H. Hu and Y. M. Lu, "Universality Laws for High-Dimensional Learning with Random Features," Submitted. arXiv:2009.07669 [cs.IT]Abstract
We prove a universality theorem for learning with random features. Our result shows that, in terms of training and generalization errors, the random feature model with a nonlinear activation function is asymptotically equivalent to a surrogate Gaussian model with a matching covariance matrix. This settles a conjecture based on which several recent papers develop their results. Our method for proving the universality builds on the classical Lindeberg approach. Major ingredients of the proof include a leave-one-out analysis for the optimization problem associated with the training process and a central limit theorem, obtained via Stein's method, for weakly correlated random variables.
O. Dhifallah and Y. M. Lu, "A Precise Performance Analysis of Learning with Random Features," Submitted. arXiv:2008.11904 [cs.IT]Abstract
We study the problem of learning an unknown function using random feature models. Our main contribution is an exact asymptotic analysis of such learning problems with Gaussian data. Under mild regularity conditions for the feature matrix, we provide an exact characterization of the asymptotic training and generalization errors, valid in both the under-parameterized and over-parameterized regimes. The analysis presented in this paper holds for general families of feature matrices, activation functions, and convex loss functions. Numerical results validate our theoretical predictions, showing that our asymptotic findings are in excellent agreement with the actual performance of the considered learning problem, even in moderate dimensions. Moreover, they reveal an important role played by the regularization, the loss function and the activation function in the mitigation of the "double descent phenomenon" in learning.
H. Hu and Y. M. Lu, "Asymptotics and Optimal Designs of SLOPE for Sparse Linear Regression," Submitted. arXiv:1903.11582 [cs.IT]Abstract
In sparse linear regression, the SLOPE estimator generalizes LASSO by assigning magnitude-dependent regularizations to different coordinates of the estimate. In this paper, we present an asymptotically exact characterization of the performance of SLOPE in the high-dimensional regime where the number of unknown parameters grows in proportion to the number of observations. Our asymptotic characterization enables us to derive optimal regularization sequences to either minimize the MSE or to maximize the power in variable selection under any given level of Type-I error. In both cases, we show that the optimal design can be recast as certain infinite-dimensional convex optimization problems, which have efficient and accurate finite-dimensional approximations. Numerical simulations verify our asymptotic predictions. They also demonstrate the superiority of our optimal design over LASSO and a regularization sequence previously proposed in the literature.
Y. M. Lu, "Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Gaussian and Random Orthogonal Ensembles," IEEE Transactions on Information Theory, vol. 67, no. 12, pp. 8264-8272, 2021. arXiv:2101.07464 [cs.IT]Abstract
This paper proposes a new algorithm, named Householder Dice (HD), for simulating dynamics on dense random matrix ensembles with translation-invariant properties. Examples include the Gaussian ensemble, the Haar-distributed random orthogonal ensemble, and their complex-valued counterparts. A "direct" approach to the simulation, where one first generates a dense n×n matrix from the ensemble, requires at least O(n2) resource in space and time. The HD algorithm overcomes this O(n2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(nT2), respectively, with T being the number of iterations. When T≪n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint. Numerical results demonstrate the promise of the HD algorithm as a new computational tool in the study of high-dimensional random systems.
A. Maillard, F. Krzakala, Y. M. Lu, and L. Zdeborova, "Construction of optimal spectral methods in phase retrieval," in Mathematical and Scientific Machine Learning, 2021. arXiv:2012.04524 [cs.IT]Abstract
We consider the phase retrieval problem, in which the observer wishes to recover a n-dimensional real or complex signal X⋆ from the (possibly noisy) observation of |ΦX⋆|, in which Φ is a matrix of size m×n. We consider a \emph{high-dimensional} setting where n,m→∞ with m/n=O(1), and a large class of (possibly correlated) random matrices Φ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal X⋆ which can be then used as initialization for a subsequent algorithm, at a low computational cost. In this paper, we extend and unify previous results and approaches on spectral methods for the phase retrieval problem. More precisely, we combine the linearization of message-passing algorithms and the analysis of the \emph{Bethe Hessian}, a classical tool of statistical physics. Using this toolbox, we show how to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix Φ, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).
O. Dhifallah and Y. M. Lu, "On the Inherent Regularization Effects of Noise Injection During Training," in International Conference on Machine Learning (ICML), 2021. arXiv:2102.07379 [cs.LG]Abstract
Randomly perturbing networks during the training process is a commonly used approach to improving generalization performance. In this paper, we present a theoretical study of one particular way of random perturbation, which corresponds to injecting artificial noise to the training data. We provide a precise asymptotic characterization of the training and generalization errors of such randomly perturbed learning problems on a random feature model. Our analysis shows that Gaussian noise injection in the training process is equivalent to introducing a weighted ridge regularization, when the number of noise injections tends to infinity. The explicit form of the regularization is also given. Numerical results corroborate our asymptotic predictions, showing that they are accurate even in moderate problem dimensions. Our theoretical predictions are based on a new correlated Gaussian equivalence conjecture that generalizes recent results in the study of random feature models.
O. Dhifallah and Y. M. Lu, "Phase Transitions in Transfer Learning for High-Dimensional Perceptrons," Entropy, Special Issue "The Role of Signal Processing and Information Theory in Modern Machine Learning", vol. 23, no. 4, 2021. arXiv:2101.01918 [cs.LG]Abstract
H. Hu and Y. M. Lu, "The Limiting Poisson Law of Massive MIMO Detection with Box Relaxation," IEEE Journal on Selected Areas in Information Theory, vol. 1, no. 3, pp. 695-704, 2020. arXiv:2006.08416 [cs.IT]Abstract
Estimating a binary vector from noisy linear measurements is a prototypical problem for MIMO systems. A popular algorithm, called the box-relaxation decoder, estimates the target signal by solving a least squares problem with convex constraints. This paper shows that the performance of the algorithm, measured by the number of incorrectly-decoded bits, has a limiting Poisson law. This occurs when the sampling ratio and noise variance, two key parameters of the problem, follow certain scalings as the system dimension grows. Moreover, at a well-defined threshold, the probability of perfect recovery is shown to undergo a phase transition that can be characterized by the Gumbel distribution. Numerical simulations corroborate these theoretical predictions, showing that they match the actual performance of the algorithm even in moderate system dimensions.
Y. M. Lu and G. Li, "Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation," Information and Inference: A Journal of the IMA, vol. 9, no. 3, pp. 507-541, 2020. arXiv:1702.06435 [cs.IT]Abstract
We study a spectral initialization method that serves as a key ingredient in recent work on using efficient iterative algorithms for estimating signals in nonconvex settings. Unlike previous analysis in the literature, which is restricted to the phase retrieval setting and which provides only performance bounds, we consider arbitrary generalized linear sensing models and present a precise asymptotic characterization of the performance of the spectral method in the high-dimensional regime. Our analysis reveals a phase transition phenomenon that depends on the sampling ratio. When the ratio is below a minimum threshold, the estimates given by the spectral method are no better than a random guess drawn uniformly from the hypersphere; above a maximum threshold, however, the estimates become increasingly aligned with the target signal. The computational complexity of the spectral method is also markedly different in the two phases. Worked examples and numerical results are provided to illustrate and verify the analytical predictions. In particular, simulations show that our asymptotic formulas provide accurate predictions even at moderate signal dimensions.
B. Aubin, Y. M. Lu, F. Krzakala, and L. Zdeboravá, "Generalization error in high-dimensional perceptrons: Approaching Bayes error with convex optimization," in Conference on Neural Information Processing Systems (NeurIPS), 2020. arXiv:2006.06560 [stat.ML]Abstract
We consider a commonly studied supervised classification of a synthetic dataset whose labels are generated by feeding a one-layer neural network with random iid inputs. We study the generalization performances of standard classifiers in the high-dimensional regime where α=n/d is kept finite in the limit of a high dimension d and number of samples n. Our contribution is three-fold: First, we prove a formula for the generalization error achieved by ℓ2 regularized classifiers that minimize a convex loss. This formula was first obtained by the heuristic replica method of statistical physics. Secondly, focussing on commonly used loss functions and optimizing the ℓ2 regularization strength, we observe that while ridge regression performance is poor, logistic and hinge regression are surprisingly able to approach the Bayes-optimal generalization error extremely closely. As α→∞ they lead to Bayes-optimal rates, a fact that does not follow from predictions of margin-based generalization error bounds. Third, we design an optimal loss and regularizer that provably leads to Bayes-optimal generalization error.
F. Mignacco, F. Krzakala, Y. M. Lu, and L. Zdeboravá, "The role of regularization in classification of high-dimensional noisy Gaussian mixture," in International Conference on Machine Learning (ICML), 2020. arXiv:2002.11544 [stat.ML]Abstract
We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number n of samples and their dimension d go to infinity while their ratio is fixed to α=n/d. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances. We also illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.
W. Luo, W. Alghamdi, and Y. M. Lu, "Optimal Spectral Initialization for Signal Recovery with Applications to Phase Retrieval," IEEE Transactions on Signal Processing, vol. 67, no. 9, pp. 2347-2356, 2019. arXiv:1811.04420 [cs.IT]Abstract
We present the optimal design of a spectral method widely used to initialize nonconvex optimization algorithms for solving phase retrieval and other signal recovery problems. Our work leverages recent results that provide an exact characterization of the performance of the spectral method in the high-dimensional limit. This characterization allows us to map the task of optimal design to a constrained optimization problem in a weighted $L^2$ function space. The latter has a closed-form solution. Interestingly, under a mild technical condition, our results show that there exists a fixed design that is uniformly optimal over all sampling ratios. Numerical simulations demonstrate the performance improvement brought by the proposed optimal design over existing constructions in the literature. In a recent work, Mondelli and Montanari have shown the existence of a weak reconstruction threshold below which the spectral method cannot provide useful estimates. Our results serve to complement that work by deriving the fundamental limit of the spectral method beyond the aforementioned threshold.
Y. Chi, Y. M. Lu, and Y. Chen, "Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview," IEEE Transactions on Signal Processing, vol. 67, no. 20, pp. 5239-5269, 2019. arXiv:1809.09573 [cs.LG]Abstract
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently.
In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated thinking of optimization and statistics leads to fruitful research findings.
C. Wang and Y. M. Lu, "The scaling limit of high-dimensional online independent component analysis," Journal of Statistical Mechanics (Special Issue on Machine Learning), vol. 2019, 2019. Publisher's VersionAbstract
We analyze the dynamics of an online algorithm for independent component analysis in the high-dimensional scaling limit. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measure of the target feature vector and the estimates provided by the algorithm will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE, which involves two spatial variables and one time variable, can be efficiently obtained. These solutions provide detailed information about the performance of the ICA algorithm, as many practical performance metrics are functionals of the joint empirical measures. Numerical simulations show that our asymptotic analysis is accurate even for moderate dimensions. In addition to providing a tool for understanding the performance of the algorithm, our PDE analysis also provides useful insight. In particular, in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically 'decoupled', with each coordinate independently solving a 1D effective minimization problem via stochastic gradient descent. Exploiting this insight to design new algorithms for achieving optimal trade-offs between computational and statistical efficiency may prove an interesting line of future research.
JSTAT_2019.pdf
C. Wang, H. Hu, and Y. M. Lu, "A Solvable High-Dimensional Model of GAN," in Proc. Thirty-third Conference on Neural Information Processing Systems (NeurIPS), 2019. arXiv:1805.08349 [cs.LG]Abstract
Despite the remarkable successes of generative adversarial networks (GANs) in many applications, theoretical understandings of their performance is still limited. In this paper, we present a simple shallow GAN model fed by high-dimensional input data. The dynamics of the training process of the proposed model can be exactly analyzed in the high-dimensional limit. In particular, by using the tool of scaling limits of stochastic processes, we show that the macroscopic quantities measuring the quality of the training process converge to a deterministic process that is characterized as the unique solution of a finite-dimensional ordinary differential equation (ODE). The proposed model is simple, but its training process already exhibits several different phases that can mimic the behaviors of more realistic GAN models used in practice. Specifically, depending on the choice of the learning rates, the training process can reach either a successful, a failed, or an oscillating phase. By studying the steady-state solutions of the limiting ODEs, we obtain a phase diagram that precisely characterizes the conditions under which each phase takes place. Although this work focuses on a simple GAN model, the analysis methods developed here might prove useful in the theoretical understanding of other variants of GANs with more advanced training algorithms.
L. Saglietti, Y. M. Lu, and C. Lucibello, "Generalized Approximate Survey Propagation for High-Dimensional Estimation," in Proc. International Conference on Machine Learning (ICML), 2019. arXiv:1905.05313 [cond-mat.dis-nn]Abstract
In Generalized Linear Estimation (GLE) problems, we seek to estimate a signal that is observed through a linear transform followed by a component-wise, possibly nonlinear and noisy, channel. In the Bayesian optimal setting, Generalized Approximate Message Passing (GAMP) is known to achieve optimal performance for GLE. However, its performance can significantly degrade whenever there is a mismatch between the assumed and the true generative model, a situation frequently encountered in practice. In this paper, we propose a new algorithm, named Generalized Approximate Survey Propagation (GASP), for solving GLE in the presence of prior or model mis-specifications. As a prototypical example, we consider the phase retrieval problem, where we show that GASP outperforms the corresponding GAMP, reducing the reconstruction threshold and, for certain choices of its parameters, approaching Bayesian optimal performance. Furthermore, we present a set of State Evolution equations that exactly characterize the dynamics of GASP in the high-dimensional limit.
H. Hong and Y. M. Lu, "Asymptotics and optimal designs of SLOPE for sparse linear regression," in Prof. International Symposium on Information Theory (ISIT), 2019. Longer version with full technical detailsAbstract
In sparse linear regression, the SLOPE estimator generalizes LASSO by assigning magnitude-dependent regular- izations to different coordinates of the estimate. In this paper, we present an asymptotically exact characterization of the performance of SLOPE in the high-dimensional regime where the number of unknown parameters grows in proportion to the number of observations. Our asymptotic characterization enables us to derive optimal regularization sequences to either minimize the MSE or to maximize the power in variable selection under any given level of Type-I error. In both cases, we show that the optimal design can be recast as certain infinite-dimensional convex optimization problems, which have efficient and accurate finite-dimensional approximations. Numerical simulations verify our asymptotic predictions. They also demonstrate the superi- ority of our optimal design over LASSO and a regularization sequence previously proposed in the literature.
G. Baechler, M. Kreković, J. Ranieri, A. Chebira, Y. M. Lu, and M. Vetterli, "Super resolution phase retrieval for sparse signals," IEEE Transactions on Signal Processing, vol. 67, no. 18, 2019. arXiv:1808.01961 [cs.IT]Abstract
In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts to recover the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. Solving the phase retrieval problem is equivalent to recovering a signal from its auto-correlation function. In this paper, we assume the original signal to be sparse; this is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. We propose an algorithm that resolves the phase retrieval problem in three stages: i) we leverage the finite rate of innovation sampling theory to super-resolve the auto-correlation function from a limited number of samples, ii) we design a greedy algorithm that identifies the locations of a sparse solution given the super-resolved auto-correlation function, iii) we recover the amplitudes of the atoms given their locations and the measured auto-correlation function. Unlike traditional approaches that recover a discrete approximation of the underlying signal, our algorithm estimates the signal on a continuous domain, which makes it the first of its kind. Along with the algorithm, we derive its performance bound with a theoretical analysis and propose a set of enhancements to improve its computational complexity and noise resilience. Finally, we demonstrate the benefits of the proposed method via a comparison against Charge Flipping, a notable algorithm in crystallography.
D. Simon, J. Sulam, Y. Romano, Y. M. Lu, and M. Elad, "MMSE Approximation For Sparse Coding Algorithms Using Stochastic Resonance," IEEE Transactions on Signal Processing, vol. 67, no. 17, 2019. arXiv:1806.10171 [eess.SP]Abstract
Sparse coding refers to the pursuit of the sparsest representation of a signal in a typically overcomplete dictionary. From a Bayesian perspective, sparse coding provides a Maximum a Posteriori (MAP) estimate of the unknown vector under a sparse prior. Various nonlinear algorithms are available to approximate the solution of such problems.
In this work, we suggest enhancing the performance of sparse coding algorithms by a deliberate and controlled contamination of the input with random noise, a phenomenon known as stochastic resonance. This not only allows for increased performance, but also provides a computationally efficient approximation to the Minimum Mean Square Error (MMSE) estimator, which is ordinarily intractable to compute. We demonstrate our findings empirically and provide a theoretical analysis of our method under several different cases.
O. Dhifallah, C. Thrampoulidis, and Y. M. Lu, "Phase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms," Technical Report, 2018. arXiv:1805.09555 [cs.IT]Abstract
We study algorithms for solving quadratic systems of equations based on optimization methods over polytopes. Our work is inspired by a recently proposed convex formulation of the phase retrieval problem, which estimates the unknown signal by solving a simple linear program over a polytope constructed from the measurements. We present a sharp characterization of the high-dimensional geometry of the aforementioned polytope under Gaussian measurements. This characterization allows us to derive asymptotically exact performance guarantees for PhaseMax, which also reveal a phase transition phenomenon with respect to its sample complexity. Moreover, the geometric insights gained from our analysis lead to a new nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm, which we call PhaseLamp. We show that this new algorithm has superior recovery performance over the original PhaseMax method. Finally, as yet another variation on the theme of performing phase retrieval via polytope optimization, we propose a weighted version of PhaseLamp and demonstrate, through numerical simulations, that it outperforms several state-of-the-art algorithms under both generic Gaussian measurements as well as more realistic Fourier-type measurements that arise in phase retrieval applications.
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