov Decision Processes (POMDP) because this is a familiar and convenient mathematical abstraction. We do not however take any stance on approaches used to solve the problem. Let s \u2208S denote a state and state space, o \u2208O denote an observation and observation space, a \u2208A denote an action and action space, T (st, at, st+1) = Pr(st+1 | st, at) denote the transition probability, g \u2208G denote a goal and goal space, and \u03c0(at; g) = Pr(at | o0, . . . , ot\u22121, a0, . . . , at\u22121, g) denote the agent\u2019s goal-conditioned policy. Articulated Rigid-body POMDPs. We will restrict our attention to worlds comprising a collection of rigid bodies and kinematic trees thereof. Speci\ufb01cally, robots and objects are modeled as a tree of rigid parts (legs, wheels, etc.) connected via joints that determine degrees of freedom (see Section 3.3.2 in [47] for a refresher). Let SO(3) denote the special orthogonal group, the space of 3D rotations, and SE(3) = R3 \u00d7 SO(3) denote the special Euclidean group, the space of rigid-body poses (3D locations and rotations). In Rearrangement, the world state space is factorized \u2013 i.e., can be written as the Cartesian product of the rigid-body pose spaces corresponding to each of the n parts: S = S1 \u00d7 S2 . . . Sn (1a) where Si = SE(3) = R3 \u00d7 SO(3) \u2200i. (1b) Notice that the expression above does not account for the constraints imposed by the joints in the kinematic chains/trees and thus not all con\ufb01gurations in this state space may be achievable. Finally, we note that the above state-space speci\ufb01cation ostensibly appears to exclude a number of important variables \u2013 any form of dynamics (velocities, acceleration, etc.), physical properties (mass, friction coef\ufb01cients, etc.), or any notion of time at all. However, as we describe next, the only role of this state space is to specify the problem (in terms of the initial state, goal state(s), and the goal speci\ufb01cations). We do not take any stand on the intermediate states that a solution to this problem may pass through. Initial and Goal State(s). In Rearrangement, we will be concerned about two special states \u2013 1. an initial or starting state s0 where the agent and environment \ufb01nd themselves at the beginning of the task, and 2. a desired goal state s\u2217that the agent attempts to rearrange the environment into. Importantly, s\u2217may not be unique, but is rather an element from a set of acceptable goal states, i.e. s\u2217\u2208S\u2217\u2286S. The exact speci\ufb01cation of S\u2217 4 \fwill depend on the task at hand, but one convenient characterization may be via a \ufb01nite set of predicates Pi(\u00b7). Thus, S\u2217is the set of states s\u2217for which all Pi(s\u2217) hold true (\u2200i). Goal Speci\ufb01cation. Let \u03c6 : S \u00d7 2S 7\u2192G denote a goalspeci\ufb01cation function. Speci\ufb01cally, given a starting state s0 and a set of acceptable goal states S\u2217, this function generates the goal speci\ufb01cation g = \u03c6(s0, S\u2217) for the agent. Since noisy and incomplete onboard perception is a \ufb01rstclass citizen in Embodied AI, the agent will typically not have access to the state space or any goal state s\u2217. Instead, the agent must operate solely from observations o and goal speci\ufb01cation g. Note that this emphasis on partial visibility is only for the agent; the \u2018experiment designer\u2019 will typically have access to S\u2217and use it for evaluation (and potentially training of the agent). This is a reasonable assumption for experiments conducted in simulation but may be infeasible for real hardware experiments. Rearrangement Task De\ufb01nition. With this notation in place, we can formally de\ufb01ne Rearrangement. Given a goal speci\ufb01cation g = \u03c6(s0, S\u2217), an agent must transform an environment from an initial starting state s0 to a goal state s\u2217\u2208S\u2217, operating purely based on sensory observations o \u2208O. This abstract de\ufb01nition covers many aforementioned examples as special cases, instantiated by picking appropriate choices of state, observation, and action space, and a goal speci\ufb01cation. For instance, the state space in setting the table, cleaning the bedroom, loading the dishwasher, picking and placing orders in a ful\ufb01llment center, and rearranging the furniture can all be de\ufb01ned (to a \ufb01rst degree of approximation) as the product space of rigid-body pose spaces corresponding to each object. We now describe a number of goal speci\ufb01cation mechanisms. \u2022 GeometricGoal. Consider a single object to be moved/rearranged. In this setting, g provides a geometric speci\ufb01cation of the transformation that this object undergoes from s0 to s\u2217. This could be at various levels of detail \u2013 (e.g. the coordinates of the center of mass of the object in the goal state relative to the start state), via 3D bounding box transformations (e.g. coordinates of an axis-aligned or oriented bounding box around the object in the goal state in the coordinate system of the object in the start state), or via a full rigid-body pose transformations for articulated objects. In multi-object rearrangement, g can be a tuple of geometric con\ufb01gurations of each object to be moved. Note that g is likely to be much lower-dimensional than s, i.e. not all objects, agents, or places in the environment may be speci\ufb01ed in the goal g. \u2022 ImageGoal. In this setting, g provides a visual rendering or representation of s\u2217, e.g. a 3rd person (say overhead, orthographic, perspective or isometric) image of the environment in the goal state. While this is a convenient and informative way to specify the goal state \u2013 simply by taking a picture \u2013 it is important to note that a goal image may be underspeci\ufb01ed. For instance, if certain objects are not visible in the image goal, there are multiple possible valid goal placements where they may be out of view. A goal image may also limit our ability to specify that we want certain objects to be placed or contained inside other objects. One natural generalization of ImageGoal is VideoGoal, where the agent receives a sequence of images depicting the goal state. The camera pose corresponding to these images could be strategically chosen by the experiment designer to disambiguate the underspeci\ufb01cation associated with ImageGoal. \u2022 LanguageGoal. In this setting, g provides a linguistic description of the environment in the goal state (e.g. \u2018move the chair to the right of the sofa\u2019, \u2018move the blue block to the right edge of the table\u2019). Note that language is typically underspeci\ufb01ed and there may be several goal states that ful\ufb01ll a given language goal. This will require care in evaluation. \u2022 ExperienceGoal. In this setting, we side-step the problem of designing a goal speci\ufb01cation function \u03c6(\u00b7) by immersing the agent in the environment in the goal condition s\u2217(under a time/interaction budget) and letting the agent build whatever representation it deems appropriate to bring the environment back to this state. Thus, the goal is essentially to \u2018make it like it was before\u2019. This captures the scenario of asking a home robot to clean up the kitchen after a particularly messy session, where the robot has already experienced the kitchen in the clean state. This idea may be generalized by also immersing the agent in \u2018non-goal\u2019 environments, so that it can distinguish between important and unimportant attributes of the environment. \u2022 PredicateGoal. In this setting, g is a set of predicates that should be satis\ufb01ed in the goal state (e.g. on(plate1, table1)). A PredicateGoal can be precisely speci\ufb01ed and evaluated by de\ufb01ning the predicates (e.g. supported by, inside, is on), the symbols on which they act (e.g. plate, pizza, table, microwave), and their grounding to objects and relations. Symbols should map to objects or sets of objects, and each predicate should be evaluated by a func5 \fFigure 3. Magic pointer abstraction in Habitat [62] (left), using a mouse pointer, and AI2-THOR [44] (right) using a raycast from the camera origin to the point of interaction (cyan markings are just for visualization purposes and not visible to the agent). tion taking as input a state. A predicate may be implemented in a number of ways \u2013 geometric thresholding of positions of objects through a program, running a neural network classi\ufb01er on the current state, or programatically checking the logical state of an object (e.g. is the microwave on or off). PredicateGoal forms a common substrate on top of which other goal types can be interpreted (e.g. by conversion of a LanguageGoal to a PredicateGoal). PredicateGoal is not necessarily a natural interface for humans, but it can be a useful interface for systems. 3.1. Embodiment: Actuators and Sensors Manipulating and rearranging objects requires a simulated physical embodiment, leaving a number of open parameters, such as the action representation, the degree to which the manipulation interaction itself is simulated or abstracted away, and the particular capabilities afforded to the agent. The choice of embodiment in a given environment will inevitably in\ufb02uence the types of methods that are effective. We therefore do not prescribe a single speci\ufb01c embodiment, but instead recommend that benchmark and algorithm designers provide a clear and reproducible description, while we provide a brief discussion of several reasonable choices. Generally, the action and perception representation is known to have a large impact on the performance of some types of algorithms, such as reinforcement learning, and being overly prescriptive with the embodiment may result in excessive focus on the particular challenges of a speci\ufb01c embodiment rather than the broader challenges inherent in Embodied AI. The particular choice of embodiment in a given environment will fall on a spectrum between the most abstract and the most realistic. We provide a few examples of points on this spectrum, starting with the most abstract and ending with the most concrete. Although we recommend using the most realistic actuator and sensor embodiments whenever possible, the more abstract representations may nonetheless present a useful simpli\ufb01cation of the problem to enable more targeted algorithmic progress in speci\ufb01c areas. Actuation: magic pointer abstraction. One way to strike a compromise between abstraction and rich physical interaction is to borrow a commonly used metaphor in video games: instead of controlling a realistic simulated body, the agent navigates the environment and continuously controls its viewing direction (i.e., pitch and yaw) and, optionally, a virtual \u201cmouse pointer\u201d that can move on the screen (see Figure 3). The agent can trigger a \u201cpick\u201d action that results in a ray cast a short distance in front of the agent, picking the closest object that intersects this cast ray. In the view direction abstraction (Figure 3, right), the ray passes through the speci\ufb01ed point of interaction, while in the case of a virtual mouse pointer (Figure 3, left) it passes through the location of the mouse pointer on the screen. This object is now \u201cheld\u201d by the agent, at which point it may be freely rotated and, with another action, placed back into the environment. The release of the object may be simulated as a fall from a certain height in front of the agent, or being placed in the same pose (relative to the agent) as the object was during \u201cpick-up\u201d. Optionally, this embodiment could provide an option for the agent to \u201cstow\u201d the currently held object into a virtual \u201cbackpack\u201d (of \ufb01xed capacity). In contrast to the \u201cdiscrete object grasping\u201d abstraction, this more complex embodiment allows more precise relocation of objects, for example in settings where we might want to reposition a variety of small items on a table. It also provides for the ability to carry out rudimentary physical interactions, since the object held in front of the agent can still interact physically with other objects in the environment. On the other hand, it also requires more precision: instead of simply standing in front of an object, the agent 6 \fFigure 4. Full physical simulation. A simulated robot interacting with the environment in the SAPIEN [76] framework. Full physical simulation provides the highest \ufb01delity in terms of modeling object interaction, but also a more challenging setting necessitating continuous feedback control. must choose a point on the image to interact with. This might be useful for example for inserting one object into another, rearranging deformable objects, or other physical interactions. However, this abstraction still omits most of the nuances of physical object interaction and controlling a simulated body. This abstraction is used in the AI2-THOR environment discussed in Section 3.2. Actuation: kinematic articulated arm with abstracted grasping. The next step along the spectrum from abstraction to realism is a simulated kinematic arm. In this setting, the agent controls a realistic model of a robotic arm, either via Cartesian end-effector control or joint space control (Cartesian 6-DoF end-effector should be suf\ufb01cient in practice for most applications), but grasping of objects is abstracted away, such that any object that is located close enough to the end effector is automatically (virtually) \u201cgrasped\u201d when the agent issues a \u201cpick\u201d command. This abstraction strikes a compromise between requiring control of an actual robotic body and avoiding some of the intricate dynamic complexities of object interaction, such as inertia and contact forces. This makes it appropriate for simpli\ufb01ed manipulation scenarios and methods focusing on perception, while still providing some of the challenges and geometric constraints of a physical body. Actuation: full physical simulation. A full physical simulation of a robotic manipulator can provide the highest degree of \ufb01delity (see Figure 4), but also the greatest challenge in terms of requiring any algorithm to perform both effective closed-loop control and perception. Unlike the kinematic, abstracted grasping setting in the previous paragraph, a full physical simulation requires controlling a robotic arm to actually grasp objects physically, using accurately simulated contact forces. In this setting, the agent\u2019s actions directly map to actuation commands for simulated robotic joints. A number of choices must be made in instantiating such an embodiment: (1) end-effector Cartesian or jointspace arm control; (2) position control or force/torque control; (3) the type of gripper control afforded to the agent. A reasonable balance of \ufb01delity and simplicity is to use (1) Cartesian-space control, (2) position control, (3) binary (open/close) gripper commands. However, other variants may also be used, and it is reasonable to consider this choice as part of the algorithm designer\u2019s prerogative, since all combinations are realistic for use on real-world robotic systems. A full physical simulation provides challenges that are most representative of those encountered in the real world, and is the only way to perform more nuanced physical object interaction. By simulating the low-level control stack (e.g. PID or impedance control), appropriate actuation noise would also be simulated. This modality also poses additional perception challenges, since the robot must now determine not only which object to pick up, but also how to physically use its end-effector to pick up or manipulate the object successfully. Of course, this added complexity also places a heavier burden on algorithm designers. The SAPIEN and RLBench environments discussed in Section 3.2 use this embodiment, as do most robotics simulation packages in use today. Sensors: ground truth positions abstraction. In some cases, researchers may choose to focus on planning and control without considering perception, in which case the coarsest abstraction of the perception problem is to utilize ground-truth positions and orientations of objects directly as input. This representation greatly simpli\ufb01es the task and allows researchers to study the control problem in isolation. While we urge a combined and integrated effort to tackle perception and control jointly, we also recognize that some researchers may end up addressing these topics separately. In this case, reasonable representations could include positions and orientations of nearby objects, annotated with their identity and parameters of their geometric shape. Such positions should be provided with realistic noise, to ensure that any control or planning strategy is robust to perceptual uncertainty. Sensors: intermediate representations. An intermediate abstraction of onboard sensing is to assume mid-level visual processing, as would be typical in a modern perception stack, and directly utilize segmented images and/or depth maps [81]. Many simulators can natively produce segmented images and depth maps, making this representation easy to obtain. This level of abstraction has a number of appealing properties: it allows researchers to avoid the need for training large pixel-level deep networks for processing raw pixels, abstracts away variation in lighting and appearance, and at the same time provides a realistic interface to current computer vision tools, since pixel-level segmentation and depth estimation are heavily studied topics. In this case, suitable noise should be added to the intermediate representations, including noise in the depth readings 7 \ffor depth maps and noise in the segmentation, including unlabeled pixels and pixels with erroneous labels. Sensors: full simulated perception. A full simulation of the robot\u2019s onboard sensors provides the highest \ufb01delity of evaluation for sensing. In this case, simulated sensors might include RGB cameras, depth from RGBD sensors or LiDAR, as well as less common sensors, such as microphones and touch sensors. The sensors should be simulated with realistic noise and uncertainty and imperfect calibration (e.g. camera calibration with small added noise). This representation provides the most realistic simulation of real-world deployment and we recommend this as the default mode of operation. 3.2. Task Characterization The complexity of a rearrangement task is characterized by several dimensions that can serve to de\ufb01ne a taxonomy of rearrangement tasks. We group these axes of complexity into agent-centric and environment-centric dimensions (see Figure 5). Agent mobility. Examples of agents that can perform rearrangement are \ufb01xed-base manipulator arms and mobile robots with manipulators. The mobility characteristics of the agent restrict the rearrangement scenarios that can be performed (e.g. a \ufb01xed robotic manipulator cannot move objects long distances). The mobility of the agent also determines the available action space for navigation. The action space can range from a set of discrete actions (\u2018turnleft\u2019, \u2018turn-right\u2019, \u2018go-forward\u2019), to parameterized actions, to fully continuous control for each motor. Although the locomotion problem itself can be made quite complex, with the addition of fully simulated legged locomotion and other intricacies, this dimension of the problem can also be reasonably abstracted away, for example by assuming that the robot\u2019s embodiment consists of a holonomic base. We will assume that mobility is accomplished by means of such a holonomic base for the purpose of this article. Agent manipulation. Examples of manipulator types include: the \u201cmagic pointer\u201d abstraction, \u201csticky mittens\u201d [56], suction-based grippers [10], needle/stick force applicators, parallel jaw grippers, and 5-\ufb01nger humanoid grippers (see relevant survey in [71]). Earlier in this document, we elaborated on several concrete examples of agent manipulation (discrete grasping, magic pointer, and kinematic articulated arm). We do not take a speci\ufb01c stance on manipulator types, though some manipulators are easier to simulate with current simulation platforms. The spectrum of manipulation capabilities also forms a natural abstraction for research on rearrangement at higher levels (i.e., planning) vs. lower levels (i.e., control). Agent sensory suite. We encourage broad investigation into sensors of different types and modalities. A variety of sensor types are common: color vision cameras, depth sensors, contact/collision sensors, tactile sensors [50], LiDAR, microphones, and elastomeric sensors such as GelSight [38, 80] are prominent examples. Speci\ufb01c rearrangement scenarios may bene\ufb01t from long-range visual sensing (e.g. navigation-heavy rearrangement) or shorter-range haptic sensing (e.g. single object grasping). When using full simulated perception, we recommend that sensors should simulate noisy and incomplete onboard perception. Odometry may be provided by the simulator, but with suitable noise and drift, necessitating realistic closed-loop corrections. Environment interactability. The level of abstraction for the objects undergoing manipulation is another dimension of complexity. In the simplest case, manipulated objects are rigid bodies with no additional state that the agent can manipulate. Depending on the dexterity of the manipulator, the object may be rotated while it is held. The agent may have a stowing capacity in which case the object can be stowed away from the manipulator. A common abstraction is a virtual \u201cbackpack\u201d with in\ufb01nite or limited capacity stated in number of items, volume, or weight. More complex scenarios may involve object articulation states (e.g. books that may open and close). Environment complexity. The dif\ufb01culty of the rearrangement task depends on what is to be rearranged, the target con\ufb01guration, and the structure of the environment. Important parameters include the number of objects to be rearranged, the number of distractor objects, the degree of occlusion or containment in source and target con\ufb01gurations, whether the space the agent is moving in is relatively open or highly cluttered, and whether ordering is important (e.g. stacking objects in order). We focus on rearrangement of sets of piecewise rigid bodies in scenarios that may include containment and ordering constraints. Note that this includes many common articulated objects such as cabinets. Environment dynamicity. The degree to which the environment is dynamic forms another dimension of complexity. Important parameters include whether the environment can change without the agent taking an action (e.g. oscillating pedestal fan), whether there are unrecoverable states (e.g. plates can break if dropped), and whether objects in the environment are subject to perturbations unrelated to the agent\u2019s actions (e.g. wind or vibrations). 3.3. Task Generalization Spectrum In addition to the above dimensions of task complexity, there exists a spectrum of task generalization settings. Dif8 \fAgent Environment Mobility Manipulation Sensory suite Complexity Dynamicity Interactability Figure 5. Dimensions of complexity characterizing the rearrangement task. Several parameters of the agent and the environment determine the complexity of the rearrangement task. On the agent side: mobility (e.g. \ufb01xed base, wheeled, bipedal), manipulator (e.g. force applicator, parallel jaw gripper, humanoid hand), and sensory suite (e.g. color camera, depth sensor, LiDAR). On the environment side: interactability (e.g. rearrangement of rigid bodies, containers with rigid bodies, and articulated rigid bodies), complexity (e.g. two objects, a full bookcase, a cluttered kitchen), and dynamicity (e.g. object fracture, dynamic objects, and dynamic objects changing the state of other objects). ferent points on this spectrum may involve generalization to novel objects, novel environments, novel source and target arrangements of the objects, as well as potentially novel agent actuation and sensing con\ufb01gurations. On one extreme of the spectrum (\u2018weak generalization\u2019) a system is evaluated with known objects in known environments, with only the arrangements of the objects being novel (i.e., there is a closed set of objects and environments shared between training and evaluation). On the other side of the spectrum (\u2018strong generalization\u2019), the agent is tasked with rearranging new objects in new environments, never encountered during training. We recommend strong generalization as the default mode of operation. Regardless of where on the generalization spectrum a particular rearrangement problem falls, it is important to explicitly state the degree of overlap between prior experience or built-in knowledge (e.g. object CAD models, grasp demonstrations for objects), and the test-time environment. It is also possible to allow limited exploration of an unseen test-time environment before the task commences. An interesting question for future work lies in the quanti\ufb01cation of these different forms of prior experience and their impact on task performance under different generalization settings. 3.4. Comparisons and Evaluation The discussion in this section lays out a large number of potential decisions that can be made in evaluating an embodied system. From the choice of action abstraction, to the presence of dynamic and interactive components in the environment, these design choices create a spectrum of different levels of realism, and therefore a spectrum of dif\ufb01culties. Naturally, any comparative evaluation that aims to compare different methods must take these differing dif\ufb01culty levels into account: a method that uses the magic pointer abstraction is not directly comparable to one that uses a full physics simulation, and fair comparisons can only be performed at similar levels of abstraction and in similarly challenging environments. However, by reporting on these design dimensions and making clear the conditions under which each algorithm is evaluated, we can move closer to a setting where researchers can begin to put the performance of different methods in context, with evaluations that are at least conducted on tasks that vary along common axes of variation (action abstraction, sensor suite, interactability, complexity, and dynamicity). 4. Evaluation A rearrangement task requires an agent (or agents) to accomplish a many-to-many transformation of a scene, and evaluation of success will necessarily therefore require the design of composite metrics which capture the quality of an overall performance. We believe that simplicity and the desire for rank-ordering make it valuable to focus on a single primary metric, and we propose that task completion, measured as a percentage, has high generality and \ufb02exibility for that purpose. We de\ufb01ne completion as the percentage of binary target state tests concerning objects of interest which are successfully passed by the agent, while not doing any harm to other parts of the scene. We discuss the details required by this metric in this section. Most tasks will have a natural range of dif\ufb01culty among the objects which make them up, and so completion percentage will appropriately represent progression in agent performance. As tasks increase in complexity, requiring 9 \fmore objects to be moved, this metric will become increasingly continuous. Besides the primary completion metric, we strongly believe that secondary metrics should also be reported, and in fact that these will often be crucial in determining whether an agent has performance which is on the path to useful Embodied AI applications. While there are many possibilities, we believe that the most important additional metrics should capture aspects of the agent\u2019s ef\ufb01ciency. We make several concrete suggestions below. 4.1. Primary Metric: Task Completion A rearrangement task is de\ufb01ned in terms of goal locations for movable objects in a simulated 3D environment. As explained in Section 3, within our current scope we assume that all objects can be considered as rigid, or to consist of articulated rigid parts. Therefore the world state space S = S1 \u00d7 S2 . . . Sn is speci\ufb01ed by the set of body pose spaces Si = SE(3) for the n objects or parts present in a scene. A particular current scene state is s \u2208S, and the goal state for a task is denoted s\u2217. When an agent has \ufb01nished working on a task, we must therefore determine our completion metric by comparing the \ufb01nal state s of all objects with goal state s\u2217. The most straightforward approach is to fully decompose s\u2217in terms of a target pose s\u2217 i \u2208SE(3) for each individual object in absolute scene coordinates. The placement accuracy of each object would be evaluated via a norm N(si, s\u2217 i ) on the difference between each object\u2019s \ufb01nal and target pose. The details of this norm can be designed to match the emphasis of a task: \u2022 The norm could ignore rotation and be simply the translation between centers of mass. \u2022 The norm could combine translation distance with a measure of rotation difference, for instance the angle between the two poses expressed in axis-angle form. \u2022 An elegant metric to unify translation and rotation which is familiar from 2D object detection is the 3D intersection-over-union (IoU) between the overlapping volumes or convex hulls. Having decided on an error metric for individual objects, these values for multiple objects must be combined to obtain an overall score. It is tempting to average the individual object distances to obtain a mean accuracy measure. We believe that a better approach is to de\ufb01ne a threshold on the distance for each object, make a binary test for each one, and then to report an overall completion percentage which is the fraction of objects passing the threshold tests. This completion metric can take account of the fact that tasks will be heterogeneous, and there may be very different sensible tolerances on the locations of different objects. For instance, placing a book anywhere on a table may be acceptable, whereas a \ufb02ower must be placed within a narrow vase. We believe that completion also has desirable robustness properties which make sense in terms of task progression, where one or two misplaced objects out of many will not have a ruinous impact on an overall score, even if those objects are very far from their target locations. A crucial \ufb01nal issue to address with the completeness metric concerns its scope, because in most environments there could be many objects present which are not related to the task at hand. It seems clear that completion should only concern tests on objects of direct relevance, rather than all objects. However, should we report an agent as successful if it achieves the correct state of the target objects but at the cost of making a mess of the rest of the scene, or even breaking other objects? We argue no, since our goal is the development of methods with value in real-world robotics, and a useful robot must understand the scope of the interactions it should attempt. We believe that this is best addressed by requiring an agent to additionally pass a do no harm test in order to achieve a non-zero completion score. This test is a single binary predicate test which can be de\ufb01ned by the task designer in any way they choose. A simple choice consists of a logical \u2018and\u2019 of state tests on all non-task objects in the scene, checking that they have not been moved. The movement threshold for each object could be a simple rule such as requiring an IoU above a threshold between every object\u2019s starting and \ufb01nal poses. More sophisticated \u2018do no harm\u2019 tests could be tuned per object (since it may not matter if some non-task objects are moved) or could test other properties than pure object pose, such as the maximum acceleration or forces experienced by objects within the simulator. General evaluation in terms of scene predicates. A more general formulation of completeness will not necessarily take the form of one binary threshold test per object or part, but would de\ufb01ne a set of predicates Pj(s, s\u2217), each of which is a binary threshold test involving the individual \ufb01nal and target states of any number of objects. Overall completeness would be the percentage of these predicate tests which have been passed. The most obvious use for more general predicates is to allow for tests of the relative poses of objects. For instance, if a saucer must be placed on a table, and a cup on the saucer, one predicate would set a threshold on the relative pose of the saucer and the table, and another on the relative pose of the cup and saucer. As with individual object pose tests, these predicates could use just relative translation, or also angular measures. Each predicate test would be given a suitable speci\ufb01c success threshold. 10 \fOur focus in rearrangement is physical object manipulation, but de\ufb01ning completion in terms of a set of general logical predicate tests does in principle allow other nonphysical binary tests to be part of an overall performance measure \u2014 e.g. whether a logical switch in the simulator has been touched to put it into the correct on/off state. Evaluation programs. Evaluation of task completion should be carried out and reported automatically by the task simulator, and must therefore be implemented by an evaluation program. This program is passed the \ufb01nal and target state con\ufb01gurations once an agent has \ufb01nished working, and performs the required threshold tests to determine the overall completion percentage. The designers of agents should have a clear understanding of how evaluation is to take place, and therefore we propose that evaluation programs should be de\ufb01ned in a clear and public domain-speci\ufb01c language, which may eventually become a standard across different simulators. Generally it is not appropriate that all of the parameters of the precise tests which make up a particular task should be published, because an important aspect of rearrangement agents is to be able to deduce the goal state of a task from the speci\ufb01cation which could be provided in different forms as detailed in Section 3. General information such as IoU thresholds could be made available, or it could be considered part of the task for the agent itself to use background semantic knowledge to deduce suitable thresholds for each type of object. Emergent properties of simple evaluation metrics. Although the formulation of completion in terms of general predicate tests allows arbitrary \ufb02exibility in task speci\ufb01cation, we would like to point out some of the interesting emergent properties of simple completion speci\ufb01cations. These arise due to the properties of a physically simulated environment with a limited scope and range of possibilities. At \ufb01rst thought, it might seem that target locations speci\ufb01ed in relative terms will quickly be needed. For instance, if a saucer must be placed on a table, a cup on the saucer, and a spoon in the cup, the target location of each object could be speci\ufb01ed relative to the previous one. However, a well-de\ufb01ned threshold test in absolute coordinates could still capture the situation well, and in particular translationonly pose tests are often suf\ufb01cient. If the cup needs to achieve a height coordinate a few millimetres above the top of the table, in a task and environment where only a few objects are present the only way to achieve this might be to place it on the saucer, and if the centre of mass of the spoon needs to be somewhat higher still the only way to achieve this might be to place it in the cup. Similarly, if a key is to be placed in a lock, there will be no need to specify its target orientation, because within the physics simulator the only way to get the centre of mass of the key to the correct location will be at the orientation which \ufb01ts into the lock. When many objects must be packed into a tight space, but their order is unimportant, a simple threshold which is the same for all of the objects may work. For instance, if many books should be placed onto a bookshelf, each can be tested with a threshold based on the size of the whole shelf. The dif\ufb01culty of placing many books in sequence will naturally increase as they must be \ufb01t into the smaller and smaller space that physics allows. We believe that many rearrangement tasks where the placement of multiple objects involves physical coupling will share these emergent properties of incremental dif\ufb01culty from simple tests. Another example would be building a tower from blocks, where each block is tested in terms of target height. The \ufb01nal blocks will only be able to pass their evaluation tests if the lower blocks have been placed into a stable tower, and the early blocks must be placed precisely if a tall tower is to be built. Predicates based on natural language. It is worth noting that the need for an automatic evaluation program poses some challenges for rearrangement tasks speci\ufb01ed with free-form natural language. Effectively this means that we need a program that can recognize whether the state achieved by an agent satis\ufb01es a natural language predicate. In many cases, such a program can be dif\ufb01cult to handcraft. For example, the meaning of \u201con\u201d may not be easily reducible to simple rules of geometric relations \u2013 \u201ca cup on a table\u201d would be very different from \u201cclock on the wall\u201d. Thus even though simple evaluation programs are preferred, our formulation allows the evaluation program to take a more complex and less interpretable form, such as a neural network that has been trained to classify the \u201con\u201d relationship. Towards evaluation of real-world robotic agents. Despite our focus in this report on rearrangement in simulated environments, our general interests in Embodied AI mean that we have a long-term interest in whether the methods we propose are also applicable to real-world robotic agents. With regard to evaluation and the completion metric, this is certainly the case in principle, though automatic and accurate evaluation of real-world rearrangement presents huge technical challenges in anything but trivial tasks. Arbitrary objects would need to be tracked or instrumented in dif\ufb01cult situations of occlusion and contact. Building such systems will certainly be an area of important future research. 4.2. Secondary Metrics Although we propose task completion as a uni\ufb01ed primary metric, we believe that rearrangement simulators should also report additional measures of performance 11 \fwhich are important in judging whether an agent has realworld value. In particular, we argue for metrics which measure ef\ufb01ciency, both of the performance of the agent in the simulated environment and of its computational requirements. While multiple metrics can always be combined into single values via weighted addition or other formulae, this must be based on a choice of the relative importance of the different factors, and we believe that it is better to present the metrics separately so that potential users can make choices between agents based on their own criteria. An agent will usually have settings which allow performance metrics to be traded off against each other, and a sampling of these settings will produce a multi-objective performance manifold which can be displayed with a Pareto Front. Useful secondary metrics include: \u2022 Simulation time taken for an agent to report completion and stop action: this is in units of the time de\ufb01ned within the simulation, or the number of simulation \u2018ticks\u2019 required, with respect to which the agent program can take actions at a constant de\ufb01ned rate. \u2022 Simulated energy required by the agent to take all of the actions within an episode. Again this is purely measured within the physics simulation, via integration of all of the virtual physical work done by the agent to move objects and its own body, and should be something that a good simulator is able to calculate. We believe that minimising simulated energy is ultimately a powerful and highly general metric that encompasses many aspects of ef\ufb01ciency and smoothness in an agent\u2019s actions. \u2022 Computational measures of the agent program. How many FLOPS and how much memory does it need per \u2018tick\u2019 of the simulation, or to complete a whole episode? One way to measure this could be if the agent program is running on the same machine as the simulator, and requesting tick updates of the simulator when it is ready to deal with them: at what factor of \u2018realtime\u2019 in the simulator is the agent able to run? These measures of ef\ufb01ciency are especially critical as the aim of building intelligent rearrangement agents is ultimately progress in real-world Spatial AI, where agents which must run on the limited embedded computation platforms in robots or other devices, although these platforms will surely be cloud-connected in general and some highlatency computation could be carried out remotely [19]. Later more sophisticated computational measures should also encompass the degree to which an agent\u2019s computation and storage can be parallelized, distributed or layered in terms of latency. 5. Experimental Testbeds Here we summarize a set of experimental testbeds for rearrangement that we contribute with this report (see Figure 6). These testbeds span a spectrum of environment complexities, as well as agent navigation and manipulation capabilities. We order the testbeds roughly by environment scale and navigational requirements (in addition to manipulating objects). T1: bimanual sweeping. A pair of \ufb01xed-base robot arms sweep \u2018trash\u2019 objects (simulated as small cubes) from the \ufb02oor into a \u2018trash bin\u2019 container. Four third-person view camera and depth sensor pairs are positioned to observe the arms, and the position and velocity of the arm kinematic chain are also available. Both of the arms have spatulas as end effectors to sweep cubes from the \ufb02oor into the trash bin. This scenario is implemented in SAPIEN [76]. See Appendix A.1 for details. T2: table organization. A \ufb01xed-base robot arm (Universal Robots UR5e) with a parallel-jaw gripper (Robotiq 2F85) is tasked with rearranging a set of tabletop objects. The objects start in a random con\ufb01guration and need to be rearranged into a speci\ufb01c state on the table. A stationary third-person and wrist-mounted camera and depth sensor are available for perception. This task is also implemented in SAPIEN. See Appendix A.2 for details. T3: storing groceries. A \ufb01xed robotic arm manipulator (Franka Panda arm with Franka gripper) is tasked with picking up a randomly scattered set of grocery objects on a table and placing them into a constrained shelf space. The target location for all of the objects is de\ufb01ned as the same volume above the shelf, rather than a speci\ufb01c goal for each object. This leads to interesting emergent dif\ufb01culty as objects will be increasingly dif\ufb01cult to place within the volume as the shelf becomes more occupied, and the best solutions require planning of the best order and placement for moving all of the objects. The sensory suite includes color camera and depth sensors mounted on the wrist, hand and over-theshoulder, as well as proprioceptive sensors including joint encoders and forces. This scenario is instantiated in the RLBench task suite [37]. See Appendix A.3 for details. T4: room rearrangement. This scenario involves rearranging randomly placed household objects in a room and changing the state of the objects, such as opening/closing a cabinet. An example is shown in Figure 1. Identifying objects that have changed, inferring the state of the objects, planning a path for reaching and manipulating objects (e.g. manipulating an object might require moving a blocking object) are among the challenges of performing this task. The 12 \fFigure 6. Suite of experimental testbeds. We contribute a set of experimental testbeds for rearrangement, spanning a spectrum of environment complexities, and agent navigation and manipulation capabilities. From left to right: T1: bimanual sweeping, T2: table organization, T3: storing groceries, T4: room rearrangement, T5: house cleanup. agent uses a \u2018magic pointer\u2019 style manipulation. The level of the dif\ufb01culty of the task varies depending on the number of objects changed, their con\ufb01guration in the scene and the complexity of actions required to recover the goal con\ufb01guration. The scenario is instantiated in AI2-THOR [44]. Refer to Appendix A.4 for the details of this rearrangement scenario. T5: house cleanup. In this scenario, a mobile agent with a \u2018magic pointer\u2019-style manipulator is tasked with cleaning up a house. The agent must \ufb01nd randomly placed household objects, pick them up, and move them to a different speci\ufb01ed location. The agent carries camera and depth sensors for perception. As the task involves relocating objects between rooms, it involves longer-range navigation. This scenario is implemented in Habitat [62] (see Appendix A.5). 6. Why Rearrangement? The spectrum of possible rearrangement scenarios can be used to exercise and evaluate a broad set of agent abilities (see Figure 7). Different research communities may focus on analyzing and evaluating subsets of these abilities. Here, we summarize several types of abilities that the rearrangement task can evaluate. Perception. Evaluation of the agent\u2019s performance on subtasks such as object detection [48], rigid object state estimation and tracking [78], and localization [27] of the agent relative to objects connects with a breadth of research problems in the vision community. Perceptual abilities underlying these subtasks are important for an agent to identify object instances that can be moved, judge whether it is in a position that affords manipulation of an object, and to estimate the current state of the object (e.g. is the cabinet door open or closed?). Evaluating whether a target rearrangement has been achieved may also involve perception. Perceptual abilities underlie both traditional vision tasks as well as emerging tasks such as prediction of object physical properties to enable interaction with the object (e.g. \u201cis the bottle likely to break if it is dropped?\u201d or \u201cis the box light enough to be picked up or does it need to be pushed?\u201d), or prediction of affordance heatmaps for objects [55]. The degree to which these abilities are stressed is linked to the choice of agent sensors and environment complexity, and how sensory information can be used for action prediction. Navigation. Controlling an agent to achieve locomotion is required when performing rearrangement within larger environments. Consequently, rearrangement can be used to investigate agent navigation capabilities, a topic that is of interest to both the robotics and vision communities [8]. The formalization of navigation tasks for Embodied AI agents has been described in more depth by Anderson et al. [1]. We can view the rearrangement task as a generalization of the navigation task as the agent is required to change its own state relative to the environment (i.e. to navigate), as well as to change the state of the environment (i.e. to manipulate). Different choices of agent mobility and environment complexity lead to varying degrees of focus on collisionfree navigation and path ef\ufb01ciency within a rearrangement task. Manipulation. To successfully rearrange objects, an agent must grasp and manipulate the objects. Grasping and manipulation are rich research areas within robotics [54, 51]. Different choices on the manipulation complexity axis lead to different levels of abstraction for manipulation. High-level abstractions such as the \u201cmagic pointer\u201d manipulator reduce the importance of manipulation and may be more appropriate for focusing on perception. At a low level of abstraction, the rearrangement task can be used to investigate control policies for trajectory and grasp planning. Different choices of the number and types of objects to be rearranged (e.g. degree of instance variation in geometry and appearance), and varying the complexity of the environment (e.g. fairly open spaces to cluttered environments) can control the focus on manipulation. Memory. The rearrangement task also provides a testbed for investigating different types of memory and representation. As rearrangement consists of a sequence of object manipulations and/or agent navigations, the ability to store 13 \fPerception                            =  Navigation Manipulation Memory Planning    (1)                       (2) Communication Figure 7. Agent abilities harnessed by rearrangement. Illustration of the broad spectrum of abilities the rearrangement task can exercise. and recall information about the environment and the self is important. Investigating agent memory representations is a topic of great interest in the machine learning and Embodied AI communities [3, 17, 19]. Simultaneous Localization and Mapping (SLAM) research in robotics has produced algorithms which can incrementally and consistently build 2D or 3D scene representations from raw data from cameras and other sensors. These representations have gradually improved in geometric accuracy and completeness, and some SLAM systems now incorporate recognition and aim to build an explicit graph of interacting object instances which can be used to simulate and plan robot manipulation [74]. An important research issue going forward is to what extent challenging rearrangement tasks require agents with the ability to build this kind of explicit 3D scene representation designed into them, versus what can be achieved with the implicit representations used by black-box agents trained via machine learning. Planning. The rearrangement task requires planning a sequence of actions and reasoning about the pre-conditions and post-conditions for speci\ufb01c actions. For example, to set a dinner table, the agent needs to: go to the kitchen, open the cabinet door, get plates, place the bigger plate on the table \ufb01rst, and then place the smaller plate on top of the bigger plate. Doing the place settings for four people requires hierarchical decision making and sub-task ordering. Agent architectures that can successfully plan for such structured decision-making scenarios are an emerging focus in Embodied AI. Communication. The rearrangement task allows studying grounding of language to perception and action, a topic relevant to natural language researchers interested in Embodied AI [7]. By specifying the \u201cLanguageGoal\u201d as either a series of instructions or a description of the \ufb01nal state of the environment, the task allows researchers to go beyond grounding nouns such as \u201cchair\u201d and \u201ctable\u201d, to spatial relations between objects and how they are expressed in different languages (\u201ccup in cabinet\u201d, \u201cpainting on the wall\u201d), to actions (\u201cpick up\u201d vs \u201cput down\u201d vs \u201copen\u201d), as well as higher-level concepts such as \u201cset the table for four\u201d and \u201cclear the table and wash the dishes\u201d. A successful agent will also need to retain \u201ccommon sense\u201d and implicit knowledge about target arrangements that may not be precisely speci\ufb01ed in the language (e.g. what objects and where they should be placed for \u201cset the table for four\u201d). The rearrangement task can also be used to study communication for coordination between multiple agents and emergent behaviors. In summary, the rearrangement task provides a uni\ufb01ed platform for investigating methods to endow agents with the above agent capabilities. By making different choices on the complexity axes, we can emphasize speci\ufb01c capabilities (e.g. more abstract agent control schemes to focus more on planning) or combinations of capabilities. 7. Discussion Comparison to existing benchmarks. The rearrangement task presents a signi\ufb01cantly more dif\ufb01cult challenge than prior navigation-based benchmarks for Embodied AI. Speci\ufb01cally, the degree of physical interaction with the environment is inherently much greater in rearrangement. In fact, the mobile variant of the rearrangement task can be viewed as encompassing prior work on PointGoal navigation [1]. Furthermore, rearrangement requires more precise reasoning about objects, including their physical properties (e.g. placing one object inside another, pouring liquids, grouping similar items) and geometric constraints (e.g. arranging books on a bookshelf). As a result, the embodied agents developed for the rearrangement task will need 14 \fto be far more sophisticated than those of today, helping to advance the development of more effective world representations, sequential decision-making algorithms, perception techniques, task-oriented grasping methodologies, and physics simulations. Complex tasks and processes. Example tasks described in this document have largely consisted of rearrangement tasks in which the end goal is speci\ufb01ed directly, without guidance as to the intermediate stages that the environment must go through to reach the goal. However, many realworld tasks consist of complex processes that can be broken down into multiple sequential subtasks or subgoals. For example, cleaning the living room may consist of multiple subgoals corresponding to different parts of the room, or putting away groceries may be broken down by items that belong in the pantry, refrigerator, freezer, and counter. In the context of complex tasks, rearrangement can be viewed as sequentially addressing each individual subgoal. Many avenues for future work exist in this area, particularly in ordering subgoals and resolving dependencies between subgoals. Application to physical robotic systems. The ultimate goal of Embodied AI is to develop systems that perceive and act in physical environments \u2014 i.e. physical robots in physical worlds. We believe that the simulation environments, tasks, and evaluation procedures proposed in this document directly aid this goal. Development in simulation presents a number of important bene\ufb01ts, including access to orders of magnitude more data, more precise evaluation techniques, and more ef\ufb01cient experimental procedures, which will all fuel more rapid research progress. However, this work must take into account a number of potential downsides to simulation-based development \u2013 mainly the potential development of techniques whose performance in simulation does not transfer to real-world domains. To address the potential disconnect between simulated and real-world performance, we propose that physical robot variants of the rearrangement tasks be also developed in parallel. Robust evaluation of performance (i.e. automated techniques for assessing the difference between the current state and the goal) remains the most challenging component of such an implementation. To aid progress in this area, early variants of the rearrangement task could be based on either simple table-top domains, or on instrumented environments. Evaluation could be based on standard object datasets, similar to the YCB dataset [12]. While more general and scalable evaluation techniques will be needed in the long term, in the short term such experimental domains can be used to ensure strong correlation between simulated and real-world experiments, such that progress in simulation effectively translates to, or is predictive of, performance on real-world systems. Simulation \ufb01delity. The \ufb01delity of the simulator in terms of both the underlying physics and simulated sensing modalities will signi\ufb01cantly in\ufb02uence task complexity. When investigating the rearrangement task in simulation, with the aim to transfer a learned model to the real world, the gap between simulation and reality will impact the additional effort to achieve successful transfer. Actuation and sensing noise can be simulated to better approximate realworld robots, adding complexity to the task. We believe that current simulation platforms are well-equipped to simulate rigid body dynamics, optionally with noise models on important physical parameters. Ef\ufb01cient and accurate simulation of deformable objects and phenomena such as \ufb02uids remains challenging and is a good direction for future work. Future extensions. In this report, we propose a number of concrete instantiations of the rearrangement task. Importantly however, the proposed task leaves room for further extensions that would increase complexity and more comprehensively exercise the intelligence of the agent. Below, we discuss several extensions to the rearrangement task that are beyond the immediate horizon but that we believe are promising directions for future work. \u2022 Deformable objects: The ability to manipulate and rearrange deformable objects (e.g. clothing, towels, curtains, or bed sheets) has many practical applications in everyday environments. However, deformable objects introduce many challenges, including simulating deformations, de\ufb01ning appropriate goal speci\ufb01cations, and evaluating the similarity between the end result and the goal. These challenges are signi\ufb01cant, and thus preclude the inclusion of deformable objects in our initial formalization and instantiation of Rearrangement. This is an obvious practical extension for future years. \u2022 Transformation of object state: Many everyday tasks involve the transformation of object state. Examples include pouring water into a glass, chopping vegetables, heating a skillet, washing a dish, turning on an oven, or whisking an egg. These state changes are fundamentally different from rigid-body transformations. They i) require the incorporation of deeper world knowledge, ii) rely on causal reasoning methods to interpret, iii) involve the execution of complex actions beyond pick-and-place, and iv) necessitate separate procedures to specify and evaluate such scenarios. As a result, we do not address such tasks within the scope of the presented formulation, although the predicate-based evaluation that we propose allows for natural future extensions of the rearrangement task in this important direction. \u2022 Multi-agent rearrangement: In this report, we focused on single agents performing the rearrangement task. 15 \fWe believe that extending the rearrangement task to multiple agents is natural and interesting. There are three types of agents that may be present: cooperative agents, adversarial agents, and non-participating agents. Cooperative agents can coordinate to achieve the task more ef\ufb01ciently than a single agent. Certain scenarios may require two agents to cooperate to accomplish a subgoal, such as moving a heavy sofa. Adversarial agents can actively prevent the agent performing rearrangement from achieving the task. These agents may either provide inaccurate or false information, or they may actively disrupt the agent or the environment. Non-participating agents are present but do not actively work towards or against the rearrangement-performing agent(s). The number of agents of each type may vary. There is also a spectrum of communication mechanisms that may be available to agents. \u2022 Interactions with human users: All examples in our work are ultimately driven by the desire to develop systems that are able to assist users in a wide variety of everyday tasks. Toward this goal, future work should examine human interactions with rearrangement systems. Example topics include intuitive real-time interactions, dialog-based interfaces, and active learning from human input, among others. We leave these and other interesting extensions of rearrangement as promising directions for future work. Acknowledgments We thank Ankur Handa, Camillo J. Taylor, Deepak Pathak, Dieter Fox, Dmitry Berenson, George Konidaris, Jana Kosecka, Josh Tenenbaum, Ken Goldberg, Kostas Daniilidis, Kristen Grauman, Leslie Kaelbling, Lucas Manuelli, Matthew T. Mason, Niko S\u00a8 underhauf, Oliver Brock, Pete Florence, Peter Corke, Pieter Abbeel, Raia Hadsell, Richard Newcombe, Russ Tedrake, Saurabh Gupta, Shubham Tulsiani, Siddhartha Srinivasa, Stefanie Tellex, Tom\u00b4 as Lozano-P\u00b4 erez, and Vincent Vanhoucke for their feedback on a draft of this report. We thank Joseph Lim for participating in early discussions. We also thank the AI2-THOR, Habitat, RL-Bench, and SAPIEN teams for releasing the experimental testbeds described in this report."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2006.13171v2",
+                    "title": "ObjectNav Revisited: On Evaluation of Embodied Agents Navigating to Objects",
+                    "abstract": "We revisit the problem of Object-Goal Navigation (ObjectNav). In its simplest\nform, ObjectNav is defined as the task of navigating to an object, specified by\nits label, in an unexplored environment. In particular, the agent is\ninitialized at a random location and pose in an environment and asked to find\nan instance of an object category, e.g., find a chair, by navigating to it.\n  As the community begins to show increased interest in semantic goal\nspecification for navigation tasks, a number of different often-inconsistent\ninterpretations of this task are emerging. This document summarizes the\nconsensus recommendations of this working group on ObjectNav. In particular, we\nmake recommendations on subtle but important details of evaluation criteria\n(for measuring success when navigating towards a target object), the agent's\nembodiment parameters, and the characteristics of the environments within which\nthe task is carried out. Finally, we provide a detailed description of the\ninstantiation of these recommendations in challenges organized at the Embodied\nAI workshop at CVPR 2020 http://embodied-ai.org .",
+                    "authors": "Dhruv Batra, Aaron Gokaslan, Aniruddha Kembhavi, Oleksandr Maksymets, Roozbeh Mottaghi, Manolis Savva, Alexander Toshev, Erik Wijmans",
+                    "published": "2020-06-23",
+                    "updated": "2020-08-30",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.RO"
+                    ],
+                    "main_content": "Introduction Life is a treasure hunt. Olivia Wyndham Imagine walking up to a home robot and asking \u2018Hey \u2013 can you go check if my laptop is on my desk? And if so, bring it to me.\u2019 In order to be successful, such a robot would need a wide range of skills \u2013 visual perception, language understanding, episodic memory construction (what is where?), reasoning/planning, navigation, etc. It is clear that one essential capability is the ability to search for and navigate to an object (\u2018\ufb01nd laptop\u2019) \u2013 that is the focus of this document. Internet to Embodied AI. Stepping back, the AI research community is undergoing a paradigm shift from tasks involving static datasets of images or pre-recorded experiment trials to tasks that rely on the deployment of an active, embodied agent in the world (i.e. a mobile robot platform) or in simulation (i.e. a virtual robot). Consequently, we have seen a surge of activity in Embodied AI \u2013 a term referring to AI agents that act within the constraints of their embodiment in either physical or virtual three-dimensional environments. This umbrella term encompasses a variety of tasks such as navigation in realistically structured interiors, manipulation of objects, and execution of natural language instructions. At a time of such vigorous interest, the need to unambiguously de\ufb01ne and systematically evaluate such tasks emerges as a fundamental requirement for research progress. Anderson et al. [2] recently categorized embodied navigation tasks via the description of the navigation goal: PointGoal (a point in space), ObjectGoal (a semantically distinct object instance), and AreaGoal (a semantically distinct area). In addition, this effort resulted in a set of recommendations, such as the \u2018success weighted by inverse path length\u2019 (SPL) metric for evaluation and the use of a \u2018stop\u2019 action by agents to indicate completion of an episode. The effort of Anderson et al. [2] was highly fruitful and led to broad convergence \u2013 as evidenced by studies [3, 4] and challenges [5] organized on the task. However, as the community begins to show increased interest in ObjectGoal navigation, a number of different often-inconsistent interpretations of this task are emerging. In particular, there are a number of seemingly low-level but highly consequential details that can vary \u2013 such as the de\ufb01nition of what constitutes success when navigating towards a target object, the details of the agent\u2019s embodiment parameters (in terms of possible actions and observations), and the characteristics of the environments within which the task is carried out. We believe the community would bene\ufb01t from \ufb01nergrained characterization of the ObjectGoal task. Therefore, we established a multi-institutional working group of researchers in this area. This document summarizes our consensus recommendations. Speci\ufb01cally, we provide a set of recommendations on ObjectNav augmenting those in Ander1 arXiv:2006.13171v2  [cs.CV]  30 Aug 2020 \fTarget Object\u2019s Center 1 m Success area Navigable  points with  object visible Shortest path  (geodesic distance) Figure 1: ObjectNav Evaluation: The goal object, in this case a monitor, is highlighted in pink. We consider all points within 1m to the object surface (highlighted as a blue circle) as potential success goal poses. From this area, we consider only navigable locations from where the object can be seen with proper orientation of the camera (highlighted in green). These points cumulatively form the \u2018success zone\u2019, locations where if the agent called \u2018stop\u2019 the episode would be considered successful. The blue arrows shows the shortest path from the agent\u2019s (random) starting location and the closest point in the success zone. son et al. [2] and a detailed description of the instantiation of these recommendations in challenges organized at the Embodied AI workshop at CVPR 2020 [1]. 2. ObjectNav Task De\ufb01nition In its simplest form, ObjectNav is de\ufb01ned as the task of navigating to an object (speci\ufb01ed by its category label) in an unexplored environment. In particular, the agent is initialized at a random pose in an environment and asked to \ufb01nd an instance of an object category, e.g. \u2018chair\u2019, by navigating to it. This task is ostensibly simple to de\ufb01ne. However, this high-level description is signi\ufb01cantly under-speci\ufb01ed and different instantiations can have different experimental outcomes. In particular, one ought to precisely de\ufb01ne: Object Finding & Evaluation: Why are we trying to \ufb01nd objects and what constitutes a successful navigation? Embodiment: What is the agent speci\ufb01cation, in terms of actions and observations? Environments: What environments (object categories, scenes) should be used? In the following, we provide general guidelines for the above details, with concrete instantiations in Habitat and RoboTHOR 2020 challenges in Sec. 3 and Sec. 4 respectively. 2.1. Object Finding and Evaluation It is important to start with the question \u2013 why is the agent trying to \ufb01nd objects? Our consensus stand is that \ufb01nding (or navigating to) objects is the \ufb01rst step to interacting with objects. It is a fundamental perception and navigation milestone that forms a core building block of any intelligent embodied agent interacting with the world (opening containers, picking up objects, opening doors along the way, clearing obstacles out of a navigation path, responding to questions [6,7], etc.). In order to enable self-contained evaluation disentangled from any speci\ufb01c downstream task, we propose evaluation protocols that measure the agent\u2019s ability to navigate to the object and not object detection or framing of the object in the agent\u2019s view at the end of the episode. At a high-level, we recommend a similar breakdown of evaluation as advocated in prior work [2,3,5] \u2013 measuring success (did the agent navigate to the goal object?) and ef\ufb01ciency (how ef\ufb01cient was the path it took compared to an optimal path?). These notions are re\ufb02ected in our ObjectNav success criterion recommendation (see Fig. 1). An agent has successfully navigated to the goal object iff all of the following conditions are satis\ufb01ed: \u2022 Intentionality: The agent emits a STOP action at a location, indicating that it \u2018believes\u2019 it has reached the object. 2 \f\u2022 Validity: The agent is at a location that is navigable given the agent\u2019s dimensions and the environmental constraints (true by construction during actual navigation but not necessarily true for a randomly sampled point on the mesh). \u2022 Proximity: The agent\u2019s location, represented as a point (e.g. center of mass), is within a Euclidean distance shell around the object surface. The exact size of this shell will vary depending on the size of the robot and its reach; a reasonable starting point may be r = 1m, which may be tightened over the years as the community makes progress on the task. The object surface may be approximated by an axis aligned or oriented 3D bounding box for computation ef\ufb01cacy. \u2022 Visibility: The object is within the \ufb01eld of view of the agent. The exact speci\ufb01cation of \u2018within\u2019 will depend on the agent\u2019s sensor speci\ufb01cation (\ufb01eld of view, pose), actions space (whether direct changes to camera roll/pitch/yaw are allowed), and the kinds of objects being considered (small table-top objects or larger furniture objects). If visibility criteria is enforced, we recommend starting with a fairly lenient criteria of at least 1 pixel corresponding to the object within frame, which may be tightened over the years. An alternative is to not require the agent to be actually viewing the object at the stopping location and operate under oracle-visibility \u2013 i.e., assume access to an oracle that is able to optimally frame the object in the agent\u2019s camera view without changing the agent\u2019s location \u2013 i.e. by in-place turning the agent, making changes to roll/pitch/yaw of the camera, etc. In this setting, the goal of the agent is to reach locations from where the the goal object can be viewed, but not necessarily visible from the pose the agent stops in. Oracle-visibility is a proxy for \u2018the agent is close enough to interact with the object and the navigation module may now hand control over to an interaction module\u00e2 \u02d8 A\u00b4 Z. This criterion is more suitable for scenarios that the focus is on path \ufb01nding instead of both path \ufb01nding and object framing. The blue arrow in Fig. 1 shows the shortest path from the agent\u2019s (random) starting location and the closest point in the success zone. ObjectNav-SPL may be de\ufb01ned analogous to PointNav-SPL. The key difference is that the shortest path is computed to the object instance closest to the agent start location. Thus, if an agent spawns close to \u2018chair1\u2019 but stops at a distant \u2018chair2\u2019, it will succeed (because it found a \u2018chair\u2019), but at a low SPL (because the agent path is much longer compared to the oracle path). Concretely, for the ith episode, Si denotes navigation success (Si = 1 if and only if the agent has succeeded as de\ufb01ned above), li denotes the length of the shortest path to the closest instance of the object, and pi denotes the length of the actual path taken by the agent. The metric then reads: SPLi = Si \u00b7 li max(pi, li) (1) The \ufb01nal evaluation metric is the average over all episodes: SPL = (1/N) P i SPLi. 2.1.1 Known Issues with SPL We note that there are a few issues in the SPL metric that appear to be understood by domain experts but may not be widely known. We acknowledge and describe them below for the sake of broad dissemination: \u2022 All failures are not equal. A navigation episode is deemed unsuccessful if the agent stops outside a predetermined radius r around the goal. However, this results in no distinction between gross failures and minor mistakes, and gives no indication of partial progress. An agent navigating 30m to reach the goal and stopping (r+\u03f5)-distance from it vs one that refuses to move from the start position are equally unsuccessful, both achieving an SPL of 0. \u2022 High Variance. The binary nature of success introduces high variance in the average SPL computation, SPL = (1/N) P i SPLi. As a thought experiment, consider N nearly identical episodes \u2013 i.e., li max(pi,li) are nearly identical \u2200i. However, due to (say) actuation noise, the agent\u2019s stopping behavior is stochastic \u2013 sometimes the agent stops just outside the success criteria (r + \u03f5) and sometimes just inside (r \u2212\u03f5); i.e., Si can be modeled as a Bernoulli random variable with probability p. In this scenario, the paths traversed by the agent are nearly identical so intuitively this should result in V ar(SPL) \u22480. However, in fact, SPL has a high variance of p \u00b7 (1 \u2212p) \u00b7 \u0002 (1/N) P i SPLi \u00032. While this is an example exaggerated to illustrate a point, we do indeed observe in practice high variance in SPL estimates, particularly in settings with noisy actuations. This results in the need to evaluate on an unnecessarily large set of episodes to gain con\ufb01dence in the SPL sample estimates. \u2022 No penalty for turning. SPLi only accounts for the distance traveled by the agent along the path it takes. On the one hand, this is a good idea because the length of this path is not tied to the agent\u2019s action space and can be compared across different robots. However, one potentially unintended side-effect is that pi is unaffected by an agent rotating/turning in-place. This means 3 \f# of scenes # of target categories # of instances per category # of instances per scene per category Sim/Real min/max geodesic distance (m) # of episodes Habitat [3] 90 21 \u223c397 4 Sim 1/30 2 803K RoboTHOR [8] 89 12 \u223c20 1 Both 0.71/16.8 34K Table 1: Environments. We provide statistics for different environments that are used for ObjectNav challenges. that SPL does not penalize the agent turning in place. Thus, an agent could always construct a panorama by turning 360\u25e6before moving. An alternative would be to measure the number of actions, or energy used by the agent [9]. A set of path comparison metrics in the context of VLN are presented by Jain et al. [10]. \u2022 No cross-dataset comparison. Different length paths provide different degrees of sloppiness available to the agent \u00e2 \u02d8 A\u02c7 T short paths require a strict adherence to achieve high SPL, while long paths allow signi\ufb01cant deviations while still achieving high SPL. See e.g. Fig. 6 in [11]. Thus, SPL cannot be used for comparisons across different datasets or different portions of a dataset with signi\ufb01cantly different optimal path lengths. A similar issue exists for other tasks such as object detection, when for example performances on small and large objects are compared. In this document, we acknowledge these shortcomings of SPL but consider proposing a new navigation evaluation metric as orthogonal to characterizing ObjectNav. Our recommendation is that whatever new metrics the community develops in the future should replace SPL for ObjectNav as well. 2.2. Embodiment Embodiment is an important part of the task insofar as we are to execute a series of realistic control actions to navigate to the object. As such, we strive to have actions that do not trivialize the problem from a decision-making standpoint (e.g. \u2018teleport to kitchen\u2019), while at the same time we leave low-level control challenges (e.g. torque-based control of joints of an articulated robot) out of scope. The motivation behind this design decision is to bring focus on the visual and decision making aspects of the problem. In particular, we recommend using a discretized version of differential drive with a set of discrete actions. As such, we come short of recommending modeling the full kinematics and dynamics of a robotic system. However, we do not exclude this possibility in future incarnations of the ObjectNav task. On the observation side, we recommend modeling realistic sensors, as commonly used both in the robotics and computer vision communities, without adding the challenge of multi-modal sensor fusion. For this, we recommend using color camera and depth sensors, which may be assumed to be synchronized, and localization sensors (e.g. noisy egomotion sensors or GPS+Compass). 2.3. Environments First and foremost, a target object is to be visually recognized while an agent is moving around the environment (as opposed to being perceived by non-visual sensors). For a complete task de\ufb01nition, a set of object classes should be pre-speci\ufb01ed (leaving open-world recognition out of scope, at least initially). Note that we recommend working with classes and not instances \u2013 i.e. \u2018\ufb01nd a chair\u2019 not \u2018\ufb01nd this speci\ufb01c chair\u2019 \u2013 because this gives a more general understanding of recognition and navigation capabilities across common objects. While we take no philosophical stand on outdoor vs indoor navigation, we start with establishing benchmarks on indoor navigation due to the datasets available and so that the set of object classes is more limited than the one used on general object detection, e.g. COCO [12]. To the extent possible, environments should represent the diversity and complexity of the real world. We contend that it might be useful for sim2real transfer for the distribution of environments to be semantically representative of the real world \u2013 i.e., object and spaces should be laid out in simulation as we might expect them to be in the real world. As such, we recommend using large sets of photo-realistic scans of real environments. These should have real sizes, layouts, and most importantly represent real interiors with high visual \ufb01delity. Note that ObjectNav is performed in novel (previously unseen) environments. As such, it is important that the environments used for training and testing come from the same distribution of buildings, e.g. homes. All of these consideration overwhelmingly favor 3D scans of real spaces, rather than synthetically generated environments, often based on video game engines, such as [13\u201315]. Currently, there are several existing 3D environment datasets which are appropriate for ObjectNav. Matterport3D. Matterport3D (MP3D) [16] consists of scans of 90 indoor spaces, with a standard split of train/val/test prescribed by Chang et al. [16]. The scenes include not only private homes, but community and com4 \fYour Agent Find stool Observations RGB Depth Goal Action Photo-realistic 3D scene GPS &  Compass Figure 2: Habitat 2020 Challenge agent sensor and action space illustration. mercial buildings such as churches, universities, libraries, and of\ufb01ce spaces. MP3D contains 40 annotated categories, including objects (e.g. doors), and \ufb02oor-plan elements (e.g. counter tops). A description of categories appropriate for ObjectNav is provided in Sec. 3. Gibson. Another set of virtualized 3D spaces is Gibson [17]. This dataset currently contains 572 scanned spaces, of which a subset has been semantically labeled with objects classes. In particular, Armeni et al. [18] provide object segmentations and labels on 35 homes. This dataset provides a 25/5/5 split for training, validation, and testing. Note that many of the homes have multiple \ufb02oors, which could be treated as disjoint spaces. We suggest using objects like \u2018bed\u2019, \u2018chair\u2019, \u2018microwave\u2019, \u2018refrigerator\u2019, \u2018table\u2019, \u2018toilet\u2019, \u2018oven\u2019, \u2018tv\u2019, \u2019sofa\u2019 as ObjectNav goals. AI2-THOR. AI2-THOR [19] consists of a set of near-photorealistic scenes that are designed by 3D artists. AI2-THOR currently includes two frameworks iTHOR and RoboTHOR. iTHOR is a synthetic platform mainly designed for interacting with objects and changing their state based on the physical properties of the objects and scenes. RoboTHOR consists of a set of scenes in simulation as well as the real world. The main purpose of RoboTHOR is to evaluate the generalization of models from simulation to the real world. In total, iTHOR and RoboTHOR currently have 209 scenes and 115 object categories which can be repositioned programatically to generate new con\ufb01gurations in the scenes. Both iTHOR and RoboTHOR have been used as platforms for the ObjectNav task [8,20,21]. 3. Habitat 2020 Challenge Habitat [3] is a simulation platform for research in embodied AI. Habitat enables training of such embodied AI agents (virtual robots and egocentric assistants) in a highly photorealistic & ef\ufb01cient 3D simulator, before transferring the learned skills to reality. The Habitat challenge is an annual autonomous navigation challenge that aims to benchmark and accelerate progress in embodied AI. In this section, we describe the choices made by the organizers of Habitat Challenge 2020 for the ObjectNav track. Scenes. The Habitat 2020 challenge uses 90 scenes from the Matterport3D dataset described in [16]. The choice was motivated by the desire to have photo-realistic virtualizations of real environments along with rich semantic segmentations. These spaces have about 2 \ufb02oors and 15 rooms, on average. We use the standard train/val/test splits as prescribed by Chang et al. [16]. Note that as in recent works, there is no overlap between train, val, and test scenes. Objects. As described in the previous section, MP3D contains 40 annotated categories including objects (e.g. doors), and architectural elements (e.g. counter tops). We manually select a subset of object categories by excluding categories that are not visually well-de\ufb01ned (e.g. doorways, windows, and architectural elements such as walls, \ufb02oors, and ceilings). In total, our subset contains 21 categories with 8,349 instances in 71 environments, with an average of 397.5/117.59 categories/instances per environment for the training split. We provide additional statistics across dataset splits in Figure 3. The full list of object categories we use is \u2013 \u2018chair\u2019, \u2018table\u2019, \u2018picture\u2019, \u2018cabinet\u2019, \u2018cushion\u2019, \u2018sofa\u2019, \u2018bed\u2019, \u2018chest_of_drawers\u2019, \u2018plant\u2019, \u2018sink\u2019, \u2018toilet\u2019, \u2018stool\u2019, \u2018towel\u2019, \u2018tv_monitor\u2019, \u2018shower\u2019, \u2018bathtub\u2019, \u2018counter\u2019, \u2018\ufb01replace\u2019, \u2018gym_equipment\u2019, \u2018seating\u2019, \u2018clothes\u2019. ObjectNav Episode Dataset. We create an ObjectNav episode dataset using the above categories. An episode is de\ufb01ned by the unique id of the scene, the starting position (x, y, z) and orientation of the agent (\u03b8), and the goal category label (e.g. \u2018chair\u2019). Additional metadata such as the Euclidean and geodesic distance along the shortest path (GDSP) from the start position to the goal position, and all valid success zone locations with viewpoints are also included. To determine locations that form the success zone, we follow the recommendations laid out in Sec. 2 \u2013 the location must be navigable (true by construction during actual navigation but not necessarily true for a randomly sampled point on the mesh), the location must be within 1m of the oriented bounding box of the object instance or inside of the bounding box, and object must be visible from that location under oracle-visibility. To make the computation of success and distance to goal ef\ufb01cient, we provide such positions (in the success zone of each goal object) within the episode speci\ufb01cation, and call them \u2018valid viewpoints\u2019. These viewpoints were sampled with an Ragent 2 grid on the \ufb02oor. If no locations on the \ufb02oor satisfy these constraints for a particular object instance, we do not include that instance in our dataset. Finally, we restricted the maximum action length of the shortest action path to 750 and checked the ratio of geodesic to Euclidean distance is at least 1.05 to rule out 5 \fFigure 3: Statistics of ObjectNav episodes in the Habitat 2020 Challenge (top: train, bottom:val). Within each row from left: distribution over Euclidean distance between start and goal, distribution over geodesic distance along shortest path between start and goal, and distribution over the ratio of geodesic to Euclidean distance. easy episodes. Embodiment. The experiments by Savva et al. [3] and the Habitat 2019 Challenge [5] modeled the agent as an idealized cylinder of radius Ragent = 0.1m and height 1.5m. As shown in Fig. 2, we con\ufb01gure the agent to match the LoCoBot robot as closely as possible. Speci\ufb01cally, we con\ufb01gure the simulated agent\u2019s base radius and height to be 0.18m and 0.88m respectively. Action Space. The action space for the agent consists of 6 actions: move-forward 0.25m, turn-left 30\u25e6, turn-right 30\u25e6, look-up 30\u25e6, look-down 30\u25e6and STOP. Note that in contrast to [3,5], we increased the angles associated with turn-left and turn-right actions from 10\u25e6to 30\u25e6degrees due to the concerns described in [4]. Sensor Speci\ufb01cation. The agent is equipped with an RGBD camera and a GPS+Compass sensor; both sensors are \u2018idealized\u2019 in the sense that no sensor-noise is injected beyond the reconstruction artifacts that may exist already in the underlying 3D data. We attempt to match the camera speci\ufb01cation in simulation to the Azure Kinect camera. Speci\ufb01cally, we set the camera\u2019s horizontal \ufb01eld of view to 79\u25e6, resolution to VGA (height \u00d7 width = 480 \u00d7 640 pixels), and depth sensing clipped to a minimum and maximum of 0.5m and 6m respectively. The GPS+Compass sensor provides relative location (x, y, z) and heading (azimuth angle) w.r.t. the agent\u2019s start location (origin) and heading (0\u25e6). Thus, the agent\u2019s spawn location and pose establishes a coordinate system for that episode. Collision Dynamics and \u2018Sliding\u2019. In prior work [3, 5], when the agent takes an action that results in a collision, the agent slides along the obstacle as opposed to stopping. 6 \fSt St+1 0.28m 0.43m SPL Path Sliding path Figure 4: Sliding behavior leading to \u2018cheating\u2019 agents. At time t, the agent at st executes a forward action, and slides along the wall to state st+1. The resulting straight-line path (used to calculate SPL) goes outside the environment. Figure and caption from [4]. This behavior is prevalent in video game engines as it allows for smooth human control; it is also enabled by default in MINOS [22], Deepmind Lab [13], AI2 THOR [19], and Gibson [17] v1. Recent work by Kadian et al. [4] has shown that this behavior enables \u2018cheating\u2019 by learned agents. Speci\ufb01cally, as illustrated by an example in Fig. 4, the agent exploits this sliding mechanism to take an effective path that appears to travel through non-navigable regions of the environment (like walls). Such policies fail disastrously in the real world where the robot bump sensors force a stop on contact with obstacles. To rectify this issue, we con\ufb01gure Habitat-Sim to disable sliding on collisions. Success criteria. An agent navigation trajectory is considered successful if the STOP action was called within 0.1m geodesic distance from a valid \u2018viewpoint\u2019 of the goal object as de\ufb01ned in the Episode Dataset description (see Fig. 1). A complete task con\ufb01guration is available at: https:// github.com/facebookresearch/habitat-challenge/ blob/challenge-2020/habitat-challenge-data/ challenge_objectnav2020.local.rgbd.yaml which we hope will help prevent inconsistencies. 3.1. Challenge Phases Minival phase The purpose of this phase is sanity checking \u2013 to con\ufb01rm that an EvalAI remote evaluation matches the results achieved by participants locally. Each team is allowed a maximum of 30 submissions per day for this phase. Test Standard phase The purpose of this phase is to serve as the public leaderboard establishing the state of the art. This is what should be used to report results in papers. Each team is allowed a maximum of 10 submissions per day for this phase. Test Challenge phase This phase will be used to decide challenge winners. Each team is allowed a total of 5 submissions until the end of the challenge submission phase. Figure 5: RoboTHOR scenes in simulation and the physical world have identical object layouts. The disparities in appearance and control dynamics make the transfer from simulation to the real world challenging. Results on this split will be made public at the Embodied AI workshop at CVPR. 4. RoboTHOR 2020 Challenge RoboTHOR [8] is a platform to develop and test embodied AI agents with corresponding environments in simulation and the physical world (Fig. 5). The aim of the RoboTHOR 2020 challenge is to develop methods in simulation that generalize to real world environments. The complexity of environments in RoboTHOR along with disparities in appearance and control dynamics between simulation and reality pose signi\ufb01cant and novel challenges to participants. Scenes. RoboTHOR includes 89 scenes in total: 75 simulated scenes used for training, 4 real/simulated scenes (referred to as test-dev) used for validation, and 10 real scenes used for the challenge (referred to as test-challenge). While the overall dimensions of all simulation environments in RoboTHOR is consistent, there is no overlap between the interior wall layouts across these three sets. This imposes a further challenge on agents generalizing to unseen layouts in the validation and challenge scenes. Objects. Scenes in RoboTHOR are populated with objects drawn from an object asset library. There are 731 unique object instances in RoboTHOR across 43 object categories: 11 classes of furniture (e.g. sofa and TV stand) and 32 classes of small objects (e.g. mug and laptop). 14 of these 43 categories are guaranteed to be found in all scenes. We use 12 of these categories as our test targets (we ignore very small objects such as remote control). We distribute the objects in the scenes as uniformly as possible to reduce the location bias of objects. Importantly, object instances in the train, validation and challenge sets have no overlap. This requires agents to generalize to unseen object instances when evaluating on the validation and challenge environments. 7 \f0 5 10 15 0 500 1000 1500 2000 2500 RoboTHOR Number of episodes 0 2 4 6 8 0 500 1000 1500 2000 0.25 0.50 0.75 1.00 0 2000 4000 6000 8000 10000 12000 Figure 6: Target distance distribution in RoboTHOR. The distribution of Geodesic distances (left), Euclidean distances (middle) and their ratio (right) from the start locations to the targets. Agent. We use the LoCoBot robot for this challenge with a few physical modi\ufb01cations. We made the agent taller (0.88m) to be able to observe objects on the tables and high surfaces. We have also replaced the camera by an Azure Kinect camera which has a wider \ufb01eld of view of 79\u25e6and captures higher quality images. The agent in simulation exactly matches the modi\ufb01ed LoCoBot. The API for controlling the robot is identical across both simulated and the real platforms. The sensors available during training are RGB and depth images as well as localization and orientation sensors. We consider two sensor settings during evaluation: (1) RGB image only and (2) RGB and depth images. The action set is identical to the one de\ufb01ned in Section 3. Object visibility. We consider a navigation episode successful if the target object is within 1m and is in the \ufb01eld of view of the camera. In simulation this information is provided by the agent controller. In the real experiments, we localize the agent by a set of Super-NIA-3D localization modules to measure the distance and manually check if the object is visible in the image. 4.1. Challenge Phases The RoboTHOR challenge has three phases: (1) Train in simulation (2) Test-Dev in simulation and real (3) Testchallenge in real. In all phases, the robot starts at a randomly chosen location and it is provided with the target object category. Phase 1. This phase involves only training using the 75 simulated training scenes. In our setup, 60 scenes are used for training the models, and the rest of the scenes are used for validation. We provide a dataset of starting points and targets. The dataset is distributed across easy, medium and hard cases, where the dif\ufb01culty is chosen based on the geodesic distance to the target. The distribution of the distances are shown in Fig. 6. The participants can use these points or their generated points for training. Phase 2. This phase involves testing the models on 4 scenes in the physical space and their counterparts in simulation. This enables the participants to check how well the discrepancies in simulation and the real world are handled by their model. Phase 3. The models are tested on a blind test set of 10 real scenes. These scenes do not have any overlap with the training and validation scenes in terms of wall layout and object instances. The target categories are the same as the ones used for training and validation. This phase evaluates how well the models generalize (1) from simulation to real, (2) to unseen scenes and targets. 5."
+                }
+            ]
+        },
+        "edge_feat": {}
+    }
+}
\ No newline at end of file