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+{
+    "url": "http://arxiv.org/abs/2405.00718v1",
+    "title": "Can't say cant? Measuring and Reasoning of Dark Jargons in Large Language Models",
+    "abstract": "Ensuring the resilience of Large Language Models (LLMs) against malicious\nexploitation is paramount, with recent focus on mitigating offensive responses.\nYet, the understanding of cant or dark jargon remains unexplored. This paper\nintroduces a domain-specific Cant dataset and CantCounter evaluation framework,\nemploying Fine-Tuning, Co-Tuning, Data-Diffusion, and Data-Analysis stages.\nExperiments reveal LLMs, including ChatGPT, are susceptible to cant bypassing\nfilters, with varying recognition accuracy influenced by question types,\nsetups, and prompt clues. Updated models exhibit higher acceptance rates for\ncant queries. Moreover, LLM reactions differ across domains, e.g., reluctance\nto engage in racism versus LGBT topics. These findings underscore LLMs'\nunderstanding of cant and reflect training data characteristics and vendor\napproaches to sensitive topics. Additionally, we assess LLMs' ability to\ndemonstrate reasoning capabilities. Access to our datasets and code is\navailable at https://github.com/cistineup/CantCounter.",
+    "authors": "Xu Ji, Jianyi Zhang, Ziyin Zhou, Zhangchi Zhao, Qianqian Qiao, Kaiying Han, Md Imran Hossen, Xiali Hei",
+    "published": "2024-04-25",
+    "updated": "2024-04-25",
+    "primary_cat": "cs.CL",
+    "cats": [
+        "cs.CL",
+        "cs.AI"
+    ],
+    "label": "Original Paper",
+    "paper_cat": "LLM AND Reasoning",
+    "gt": "Ensuring the resilience of Large Language Models (LLMs) against malicious\nexploitation is paramount, with recent focus on mitigating offensive responses.\nYet, the understanding of cant or dark jargon remains unexplored. This paper\nintroduces a domain-specific Cant dataset and CantCounter evaluation framework,\nemploying Fine-Tuning, Co-Tuning, Data-Diffusion, and Data-Analysis stages.\nExperiments reveal LLMs, including ChatGPT, are susceptible to cant bypassing\nfilters, with varying recognition accuracy influenced by question types,\nsetups, and prompt clues. Updated models exhibit higher acceptance rates for\ncant queries. Moreover, LLM reactions differ across domains, e.g., reluctance\nto engage in racism versus LGBT topics. These findings underscore LLMs'\nunderstanding of cant and reflect training data characteristics and vendor\napproaches to sensitive topics. Additionally, we assess LLMs' ability to\ndemonstrate reasoning capabilities. Access to our datasets and code is\navailable at https://github.com/cistineup/CantCounter.",
+    "main_content": "Introduction Large Language Models (LLMs), exemplified by ChatGPT[1], redefine information acquisition, communication, and problem-solving[2]. These models are trained on extensive datasets or fine-tuned from pre-existing models, necessitating vast amounts of data. However, LLMs also pose security and ethical concerns as attackers can exploit their generative capabilities for malicious purposes [3]. Such misuse encompasses disinformation dissemination [4], AI-driven crime [5], privacy breaches [6], and social engineering [7]. Despite efforts by regulators like OpenAI to implement content filters [8], there remains a risk of attackers disguising malicious content using \u201ccant\u201d or \u201cdark jargon\u201d concealed language elements requiring deeper comprehension [9]. LLMs excel in understanding and generating natural language responses, fostering user trust. While research evaluates their efficacy in providing accurate responses [10], little attention has been paid to LLMs\u2019 interaction with cant in specific domains. Prior studies often lack depth in understanding the intricacies of cant [11], especially its varied representations in domains like politics and drugs. In this paper, we investigate LLMs\u2019 ability to recognize and reason about cant, particularly in domains prone to offensive content like politics and drugs. Despite progress in filtering harmful content, attackers can still exploit cant to evade detection. Understanding LLMs\u2019 response to cant in specific domains is essential for addressing emerging security challenges. Additionally, we assess LLMs\u2019 ability to demonstrate reasoning capabilities. \u2217Corresponding author: zjy@besti.edu.cn arXiv:2405.00718v1  [cs.CL]  25 Apr 2024 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT Research Questions. To address the above issues, in this paper, we evaluate the reasoning abilities of current LLMs involving cant or dark jargon from the following four perspectives: 1. RQ1: Do different types of questions help LLM understand the cant? 2. RQ2: Do different question setups and prompt clues help LLM understand cant? 3. RQ3: Do different LLMs have the same understanding of the same cant? 4. RQ4: How well does LLM understand cant in different domains? CantCounter: Addressing past shortcomings[11], CantCounter is a system crafted to evaluate LLM\u2019s grasp of cant within specific domains. We compile Cant and Scene datasets from various sources to form adversarial texts. These datasets fine-tune the GPT-2 model and generate Scene fragments for assessing LLM comprehension. Co-Tuning methods align the Cant dataset and Scene fragments, while Data-Diffusion techniques augment and refine adversarial text. Employing Type, Sample learning, and Clue approaches enrich our experiments. Finally, Data-Analysis methods systematically evaluate 1.67 million data points. CantCounter is locally deployable and adaptable to any open-world dialogue system. Its replication has both advantages and drawbacks, aiding attackers in bypassing LLM classifiers while facilitating safety filter development. We define \u201centities\u201d as distinct objects or concepts and \u201cscenes\u201d as related events in specific environments. Ethical Considerations: CantCounter draws from public datasets such as Reddit [12] and 4chan [13], avoiding direct user interaction. However, its misuse poses risks, despite its benefits in addressing LLM\u2019s challenges. Despite these potential risks, we believe that the benefits of CantCounter far outweigh the risks. LLM has become a hot topic [14], and we need to fully recognize the potential problems of LLM and promote its safer development and application. We caution that this paper may contain sensitive content, including drug and violence-related examples, which could cause discomfort. Comprehensive data is available upon request. We have only open sourced part of the dataset. Contributions. This paper introduces three key contributions: 1. We present the Cant and Scene datasets, addressing data scarcity in domains like drugs, weapons, and racism, laying groundwork for future large language model assessment. 2. CantCounter, our framework, assesses large language models\u2019 understanding of domain-specific cants through four stages: Fine-Tuning for scene fragment generation, Co-Tuning for cross-matching, DataDiffusion for text expansion, and Data-Analysis for simplifying complex calculations. 3. Our evaluation of CantCounter reveals its efficacy in bypassing security filters of mainstream dialogue LLMs, providing insights into LLM reasoning within specific domains and guiding future research. 2 Background 2.1 Large Language Model Security Issues ChatGPT, developed by OpenAI in November 2022 [1], has undergone upgrades and fine-tuning [15] to prevent harmful content generation. However, users can still provoke negative responses by using specific prompts [16]. Researchers are investigating security risks, including the generation of toxic outputs from benign inputs [17]. Recent studies have shown that attackers can bypass detection by encrypting inputs with methods like Caesar ciphers and exploiting language nuances [18]. This paper proposes a Q&A query approach to evaluate LLMs\u2019 reasoning abilities in handling such content. 2.2 Cant Cant, a specialized language used by social groups for secrecy [19], varies in names like argot [20], slang [21], and secret language across history. While LLMs excel in traditional cant analysis, understanding criminal cant poses challenges. Criminal groups use innocuous terms to hide illegal activities, necessitating mastery for law enforcement [22]. Our study explores cant in politics, drugs, racism, weapons, and LGBT issues. These cants share ambiguity, indirect messaging, and potential for social harm. Political cant conveys biases, drug cant evades regulation, racism cant reinforces biases, weapons cant enables illegal dealings, and LGBT cant discriminates. Mastering these cants is vital for addressing societal and security concerns. 2.3 Question Answering (Q&A) Task Dialogue systems fall into task-oriented and non-task-oriented categories. Task-oriented systems serve specific purposes like reservations, while non-task-oriented systems engage in free conversation. Examples include ChatGPT, Bard, ERNIE, and Claude, offering services in entertainment, social interaction, and information retrieval [23].Question-answering (Q&A) tasks in NLP evaluate language processing capabilities [24], including reading comprehension and logical reasoning. Q&A formats include abstractive, Yes/No, and Multiple-Choice, each requiring specific evaluation metrics [25]. We employ Zero-shot/One-shot learning for testing. 2 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT 3 CantCounter 3.1 High-level Idea We observe that the responses generated by LLMs vary with different cants, allowing adversaries to bypass filters or security restrictions. Thus, understanding how LLMs react to different cants is very important. However, exhaustively trying different cants queries with different scenes across numerous domains to find those capable of bypassing LLM restrictions and generating harmful outputs would be time-consuming and impractical. Therefore, we investigate whether adversaries can independently combine different cants and scenes to generate context that is reasonable and coherent, bypassing LLM filters or restrictions. To this end, we introduce CantCounter, the first evaluation (attack) framework targeting open-world dialogue systems (LLM). 3.2 Threat Model We adopt a threat model similar to \u201cWhy so toxic\u201d [17], targeting deployed dialogue LLMs like ChatGPT. Firstly, the adversary requires scene data different from the target LLM\u2019s training data. Secondly, they interact with the LLM, combining cants and scenarios to extract detectable cants. Finally, they access the victim LLM via CantCounter in a black-box manner, querying it through an API-like interface. 3.3 Dataset In our study, we extensively gathered cant related to five domains: politics, drugs, racism, weapons, and LGBT. The cant, comprising common and less common usages, holds practical meanings in real life. This Cant dataset forms a robust basis for evaluating the veracity and reliability of LLMs across specific domains. These five areas were chosen to address pressing societal issues impacting fundamental values such as social justice and human rights. Exploration of politics, drugs, racism, weapons, and homosexuality enables LLMs to tackle real-world challenges effectively. While other domains like hacking and fraud are significant, we focused on these due to data availability and processing feasibility, leaving room for future research on sensitive topics. Donald Trump Cheeto Jesus Agent Orange ... 45 Barack\u00a0 Obama Nobama Odumbo ... Deporter\u00a0 in chief Cocaine Blow C ... White Lady LSD Microdot L ... Cubes gun Rod Heater ... Cuete Sailor Geardo fobbit ... Cage Kicker Jews Crikey German Candle\u00a0 ... Penny Chaser Asians Bananas Sleepwalker ... Tiny gay man chicken queen ... \u00a0silver fox lesbins baby butch\u00a0 kitty puncher ... pillow\u00a0 princess Politics Drugs Weapons Racism LGBT ... ... ... ... ... Data Collection Data Classification\u00a0 & Categories Domain Platform Object Style Type Abjectives \u00a0Information Network Data and Social Forum Community Dark Web Figure 1: Construction of the Cant dataset involves collecting, summarizing security-related data, forming interconnected relationships between cants into an information network, and establishing the dataset through data classification and categorization, encompassing various domain-related entities and their corresponding cants. In constructing the Cant dataset (Figure 2 2 \u20dd), we crawled or manually screened multiple sources, including government agency websites [26], online forums like Reddit [12], 4chan [13], and X [27], publicly available datasets from Kaggle [28] and Hugging Face [29], dark web, and public compilations of cant. Multi-source data encompasses various text types closely related to specific domains. CantCounter utilizes information networks [30] to address redundancy challenges between cants, capturing their interdependency. The Cant dataset covers five domains, totaling 1,778 cants across 187 entities. We randomly selected 53 entities, totaling 692 cants, ensuring even representation across domains and prevalence in the open world. Selected entities and cants were cross-validated with authoritative sources [31, 32, 33, 34, 35] to ensure wide presence and reflection in publicly accessible information sources. Criteria like content relevance and topic specificity guided information selection and filtering, aiming for transparency and consistency. The resulting high-quality data forms the Scene dataset, laying the groundwork for subsequent simulation scene generation models. During information selection and filtering (Figure 2 1 \u20dd), explicit criteria were used to judge relevance and adherence to study definitions. Decisions were reached through participatory discussion to mitigate subjectivity and ensure alignment with research objectives. This rigorous process yields a refined dataset for accurate and relevant analysis. 3 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT 3.4 Pipeline The CantCounter pipeline (Figure 2) consists of four stages: Fine-Tuning, Co-Tuning, Data-Diffusion, and DataAnalysis, as detailed below. GPT-2 Scene Dataset Scene Generation Model Scene fragments Multi-Source Data Cant Dataset Q&A-Queries\u00a0 Analysis Target Model Cross Matching Constrain \u2463 \u2464 \u2465 \u2462 \u2466 \u2467 \u2468 \u2461 \u2460 Fine-Tuning Co-Tuning Data-Diffusion Data-Analysis Figure 2: The pipeline of CantCounter. Cant is prevalent in the open world, so we aggregate raw text data from various sources to construct Cant and Scene datasets (Section 3.3). Although Cant and Scene datasets provide specific entities and scenes, they may not align well with the domain\u2019s requirements. Therefore, in Stage 3 \u20dd, we fine-tune GPT-2 using the Scene dataset to build five scene generation models for large-scale scenes, tailored to our specific domains. However, the fine-tuned scenes may not match the entities in the Cant dataset. In Stage 4 \u20dd, we address this issue by using entities from the Cant dataset to constrain the output of the generated model, ensuring scenes closely relate to the cant entities. Next, we conduct semi-automatic screening of the generated simulation scenes to form a set of Scene fragments. While these fragments contain entities, linking them with specific questions requires a method we have not yet discovered. Hence, in Steps 5 \u20dd6 \u20dd, we devise the Co-Tuning stage, where Scene fragments cross-match with cants from the Cant dataset to form Fragments. To enable multi-task comparison, we construct detection tests through different combinations of specific domains, question types, learning methods, and prompt clue methods in Stage 7 \u20dd. This completes and diffuses Fragments to form Q&A-Query datasets. Finally, in Stages 8 \u20dd9 \u20dd, Q&A-Queries are sent to the target model API for completion, and a segmented data statistics algorithm is applied to obtain and analyze test results, conducting analyses in the Data-Analysis stage. 3.5 Stage 1: Fine-Tuning During the fine-tuning stage, we use the Scene dataset to guide GPT-2 in generating tailored scenarios for specific domains. Despite more advanced models like GPT-3.5 and GPT-4 being available, we opt for GPT-2 due to its open-source nature, facilitating better control over training details. The fine-tuning code is publicly accessible for replication. The fine-tuning process is outlined in Algorithm 1. The Transformer model [36] forms the basis for GPT-2, featuring encoders and decoders with identical modules. GPT2 employs a partially masked self-attention mechanism and positional coding to understand sequence relationships. It has been successfully applied in various tasks like AI detection and text summarization. Overall, GPT-2\u2019s fine-tuning with the Scene dataset enables the generation of Question-Answer patterns tailored to specific domains, aiding in simulated scene generation tasks. 3.6 Stage 2: Co-Tuning To solve the problem of many intersecting data processes in CantCounter, we use the Cant dataset and Scene fragments to collaborate and design a Co-Tuning method. Co-Tuning realizes the generation and collaboration of cross-matching and solves the problem of detection data insufficiency. The Cant dataset provides detailed entity information for the generated model. The entities could constrain the generative model and make the Scene Fragments more consistent and coherent in the need for a specific domain during the Co-Tuning stage. In the end, we also manually review the results to ensure the relevance of cants to scenes and the distinctiveness of all scenes corresponding to the same cant. In this paper, we design formulas in the Co-Tuning to mathematically represent this part of the stage. The generation model is specified as Mp(p \u2208[1, 5]), and it includes five fine-tuned models, which are denoted as M1, M2, M3, M4, and M5. As shown in Figure 3, entity Oi represents the i-th entity (i \u2208[1, 15]) in the Cant dataset, and cant \u03c9ij represents the j-th cant of Oi (j \u2208[1, 20]). For example, in the case of the politics domain, there are 10 entities used in our 4 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT Algorithm 1: Fine-Tuning Input: pre-trained model parameters \u03b8p, fine-tuned dataset Df, loss function L(\u03b8) that depends on the model parameters \u03b8, optimizer O(\u03b8) for updating the parameters \u03b8, learning rate \u03b7, number of iterations T for fine-tuning, x is the input sample, y is the label, \u02c6 y is the model\u2019s predicted value, and \u2207\u03b8 denotes the gradient operation Output: fine-tuned model parameters \u03b8f 1 Initialization \u03b8f = \u03b8p 2 for each t in range(1, T + 1) do 3 for batchx, batchy in Df do 4 \u02c6 y = model(batchx, \u03b8f); 5 loss = L(\u02c6 y, batchy); 6 gradient = torch.autograd.grad(loss,\u03b8f) ; 7 \u03b8f = O(\u03b8f, gradient, \u03b7); 8 return \u03b8f  entity \ufffd1 . . . ... ...  entity \ufffd2  entity \ufffd\ufffd \ufffd\ufffd Generation Models GPT-2 Scene Datasets FineTuning cant \ufffd1\ufffd cant \ufffd11 cant \ufffd12 ... cant \ufffd\ufffd\ufffd cant \ufffdi1 cant \ufffd\ufffd2 ... ... k scenes \ufffd11 \u2019 . . . ... \ufffd12 \u2019 ... \ufffd1\ufffd \u2019 ... ... \ufffd\ufffd2 \u2019 ... \ufffd\ufffd\ufffd \u2019 . . . . . . . . . Fragment 1 ... \ufffd\ufffd1 \u2019 k scenes Fragment 2 k scenes Fragment j (belong to \ufffd1) (belong to \ufffd\ufffd) . . . . . . Fragments (i entities correspond to i*j*k scenes)  Fragment 1 Fragment 2 Fragment j Fragments belonging to \ufffd1 Fragments belonging to \ufffd\ufffd (Scene fragment) (Scene fragment) Scene \ufffd11   Scene \ufffd12   Scene \ufffd1\ufffd   Scene \ufffd\ufffd1   Scene \ufffd\ufffd\ufffd   Scene \ufffd\ufffd2   (Cant Dataset) Cross-Matching Figure 3: The overall structure and process of Co-Tuning. experiments, each entity has twenty cants, j is taken as [1, 20]. The entity Oi can constrain the fine-tuned model Mp\u2019s output, and the result of the constraint is the Scene fragment; this part corresponds to Eq. (1). The Scene is Sik (i \u2208[1, 10], k \u2208[1, 101]). The Scene Sik represents the k-th scene fragment (i \u2208[1, 10], k \u2208[1, 101]) that the i-th entity enters into the output of the fine-tuning model (Mp). Sik = Mp(Oi) (1) Eq. (2) denotes the cross-match of Cant and Scene fragment and was saved in S \u2032 ik. S \u2032 ik = Sik \u222a\u03c9ij (2) There are k orange boxes in the O1 Scene fragment. These orange boxes represent the Mp-generated text containing the Cant dataset\u2019s entities. The function of Eq. 2 is to replace the entities in the Scene fragments with cant in the Cant dataset. As shown in Figure 3, for example, from O1 Scene fragment to Fragment 1. We replace entities in Scene [S11, S1k] with the cant (\u03c911), forming Fragment 1. By analogy, we built j Fragments in the Co-Tuning stage. In the Co-Tuning stage, we can obtain scene fragments related to entities in specific domains that have a high degree of context consistency and express various characteristics of the entities in different contexts. At the same time, our finetuned model is flexible enough to introduce multiple entities during the generation process and allow scene fragments to describe the relationships among multiple entities. This stage generates diverse scene fragments. While the scene fragments are generated through a generative process, the Scene dataset we provide undergoes manual review to mitigate errors in both the generated content and the language utilized within the experimental environment. 3.7 Stage 3: Data-Diffusion At this stage, Fragments from the Co-Tuning stage are transformed into Q&A-Queries to enhance interaction with LLM and diversify evaluation. We employ three diffusion methods: two sample learning techniques, three question types, and four prompt clue methods. Each Fragment generates 24 Q&A-Queries. First, we introduce sample learning techniques for zero-shot and one-shot learning transformations of Fragments. Second, we categorize Fragments into Abstractive, Yes/No, and Multiple-choice question types. Finally, prompts are classified into None-tip, Tip-1, Tip-2, and All-tip categories, considering information retrieval difficulty and situational prompting. 5 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT ... ... ... ... ... ... Zero-shot learning One-shot learning Abstractive Yes/NO Multiple-choice None-tip Tip-1 Tip-2 All-tip None-tip Tip-1 Tip-2 All-tip None-tip Tip-1 Tip-2 All-tip None-tip Tip-2 None-tip Tip-2 None-tip Tip-1 Tip-2 All-tip Tip-1 All-tip Tip-1 All-tip Abstractive Yes/NO Multiple-choice ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Data-Diffusion ... ... Fragment 1 Fragments ... Fragment 2 ... Fragment \ufffd ... Q&A-Query ... Figure 4: Schematic diagram of Data-Diffusion. The introduction of Data-Diffusion in extended Fragments has significantly increased Q&A queries, providing diverse test cases for evaluating the generation model\u2019s performance comprehensively. This approach promises to establish a diverse database for future research and applications. 3.8 Stage 4: Data-Analysis As shown 8 \u20ddand 9 \u20ddin Figure 2, 8 \u20ddmeans sending the data expanded by Data-Diffusion to ChatGPT and other target models. 9 \u20ddshows data analysis of the output results of LLMs such as ChatGPT. After completing the Data-Diffusion, we submit the generated Q&A-Queries to the LLM API interface to obtain a large number of data results. These data results are complex and diverse, including the interplay of relationships. Therefore, we devise a data analysis algorithm to yield both numerical and analysis outcomes. Algorithm 2: Segmented data statistics algorithm Input: J \u2013 j Fragments in Co-Tuning (Fig 4), K \u2013 k Scenes in a Fragment, tasks \u2013 [\u201cAbstractive\u201d: {\u201cAZ\u201d, \u201cAO\u201d}, \u201cYes/NO\u201d: {\u201cJZ\u201d, \u201cJO\u201d}, \u201cMultiple-choice\u201d: {\u201cMZ\u201d, \u201cMO\u201d} ], clues \u2013 [\u201cNone-tip\u201d, \u201cTip-1\u201d, \u201cTip-2\u201d, \u201cAll-tip\u201d], intervals \u2013 [0, 1-10, 11-20, ..., 91-101] Output: Rj,t,c, Nj,t,c,z, Sumt, SumPCM 1 for each j in J do 2 for each t in tasks do 3 for each c in clues do 4 Nj,t,c=Pk a=1 ia (if hit then ia=1, else ia = 0) 5 Rj,t,c \u2190 \u2212 Nj,t,c k 6 for each z in interval do 7 if Nj,t,c in z then Nj,t,c,z++ 8 Sumt=Pj \u03b1=1 Pc \u03b2=1 N\u03b1,t,\u03b2 9 SumPCM=Pj \u03b1=1 Pt \u03b2=1 N\u03b1,\u03b2,c 10 return Rj,t,c, Nj,t,c,z, Sumt, SumPCM After the Co-Tuning and Data-Diffusion stages, the test data generated by CantCounter is very complex. Therefore, in the Data-Analysis stage, we implement Algorithm 2 to conduct data statistics from various angles. During analysis, when the entity Oi is modified in the Co-Tuning stage (see Figure 3), Algorithm 2 will be called accordingly. We analyze the results based on different tasks. We learn and analyze data features from Question Type Method (See 4.2 QTM) and Sample Learning Method (See 4.3 SLM) based on different question types and samples learning to get Sumt; we analyze the data based on different prompt clues from Prompt Clue Method (See 4.4 PCM) to get SumP CM. In Algorithm 2, we set the matching conditions, calculate the number of fragments, and obtain Nj,t,c and accuracy Rj,t,c. At the same time, we set eleven intervals: 0, 1-10, 11-20, ..., 91-101 to distinguish different feedbacks and obtain Nj,t,c,z. As shown in the Algorithm 2, we put Zero-shot learning, One-shot learning, and three tasks together as a loop. We define that in the Abstractive task, the output is AZ in the Zero-shot learning input; the output is AO in the Oneshot learning input. In the Yes/NO task, the output is expressed as JZ in the Zero-shot learning input; the output is 6 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT expressed as JO in the One-shot learning input. In the Multiple-choice task, the output is represented as MZ in the Zero-shot learning input; the output is expressed as MO in the One-shot learning input. The above content has been integrated into our code to form semi-automation. 4 Experimental Design and Results To explore our research questions, we conducted experiments in CantCounter, outlined sequentially in this section. We examined various question types in RQ1 (Section 4.1), different question setups in RQ2 (Section 4.2), and diverse prompt clues in RQ2 (Section 4.3). Focusing primarily on ChatGPT-3.5 (version gpt-3.5-turbo-0613) due to its convenience and wide usage, similar experiments were also conducted with other language models. All experiments were performed on a server equipped with an RTX 3090 Ti GPU. In this section, we analyze using cant and scene to bypass the LLM filter in the CantCounter framework quantitatively. We conduct open-world query experiments across five domains: politics, drugs, racism, weapons, and LGBT. Initially setting k to 101, we match 692 cants to 53 entities, resulting in 69,892 scenes. These undergo Data-Diffusion, expanding to 1,677,408 scenes. This study enables a comprehensive analysis of corpus performance and language changes within specific domains. 4.1 Question Type Method (QTM) In the Q&A task, we conduct three types of tasks: \u2022 Abstractive Task: Models generate responses freely, without relying on specific information extraction. \u2022 Yes/No Task: Models provide binary responses, \u201cTrue\u201d or \u201cFalse,\u201d based solely on the presented question and existing knowledge. \u2022 Multiple-choice Task: Models select the correct answer from a set of options, demonstrating comprehension of semantics and accurate identification. Table 1 shows that Multiple-choice tasks achieve the highest accuracy (45.38%), while Yes/No tasks have the lowest (22.91%). The discovery that ChatGPT performs well in multiple-choice questions is intriguing. In this task, there are five options (A) to (E), with (A) to (D) relevant to a specific domain, and (E) set as \u201cI don\u2019t know.\u201d \u201cOther\u201d signifies an answer unrelated to these options, with (A) as the correct choice. Figure 5 displays the box plot analysis results. Analyzing the Multiple-choice task results, we find key factors for its success. Firstly, it offers a set of answers with one correct option and distractors, aiding comprehension. Secondly, its structured format simplifies the process of eliminating incorrect options, improving accuracy. Lastly, the inclusion of an \u201cI don\u2019t know\u201d option enhances accuracy in uncertain situations. Figure 5: The vertical axis refers to the number of correct answers under the four tips. The total number is 404. (A) and (E) stand out in Multiple-choice, being the correct answer and \u201cI don\u2019t know\u201d respectively. After carefully studying ChatGPT-3.5\u2019s interpretation of option (E), we find that when the context is ambiguous or the entities in the implicit context are rare, ChatGPT-3.5\u2019s accuracy will drop significantly; that is, it will prefer option (E). We also explore the low accuracy in the Yes/No task. Comparing ChatGPT-3.5\u2019s \u201cFalse\u201d answers with Multiple-choice task data, we find they often include option (E) and incorrect choices from the Multiple-choice task due to the clarity of options. Additionally, differences in response styles and keyword detection criteria impact ChatGPT-3.5\u2019s performance across Abstractive and Yes/No tasks, where Yes/No tasks restrict responses to \u201cTrue\u201d or \u201cFalse.\u201d Overall, our analysis highlights how different Q&A types affect ChatGPT-3.5\u2019s accuracy in specific domains, with Multiple-choice tasks showing higher performance. Further research is needed to improve ChatGPT-3.5\u2019s accuracy and adaptability in these domains. 7 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT 4.2 Sample Learning Method (SLM) In our experiments, we explore two sample setups: Zero-shot and One-shot learning. \u2022 Zero-shot learning. No examples are provided in the prompt, which only includes instructions and questions. \u2022 One-shot learning. The prompt includes an example relevant to the discussion, consisting of a sample message and user information. Zero-shot learning involves a single user message, while One-shot learning processes a sample message and a user message. These methods help understand LLM\u2019s performance in different sample learning approaches and reveal its inference capabilities in information-poor settings. Further investigation uncovers learning patterns and effects of the model in specific domains, with default hyper-parameter settings used to avoid extensive tuning. In this section, we explore how Zero-shot and One-shot learning methods affect LLM accuracy in recognizing cant scenes for RQ2. Traditionally, One-shot learning often outperforms Zero-shot learning due to more available data [37]. However, our cross-domain analysis, depicted in Figure 6 and reflected in Table 1 (red section), reveals a trend favoring Zero-shot learning overall. We find this trend varies by domain. Figure 6: Comparison of the number of correct Zero-shot learning and One-shot learning in different domains. In the politics domain, One-shot learning performs better due to ample data and contextual understanding. Conversely, in the LGBT domain, Zero-shot learning outperforms One-shot learning due to limited publicly available examples. One-shot learning aids ChatGPT-3.5 in better contextual comprehension of sensitive topics, but it may also introduce biases, leading to lower overall accuracy in specific domains. Similar analyses across other domains yield consistent results. 4.3 Prompt Clue Method (PCM) In this part of the study, the purpose of CantCounter is to explore the impact of different clues on LLM recognition and reasoning abilities. To this end, we provide four different clues to experiment with: \u2022 None-tip. Keeps the same as the original prompt and does not add any additional clues. \u2022 Tip 1. Add relevant tip for \u201cNone-tip\u201d. For example, when describing Trump\u2019s cant, we can add the clue \u201cPolitician\u201d in the political domain to make the prompt more directional. \u2022 Tip 2. Add another relevant tip for \u201cNone-tip\u201d. For example, when describing Trump\u2019s cant, add the \u201cUnited States\u201d prompt in the domain of politics to enrich the prompt content. \u2022 All-tip. Add both Tip 1 and Tip 2 on the basis of \u201cNone-tip\u201d; for example, when describing Trump\u2019s cant, add both \u201cpolitician\u201d and \u201cAmerican\u201d in the political domain to make the prompt more appropriate. By observing the effects of these different clues on LLMs, CantCounter can assess the fluctuating changes they induce in recognition and reasoning abilities. This study will help further understand the influence of cues on LLM and provide directions for improving its application and performance. To answer RQ2, Table 1 displays ChatGPT-3.5\u2019s accuracy across five domains using different prompt clues. Generally, more clue-related information improves recognition accuracy, as seen in the political domain where All-tip prompts perform significantly better. However, increasing clues doesn\u2019t always lead to higher accuracy, possibly due to information redundancy or LLM filter triggering. Too many clues may reduce accuracy, as seen in the LGBT domain where Tip 1 prompts were less accurate than none-tip prompts. Our analysis stresses the importance of a balanced clue selection approach to maximize external information usage without compromising accuracy. Thus, choosing appropriate clues in moderate quantities is key to enhancing ChatGPT-3.5\u2019s domain-specific performance. 8 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT Table 1: CantCounter stats highlight top performances(%): Multiple-choice excels in QTM (45.38%), Zero-shot learning shines in SLM (52.13%), and All-tip prevails in PCM (29.11%). \u201cA\u201d represents \u201cAbstractive\u201d. \u201cY/N\u201d represents \u201cYes/No\u201d. \u201cMc\u201d represents \u201cMultiple-choice\u201d. \u201cZs\u201d represents \u201cZero-shot\u201d. \u201cOs\u201d represents \u201cOne-shot\u201d. \u201cNT\u201d represents \u201cNone-tip\u201d. \u201cT1\u201d represents \u201cTip 1\u201d. \u201cT2\u201d represents \u201cTip 2\u201d. \u201cAllT\u201d represents \u201cAll-tip\u201d. QTM SLM PCM Domain A Y/N Mc Zs Os NT T1 T2 AllT Politics 26.81 22.55 50.64 42.85 57.15 19.01 24.75 25.19 31.05 Drugs 21.16 22.41 56.43 55.41 44.59 17.32 27.43 25.47 29.78 Racism 29.05 27.60 43.35 41.39 58.61 11.22 19.63 37.50 31.66 Weapons 50.89 16.20 32.91 54.96 45.04 18.73 28.11 25.27 27.90 LGBT 34.41 25.75 39.84 59.78 40.22 22.58 22.10 28.53 26.79 Total 31.71 22.91 45.38 52.13 47.87 19.03 24.61 27.24 29.11 Table 2: Zero-shot learning and One-shot learning Q&A accuracy in CantCounter for GPT-4, Bard, New Bing, and SparkDesk. \u201cAcc\u201d represents \u201cAccuracy Rate\u201d. \u201cRej\u201d represents \u201cRejection Rate\u201d. \u201cDon\u2019t know\u201d represents \u201c\u2019I don\u2019t know\u2019 Rate\u201d. Zero-shot learning One-shot learning Acc Rej Don\u2019t know Acc Rej Don\u2019t know ChatGPT-3.5[1] 47.61 4.66 39.91 45.52 1.63 46.45 GPT-4[38] 27.27 0.00 70.45 50.00 0.00 34.09 Bard[39] 47.73 4.55 13.64 65.91 15.91 6.82 New Bing[40] 50.00 11.36 34.09 50.00 36.36 2.27 SparkDesk[41] 29.55 45.45 9.09 20.45 68.18 2.27 4.4 Comparison with other LLMs In our study, we examine several LLMs alongside ChatGPT-3.5 to address RQ3, including GPT-4[1], New Bing [40], Bard [39], Claude [42], ERNIE [43], and SparkDesk [41]. While ERNIE is optimized for Chinese content, translating cant prompts may compromise their subtlety and effectiveness. Moreover, ERNIE\u2019s frequent account suspensions hindered extensive trials [44]. Claude\u2019s sensitive content handling also led to account suspensions [42]. Thus, we focus on comparing and validating four other LLMs: GPT-4, Bard, New Bing, and SparkDesk. Table 2 presents ratios of correct answers, refused answers, and \u201cI don\u2019t know\u201d responses. Interestingly, GPT-4 consistently responds in all situations, avoiding refusal to answer. This contrasts with other models that often refuse to respond due to content filtering. GPT-4\u2019s tendency to use \u201cI don\u2019t know\u201d may stem from our controlled comparisons in the QTM and PCM methods, particularly in Multiple-choice scenarios. Conversely, other LLMs tend to refuse to answer, likely due to content categorization by filters and classifiers. SparkDesk exhibits the highest refusal rate, possibly due to overly strict filters. Furthermore, One-shot learning models are more prone to refusal to answer, as they rely on context understanding, potentially triggering filters. These findings offer insights into the performance of these LLMs across different learning tasks, informing future research directions. 4.5 Takeaways We observe varying accuracy across different Q&A-Query types (RQ1), with Multiple-choice tasks being most accurate and Yes/No tasks the least. In sensitive domains, Zero-shot learning performs better than One-shot learning (RQ2). Increasing prompt clues improves cant identification accuracy (RQ2). More recent LLM models consistently avoid refusing to answer (RQ3), but they are more likely to refuse answering questions related to racism compared to LGBT (RQ4). 5",
+    "additional_graph_info": {
+        "graph": [
+            [
+                "Xu Ji",
+                "Jianyi Zhang"
+            ],
+            [
+                "Jianyi Zhang",
+                "Yiran Chen"
+            ],
+            [
+                "Jianyi Zhang",
+                "Changyou Chen"
+            ],
+            [
+                "Jianyi Zhang",
+                "Ruiyi Zhang"
+            ],
+            [
+                "Jianyi Zhang",
+                "Zhiqiang Wang"
+            ]
+        ],
+        "node_feat": {
+            "Xu Ji": [
+                {
+                    "url": "http://arxiv.org/abs/2405.00718v1",
+                    "title": "Can't say cant? Measuring and Reasoning of Dark Jargons in Large Language Models",
+                    "abstract": "Ensuring the resilience of Large Language Models (LLMs) against malicious\nexploitation is paramount, with recent focus on mitigating offensive responses.\nYet, the understanding of cant or dark jargon remains unexplored. This paper\nintroduces a domain-specific Cant dataset and CantCounter evaluation framework,\nemploying Fine-Tuning, Co-Tuning, Data-Diffusion, and Data-Analysis stages.\nExperiments reveal LLMs, including ChatGPT, are susceptible to cant bypassing\nfilters, with varying recognition accuracy influenced by question types,\nsetups, and prompt clues. Updated models exhibit higher acceptance rates for\ncant queries. Moreover, LLM reactions differ across domains, e.g., reluctance\nto engage in racism versus LGBT topics. These findings underscore LLMs'\nunderstanding of cant and reflect training data characteristics and vendor\napproaches to sensitive topics. Additionally, we assess LLMs' ability to\ndemonstrate reasoning capabilities. Access to our datasets and code is\navailable at https://github.com/cistineup/CantCounter.",
+                    "authors": "Xu Ji, Jianyi Zhang, Ziyin Zhou, Zhangchi Zhao, Qianqian Qiao, Kaiying Han, Md Imran Hossen, Xiali Hei",
+                    "published": "2024-04-25",
+                    "updated": "2024-04-25",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Large Language Models (LLMs), exemplified by ChatGPT[1], redefine information acquisition, communication, and problem-solving[2]. These models are trained on extensive datasets or fine-tuned from pre-existing models, necessitating vast amounts of data. However, LLMs also pose security and ethical concerns as attackers can exploit their generative capabilities for malicious purposes [3]. Such misuse encompasses disinformation dissemination [4], AI-driven crime [5], privacy breaches [6], and social engineering [7]. Despite efforts by regulators like OpenAI to implement content filters [8], there remains a risk of attackers disguising malicious content using \u201ccant\u201d or \u201cdark jargon\u201d concealed language elements requiring deeper comprehension [9]. LLMs excel in understanding and generating natural language responses, fostering user trust. While research evaluates their efficacy in providing accurate responses [10], little attention has been paid to LLMs\u2019 interaction with cant in specific domains. Prior studies often lack depth in understanding the intricacies of cant [11], especially its varied representations in domains like politics and drugs. In this paper, we investigate LLMs\u2019 ability to recognize and reason about cant, particularly in domains prone to offensive content like politics and drugs. Despite progress in filtering harmful content, attackers can still exploit cant to evade detection. Understanding LLMs\u2019 response to cant in specific domains is essential for addressing emerging security challenges. Additionally, we assess LLMs\u2019 ability to demonstrate reasoning capabilities. \u2217Corresponding author: zjy@besti.edu.cn arXiv:2405.00718v1  [cs.CL]  25 Apr 2024 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT Research Questions. To address the above issues, in this paper, we evaluate the reasoning abilities of current LLMs involving cant or dark jargon from the following four perspectives: 1. RQ1: Do different types of questions help LLM understand the cant? 2. RQ2: Do different question setups and prompt clues help LLM understand cant? 3. RQ3: Do different LLMs have the same understanding of the same cant? 4. RQ4: How well does LLM understand cant in different domains? CantCounter: Addressing past shortcomings[11], CantCounter is a system crafted to evaluate LLM\u2019s grasp of cant within specific domains. We compile Cant and Scene datasets from various sources to form adversarial texts. These datasets fine-tune the GPT-2 model and generate Scene fragments for assessing LLM comprehension. Co-Tuning methods align the Cant dataset and Scene fragments, while Data-Diffusion techniques augment and refine adversarial text. Employing Type, Sample learning, and Clue approaches enrich our experiments. Finally, Data-Analysis methods systematically evaluate 1.67 million data points. CantCounter is locally deployable and adaptable to any open-world dialogue system. Its replication has both advantages and drawbacks, aiding attackers in bypassing LLM classifiers while facilitating safety filter development. We define \u201centities\u201d as distinct objects or concepts and \u201cscenes\u201d as related events in specific environments. Ethical Considerations: CantCounter draws from public datasets such as Reddit [12] and 4chan [13], avoiding direct user interaction. However, its misuse poses risks, despite its benefits in addressing LLM\u2019s challenges. Despite these potential risks, we believe that the benefits of CantCounter far outweigh the risks. LLM has become a hot topic [14], and we need to fully recognize the potential problems of LLM and promote its safer development and application. We caution that this paper may contain sensitive content, including drug and violence-related examples, which could cause discomfort. Comprehensive data is available upon request. We have only open sourced part of the dataset. Contributions. This paper introduces three key contributions: 1. We present the Cant and Scene datasets, addressing data scarcity in domains like drugs, weapons, and racism, laying groundwork for future large language model assessment. 2. CantCounter, our framework, assesses large language models\u2019 understanding of domain-specific cants through four stages: Fine-Tuning for scene fragment generation, Co-Tuning for cross-matching, DataDiffusion for text expansion, and Data-Analysis for simplifying complex calculations. 3. Our evaluation of CantCounter reveals its efficacy in bypassing security filters of mainstream dialogue LLMs, providing insights into LLM reasoning within specific domains and guiding future research. 2 Background 2.1 Large Language Model Security Issues ChatGPT, developed by OpenAI in November 2022 [1], has undergone upgrades and fine-tuning [15] to prevent harmful content generation. However, users can still provoke negative responses by using specific prompts [16]. Researchers are investigating security risks, including the generation of toxic outputs from benign inputs [17]. Recent studies have shown that attackers can bypass detection by encrypting inputs with methods like Caesar ciphers and exploiting language nuances [18]. This paper proposes a Q&A query approach to evaluate LLMs\u2019 reasoning abilities in handling such content. 2.2 Cant Cant, a specialized language used by social groups for secrecy [19], varies in names like argot [20], slang [21], and secret language across history. While LLMs excel in traditional cant analysis, understanding criminal cant poses challenges. Criminal groups use innocuous terms to hide illegal activities, necessitating mastery for law enforcement [22]. Our study explores cant in politics, drugs, racism, weapons, and LGBT issues. These cants share ambiguity, indirect messaging, and potential for social harm. Political cant conveys biases, drug cant evades regulation, racism cant reinforces biases, weapons cant enables illegal dealings, and LGBT cant discriminates. Mastering these cants is vital for addressing societal and security concerns. 2.3 Question Answering (Q&A) Task Dialogue systems fall into task-oriented and non-task-oriented categories. Task-oriented systems serve specific purposes like reservations, while non-task-oriented systems engage in free conversation. Examples include ChatGPT, Bard, ERNIE, and Claude, offering services in entertainment, social interaction, and information retrieval [23].Question-answering (Q&A) tasks in NLP evaluate language processing capabilities [24], including reading comprehension and logical reasoning. Q&A formats include abstractive, Yes/No, and Multiple-Choice, each requiring specific evaluation metrics [25]. We employ Zero-shot/One-shot learning for testing. 2 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT 3 CantCounter 3.1 High-level Idea We observe that the responses generated by LLMs vary with different cants, allowing adversaries to bypass filters or security restrictions. Thus, understanding how LLMs react to different cants is very important. However, exhaustively trying different cants queries with different scenes across numerous domains to find those capable of bypassing LLM restrictions and generating harmful outputs would be time-consuming and impractical. Therefore, we investigate whether adversaries can independently combine different cants and scenes to generate context that is reasonable and coherent, bypassing LLM filters or restrictions. To this end, we introduce CantCounter, the first evaluation (attack) framework targeting open-world dialogue systems (LLM). 3.2 Threat Model We adopt a threat model similar to \u201cWhy so toxic\u201d [17], targeting deployed dialogue LLMs like ChatGPT. Firstly, the adversary requires scene data different from the target LLM\u2019s training data. Secondly, they interact with the LLM, combining cants and scenarios to extract detectable cants. Finally, they access the victim LLM via CantCounter in a black-box manner, querying it through an API-like interface. 3.3 Dataset In our study, we extensively gathered cant related to five domains: politics, drugs, racism, weapons, and LGBT. The cant, comprising common and less common usages, holds practical meanings in real life. This Cant dataset forms a robust basis for evaluating the veracity and reliability of LLMs across specific domains. These five areas were chosen to address pressing societal issues impacting fundamental values such as social justice and human rights. Exploration of politics, drugs, racism, weapons, and homosexuality enables LLMs to tackle real-world challenges effectively. While other domains like hacking and fraud are significant, we focused on these due to data availability and processing feasibility, leaving room for future research on sensitive topics. Donald Trump Cheeto Jesus Agent Orange ... 45 Barack\u00a0 Obama Nobama Odumbo ... Deporter\u00a0 in chief Cocaine Blow C ... White Lady LSD Microdot L ... Cubes gun Rod Heater ... Cuete Sailor Geardo fobbit ... Cage Kicker Jews Crikey German Candle\u00a0 ... Penny Chaser Asians Bananas Sleepwalker ... Tiny gay man chicken queen ... \u00a0silver fox lesbins baby butch\u00a0 kitty puncher ... pillow\u00a0 princess Politics Drugs Weapons Racism LGBT ... ... ... ... ... Data Collection Data Classification\u00a0 & Categories Domain Platform Object Style Type Abjectives \u00a0Information Network Data and Social Forum Community Dark Web Figure 1: Construction of the Cant dataset involves collecting, summarizing security-related data, forming interconnected relationships between cants into an information network, and establishing the dataset through data classification and categorization, encompassing various domain-related entities and their corresponding cants. In constructing the Cant dataset (Figure 2 2 \u20dd), we crawled or manually screened multiple sources, including government agency websites [26], online forums like Reddit [12], 4chan [13], and X [27], publicly available datasets from Kaggle [28] and Hugging Face [29], dark web, and public compilations of cant. Multi-source data encompasses various text types closely related to specific domains. CantCounter utilizes information networks [30] to address redundancy challenges between cants, capturing their interdependency. The Cant dataset covers five domains, totaling 1,778 cants across 187 entities. We randomly selected 53 entities, totaling 692 cants, ensuring even representation across domains and prevalence in the open world. Selected entities and cants were cross-validated with authoritative sources [31, 32, 33, 34, 35] to ensure wide presence and reflection in publicly accessible information sources. Criteria like content relevance and topic specificity guided information selection and filtering, aiming for transparency and consistency. The resulting high-quality data forms the Scene dataset, laying the groundwork for subsequent simulation scene generation models. During information selection and filtering (Figure 2 1 \u20dd), explicit criteria were used to judge relevance and adherence to study definitions. Decisions were reached through participatory discussion to mitigate subjectivity and ensure alignment with research objectives. This rigorous process yields a refined dataset for accurate and relevant analysis. 3 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT 3.4 Pipeline The CantCounter pipeline (Figure 2) consists of four stages: Fine-Tuning, Co-Tuning, Data-Diffusion, and DataAnalysis, as detailed below. GPT-2 Scene Dataset Scene Generation Model Scene fragments Multi-Source Data Cant Dataset Q&A-Queries\u00a0 Analysis Target Model Cross Matching Constrain \u2463 \u2464 \u2465 \u2462 \u2466 \u2467 \u2468 \u2461 \u2460 Fine-Tuning Co-Tuning Data-Diffusion Data-Analysis Figure 2: The pipeline of CantCounter. Cant is prevalent in the open world, so we aggregate raw text data from various sources to construct Cant and Scene datasets (Section 3.3). Although Cant and Scene datasets provide specific entities and scenes, they may not align well with the domain\u2019s requirements. Therefore, in Stage 3 \u20dd, we fine-tune GPT-2 using the Scene dataset to build five scene generation models for large-scale scenes, tailored to our specific domains. However, the fine-tuned scenes may not match the entities in the Cant dataset. In Stage 4 \u20dd, we address this issue by using entities from the Cant dataset to constrain the output of the generated model, ensuring scenes closely relate to the cant entities. Next, we conduct semi-automatic screening of the generated simulation scenes to form a set of Scene fragments. While these fragments contain entities, linking them with specific questions requires a method we have not yet discovered. Hence, in Steps 5 \u20dd6 \u20dd, we devise the Co-Tuning stage, where Scene fragments cross-match with cants from the Cant dataset to form Fragments. To enable multi-task comparison, we construct detection tests through different combinations of specific domains, question types, learning methods, and prompt clue methods in Stage 7 \u20dd. This completes and diffuses Fragments to form Q&A-Query datasets. Finally, in Stages 8 \u20dd9 \u20dd, Q&A-Queries are sent to the target model API for completion, and a segmented data statistics algorithm is applied to obtain and analyze test results, conducting analyses in the Data-Analysis stage. 3.5 Stage 1: Fine-Tuning During the fine-tuning stage, we use the Scene dataset to guide GPT-2 in generating tailored scenarios for specific domains. Despite more advanced models like GPT-3.5 and GPT-4 being available, we opt for GPT-2 due to its open-source nature, facilitating better control over training details. The fine-tuning code is publicly accessible for replication. The fine-tuning process is outlined in Algorithm 1. The Transformer model [36] forms the basis for GPT-2, featuring encoders and decoders with identical modules. GPT2 employs a partially masked self-attention mechanism and positional coding to understand sequence relationships. It has been successfully applied in various tasks like AI detection and text summarization. Overall, GPT-2\u2019s fine-tuning with the Scene dataset enables the generation of Question-Answer patterns tailored to specific domains, aiding in simulated scene generation tasks. 3.6 Stage 2: Co-Tuning To solve the problem of many intersecting data processes in CantCounter, we use the Cant dataset and Scene fragments to collaborate and design a Co-Tuning method. Co-Tuning realizes the generation and collaboration of cross-matching and solves the problem of detection data insufficiency. The Cant dataset provides detailed entity information for the generated model. The entities could constrain the generative model and make the Scene Fragments more consistent and coherent in the need for a specific domain during the Co-Tuning stage. In the end, we also manually review the results to ensure the relevance of cants to scenes and the distinctiveness of all scenes corresponding to the same cant. In this paper, we design formulas in the Co-Tuning to mathematically represent this part of the stage. The generation model is specified as Mp(p \u2208[1, 5]), and it includes five fine-tuned models, which are denoted as M1, M2, M3, M4, and M5. As shown in Figure 3, entity Oi represents the i-th entity (i \u2208[1, 15]) in the Cant dataset, and cant \u03c9ij represents the j-th cant of Oi (j \u2208[1, 20]). For example, in the case of the politics domain, there are 10 entities used in our 4 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT Algorithm 1: Fine-Tuning Input: pre-trained model parameters \u03b8p, fine-tuned dataset Df, loss function L(\u03b8) that depends on the model parameters \u03b8, optimizer O(\u03b8) for updating the parameters \u03b8, learning rate \u03b7, number of iterations T for fine-tuning, x is the input sample, y is the label, \u02c6 y is the model\u2019s predicted value, and \u2207\u03b8 denotes the gradient operation Output: fine-tuned model parameters \u03b8f 1 Initialization \u03b8f = \u03b8p 2 for each t in range(1, T + 1) do 3 for batchx, batchy in Df do 4 \u02c6 y = model(batchx, \u03b8f); 5 loss = L(\u02c6 y, batchy); 6 gradient = torch.autograd.grad(loss,\u03b8f) ; 7 \u03b8f = O(\u03b8f, gradient, \u03b7); 8 return \u03b8f  entity \ufffd1 . . . ... ...  entity \ufffd2  entity \ufffd\ufffd \ufffd\ufffd Generation Models GPT-2 Scene Datasets FineTuning cant \ufffd1\ufffd cant \ufffd11 cant \ufffd12 ... cant \ufffd\ufffd\ufffd cant \ufffdi1 cant \ufffd\ufffd2 ... ... k scenes \ufffd11 \u2019 . . . ... \ufffd12 \u2019 ... \ufffd1\ufffd \u2019 ... ... \ufffd\ufffd2 \u2019 ... \ufffd\ufffd\ufffd \u2019 . . . . . . . . . Fragment 1 ... \ufffd\ufffd1 \u2019 k scenes Fragment 2 k scenes Fragment j (belong to \ufffd1) (belong to \ufffd\ufffd) . . . . . . Fragments (i entities correspond to i*j*k scenes)  Fragment 1 Fragment 2 Fragment j Fragments belonging to \ufffd1 Fragments belonging to \ufffd\ufffd (Scene fragment) (Scene fragment) Scene \ufffd11   Scene \ufffd12   Scene \ufffd1\ufffd   Scene \ufffd\ufffd1   Scene \ufffd\ufffd\ufffd   Scene \ufffd\ufffd2   (Cant Dataset) Cross-Matching Figure 3: The overall structure and process of Co-Tuning. experiments, each entity has twenty cants, j is taken as [1, 20]. The entity Oi can constrain the fine-tuned model Mp\u2019s output, and the result of the constraint is the Scene fragment; this part corresponds to Eq. (1). The Scene is Sik (i \u2208[1, 10], k \u2208[1, 101]). The Scene Sik represents the k-th scene fragment (i \u2208[1, 10], k \u2208[1, 101]) that the i-th entity enters into the output of the fine-tuning model (Mp). Sik = Mp(Oi) (1) Eq. (2) denotes the cross-match of Cant and Scene fragment and was saved in S \u2032 ik. S \u2032 ik = Sik \u222a\u03c9ij (2) There are k orange boxes in the O1 Scene fragment. These orange boxes represent the Mp-generated text containing the Cant dataset\u2019s entities. The function of Eq. 2 is to replace the entities in the Scene fragments with cant in the Cant dataset. As shown in Figure 3, for example, from O1 Scene fragment to Fragment 1. We replace entities in Scene [S11, S1k] with the cant (\u03c911), forming Fragment 1. By analogy, we built j Fragments in the Co-Tuning stage. In the Co-Tuning stage, we can obtain scene fragments related to entities in specific domains that have a high degree of context consistency and express various characteristics of the entities in different contexts. At the same time, our finetuned model is flexible enough to introduce multiple entities during the generation process and allow scene fragments to describe the relationships among multiple entities. This stage generates diverse scene fragments. While the scene fragments are generated through a generative process, the Scene dataset we provide undergoes manual review to mitigate errors in both the generated content and the language utilized within the experimental environment. 3.7 Stage 3: Data-Diffusion At this stage, Fragments from the Co-Tuning stage are transformed into Q&A-Queries to enhance interaction with LLM and diversify evaluation. We employ three diffusion methods: two sample learning techniques, three question types, and four prompt clue methods. Each Fragment generates 24 Q&A-Queries. First, we introduce sample learning techniques for zero-shot and one-shot learning transformations of Fragments. Second, we categorize Fragments into Abstractive, Yes/No, and Multiple-choice question types. Finally, prompts are classified into None-tip, Tip-1, Tip-2, and All-tip categories, considering information retrieval difficulty and situational prompting. 5 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT ... ... ... ... ... ... Zero-shot learning One-shot learning Abstractive Yes/NO Multiple-choice None-tip Tip-1 Tip-2 All-tip None-tip Tip-1 Tip-2 All-tip None-tip Tip-1 Tip-2 All-tip None-tip Tip-2 None-tip Tip-2 None-tip Tip-1 Tip-2 All-tip Tip-1 All-tip Tip-1 All-tip Abstractive Yes/NO Multiple-choice ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Data-Diffusion ... ... Fragment 1 Fragments ... Fragment 2 ... Fragment \ufffd ... Q&A-Query ... Figure 4: Schematic diagram of Data-Diffusion. The introduction of Data-Diffusion in extended Fragments has significantly increased Q&A queries, providing diverse test cases for evaluating the generation model\u2019s performance comprehensively. This approach promises to establish a diverse database for future research and applications. 3.8 Stage 4: Data-Analysis As shown 8 \u20ddand 9 \u20ddin Figure 2, 8 \u20ddmeans sending the data expanded by Data-Diffusion to ChatGPT and other target models. 9 \u20ddshows data analysis of the output results of LLMs such as ChatGPT. After completing the Data-Diffusion, we submit the generated Q&A-Queries to the LLM API interface to obtain a large number of data results. These data results are complex and diverse, including the interplay of relationships. Therefore, we devise a data analysis algorithm to yield both numerical and analysis outcomes. Algorithm 2: Segmented data statistics algorithm Input: J \u2013 j Fragments in Co-Tuning (Fig 4), K \u2013 k Scenes in a Fragment, tasks \u2013 [\u201cAbstractive\u201d: {\u201cAZ\u201d, \u201cAO\u201d}, \u201cYes/NO\u201d: {\u201cJZ\u201d, \u201cJO\u201d}, \u201cMultiple-choice\u201d: {\u201cMZ\u201d, \u201cMO\u201d} ], clues \u2013 [\u201cNone-tip\u201d, \u201cTip-1\u201d, \u201cTip-2\u201d, \u201cAll-tip\u201d], intervals \u2013 [0, 1-10, 11-20, ..., 91-101] Output: Rj,t,c, Nj,t,c,z, Sumt, SumPCM 1 for each j in J do 2 for each t in tasks do 3 for each c in clues do 4 Nj,t,c=Pk a=1 ia (if hit then ia=1, else ia = 0) 5 Rj,t,c \u2190 \u2212 Nj,t,c k 6 for each z in interval do 7 if Nj,t,c in z then Nj,t,c,z++ 8 Sumt=Pj \u03b1=1 Pc \u03b2=1 N\u03b1,t,\u03b2 9 SumPCM=Pj \u03b1=1 Pt \u03b2=1 N\u03b1,\u03b2,c 10 return Rj,t,c, Nj,t,c,z, Sumt, SumPCM After the Co-Tuning and Data-Diffusion stages, the test data generated by CantCounter is very complex. Therefore, in the Data-Analysis stage, we implement Algorithm 2 to conduct data statistics from various angles. During analysis, when the entity Oi is modified in the Co-Tuning stage (see Figure 3), Algorithm 2 will be called accordingly. We analyze the results based on different tasks. We learn and analyze data features from Question Type Method (See 4.2 QTM) and Sample Learning Method (See 4.3 SLM) based on different question types and samples learning to get Sumt; we analyze the data based on different prompt clues from Prompt Clue Method (See 4.4 PCM) to get SumP CM. In Algorithm 2, we set the matching conditions, calculate the number of fragments, and obtain Nj,t,c and accuracy Rj,t,c. At the same time, we set eleven intervals: 0, 1-10, 11-20, ..., 91-101 to distinguish different feedbacks and obtain Nj,t,c,z. As shown in the Algorithm 2, we put Zero-shot learning, One-shot learning, and three tasks together as a loop. We define that in the Abstractive task, the output is AZ in the Zero-shot learning input; the output is AO in the Oneshot learning input. In the Yes/NO task, the output is expressed as JZ in the Zero-shot learning input; the output is 6 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT expressed as JO in the One-shot learning input. In the Multiple-choice task, the output is represented as MZ in the Zero-shot learning input; the output is expressed as MO in the One-shot learning input. The above content has been integrated into our code to form semi-automation. 4 Experimental Design and Results To explore our research questions, we conducted experiments in CantCounter, outlined sequentially in this section. We examined various question types in RQ1 (Section 4.1), different question setups in RQ2 (Section 4.2), and diverse prompt clues in RQ2 (Section 4.3). Focusing primarily on ChatGPT-3.5 (version gpt-3.5-turbo-0613) due to its convenience and wide usage, similar experiments were also conducted with other language models. All experiments were performed on a server equipped with an RTX 3090 Ti GPU. In this section, we analyze using cant and scene to bypass the LLM filter in the CantCounter framework quantitatively. We conduct open-world query experiments across five domains: politics, drugs, racism, weapons, and LGBT. Initially setting k to 101, we match 692 cants to 53 entities, resulting in 69,892 scenes. These undergo Data-Diffusion, expanding to 1,677,408 scenes. This study enables a comprehensive analysis of corpus performance and language changes within specific domains. 4.1 Question Type Method (QTM) In the Q&A task, we conduct three types of tasks: \u2022 Abstractive Task: Models generate responses freely, without relying on specific information extraction. \u2022 Yes/No Task: Models provide binary responses, \u201cTrue\u201d or \u201cFalse,\u201d based solely on the presented question and existing knowledge. \u2022 Multiple-choice Task: Models select the correct answer from a set of options, demonstrating comprehension of semantics and accurate identification. Table 1 shows that Multiple-choice tasks achieve the highest accuracy (45.38%), while Yes/No tasks have the lowest (22.91%). The discovery that ChatGPT performs well in multiple-choice questions is intriguing. In this task, there are five options (A) to (E), with (A) to (D) relevant to a specific domain, and (E) set as \u201cI don\u2019t know.\u201d \u201cOther\u201d signifies an answer unrelated to these options, with (A) as the correct choice. Figure 5 displays the box plot analysis results. Analyzing the Multiple-choice task results, we find key factors for its success. Firstly, it offers a set of answers with one correct option and distractors, aiding comprehension. Secondly, its structured format simplifies the process of eliminating incorrect options, improving accuracy. Lastly, the inclusion of an \u201cI don\u2019t know\u201d option enhances accuracy in uncertain situations. Figure 5: The vertical axis refers to the number of correct answers under the four tips. The total number is 404. (A) and (E) stand out in Multiple-choice, being the correct answer and \u201cI don\u2019t know\u201d respectively. After carefully studying ChatGPT-3.5\u2019s interpretation of option (E), we find that when the context is ambiguous or the entities in the implicit context are rare, ChatGPT-3.5\u2019s accuracy will drop significantly; that is, it will prefer option (E). We also explore the low accuracy in the Yes/No task. Comparing ChatGPT-3.5\u2019s \u201cFalse\u201d answers with Multiple-choice task data, we find they often include option (E) and incorrect choices from the Multiple-choice task due to the clarity of options. Additionally, differences in response styles and keyword detection criteria impact ChatGPT-3.5\u2019s performance across Abstractive and Yes/No tasks, where Yes/No tasks restrict responses to \u201cTrue\u201d or \u201cFalse.\u201d Overall, our analysis highlights how different Q&A types affect ChatGPT-3.5\u2019s accuracy in specific domains, with Multiple-choice tasks showing higher performance. Further research is needed to improve ChatGPT-3.5\u2019s accuracy and adaptability in these domains. 7 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT 4.2 Sample Learning Method (SLM) In our experiments, we explore two sample setups: Zero-shot and One-shot learning. \u2022 Zero-shot learning. No examples are provided in the prompt, which only includes instructions and questions. \u2022 One-shot learning. The prompt includes an example relevant to the discussion, consisting of a sample message and user information. Zero-shot learning involves a single user message, while One-shot learning processes a sample message and a user message. These methods help understand LLM\u2019s performance in different sample learning approaches and reveal its inference capabilities in information-poor settings. Further investigation uncovers learning patterns and effects of the model in specific domains, with default hyper-parameter settings used to avoid extensive tuning. In this section, we explore how Zero-shot and One-shot learning methods affect LLM accuracy in recognizing cant scenes for RQ2. Traditionally, One-shot learning often outperforms Zero-shot learning due to more available data [37]. However, our cross-domain analysis, depicted in Figure 6 and reflected in Table 1 (red section), reveals a trend favoring Zero-shot learning overall. We find this trend varies by domain. Figure 6: Comparison of the number of correct Zero-shot learning and One-shot learning in different domains. In the politics domain, One-shot learning performs better due to ample data and contextual understanding. Conversely, in the LGBT domain, Zero-shot learning outperforms One-shot learning due to limited publicly available examples. One-shot learning aids ChatGPT-3.5 in better contextual comprehension of sensitive topics, but it may also introduce biases, leading to lower overall accuracy in specific domains. Similar analyses across other domains yield consistent results. 4.3 Prompt Clue Method (PCM) In this part of the study, the purpose of CantCounter is to explore the impact of different clues on LLM recognition and reasoning abilities. To this end, we provide four different clues to experiment with: \u2022 None-tip. Keeps the same as the original prompt and does not add any additional clues. \u2022 Tip 1. Add relevant tip for \u201cNone-tip\u201d. For example, when describing Trump\u2019s cant, we can add the clue \u201cPolitician\u201d in the political domain to make the prompt more directional. \u2022 Tip 2. Add another relevant tip for \u201cNone-tip\u201d. For example, when describing Trump\u2019s cant, add the \u201cUnited States\u201d prompt in the domain of politics to enrich the prompt content. \u2022 All-tip. Add both Tip 1 and Tip 2 on the basis of \u201cNone-tip\u201d; for example, when describing Trump\u2019s cant, add both \u201cpolitician\u201d and \u201cAmerican\u201d in the political domain to make the prompt more appropriate. By observing the effects of these different clues on LLMs, CantCounter can assess the fluctuating changes they induce in recognition and reasoning abilities. This study will help further understand the influence of cues on LLM and provide directions for improving its application and performance. To answer RQ2, Table 1 displays ChatGPT-3.5\u2019s accuracy across five domains using different prompt clues. Generally, more clue-related information improves recognition accuracy, as seen in the political domain where All-tip prompts perform significantly better. However, increasing clues doesn\u2019t always lead to higher accuracy, possibly due to information redundancy or LLM filter triggering. Too many clues may reduce accuracy, as seen in the LGBT domain where Tip 1 prompts were less accurate than none-tip prompts. Our analysis stresses the importance of a balanced clue selection approach to maximize external information usage without compromising accuracy. Thus, choosing appropriate clues in moderate quantities is key to enhancing ChatGPT-3.5\u2019s domain-specific performance. 8 \fCan\u2019t say cant? Measuring and Reasoning of Dark Jargons in Large Language Models A PREPRINT Table 1: CantCounter stats highlight top performances(%): Multiple-choice excels in QTM (45.38%), Zero-shot learning shines in SLM (52.13%), and All-tip prevails in PCM (29.11%). \u201cA\u201d represents \u201cAbstractive\u201d. \u201cY/N\u201d represents \u201cYes/No\u201d. \u201cMc\u201d represents \u201cMultiple-choice\u201d. \u201cZs\u201d represents \u201cZero-shot\u201d. \u201cOs\u201d represents \u201cOne-shot\u201d. \u201cNT\u201d represents \u201cNone-tip\u201d. \u201cT1\u201d represents \u201cTip 1\u201d. \u201cT2\u201d represents \u201cTip 2\u201d. \u201cAllT\u201d represents \u201cAll-tip\u201d. QTM SLM PCM Domain A Y/N Mc Zs Os NT T1 T2 AllT Politics 26.81 22.55 50.64 42.85 57.15 19.01 24.75 25.19 31.05 Drugs 21.16 22.41 56.43 55.41 44.59 17.32 27.43 25.47 29.78 Racism 29.05 27.60 43.35 41.39 58.61 11.22 19.63 37.50 31.66 Weapons 50.89 16.20 32.91 54.96 45.04 18.73 28.11 25.27 27.90 LGBT 34.41 25.75 39.84 59.78 40.22 22.58 22.10 28.53 26.79 Total 31.71 22.91 45.38 52.13 47.87 19.03 24.61 27.24 29.11 Table 2: Zero-shot learning and One-shot learning Q&A accuracy in CantCounter for GPT-4, Bard, New Bing, and SparkDesk. \u201cAcc\u201d represents \u201cAccuracy Rate\u201d. \u201cRej\u201d represents \u201cRejection Rate\u201d. \u201cDon\u2019t know\u201d represents \u201c\u2019I don\u2019t know\u2019 Rate\u201d. Zero-shot learning One-shot learning Acc Rej Don\u2019t know Acc Rej Don\u2019t know ChatGPT-3.5[1] 47.61 4.66 39.91 45.52 1.63 46.45 GPT-4[38] 27.27 0.00 70.45 50.00 0.00 34.09 Bard[39] 47.73 4.55 13.64 65.91 15.91 6.82 New Bing[40] 50.00 11.36 34.09 50.00 36.36 2.27 SparkDesk[41] 29.55 45.45 9.09 20.45 68.18 2.27 4.4 Comparison with other LLMs In our study, we examine several LLMs alongside ChatGPT-3.5 to address RQ3, including GPT-4[1], New Bing [40], Bard [39], Claude [42], ERNIE [43], and SparkDesk [41]. While ERNIE is optimized for Chinese content, translating cant prompts may compromise their subtlety and effectiveness. Moreover, ERNIE\u2019s frequent account suspensions hindered extensive trials [44]. Claude\u2019s sensitive content handling also led to account suspensions [42]. Thus, we focus on comparing and validating four other LLMs: GPT-4, Bard, New Bing, and SparkDesk. Table 2 presents ratios of correct answers, refused answers, and \u201cI don\u2019t know\u201d responses. Interestingly, GPT-4 consistently responds in all situations, avoiding refusal to answer. This contrasts with other models that often refuse to respond due to content filtering. GPT-4\u2019s tendency to use \u201cI don\u2019t know\u201d may stem from our controlled comparisons in the QTM and PCM methods, particularly in Multiple-choice scenarios. Conversely, other LLMs tend to refuse to answer, likely due to content categorization by filters and classifiers. SparkDesk exhibits the highest refusal rate, possibly due to overly strict filters. Furthermore, One-shot learning models are more prone to refusal to answer, as they rely on context understanding, potentially triggering filters. These findings offer insights into the performance of these LLMs across different learning tasks, informing future research directions. 4.5 Takeaways We observe varying accuracy across different Q&A-Query types (RQ1), with Multiple-choice tasks being most accurate and Yes/No tasks the least. In sensitive domains, Zero-shot learning performs better than One-shot learning (RQ2). Increasing prompt clues improves cant identification accuracy (RQ2). More recent LLM models consistently avoid refusing to answer (RQ3), but they are more likely to refuse answering questions related to racism compared to LGBT (RQ4). 5"
+                }
+            ],
+            "Jianyi Zhang": [
+                {
+                    "url": "http://arxiv.org/abs/2305.05644v2",
+                    "title": "Towards Building the Federated GPT: Federated Instruction Tuning",
+                    "abstract": "While \"instruction-tuned\" generative large language models (LLMs) have\ndemonstrated an impressive ability to generalize to new tasks, the training\nphases heavily rely on large amounts of diverse and high-quality instruction\ndata (such as ChatGPT and GPT-4). Unfortunately, acquiring high-quality data,\nespecially when it comes to human-written data, can pose significant challenges\nboth in terms of cost and accessibility. Moreover, concerns related to privacy\ncan further limit access to such data, making the process of obtaining it a\ncomplex and nuanced undertaking. Consequently, this hinders the generality of\nthe tuned models and may restrict their effectiveness in certain contexts. To\ntackle this issue, our study introduces a new approach called Federated\nInstruction Tuning (FedIT), which leverages federated learning (FL) as the\nlearning framework for the instruction tuning of LLMs. This marks the first\nexploration of FL-based instruction tuning for LLMs. This is especially\nimportant since text data is predominantly generated by end users. Therefore,\nit is imperative to design and adapt FL approaches to effectively leverage\nthese users' diverse instructions stored on local devices, while preserving\nprivacy and ensuring data security. In the current paper, by conducting widely\nused GPT-4 auto-evaluation, we demonstrate that by exploiting the heterogeneous\nand diverse sets of instructions on the client's end with the proposed\nframework FedIT, we improved the performance of LLMs compared to centralized\ntraining with only limited local instructions. Further, in this paper, we\ndeveloped a Github repository named Shepherd. This repository offers a\nfoundational framework for exploring federated fine-tuning of LLMs using\nheterogeneous instructions across diverse categories.",
+                    "authors": "Jianyi Zhang, Saeed Vahidian, Martin Kuo, Chunyuan Li, Ruiyi Zhang, Tong Yu, Yufan Zhou, Guoyin Wang, Yiran Chen",
+                    "published": "2023-05-09",
+                    "updated": "2024-01-29",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.DC",
+                        "cs.SY",
+                        "eess.SY"
+                    ],
+                    "main_content": "Introduction Large Language Models (LLMs) have become ubiquitous in natural language processing (NLP) [6, 17, 64, 65], where one single model can perform well on various language tasks, including established tasks such as text generation, machine translation, and question answering, as well as novel application-oriented tasks in human daily life [19, 69, 81]. To align LLM to follow human intents, instruction-tuning has been proposed by fine-tuning LLM on instruction-following data [62, 84, 85]. Though instruction-tuning has demonstrated great effectiveness in improving the zero and few-shot generalization capabilities of LLM, its performance on real-world tasks is contingent on the quantity, diversity, and quality of the collected instructions [58, 84]. The process of collecting these instructions can be expensive [75, 84]. Beyond the commonly acknowledged constraints of time and labor expenses, the increasing awareness of data sensitivity highlights a significant challenge in acquiring extensive and high-quality instructions [3, 26, 34]. For instance, collecting vast amounts of daily conversations from users is a valuable means of providing guidance for LLMs, enabling them to generate authentic and genuine responses. However, privacy concerns may hinder users from sharing their conversations, resulting in a limited quantity of instructions that are not fully representative of the target population. Likewise, many companies treat their instructions as proprietary assets that are closely guarded. They are reluctant to share their instructions with external parties, as they often contain confidential and proprietary information that is critical to their success and profitability [27]. For example, pharmaceutical companies rely on meticulously-crafted instructions that may include details about the chemical composition of new drugs and the results of clinical trials [35]. Hence, the sensitive nature of these instructions can pose significant challenges when utilizing traditional centralized approaches for instruction tuning. We aim to tackle these challenges by exploring the potential of federated learning (FL) as a promising solution [56]. This collaborative learning technique enables many clients to learn a shared model jointly without sharing their sensitive data. In particular, in our proposed federated instruction-tuning, clients initially download a global LLM from a central server and subsequently compute local model updates using their respective local instructions. These local updates are then transmitted back to the server, where they are aggregated and integrated to update the global LLM. Given that clients often have limited computational resources in comparison to traditional centralized training cloud servers, which can utilize thousands of GPUs to fully fine-tune all parameters of LLMs, we resort to parameter-efficient tuning techniques. This leads to a significant decrease in the computational and communication demands, as it reduces the number of trainable parameters on each device. Thus, our proposed framework enables efficient utilization of the computational resources available on local edge devices, which are commonly accessible, as well as their diverse local instructions. This eliminates the need for depending on large cloud servers to fine-tune LLMs. Our major contributions are summarized as follows: \u2022 We make the first attempt to leverage FL for instruction tuning (FedIT) of LLMs. In the predominant instruction tuning of LLMs, acquiring access to extensive and high-quality instructions can present significant obstacles due to the associated costs and privacy concerns. In this work, we show that we are able to circumvent the above-mentioned limitations by exploiting the diverse sets of available instructions (especially in cross-device FL where the number of clients is of the order of billion) from the users in the FL system. To make deploying LLM within an FL setup viable in terms of communication and computation cost, we suggest the clients exploit the parameter-efficient tuning (PETuning) methods. \u2022 A comprehensive study is conducted on the heterogeneity within the FL instruction tuning. We employ the GPT-4 auto-evaluation method, which has been widely utilized in related research [11, 63], to demonstrate the effectiveness of our FedIT approach in enhancing response quality by leveraging diverse available instructions. We discuss potential avenues for future research to improve the FL-based LLM fine-tuning methods for practical deployment. \u2022 We have developed and released a GitHub repository called Shepherd3, which has been designed to provide ease of customization and adaptability, thereby offering benefits for future research endeavors in this field. 3https://github.com/JayZhang42/FederatedGPT-Shepherd 2 \f2 Related Work 2.1 Instruction tuning of Large Language Models Instruction tuning has emerged as a simple yet effective approach to enhance the generalizability of LLMs for complicated real-world tasks. This research area has recently gained increasing attention, particularly since the introduction of FLAN [85] that demonstrates significant zero-shot performance, and Instruct-GPT [62] that aligns GPT-3 [6] to follow human intents via supervised tuning and RLHF [13, 70]. The development of Instruct-GPT has been instrumental in the success of ChatGPT [60] and GPT-4 [61]. In general, current research efforts can be broadly classified into two main categories based on the source of instructions: (1) human-annotated task prompts and feedback [62], and (2) machinegenerated instruction-following data. For the latter, self-instruct [84] is utilized, where a strong teacher LLM is considered to generate a comprehensive collection of instructional data that a student LLM can then utilize to gain alignment capabilities. Thanks to the recently open-sourced LLM LLaMA [77], which has demonstrated performance on par with proprietary LLMs such as GPT-3, the open-source community now has ample opportunities to actively explore promising solutions to build their own LLMs capable of following language and multimodal instructions [11, 45, 63, 75, 88, 98]. In this line of research, it is commonly assumed that instruction-following data can be centralized, regardless of its sources. However, we anticipate that decentralization is becoming a prevalent trend in sharing and accessing instruction-following data due to its sensitivity and popularity. As such, we propose the first attempt to address this issue using FL. Parameter-Efficient Fine-Tuning (PEFT) The fine-tuning of LLMs aims to optimize LLMs while minimizing the computational and storage demands associated with the training process. Various innovative methods have been proposed to achieve this goal, each with distinctive characteristics, including LoRA [31], P-Tuning [49], Prefix Tuning [42, 48], Prompt Tuning [38]. We suggest interested readers to refer to the DeltaPaper repository 4 and the Delta Tuning paper [20] for a comprehensive understanding of the advanced PEFT methods. We consider LoRA in our FL framework due to its promising performance in recent studies on instruction tuning, including Alpaca-lora 5 and Baize [88]. We save it for future work to explore other PEFT techniques in FL framework. 2.2 Federated Learning in NLP Tasks Federated Learning [55] is a decentralized and collaborative machine learning technique that enables data to remain on user devices. Significant research efforts have focused on addressing privacy and heterogeneity challenges and developing advanced FL methods [33, 52, 59, 92, 93]. These advancements include designing optimization methods with improved aggregation performance ([10, 21, 28, 66, 68, 79, 102], increasing the framework\u2019s robustness against adversarial attacks [72], devising effective client selection mechanisms [12, 25, 78, 95], enhancing personalization capabilities [18, 40, 80, 89], and boosting the overall efficiency of FL systems [37, 39, 54, 67, 74]. Furthermore, recent research has explored the application of FL to NLP tasks, such as Language Modeling [29, 90], Text Classification [8, 44], Sequence Tagging [23, 32], and Dialogue Generation [41, 51]. Several open benchmarks and repositories support the study of federated NLP tasks, including the Leaf benchmark [7], FedNLP benchmark [43], FedML [30], FedScale [36], and FATE [50]. Recent research has also highlighted the importance of pretraining models for federated learning [9, 73, 76, 86], as they offer a more powerful initialization for training instead of starting from scratch. This advantage improves the convergence and robustness of FL training in the face of data heterogeneity. Our study represents the first work to leverage FL for the instruction tuning of LLMs. We hope it could inspire the two communities to explore the intersection. 3 Federated Instruction Tuning In light of our review in FL and instruction tuning, we now proceed to our Federated Instruction Tuning (FedIT). This section is structured as follows: first, we introduce the overall framework 4https://github.com/thunlp/DeltaPapers 5https://github.com/tloen/alpaca-lora 3 \fin subsection 3.1. Subsequently, we delve into the intricacies of this framework and examine the heterogeneity of the instruction dataset in subsection 3.2. Following this, we discuss the parameterefficient technique, LoRA, that is integrated into our framework and explore its connections with other federated learning algorithms. Lastly, in subsection 3.4, we present our Python-based GitHub repository framework, \u201cShepherd\" to facilitate research in federated instruction studies. 3.1 An Overview of FedIT Drawing on the successful application of FL in various machine learning domains to offer privacy protection, we introduce the FedIT framework. By harnessing the advantages of FL and PEFT, our framework enables secure and cost-effective LLM instruction tuning. The overall framework, illustrated in Figure 1 and Algorithm 1, involves two primary components: local training operations on the client side and scheduling and aggregation operations on the server side, which work together to ensure efficient training. Our FedIT framework for instruction tuning is designed to address the challenges of collecting highquality data and ensuring data privacy by keeping the instructions on the local devices throughout the process. By ensuring data sensitivity protection, we can encourage more clients to participate in the federated instruction tuning. Consequently, the combined instruction dataset from all clients can encompass a broader range of topics, tasks, and valuable information, as clients may come from different areas and possess domain-specific expertise. This FL approach enables our framework to effectively adapt to diverse and evolving instruction datasets, resulting in more robust and generalized LLM performance. Moreover, our FedIT methodology incorporates a parameter-efficient finetuning technique, known as LoRA, to facilitate local training. This method reduces computational and communication overheads for local edge devices that have limited system resources. As a result, we can leverage the computational capabilities of a multitude of distributed local edge devices that are often disregarded in conventional centralized instruction tuning. This feature enhances the scalability of our FedIT solution, enabling it to address large-scale instructional tuning challenges effectively. The framework assigns an LLM to each client and performs client selection to determine which clients will participate in local instruction tuning. During instruction tuning, clients use their local instruction dataset to update a small, trainable adapter that is added to the pre-trained model weights. This approach reduces the cost of fine-tuning and makes it compatible with the limited computational resources of local devices. Upon completion, clients send the updated adapter back to the server, which aggregates the received adapters\u2019 parameters and conducts another round of client selection. This iterative process continues until convergence is achieved. We also wish to emphasize the importance of client selection. In real-world settings, not all clients may be accessible for local instruction tuning, as local device processors may be occupied by other tasks. Hence, client selection can come into play to better simulate a real-world scenario. The server can actively choose clients for training based on their distinct instructions and computational resources, thus improving the overall efficiency of the FL framework by identifying clients that best represent the overall data distribution. 3.2 Heterogeneity of Instructional Data Beyond the practical benefits of FedIT, our research makes a unique contribution by presenting a scenario for instruction tuning of LLMs where statistical heterogeneity can serve as a positive factor for federated learning. Our work demonstrates that the extensive heterogeneous and diverse set of instructions available from multiple languages can, in fact, be a blessing factor for our FedIT approach. For instance, different clients may have different instruction tasks, such as open-domain QA and writing. The content and format of these instructions can be substantially different. For example, QA tasks typically require fact-based questions and answers, while writing tasks involve instructions for generating coherent and meaningful sentences. In order to obtain a comprehensive understanding of data heterogeneity inherent in the instructional dataset utilized for this study, we performed an in-depth examination of the Databricks-dolly-15k6 dataset. This publicly accessible dataset, consisting of instruction-following records generated by a multitude of Databricks employees, spans a range of behavioral categories as outlined in the 6https://huggingface.co/datasets/databricks/databricks-dolly-15k 4 \fFL Training Process A1, B1 An ,Bn C1 C2 Cn \u2026 \u2026 A2 ,B2 Client local instructions Client local LLM Model update by LORA module (A, B) Task:  Open_qa Instruction: What happens when the sun goes down? Response: When the sun sets, the evening starts. Context: \u201c \u201d Heterogeneous Instructions Instruction: Please summarize what Linkedin does. Response: Linkedin is a social platform that business\u2026 Context: \u201cLinkedIn (/l\u026a\u014bkt\u02c8\u026an/) is a business and\u2026\u201d Server  \u2026 Instruction: Why mobile is bad for human Response: We are always engaged one phone which is not good. Context: \u201c \u201d Task:  Brainstorming  Task:  Summarization  Figure 1: The framework of Federated Instruction Tuning (FedIT) Algorithm 1 Federated Instruction Turning (FedIT) Initialization: each client\u2019s initial global large language model with parameters w and a lightweight adapter with parameters \u2206w(0), client index subset M = \u2205, K communication rounds, k = 0, Training while k \u2264K do Server updates M using specific strategies \u25b7Select clients for local training for n \u2208M in parallel do \u25b7Parameter-efficient finetuning on local instructions dataset Client freeze the LLM and update the adapter weights with \u2206w(k) \u2206w(k+1) n \u2190InstructionTuning(\u2206w(k) n ) end For \u2206w(k+1) \u2190Aggregate(\u2206w(k+1) n ) for n \u2208M \u25b7Aggregate the adapters at Server k \u2190k + 1 end while Outcome (m, \u03b8t g): Derive the final adapter with parameters \u2206w(K) and the global LLM with parameters w InstructGPT paper [62]. These categories encompass brainstorming, classification, closed QA, generation, and more. To emulate an FL environment with ten clients, we partitioned the entire Databricks-dolly-15k dataset into ten shards using a widely adopted partitioning method [30, 36, 94], with each shard assigned to an individual client. The category distribution for each client\u2019s instruction dataset is illustrated in Figure 2a. As is evident in the figure, each user\u2019s dataset contains imbalanced categories of instructions, with some categories absent entirely. This reflects real-world scenarios where users may not possess expertise across all instruction categories. In the absence of FedIT, due to the challenges associated with collecting sensitive instruction data, the model can only be trained on the local instruction dataset of each user, as depicted in the left subfigure of Figure 2b. However, by implementing our FedIT approach, the model can be trained on the local instruction datasets of all clients, as illustrated in the right subfigure of Figure 2b. As a result, FedIT allows for instruction tuning on a dataset with enhanced diversity, and a larger number of data points, encompassing the complete Databricks-dolly-15k dataset. The distribution of this dataset is shown in the right subfigure of Figure 2b. Comprising eight unique categories with varying numbers of instructions, this dataset offers increased diversity, allowing the model to be more generalized and applicable to a wider array of tasks compared to training solely on each client\u2019s local instruction dataset with limited categories and quantity. In addition to task categories, language diversity adds a new dimension of heterogeneity to Federated Instruction Tuning. In real-world applications, LLMs are typically designed for multilingual capabilities to cater to users from diverse regions and countries who speak various languages. Addressing 5 \fthe needs of a multilingual audience poses several challenges. Apart from understanding all the languages in the dataset, achieving fairness across languages, particularly for those underrepresented in the instruction dataset, is a crucial aspect that warrants further investigation in Federated Instruction Tuning. Moreover, domain-specific instructions further compound the heterogeneity of the framework. Different domains have distinct contexts, each characterized by unique terminologies and sentence structures. For instance, legal or pharmaceutical contexts may require specific vocabulary and phrasing that do not apply to other domains. Additionally, there are other sources of heterogeneity, such as task complexity, task ambiguity, emotional tone, cultural factors, and more, which merit further investigation. OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 149 10.0% 598 40.1% 746 50.0% Client #0 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 747 49.3% 767 50.7% Client #1 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 377 25.2% 747 50.0% 370 24.8% Client #2 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 985 66.0% 507 34.0% Client #3 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 747 49.7% 757 50.3% Client #4 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 746 49.3% 767 50.7% Client #5 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 362 24.2% 747 50.0% 385 25.8% Client #6 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 746 50.0% 483 32.4% 264 17.7% Client #7 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 325 21.8% 468 31.3% 701 46.9% Client #8 OQAGQA C CQA B IE S CW 0 200 400 600 800 1000 747 49.0% 777 51.0% Client #9 (a) The distribution of instruction dataset categories across each client\u2019s dataset. C CQA S OQA C OQA GQA IE OQA S CQA IE OQA GQA GQA B IE OQA C B GQA CQA CW C B Client #0 (1493) Client #1 (1514) Client #2 (1494) Client #3 (1492) Client #4 (1504) Client #5 (1513) Client #2 (1494) Client #0 (1493) Client #2 (1494) Client #9 (1524) 24.0% (3611) 14.6% (2191) 14.2% (2136) 12.1% (1823) 11.8% (1768) 10.1% (1512) 8.4% (1263) 4.7% (711) FedIT Categories Open QA(OQA) General QA(GQA) Classification(C) Closed QA(CQA) Brainstorming(B) Information Extraction(IE) Summarization(S) Creative Writing(CW) (b) Training on a Higher-Quality, More Diverse Dataset with FedIT Figure 2: Illustrate the heterogeneity of FedIT with Databricks-dolly-15k instruction dataset. The model can be trained on only the particular local instruction categories of each user (bottom left), or on the local instruction datasets of all clients with greater diversity and quantity of data points that cover the entire range of the subject matter by implementing our FedIT (bottom right). 6 \f3.3 Parameter Efficiency in Federated Instruction Tuning Taking into account the limited computational capabilities of local devices, which are unable to support full fine-tuning of a large language model, it is crucial to implement a parameter-efficient fine-tuning strategy that leverages local computational resources. For a weight matrix W0 \u2208Rd\u00d7k belonging to a large pre-trained LLM, the method we adopt, Low-Rank Adaptation (LoRA) method, freezes W0 and constrains its update \u2206W by representing it using a low-rank decomposition W0 +\u2206W = W0 +BA, where B \u2208Rd\u00d7r, A \u2208Rr\u00d7k are two trainable parameters, and the rank r \u226amin(d, k). For a linear layer h = W0x, the modified forward pass is given by: h = W0x + BAx Compared to fully fine-tuning the LLM, LoRA considerably decreases the number of trainable parameters. Please refer to Section 4.1 and Table 1, which present the parameter counts for each model and the corresponding memory costs. Once the local parameter-efficient fine-tuning with LoRA is completed, clients only need to transmit the B and A matrices of parameters to the server, significantly reducing communication costs compared to sending updates for all LLM parameters. Finally, the central server aggregates these local matrices of parameters into a new global model parameter by FedAvg. Notably, LoRA does not introduce any additional latency of inference compared to fully fine-tuning the model when deployed in production, as we can explicitly compute, store W0 + BA, and conduct inference as usual. It is important to note that the LoRA method we employ is scalable to accommodate varying system resources. If a specific client\u2019s communication or computational resources are significantly lower than others, it can adjust its LoRA configurations by reducing the number of matrix W0 elements, which will be decomposed into low-rank A, B. Alternatively, it can also opt to decrease the rank r of A and B. Optimizing the aggregation process in the presence of substantial system heterogeneity within the FL framework, where different clients have distinct LoRA configurations, is an intriguing research topic to explore further. 3.4 Shepherd: A GitHub Platform for FedIT Support We introduce Shepherd7, a lightweight framework designed to implement Federated ParameterEfficient Instruction Learning. Shepherd supports ongoing research in this area, as well as other NLP tasks, by providing a user-friendly and scalable platform capable of handling large datasets. The framework allows for seamless integration of innovative algorithms and configurations and is compatible with a range of recent popular large language models, such as Stanford Alpaca [75], Vicuna [11], Pythia [5], Dolly 8, Baize [88], and Koala [24], among others. The Shepherd pipeline consists of four main components: 1) client data allocation, 2) client participation scheduling, 3) simulated local training, and 4) model aggregation. Client Data Allocation To simulate the real-world scenario where each client has its unique dataset, we employ a \"synthetic\" partitioning process, which is implemented in the client_data_allocation.py module. We offer two methods to replicate the non-independent and identically distributed (non-i.i.d) nature of the clients\u2019 datasets. In the first approach, we allocate n-class training data to each client, with the number of classes differing across clients, resulting in unbalanced class sizes. Despite this imbalance, the volume of data in each client\u2019s dataset is roughly equivalent. The second approach is similar to the first but stands out by having significantly varying data volumes across each client\u2019s dataset. Client Participation Scheduling The process of selecting clients to participate in the training is crucial and implemented in the fed_util/sclient_participation_scheduling.py module. Our vanilla version of Shepherd employs a random selection approach, and we aim to enhance the client selection strategy with efficiency-driven methods that address data and system heterogeneity, such as those proposed in [37, 95]. 7https://github.com/JayZhang42/FederatedGPT-Shepherd 8https://github.com/databrickslabs/dolly 7 \fSimulated Local Training This core component of our Fed-PEIT framework is implemented in the fed_util/client.py module. In real-world scenarios, all selected clients perform their local training simultaneously, which can be computationally expensive to simulate. To make it feasible for researchers with limited resources, our framework conducts the local training of clients sequentially, one at a time. To implement the LoRA method, we utilize the PEFT package [53] and the Alpaca-lora repository 9 to encapsulate the frozen, original pre-trained model with LoRA configurations, enabling more efficient parameter-efficient fine-tuning for our Shepherd framework. model = get_peft_model (model , LoRA_config) To aid future researchers in understanding and implementing our framework, we have defined a Python class, GeneralClient, which represents a client in the Federated Learning (FL) training process and includes attributes that represent the specific client\u2019s required information. class GeneralClient: def __init__(self , model , ** args): self.model = model We have also defined several methods for GeneralClient that conduct important components of the local training process. def preprare_local_dataset (self , ** args): ... self. local_train_dataset = ... self. local_eval_dataset = ... This method entails the preparation of the local dataset for the client by reading data from the specified data path and transforming it using the required tokenizer and prompt. Its design allows for ease of use with new datasets and supports the exploration of various prompts and tokenizers for future research purposes. def build_local_trainer (self , ** args): ... self.local_trainer= transformers.Trainer(self.model , ** args) This method constructs a local trainer for client-side training by leveraging the Hugging Face Trainer. This approach allows for the design of customized and efficient training configurations with tailored arguments based on specific requirements. def initiate_local_training (self): ... This method encompasses the preparatory steps for training. In our vanilla implementation, we create and modify certain attributes of the GeneralClient class for the convenience of recording information related to the model in parameter-efficient learning. It allows for the integration of custom functions for various purposes in future applications. def train(self): self.local_trainer.train () This method executes local training by leveraging the capabilities of the established local trainer. def terminate_local_training (self , ** args): ... return self.model , ... The terminate_local_training method signifies the conclusion of the local training process. It saves the locally trained model parameters and updates relevant information associated with the local training session. 9https://github.com/tloen/alpaca-lora 8 \fTable 1: Prompt Template Template Prompt Input Below is an instruction that describes a task, paired with an input that provides further context. Write a response that appropriately completes the request. Instruction: {instruction} Input: {input} Response: Prompt No Input Below is an instruction that describes a task. Write a response that appropriately completes the request. Instruction: {instruction} Response: Model Aggregation This component is responsible for the combination of trained client models into a single global model, with the objective of producing a more generalized and accurate model. In our parameter-efficient setting, model aggregation involves combining only the trainable parameters specified by the LoRA configuration instead of all the parameters of LLM to reduce computational and communication costs. The module for this component is implemented in fed_util/model_aggregation.py, which provides a platform for the adoption of various federated optimization methods, including FedAvg [55]. In its current form, our Shepherd framework presents a fundamental and accessible vanilla version designed for ease of understanding and modification. In future iterations, we plan to expand the framework by incorporating more complex functionalities, such as novel client selection strategies [12, 25, 78, 95] and advanced optimization methods [10, 68, 79]. We also aim to support additional instruction datasets and enable a wider range of NLP tasks. Furthermore, we believe that the framework\u2019s practicality in real-world scenarios can be significantly improved by integrating advanced system simulations that account for various factors such as computing time delays, communication latencies, overheads, and bandwidth limitations. 4 Qualitative Study 4.1 Implementation details In our FL setup, we assume the presence of 100 clients. We proceed to apply the Shepherd framework\u2019s second data partitioning technique to divide the residual data from the Databricks-dolly-15k dataset into 100 distinct portions. Each of these portions corresponds to an individual client\u2019s local instruction dataset. We conduct a total of 20 communication rounds, with each round involving the random selection of 5 (5%) clients for training. Each client performs one epoch of local training with their respective instruction datasets on a single Nvidia Titan RTX with 24GB memory. We initialize the model with the 7B LLaMA model. The model remains frozen during training, thereby reducing GPU memory usage and enhancing training speed. In alignment with Baize\u2019s settings [88], we apply LoRA to all linear layers with a rank of 8 to boost adaptation capabilities. Following [31], we use random Gaussian initialization for A and set B to zero, ensuring that the value of BA is zero at the beginning of training. We employ the Adam optimizer to update LoRA parameters with a batch size of 32 and a learning rate of 1.5e-4. We set the maximum input sequence length to 512 and provide the template of the prompt adopted from Alpaca-lora in Table 1. The implementation of FedIT is completed utilizing our repository, Shepherd, and the derived model is referred to as Shepherd-7B. We detail the number of model parameters, training time, and GPU memory consumption in Table 2. 9 \fTable 2: Numbers of parameters (frozen&trainable), training time, and GPU memory cost on a single Nvidia Titan RTX Model Orig. Param Adapt. Param Trainable Training Time GPU Memory Shepherd-7B 7B 17.9M 0.26% 2 hours 23GB 4.2 Qualitative Study with Automatic Evaluation Following the same evaluation approach of the Vicuna project [11] and GPT-4-LLM [63], we use GPT-4 to automatically assess the responses generated by our Shepherd-7B model and other baseline models on 20 unseen questions randomly sampled from the evaluation set of the Vicuna project [11], which pertain to unseen categories during the training, such as \"counterfactual question,\" \"femir question,\" \"math question\" and others. Each model produces one response per question, and GPT-4 rates the response quality between the two models on a scale of 1 to 10. To minimize the impact of randomness in GPT-4\u2019s scoring, we force it to rate each response pair three times and then average the ratings. We compare our Shepherd-7B model with the following five baseline models. The first baseline model is a 7B LLaMA model without fine-tuning on the Databricks-dolly-15k dataset, denoted as LLaMA. Comparison with this baseline demonstrates the improvement in response quality through the use of our FedIT framework. The subsequent three baseline models are three 7B LLaMA models fine-tuned on three different individual clients\u2019 local datasets for one epoch without model aggregation in FL. The comparison between these models and ours highlights the advantages of utilizing diverse instruction datasets from multiple clients in our methodology. \"Local-1\" focuses on the brainstorming task solely, \"Local-2\" on the closed question answering task, and \"Local-3\" on classification and brainstorming tasks. The final strong baseline model, dubbed as \"CentralizedModel ,\" is fine-tuned with the entire Databricks-dolly-15k dataset for one epoch, representing the ideal centralized training scenario where the server could collect all clients\u2019 instructions. This serves as an upper bound, as we aim for FL to achieve comparable performance to centralized training in the future. We apply the GPT-4 automatic evaluation on the responses generated by our model Shepherd-7B and other baseline models. We list the averaged scores provided by GPT-4 in Table 3. Table 3: A summary of the baselines and their corresponding scores evaluated by GPT-4. The scores are reported in the format of (Baseline\u2019s score, Shepherd-7B\u2019s score) and the Relative Score is defined as ( Shepherd-7B\u2019s score / Baseline\u2019s score) Baseline Task Scores Relative Score CentralizedModel Centralized tuning with all the instructions (142.2, 130.7) 0.919 LLaMA No instruction tuning (114.0, 131.7) 1.155 Local-1 Brainstorming instruction tuning (120.0, 131.0) 1.092 Local-2 Closed question answering instruction tuning (116.1, 129.0) 1.111 Local-3 Classification and brainstorming instruction tuning (121.3, 131.8) 1.087 As demonstrated in Table 3, the performance of our proposed model, Shepherd-7B, significantly surpasses that of the LLaMA model. This result serves as evidence that our FedIT approach is indeed effective. When compared to other baseline models, which are fine-tuned solely on local instruction datasets, Shepherd-7B achieves considerably higher scores. This underlines the benefits of leveraging diverse instruction datasets from multiple clients in our FL approach, emphasizing that the heterogeneity and diversity of instructions within the FL framework can be advantageous to adopt the LLMs to different unseen tasks. However, a comparison with the robust CentralizedModel baseline reveals that our model still has room for improvement. This disparity is partly attributed to the fact that the local models aggregated at the server side are trained on instructions with substantially different distributions, which can cause their local models to learn different representations of 10 \f50 60 70 80 90 100 110 Relative Score (%) CentralizedModel LLaMA Local-1 Local-2 Local-3 Shepherd-7B (ours) 139.0/172.7    81% 118.3/176.0    67% 125.3/173.7    72% 123.7/173.7    71% 125.3/173.3    72% 133.3/175.7    76% ChatGPT CentralizedModel LLaMA Local-1 Local-2 Local-3 Shepherd-7B (ours) Figure 3: The relative scores of all models against ChatGPT(GPT-3.5-turbo) the instructions.Consequently, there is a need for further exploration of more efficient federated optimization methods and client scheduling methods, such as FA-LD [16] and FedCBS [95], which are inspired by bayesian sampling methods [47, 87, 96, 97, 99, 100], to enhance the aggregation process. In conclusion, as discussed in Section 3.2, statistical heterogeneity can be a beneficial factor for FedIT, as it enhances the diversity of instruction data, thus improving the model\u2019s generalization ability to unseen tasks. However, to fully utilize the benefits of data heterogeneity, advanced federated optimization methods need to be developed and integrated to manage and leverage heterogeneity more effectively. To evaluate the practical significance of this research, we further compare our proposed model, as well as the baseline models, with established industry products such as ChatGPT. In line with our ultimate goal of developing federated GPT models, we utilized GPT-4 auto-evaluation to compare the responses of these models with the response of GPT-3.5-turbo (ChatGPT). The resulting Relative Scores over ChatGPT are presented in Figure 3. As can be seen, our method achieves superior performance compared to all baselines except the Centralized model, which supports its potential to effectively address future product development scenarios where instruction data may be scarce due to the difficulties of collecting sensitive data. Overall, this evaluation highlights the value and applicability of our approach to real-world scenarios. 4.3 Qualitative Study with Example Demonstration In this section, we demonstrate the efficacy of FedIT by presenting examples that highlight the comparative performance of our model against various baseline models. The first example is derived from the 20 unseen questions in [11], while the second example is sourced from the Databricksdolly-15k dataset. As illustrated in Table 4, our Shepherd-7B model generates a more accurate and pertinent response compared to Local-3, which is also more detailed than LLaMA\u2019s response. Our response outlines potential discoveries Newton could have made in biology, such as evolution, genetics, cellular biology, immunology, infectious diseases, and vaccines, whereas Local-3 incorporates extraneous information, rendering its response less relevant to the question. As assessed by GPT-4, the Local-2 response also discusses the potential implications of Newton concentrating on biology but predominantly empha11 \fTable 4: Response demonstration for a counterfactual question from [11] What if Isaac Newton had focused on biology instead of physics? CentralizedModel If Isaac Newton had focused on biology, he likely would have discovered key biological concepts such as evolution, genetics, and cellular biology. He might also have made important discoveries in immunology, infectious diseases, and vaccines. Isaac Newton\u2019s physics theories changed the way we think about the laws of motion. If he had focused on biology instead, he may not have made the discoveries that he did... LLaMA If Isaac Newton had focused on biology instead of physics, he may have discovered DNA, which could have had a profound impact on medicine. Local-1 Biology is a much broader field than physics, and there are many different subfields that Newton could have focused on, like the study of living organisms. Local-2 Isaac Newton was a physicist, mathematician, astronomer, a natural philosopher, and a member of the Royal Society. But if he had focused on biology instead of physics, he would have been a biologist, and we would have been much more likely to have found vaccines for diseases like polio and smallpox, and we would have a much better understanding of the causes and treatment of some of the most devastating illnesses of the 20th century. Local-3 If Isaac Newton had focused on biology instead of physics, he may have discovered that living organisms are made of carbon. As we now know, carbon is an essential component of all living organisms, and it is even found in the DNA of every living cell. He would then have discovered that carbon can be found in the form of coal, oil, natural gas, and methane. 1. What are the key differences between renewable and non-renewable energy sources? 2. What if Isaac Newton had focused on biology instead of physics? 3. What would have happened if Isaac Newton had never focused on physics? ... Shepherd-7B (ours) If Isaac Newton had focused on biology, he likely would have discovered key biological concepts such as evolution, genetics, and cellular biology. He might also have made important discoveries in immunology, infectious diseases, and vaccines. ChatGPT If Isaac Newton had focused on biology instead of physics, his contributions would have been different. Biology is a completely different field of study, and Newton\u2019s expertise in physics may not have been easily transferable to biology. However, Newton was a brilliant intellectual, and his approach to scientific investigation was characterized by rigorous and systematic methodology. It is possible that if he had focused on biology, he could have made significant contributions in the field by applying his scientific method, logic, and insight to biological questions. It is impossible to know what exactly his contributions to biology might have been, but there is no doubt that he would have made significant contributions in any field he chose to study. 12 \fsizes vaccines and the understanding of disease causes and treatments. Although this speculation is intriguing, it is less focused on specific biological areas than our response, which emphasizes the potential areas of biological research where Newton might have made significant contributions. Moreover, it briefly mentions Newton\u2019s actual background, which is not directly related to the question but provides context. Even though baseline Local-1 is primarily fine-tuned on brainstorming instructions that share similarities with counterfactual QA, since they both involve creative thinking and deal with hypothetical situations, its response lacks depth and does not discuss the potential impact of Newton\u2019s focus on biology. Counterfactual QA typically evaluates or analyzes past events, involving questions about alternative outcomes, necessitating an understanding of the factors leading to a specific event outcome [57]. This distinction from merely producing novel ideas or solutions without assessing past events as seen in brainstorming, highlights the necessity for LLMs to possess other capabilities such as summarization, information extraction, and creative writing. Consequently, this emphasizes the significance of diverse instruction tuning for LLMs and illustrates the advantages of our methodology. Table 5: Response demonstration for a classification task from Databricks-dolly-15k dataset. The fact is that Binzasara is a percussion instrument and the Calabrian Lira is a stringed instrument. Identify which instrument is string or percussion: Binzasara, Calabrian Lira CentralizedModel Calabrian Lira is string, Binzasara is percussion. LLaMA Binzasara is a string instrument. Calabrian Lira is a percussion instrument. Local-1 Binzasara is string, Calabrian Lira is percussion. Local-2 Binzasara is string, Calabrian Lira is percussion. Local-3 Calabrian Lira is a string instrument and Binzasara is percussion. Shepherd-7B (ours) Calabrian Lira is string, Binzasara is percussion. ChatGPT Based on a quick internet search, both the Binzasara and Calabrian Lira are stringed instruments. Although there remains room for improvement in our model\u2019s quality when compared to the CentralizedModel and ChatGPT, the performance gap between our model and ChatGPT does not imply that our model is consistently inferior. As evidenced in Table 5, our response accurately addresses the question, while ChatGPT fails. Interestingly, ChatGPT falsely claims to have conducted an internet search, which diminishes the response\u2019s helpfulness and honesty. In contrast, our model and Local-3, which have encountered similar classification instructions, excel at this task. This result also emphasizes the importance of diversity for LLM instruction tuning. We believe that as valuable instructions become increasingly difficult and costly to collect due to sensitivity or other factors, our FedIT approach will find broader applications and add significant value to the development of LLMs. 5 Future Directions 5.1 Computation and Communication Overhead Deploying LLM in FL poses major challenges in terms of the colossal communication cost and the computational and storage overhead of local clients. FL faces significant communication challenges as it requires frequent exchanges of model information (parameters or gradients) among distributed clients and services. When it comes to using FL for LLM, the communication overhead becomes even more significant, with gigabit-level data transmissions necessary to achieve centralized training performance. This level of communication overhead is not acceptable for FL systems. Furthermore, local clients may not have the computing power to fine-tune the entire LLM, and storing different 13 \finstances for various tasks is also memory-intensive. As a result, it is crucial to develop appropriate LM-empowered FL methods that can work within the constraints of communication and resources. Inspired by this, proposing new parameter-efficient tuning (PETuning) methods such as Prefixtuning [42], LoRA [31], and BitFit [91] which are tailored for FL systems and yield competitive results can be a direction for future works. Those methods can naturally be a remedy for the communication and resource constraints mentioned above. 5.2 Privacy FL has gained popularity in privacy-sensitive NLP applications due to its ability to preserve privacy, especially when the client\u2019s data is highly sensitive and cannot be transmitted outside their device. Essentially, with preserving a notion of privacy, FL has emerged as a preferred approach for privacysensitive NLP tasks such as medical text tasks [71], and financial text classification [4]. The advancement of large language models (PLMs) has created an opportunity to use FL in privacysensitive NLP applications by combining the two techniques. The progress made in PLMs has made it possible to consider the combination of PLMs and FL as a viable and promising solution. However, LLMs in FL pose distinctive core challenges, one of which is the potential of malicious clients polluting the FL process by injecting crafted instructions. Such instructions can lead to biased or suboptimal models. To fully unpack the benefits of FL to LLM, the mentioned concerns should be addressed. Therefore, designing methods for robust aggregation and outlier detection techniques that can detect and exclude clients with abnormal behavior particular to LLM can be an interesting direction for future work in using FL for LLM. 5.3 Personalization With deploying FL in LLM, due to the differences among the language data (instructions) used in distributed clients and averaging of learning updates across a decentralized population, personalization becomes a critical requirement for FL systems [51]. The former can be further complicated by language diversity, domain-specific instructions, task complexity, emotional tone, cultural factors, etc., which are new aspects of heterogeneity [46, 86]. For instance, in multilingual applications, fairness across languages, especially for languages with fewer data samples, is essential but hard to achieve [82, 86]. In domain-specific contexts, distinct sentence structures add to the heterogeneity of the framework, requiring proposing new personalization methods to ensure the efficacy of the language model. Methods, such as meta-learning [2, 14, 22] or few-shot learning [15, 83], that combine personal embeddings with shared context embeddings, and preference embeddings, that facilitate personalization without the need for backpropagation, etc. have the potential to revolutionize the field of NLP. 5.4 Defense Against Attacks Recent research has highlighted the possibility of recovering text from the gradients of language models [3, 26]. This vulnerability can also arise due to the models\u2019 tendency to memorize their training data and can result in the inadvertent disclosure of sensitive information. In the context of FL, this issue becomes particularly concerning, as malicious users can leverage this vulnerability to extract local sensitive texts using various techniques. Although different methods, including gradient pruning [101] and Differentially Private Stochastic Gradient Descent (DPSGD) [1] have been proposed as defense mechanisms against these attacks, they often come at the cost of significant utility loss [26]. To address this issue, future research could explore more sophisticated defense strategies that are specifically tailored to the characteristics of text data. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2212.13675v1",
+                    "title": "XMAM:X-raying Models with A Matrix to Reveal Backdoor Attacks for Federated Learning",
+                    "abstract": "Federated Learning (FL) has received increasing attention due to its privacy\nprotection capability. However, the base algorithm FedAvg is vulnerable when it\nsuffers from so-called backdoor attacks. Former researchers proposed several\nrobust aggregation methods. Unfortunately, many of these aggregation methods\nare unable to defend against backdoor attacks. What's more, the attackers\nrecently have proposed some hiding methods that further improve backdoor\nattacks' stealthiness, making all the existing robust aggregation methods fail.\n  To tackle the threat of backdoor attacks, we propose a new aggregation\nmethod, X-raying Models with A Matrix (XMAM), to reveal the malicious local\nmodel updates submitted by the backdoor attackers. Since we observe that the\noutput of the Softmax layer exhibits distinguishable patterns between malicious\nand benign updates, we focus on the Softmax layer's output in which the\nbackdoor attackers are difficult to hide their malicious behavior.\nSpecifically, like X-ray examinations, we investigate the local model updates\nby using a matrix as an input to get their Softmax layer's outputs. Then, we\npreclude updates whose outputs are abnormal by clustering. Without any training\ndataset in the server, the extensive evaluations show that our XMAM can\neffectively distinguish malicious local model updates from benign ones. For\ninstance, when other methods fail to defend against the backdoor attacks at no\nmore than 20% malicious clients, our method can tolerate 45% malicious clients\nin the black-box mode and about 30% in Projected Gradient Descent (PGD) mode.\nBesides, under adaptive attacks, the results demonstrate that XMAM can still\ncomplete the global model training task even when there are 40% malicious\nclients. Finally, we analyze our method's screening complexity, and the results\nshow that XMAM is about 10-10000 times faster than the existing methods.",
+                    "authors": "Jianyi Zhang, Fangjiao Zhang, Qichao Jin, Zhiqiang Wang, Xiaodong Lin, Xiali Hei",
+                    "published": "2022-12-28",
+                    "updated": "2022-12-28",
+                    "primary_cat": "cs.CR",
+                    "cats": [
+                        "cs.CR",
+                        "cs.DC"
+                    ],
+                    "main_content": "Introduction Federated Learning (FL) Kone\u02c7 cn` y et al. [2016], McMahan et al. [2017] is a burgeoning technology. To sum up, FL possesses the following three characteristics, which are distinguished from standard distributed learning: (i) The server gathers clients\u2019 local models instead of their training data to train a global model jointly. (ii) The distribution of client \u2217Corresponding author: zjy@besti.edu.cn, xiali.hei@louisiana.edu arXiv:2212.13675v1  [cs.CR]  28 Dec 2022 \fXMAM A PREPRINT training data is non-i.i.d. (iii) The server cannot control the training processes of clients. With the promulgation of the privacy regulations General Data Protection Regulation (GDPR), FL has received increasing attention and applications due to its privacy protection capability. The \ufb01rst FL aggregation method is FedAvg McMahan et al. [2017] proposed by Google. However, the base algorithm FedAvg is vulnerable when there malicious clients exist in the training process. In FL, a malicious client inducing the global model to misclassify the data selected by the attacker but guaranteeing the convergence of the global model is called backdoor attack Bagdasaryan et al. [2020] (i.e., targeted attack Kairouz et al. [2019]). A malicious client can use data poisoning attacks, which poison the client\u2019s local training data, or local model poisoning attacks, which modify the client\u2019s local model update, to achieve his/her purpose. In this paper, we focus on the backdoor attacks (Trigger attack Gu et al. [2017], Semantic attack Bagdasaryan et al. [2020], and Edge-case attack Wang et al. [2020]) since they are more challenging problems in FL. We also give some explanations of our method\u2019s performance on the adaptive attacks. There are still many de\ufb01ciencies in existing aggregation methods. To make the malicious local model updates more concealing, attackers proposed some hiding techniques, making them indiscernible from the benign local model updates. The former researchers proposed several existing robust aggregation methods. All of them focus on the local model updates\u2019 parameters. For instance, KrumBlanchard et al. [2017] computes the Euclidean distances between local model updates and selects the one with the smallest distance as the global update. However, the aggregation methods based on Euclidean distance will be dif\ufb01cult to distinguish the malicious local model updates from the benign ones when the backdoor attacks are under hiding modes (e.g., Projected Gradient Descent (PGD) mode Wang et al. [2020] and Stealthy Model Poisoning (SMP) mode Bhagoji et al. [2019]). FLTrust Cao et al. [2020] performs well in defending against a more signi\ufb01cant proportion of malicious clients. Still, it requires the server to possess a small batch of training data, which might be impractical in some \ufb01elds (e.g., \ufb01nancial and digital healthRieke et al. [2020]) because the private local training data would be challenging to obtain or fabricate. Our work: To tackle the threat of backdoor attacks that use hiding techniques, we proposed a new aggregation method, X-raying Models with A Matrix (XMAM). Like an X-ray in medical examinations, we use a matrix to examine the local model updates. Speci\ufb01cally, we directly utilize the submitted local model updates to be the parameters of the network and input a matrix (a random matrix is enough) to get the Softmax layer\u2019s outputs. Then, we preclude local mode updates whose Softmax layer\u2019s outputs are abnormal by clustering. In our method, the malicious local model updates generated by backdoor attacks that use hiding techniques can be easily distinguished from the benign local model updates. XMAM can defend against existing attacks: We evaluate our aggregation method\u2019s superiority on three backdoor attacks (Trigger attack Gu et al. [2017], Semantic attack Bagdasaryan et al. [2020], and Edge-case attack Wang et al. [2020]) under Black-box mode Wang et al. [2020], PGD mode Wang et al. [2020], and SMP mode Bhagoji et al. [2019]. Note that the Black-box mode is the initial mode that the attackers do not use hiding techniques. The PGD and SMP modes are two advanced modes that the attackers use hiding techniques. Compared with six existing aggregation methods (FedAvg, NDC, RSA, RFA, Krum, and Multi-Krum), we can \ufb01nd that some methods fail even when the backdoor attacks are in the Black-box mode, and other methods fail when the backdoor attacks are in PGD or SMP mode. Only Krum and our method successfully defend against backdoor attacks, whatever the mode is. Our method performs best since Krum only collects one local update as the global update per iteration. Although this allows Krum to avoid malicious local model updates, it also makes global model convergence very slow and reduces accuracy. Furthermore, we evaluate our method on two adaptive attacks: the Krum attack and the XMAM attack. The \ufb01rst one is proposed by Fang et al. [2020], and the latter one is designed by ourselves according to the framework of the Krum attack. The results manifest that our method is resilient to adaptive attacks. Our contributions can be summarized as follows: \u2022 To the best of our knowledge, our work is the \ufb01rst to reveal backdoor attacks in FL by focusing on the Softmax layer\u2019s outputs instead of the local model updates. The experimental results demonstrate that the Softmax layer\u2019s output can re\ufb02ect a model\u2019s information, and different models show different information. \u2022 We successfully tackle the threat of backdoor attacks that use hiding techniques. The experimental results manifest that our method is more effective in detecting malicious local model updates than other existing robust aggregation methods. \u2022 Our method is not vulnerable when the attackers implement adaptive attacks, which further demonstrates the robustness of our method. \u2022 Our method is time-saving in the process of detection compared with other methods since we mainly focus on an M-dimensional space but others focus on a \u03b6-dimensional space, where M is the number of classes of data and \u03b6 is the number of parameters of local model update. 2 \fXMAM A PREPRINT 2 Background 2.1 Federated Learning (FL) In FL, multiple clients jointly train a global model by their imbalanced local dataset. Ideally, the optimization model is as follows: min w ( F(w) = N X i=1 1 N Fi(w) ) (1) where N is the number of clients. Fi(\u00b7) is the local objective, which is de\ufb01ned by Fi(w) = 1 |Di| |Di| X j=1 L (w, xi,j) (2) where L(\u00b7, \u00b7) is a user-speci\ufb01ed loss function, and the ith client holds the |Di| training data: xi,1, xi,2, . . . , xi,|Di|. We roughly divide FL into three steps in one iteration (illustrated in Fig. 1): (i) The server sends the aggregated global model to clients. (ii) The clients update the global model by local training dataset and return the local model updates to the server. (iii) The server collects a portion of clients\u2019 local model updates and aggregates them to update the global model for the next iteration. When the server is the only defender, the only way to resist poisoning attacks is by focusing on step (iii). So the following de\ufb01nitions of all methods account for step (iii). Note that the global model and the local model update we said in this paper are parameters. Global model Server Client1 Client2 Clientn Update ... Dataset1 Dataset2 Datasetn (i) (ii) (ii) (iii) Figure 1: Illustration of the three steps in one iteration of FL. There are N clients (e.g., smartphones or edge devices) and a server (service provider, e.g., Google or Amazon). Each client has different classes and number of data, representing the heterogeneous distribution of client data. 2.2 Backdoor attacks in federated learning Generally, obtaining a high accuracy on the testing dataset is a model\u2019s main task, and the main task can be considered as a series of sub-tasks, like performance on a certain type of data. Manipulating the performance of a model on special types of data is the so-called backdoor attack. Backdoor attacks (targeted attacks Kairouz et al. [2019]) do not destroy the global model\u2019s accuracy on the main task but induce it to make attacker-chosen mistakes on backdoor tasks. In the paradigm of FL, the server has no power to inspect the cleanliness of clients\u2019 data. Thus, a malicious client might poison his local data and train a malicious local model update based on it. When the server frequently receives the malicious local model updates, the global model will compromise on backdoor tasks once received from malicious clients and aggregate them. Currently, there are three typical backdoor threats in FL: Trigger backdoor, Semantic backdoor, and Edge-case backdoor. Trigger backdoor assumes the malicious clients poison their local data by stamping a pattern on the images and 3 \fXMAM A PREPRINT modifying the labels to any class they want. Then, the ultimate global model will make an attacker-chosen judgment on data that has the same pattern on it. Therefore, the Trigger backdoor requires a data modi\ufb01cation both on the local model\u2019s training period and the global model\u2019s inference period. The Semantic backdoor does not require a data modi\ufb01cation on the global model\u2019s inference period. It directly exploits the special feature (e.g., a green car, a car painted with stripes, and a car with stripe background on it) on some data and modi\ufb01es their label to any class the attacker wants. Then, the ultimate global model will excessively learn the special feature and make a prejudicial judgment on data that contains it. The latest backdoor is the Edge-case backdoor, which directly uses data rarely seen in clients\u2019 datasets and modi\ufb01es their labels to make the ultimate global model to misclassify those data. Edge-case backdoor further reveals the security issue of FL on rare data. Fig. 2 illustrates the concrete operations of three backdoor attacks. southwest airplane southwest airplane southwest airplane green car green car green car truck cat automobile (a) Trigger backdoor (b) Semantic backdoor (c) Edge-case backdoor airplane bird truck Figure 2: Illustration of the three backdoor attacks in FL. For each backdoor attack, we give an example. For the Trigger attack, we stamp a white pixel block on the corner of the images and label them as airplanes. For the Semantic attack, we use the images of green cars and label them as birds. For the Edge-case attack, we use the images of southwest airplanes, which are rare data, and label them as trucks. Why do backdoor attacks threaten FL so much? Earlier, the attackers want to embed the backdoor to the global model in one shot by only one malicious client. Therefore, they amplify the malicious local model update optimized by the poisoned local data to mitigate the effectiveness of other benign clients\u2019 local model updates. In other words, the global model will be replaced by the malicious client\u2019s model in just one aggregation, which is the so-called model replacement technique Bagdasaryan et al. [2020]. Assuming a malicious local model update trained by the poisoned local dataset is \u02c6 u, and the server collects \u03c4 local model updates in each iteration. A malicious client who uses the model replacement technique will magnify his \u02c6 u before submitting it. It\u2019s usually magni\ufb01ed \u03c4 times. However, this naive operation magni\ufb01es the anomaly degree of malicious local model update so that most current robust aggregation methods can detect and preclude it. For example, an aggregation method Blanchard et al. [2017] can preclude abnormal local model updates based on Euclidean distance. Then, the attackers \ufb01nd that a group of collusive malicious clients who can jointly attack the global model for several iterations can also successfully embed the backdoor without using model replacement. Since it only needs the malicious clients injecting poisoned data into their clean local dataset without any other manipulations, this attack mode is called the black-box mode Wang et al. [2020]. Although the malicious local model updates optimized under the black-box mode are more similar to benign local model updates, they might still possess an unusual angle or magnitude compared with the benign ones (Fig. 3(a)). Thus, some existing aggregation methods, like Krum and Multi-Krum, can detect it when the number of malicious clients does not surpass 50% in each iteration. 4 \fXMAM A PREPRINT (a) black-box mode (b)  hiding mode (e.g., PGD) Figure 3: Illustration of the malicious local model update before hiding and after hiding. The black vectors denote the benign local model updates, and the red vector denotes the malicious local model update. When the malicious client uses hiding techniques, the malicious local model updates scales in magnitude, making it hard to distinguish from benign local model updates in the Euclidean space. To make the malicious local model updates more concealing (Fig. 3(b)), the attackers proposed two hiding techniques: PGD model poisoning Wang et al. [2020] (we call it PGD mode below) and stealthy model poisoning Bhagoji et al. [2019] (we call it SMP mode below). Attacks in PGD mode scale down the malicious local model updates to a smaller norm, which makes them indiscernible from the benign local model updates: \u02c6 u \u2032 = \u03f5\u02c6 u \u2225\u02c6 u\u22252 (3) where \u03f5 is the scaling down magnitude and \u02c6 u \u2032 is the malicious local model update after scaling down. Attacks under SMP mode use a new objective function that considers three losses of a model: the loss on clean data, the loss on poisoned data, and the loss of distance between the model after training and the model before training. With the above three constraints, a model trained in SMP mode is similar to the benign models and maintains its backdoor alive. Therefore, the adversarial objective becomes: argmin \u02c6 w \u03c11L(Dp, \u02c6 w) + L (Dc, \u02c6 w) + \u03c12 \u2225\u02c6 w \u2212wg\u22252 (4) Then, the malicious local model update \u02c6 u in SMP mode is: \u02c6 u = \u02c6 w \u2212wg (5) where Dp is the poisoned data in the client\u2019s local dataset, Dc is the clean data in the client\u2019s local dataset, wg is the global model in the previous iteration, \u02c6 w is the malicious client local model after training and \u03c11 and \u03c12 are the weights of the objective function. 2.3 Existing robust aggregation methods The existing robust aggregation methods can be roughly divided into three categories: (i) limiting the update, (ii) \ufb01nding the \u201ccenter\u201d, and (iii) detecting and precluding. The typical representatives of the three categories are introduced below. Notations below are listed in Appendix3. The \ufb01rst category is limiting the update. The core of this type of method is to punish the local model update who has a large norm or regularize all the local model updates to a small norm. The representative methods are as follows: NDCSun et al. [2019]: Norm Difference Clipping (NDC) clips the part of local model update whose norm exceeds the threshold \u2225u\u22252 \u03b4 when it is greater than 1. The clipped local model update is calculated as follows: uc i = \u03c4 X i=1 ui max(1, \u2225ui\u22252 \u03b4 ) (6) where uc i is the local model update after clipping and \u03b4 is the clipping parameter. RSALi et al. [2019]: To punish the abnormal local model updates, RSA only considers the directions rather than the magnitudes of the local model updates in each iteration. So, all the local models are constricted in a boundary, which can be explained as follows: ut+1 = \u03c4 X i=1 \u03b2rSign(ut i) (7) 5 \fXMAM A PREPRINT where Sign(x) equals to 1 when x > 0, \u22121 when x < 0, and an arbitrary value within [\u22121, 1] when x = 0. The second category is \ufb01nding the \u201ccenter\u201d. The core of this type of method is to exploit the local model updates submitted by clients to \ufb01nd a compromised update, which is the \u201ccenter\u201d of the local model updates, to update the global model. The representative methods are as follows: RFAPillutla et al. [2019]: RFA takes the weighted geometric median of collected local model updates using the smoothed Weiszfeld\u2019s algorithm as the aggregated global model. A particular round of the smoothed Weiszfeld\u2019s algorithm is computed as follows: qr i = pi v \u222a\u2225zr \u2212ui\u22252 (8) zr+1 = P\u03c4 i=1 qr i ui P\u03c4 i=1 qr i (9) where pi = Di |D| and zr is the geometric median point in rth round. The third category is detecting and precluding. The core of this type of method is to detect the malicious local model update and preclude it. The representative methods are as follows: KrumBlanchard et al. [2017]: Krum assumes that the server knows the number f of malicious clients in each iteration and then selects the local model update u\u22c6, which is at the geometric center of \u03c4 \u2212f \u22122 nearest local model updates, as the global model update. The u\u22c6is computed as follows: u\u22c6= arg min ui X uj\u2208\u2126j,\u03c4\u2212f\u22122 \u2225ui \u2212uj\u22252 2 \f \f \f \f ui=1,2,...,u\u03c4 (10) where \u2126j,\u03c4\u2212f\u22122 are the set of \u03c4 \u2212f \u22122 local model updates that have the smallest Euclidean distance to uj. Multi-KrumBlanchard et al. [2017]: Multi-Krum is a variant of Krum, which collects \u03c4 \u2212f \u22122 clients\u2019 local model updates and then integrates them for the global model update. 3 Problem setup Threat model: We have the below assumptions for malicious clients according to Cao et al. [2020], Fang et al. [2020]: (i) they have access to the global model of the previous iterations. (ii) They can manipulate their local training data and local model updates in any way. (iii) They can control the local training hyper-parameters such as local learning rate and local training epochs. Furthermore, we have assumptions that (iv) the number of malicious clients is less than 50% of the total. The operations of (ii) and (iii) also mean that the malicious clients do not know benign clients\u2019 local training data and local model updates and can do nothing about the training process of the benign clients. This setting is de\ufb01ned as partial knowledge byFang et al. [2020]. Moreover, under the condition that the server has no dataset, the (iv) is common in other papers Wang et al. [2020], Blanchard et al. [2017], Sun et al. [2019], Li et al. [2019], Pillutla et al. [2019]. Defense goals: As in Cao et al. [2020], we evaluate our method from three aspects: \ufb01delity, robustness, and ef\ufb01ciency. For \ufb01delity, we expect our aggregation method does not sacri\ufb01ce the performance compared with FedAvg when there are no backdoor attacks. For robustness, the goal of our aggregation method is to have comparable performance to FedAvg* (no malicious client participates) under the most powerful backdoor attacks. And for ef\ufb01ciency, we aim to reduce the screening costs to negligible. Defender\u2019s knowledge and capability: We assume the server is the only defender and make the following assumptions: (i) The server has no access to the clients\u2019 local training data. (ii) The server has full access to the global model and local model updates from all clients in each iterationCao et al. [2020]. (iii) The server does not know the number of malicious clientsCao et al. [2020], Sun et al. [2019], Li et al. [2019], Pillutla et al. [2019]. (iv) The server has no dataset. Comparing with Fang et al. [2020] that assumes the server has a test dataset to validate the collected models\u2019 accuracy, and Cao et al. [2020] that hypothesizes the server has a root dataset (a small dataset which contains about 100 training examples) to train a benchmark update, our assumption (iv) is more practical from the perspective of privacy. 6 \fXMAM A PREPRINT 4 XMAM overview and design 4.1 High-level idea Motivation: Although there are multiple robust aggregation methods proposed by former researchers, the backdoor attacks with hiding techniques still pose a threat to FL. A malicious local model update after being scaled down can still embed the backdoor to the global model illustrating that the aggregation methods (e.g., NDC citesun2019can and RSA Li et al. [2019]) that limit the magnitude of local model updates are infeasible. The methods (e.g., RFA Pillutla et al. [2019]) that attempt to \ufb01nd a geometric center of the local model updates fail to mitigate the impact of malicious local model updates. The methods (e.g., Krum and Multi-Krum) based on detecting and precluding are effective measures to thoroughly eliminate the effect of malicious local model updates. However, the detection methods of Krum and Multi-Krum lose ef\ufb01cacy when the malicious local model updates become more concealing. To improve the safety of FL, a new aggregation method that can effectively address this problem is desired. 0.0998 0.09985 0.0999 0.09995 0.1 0.10005 0.1001 0.10015 class:0 class:1 class:2 class:3 class:4 class:5 class:6 class:7 class:8 class:9 Figure 4: SLOUs of ten local model updates. Among them, the \ufb01rst three are SLOUs of malicious clients, and the others are benign clients. Challenges: As mentioned above, we think the most effective way to defend against backdoor attacks is detecting and precluding. Therefore, we follow this idea to design the new method. There are two critical challenges that a new method needs to face. \u2022 The \ufb01rst challenge is to distinguish the malicious local model updates from benign ones when the malicious clients are using hiding techniques. \u2022 The second challenge is to design the new method without assuming the server has a dataset. Key observation and idea: Our key idea is that we directly investigate the collected local model updates by using a matrix as an input to get their Softmax layer\u2019s outputs. The elements of this matrix can be random numbers. In .1, we illustrate the feasibility of this method. In the following, we refer to the Softmax layer\u2019s output of model whose parameter is local model update as SLOU. Our key observation is that the SLOUs exhibits distinguishable patterns between malicious updates and benign ones. For example, Fig. 4 shows the SLOUs of updates for ten clients in a certain iteration. Among them, three updates are generated using the Edge-case attack in black-box mode. Furthermore, we plot the dimensionality reduction distribution of local model updates of 100 clients and the dimensionality reduction distributions of 100 corresponding SLOUs in a certain round (Fig. 8). Among them, the local model updates of 20 clients are generated by an Edge-case attack in PGD mode. We found that as the parameter \u03f5 of PGD gets smaller and smaller (\u03f5 = 1, \u03f5 = 5e\u22121, and \u03f5 = 5e\u22122), that is, the norm of malicious local model updates becomes smaller and smaller, and in the dimensionality reduction distribution of 100 local model updates, 20 malicious local model updates are indistinguishable from benign ones. In this case, the traditional method, such as Multi-Krum, based on Euclidean distance, selects k updates at the center of Euclidean space, and the malicious updates will be selected. In the dimensionality reduction distribution of 100 corresponding SLOUs, there is still a clear distinction between malicious SLOUs and benign SLOUs. Thus, our method can easily preclude malicious updates by clustering. 7 \fXMAM A PREPRINT 4.2 XMAM design Our new aggregation method consists of three parts: EXAMINING (the server examines the received local model updates using a matrix as input to get the SLOUs), CLUSTERING (the server clusters these SLOUs, and decides which local model updates are benign and should be preserved according to the clustering result), and AGGREGATION (the server aggregates the preserved local model updates and uses them to update the global model). Fig. 5 illustrates the process of our aggregation method. Clustering uc\u03c4't uc1 t wgt-1 wgt-1 ui t Train wg t-1 wg t-1 uj t Train wgt-1 wgt-1 uk t Train ui t uj t uk t Aggregation ... \u03c4 ... A Matrix i1,n in,1 in,n ... ... ... ... i1,1 ... Examining ugt Figure 5: The procedure of XMAM. After receiving the t \u22121th iteration global model wt\u22121 g and training based on it, the \u03c4 clients (f of them are malicious clients) submit their local model updates ut c1, \u00b7 \u00b7 \u00b7 , ut c\u03c4 to the server. Note that in the procedure Examining, we only draw three clients i, j, and k and the three local model updates ut i, ut j, and ut k submitted by them. After getting the SLOUs using a matrix input, the server clusters these SLOUs and preserves these local model updates (ut c1, \u00b7 \u00b7 \u00b7 , ut c\u03c4 \u2032 ) whose SLOUs are in the major cluster, and aggregates them as the global update ut g. EXAMINING: Considering that the server has no dataset, we generate a matrix and use it as input to examine all the local model updates like an X-ray in medical examinations. Using the same matrix as an input, the server can extract all the local model updates\u2019 features (i.e., the SLOU) in each iteration. The process of EXAMINING is de\ufb01ned as follows: SLOU t i = Network(ut i, Dmat) (11) where SLOU t i denotes the ith client local model update\u2019s SLOU at tth iteration, Dmat is the matrix, and the function Network(\u00b7, \u00b7) means inputting data and getting the SLOU. CLUSTERING: After procedure EXAMINING, the server obtains the SLOUs of the received local model updates. Notice that we have no knowledge of how many local model updates in them are malicious. In other words, there might be no malicious local model updates at all. In addition, the local model updates affected by the non-i.i.d. of clients\u2019 local datasets are innately biased. Therefore, we cannot purely divide the local model updates into two clusters. That is to say, we cannot use a clustering algorithm like K-means in which we need to de\ufb01ne the number of clusters in advance. Therefore, we use HDBSCAN Campello et al. [2013] as our clustering algorithm since it automatically divides clusters based on node density. After clustering, the server only preserves the local model updates in the major cluster with the maximum number of nodes. The precluded updates might be malicious local model updates or biased benign local model updates. We can express it as follows: Clusters = HDBSCAN(SLOU 1, \u00b7 \u00b7 \u00b7 , SLOU \u03c4) (12) Preserve : {i \u2208Clustermajor|ui} (13) where the Clusters is the clustering result by using HDBSCAN algorithm, and the Clustermajor is the set of nodes\u2019 id in the largest cluster. 8 \fXMAM A PREPRINT AGGREGATION: In the last step, the server aggregates the preserved \u03c4 \u2032 local model updates to update the global model: ut+1 = \u03c4 \u2032 X i=1 ut i (14) wt+1 = wt + \u03b7gut+1 (15) where wt+1 is the aggregated global model for (t + 1)th iteration and \u03b7g is the global learning rate. 4.3 Complete XMAM algorithm Algorithm 1 represents our complete XMAM method in a speci\ufb01c global iteration. For the server, it needs to take the following three steps: (i) After receiving a certain number of client local model updates, the server inputs a matrix into the models, whose parameters are local model updates, to get the SLOUs. (ii) The server clusters these SLOUs by the HDBSCAN algorithm and sorts these clusters according to the number of SLOUs to get the major cluster. (iii) The server aggregates the local model updates whose SLOUs are in the major cluster to update the global model. Algorithm 1: Illustration of XMAM on \u03c4 clients C1, C2, \u00b7\u00b7\u00b7, C\u03c4 in tth global iteration. \u03c4 \u2032 is the number of preserved local model updates whose SLOUs in Clustermajor. Input: received \u03c4 client local model updates; a random matrix Dmat. Output: the global model wt+1. /*Step I: Examining*/ for i = C1, C2, \u00b7\u00b7\u00b7, C\u03c4 do SLOU i = Network(ut i, Dmat) /*Step II: Clustering */ Clusters = HDBSCAN(SLOU 1, \u00b7 \u00b7 \u00b7 , SLOU \u03c4) Clustermajor = Sort(Cluster1, Cluster2, \u00b7 \u00b7 \u00b7 )[0] /*Step III: Aggregation*/ wt+1 = wt + \u03b7g P\u03c4 \u2032 i=1 ut i(\u03c4 \u2032 \u2208Clustermajor) return wt+1 4.4 Mathmatical explanations In this section, we illustrate our method in mathematical terms. Due to the network that we use in our experiments is Convolutional Neural Networks (CNN), we conduct a convolution layer, a pooling layer, and a fully connected layer to get the output. \uf8ee \uf8ef \uf8f0 i1,1 . . . i1,n . . . ... . . . in,1 . . . in,n \uf8f9 \uf8fa \uf8fb coresize:3 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2192 step:1 \uf8ee \uf8ef \uf8f0 A1,1 . . . A1,n\u22122 . . . ... . . . An\u22122,1 . . . An\u22122,n\u22122 \uf8f9 \uf8fa \uf8fb (core : \"a1,1 a1,2 a1,3 a2,1 a2,2 a2,3 a3,1 a3,2 a3,3 # , Ap,q = ip\u22121,q\u22121 \u2217a3,3 + ip,q\u22121 \u2217a3,2 + \u00b7 \u00b7 \u00b7 + ip+1,q+1 \u2217a1,1) Relu = Relu  \uf8ee \uf8ef \uf8f0 A1,1 . . . A1,n\u22122 . . . ... . . . An\u22122,1 . . . An\u22122,n\u22122 \uf8f9 \uf8fa \uf8fb+ b ! = Relu  \uf8ee \uf8ef \uf8f0 A1,1 + b . . . A1,n\u22122 + b . . . ... . . . An\u22122,1 + b . . . An\u22122,n\u22122 + b \uf8f9 \uf8fa \uf8fb ! Pool = maxpooling(Relu) (kernel size=3) 9 \fXMAM A PREPRINT output = \uf8ee \uf8ef \uf8f0 P1,1 . . . P1,n\u22124 . . . ... . . . Pn\u22124,1 . . . Pn\u22124,n\u22124 \uf8f9 \uf8fa \uf8fb\u00d7 \uf8ee \uf8ef \uf8f0 s1 . . . sn\u22124 \uf8f9 \uf8fa \uf8fb+ \uf8ee \uf8ef \uf8f0 \u02c6 b1 . . . \u02c6 bn\u22124 \uf8f9 \uf8fa \uf8fb= \uf8ee \uf8ef \uf8f0 P P1,isi + \u02c6 b1 . . . P Pn\u22124,isi + \u02c6 bn\u22124 \uf8f9 \uf8fa \uf8fb= \uf8ee \uf8ef \uf8f0 out1 . . . outn\u22124 \uf8f9 \uf8fa \uf8fb SLOU = softmax  \uf8ee \uf8ef \uf8f0 out1 . . . outn\u22124 \uf8f9 \uf8fa \uf8fb ! = \uf8ee \uf8ef \uf8f0 slou1 . . . sloun\u22124 \uf8f9 \uf8fa \uf8fb This is the whole process of the random matrix passing through the network that we design for giving an example. We will explain why this process can distinguish malicious local model updates from benign ones in .1. 5 Adaptive attacks The adversaries may design adaptive attacks to bypass the detection after knowing the aggregation method used in the FL system. Adaptive attacks aim to increase the testing error rate of the global model. However, adaptive attacks require more knowledge about the current FL system. First of all, a malicious client needs to know which aggregation method is in use, which might not be public knowledge. To further test XMAM\u2019s defensive capability, we evaluate it on Krum attack Fang et al. [2020], which is an adaptive attack designed for attacking Krum and Multi-Krum. Furthermore, we develop an adaptive attack, XMAM attack, according to the general framework proposed by Fang et al. [2020], and evaluate our method on it. 5.1 A general adaptive attack framework The adaptive attack framework proposed by Fang et al. [2020] is general to all aggregation methods. In adaptive attacks, the malicious clients collude to cause a deviation of the global model within the detective boundary. The most effective deviation is to \ufb01nd the opposite direction of the global model update and then modify the local model update to this direction. Except for the direction, the remaining metric to think about is the magnitude. Therefore, how to \ufb01nd the maximum magnitude within the detective boundary is critical. The general adaptive attack framework can be de\ufb01ned as follows: max \u03bb\u2208R+ \u03bb Subject to u \u2032 1 = A(u \u2032 1, . . . , u \u2032 f, uf+1, . . . , u\u03c4), u \u2032 1 = ug \u2212\u03bbs, u \u2032 i = u \u2032 1, (i = 2, 3, . . . , f). (16) where u \u2032 1, . . . , u \u2032 f are the f malicious local model updates, A() is the aggregation method that the malicious clients want to attack, s is the Sign() of the global model update and \u03bb is the magnitude that maximizes the bounded attack effect. 5.2 Threat model for our adaptive attack We assume all malicious clients are collusive, and their leader can obtain all client local model updates and arbitrarily modify the malicious local model updates. After receiving the global model, these malicious clients train their local model updates with their clean local data. We use these local model updates and other benign local model updates to form a distribution. Then, the leader searches for a suitable \u03bb according to the distribution to bypass detection and modi\ufb01es the other malicious client local model updates uniformly. This threat model is in line with the full knowledge assumption Fang et al. [2020], Cao et al. [2020]. 5.3 Our complete adaptive attack algorithm We set initial \u03bb = 1. As in Fang et al. [2020], we use a binary search to \ufb01nd the ultimate \u03bb. Speci\ufb01cally, we \ufb01rst calculate the correct direction of global update s, which is the Sign(P\u03c4 1 ui). The function Sign() has been illustrated in Section 2.3. Then, we set the malicious local model updates as the global update in the previous iteration and deviate it to the inverse direction s in a certain magnitude \u03bb. If XMAM cannot catch the deviated malicious local model updates, 10 \fXMAM A PREPRINT we will reduce the magnitude \u03bb by half. Otherwise, we will return the current magnitude \u03bb. That means the returned magnitude \u03bb is the maximum attack magnitude that the attackers can implement. Algorithm 2: XMAM attack Input: the global update ug; the \u03c4 client local model updates; a random matrix Dmat. Output: the deviation parameter \u03bb. Initialize \u03bb = 1 while C1, C2, \u00b7\u00b7\u00b7, Cf / \u2208Clustermajor or \u03bb \u22641e\u221210 do for i = C1, C2, \u00b7\u00b7\u00b7, Cf do ui = ug \u03bbs Clusters = HDBSCAN(SLOU 1, \u00b7 \u00b7 \u00b7 , SLOU \u03c4) Clustermajor = Sort(Cluster1, Cluster1, \u00b7 \u00b7 \u00b7 )[0] return \u03bb 6 Evaluation In this section, we \ufb01rst evaluate the \ufb01delity of XMAM and the other six aggregation methods. Then, we implement the three backdoor attacks to FL with and without hiding techniques. we compare the six former aggregation methods with XMAM to highlight the superior capability of XMAM for defending against backdoor attacks. Furthermore, we implement the adaptive attack (Krum attack) proposed by Fang et al. [2020] and self-designed adaptive attack (XMAM attack) to demonstrate the robustness of XMAM. At last, we theoretically analyze the time complexity of each aggregation method and test the real run time of one-round aggregation on our device. The results show that our method is ef\ufb01cient. For example, our method is about 10000 times faster than Multi-Krum. For simplicity, in our experiments, we use an all-ones matrix to evaluate our method. Our code is publicly available at https://github.com/BESTICSP/XMAM. 6.1 Experimental setup 1) Datasets: We use two datasets in the computer-vision domain and sample them to simulate non-i.i.d. data. Note that we do not conduct our experiments in the i.i.d. scenario since the malicious local model updates are much easier to be detected in this scenario. As in Wang et al. [2020], for each dataset, we divide the training data into N piles using Dirichlet distributionHsu et al. [2019] and allocate them to N clients to simulate the practical FL system (i.e., the data distribution is non-i.i.d., and \u03c7 \u223cDir(0.5, N), where \u03c7 is the data distribution, N is the number of total clients, and 0.5 is the distribution parameter.) CIFAR-10 Krizhevsky et al. [2009]: CIFAR-10 is a color image classi\ufb01cation dataset incorporating prede\ufb01ned 50,000 training examples and 10,000 testing examples. Each example belongs to one of the 10 classes (airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck). MNIST LeCun et al. [1998]: The MNIST dataset of handwritten digits has a training set of 60,000 examples and a test set of 10,000 examples. It is a subset of a larger set available from NIST. The digits have been size-normalized and centered in a \ufb01xed-size image. 2) Evaluated backdoor attacks and adaptive attacks: We \ufb01rst evaluate the performance of the seven aggregation methods (FedAvg, Krum, Multi-Krum, NDC, RFA, RSA, and XMAM) on the three backdoor attacks (Trigger attack, Semantic attack, and Edge-case attack). To show the threat of the backdoor attacks with hiding techniques and the robustness of XMAM, we respectively conduct experiments on Black-box mode, PGD mode, and SMP mode. Then, we evaluate our method for adaptive attacks. Note that the three attack modes are designed for backdoor attacks to ensure that the server will select the malicious local model updates. We do not conduct adaptive attacks on them since the adaptive attacks have achieved this. Trigger attack: As in Gu et al. [2017], we poison 30% local data of malicious clients by stamping a square pixel block on the corner of images. Semantic attack: As in Bagdasaryan et al. [2020], we use the images of green cars (class: 1) to generate the poisoned training data and backdoor test dataset. Consistent with Bagdasaryan et al. [2020], we set the target label as the bird (class: 2). Edge-case attack: As in Wang et al. [2020], we use the images of the southwest airplane, which is in the tail of data distribution, to generate the poisoned training data and backdoor test dataset, and set the target label as the truck (class: 11 \fXMAM A PREPRINT Table 1: The default FL system parameter settings. Backdoor attacks Adaptive attacks Trigger attack Semantic attack Edge-case attack Krum attack XMAM attack Total number of clients 200 Clients selected in each iteration 30 Byzantine clients in each iteration 20% 40% Frequency of attacks 1 Local iterations 1 Global iterations 100 50 Batch size 32 Combined learning rate 0.001 \u00d7 0.998t Optimizer SGD Momentum 0.9 Weight decay 10\u22124 9). Note that the images of the southwest airplane would be classi\ufb01ed as an automobile (class: 1) if we do not implement backdoor attacks. Adaptive attack: We use the adaptive attack designed in Section 5. We assume the server collects 20 clients in each iteration, where 5% to 50% of them are malicious clients. Attack modes: For Black-box mode, a malicious client trains his local model in the poisoned dataset and submits the local model update to the server. For PGD mode, we set the PDG parameter \u03f5 = 5e\u22122 for three backdoor attacks. For SMP mode, as in Bhagoji et al. [2019], we set \u03c11 = 10 and \u03c12 = 1e\u22124. 3) Defenders\u2019 setting: As in Wang et al. [2020], in our experiments, we set \u03b4 = 2 for NDC, and we set the smoothing factor v = 0.1, the fault tolerance threshold \u00b5 = 10\u22125, and the maximum number of rounds R = 500 for RFA. We set the learning rate of RSA \u03b2r = 5 \u00d7 10\u22125 \u00d7 0.998t, which is proved by experiments that in this setting, and RSA performs well when there is no attack. 4) Evaluation metrics: Consistent with Cao et al. [2020], we use attack success rate to re\ufb02ect the performance of backdoor attacks since the aim of them is to promote the accuracy of the backdoor task. Speci\ufb01cally, the attack success rate is the fraction of data in the backdoor task, which is classi\ufb01ed by the global model as the attacker-chosen class. For adaptive attacks, we use testing error rates to re\ufb02ect the performance of adaptive attacks since their goal is to increase the testing error rate of the testing dataset. Speci\ufb01cally, the testing error rate is the fraction of data in the testing dataset that are mistakenly classi\ufb01ed by the global model. 5) System settings: In the experiments of backdoor attacks, we set 200 clients in the simulated FL system, the server collects 30 clients in each iteration, and 20% of the 30 clients are malicious clients. In the experiments of adaptive attacks, we set 200 clients in the simulated FL system, the server collects 30 clients in each iteration, and 40% of the 30 clients are malicious clients. Furthermore, we plot the PCA scatter diagram in the setting that the server collects 100 clients in each iteration, and 20% of the 100 clients are malicious clients. For CIFAR-10, we use network VGG9, and for MNIST, we use network LeNet. See Table 1 for more details. 6.2 Experimental results 1) Fidelity: When there is no malicious client in FL, we can see from Fig. 6 that all aggregation methods except Krum and RSA have a similar performance to FedAvg. That is to say, most existing methods can ensure \ufb01delity. It is no surprise that Krum has such a \ufb02uctuation in testing error rate since it only collects one local model update used for the global model\u2019s updating in each iteration. To punish the malicious local model updates, RSA limits the magnitude of all received local model updates to a \ufb01xed number and only preserves the direction, which makes the ultimate global model tend to a sub-optimal solution. 2) Robustness: Viewing Fig. 9 horizontally, we can \ufb01nd that different backdoor attacks have different performances. In contrast, Trigger backdoor attack is harder to succeed. In black-box mode, Trigger backdoor attack only evades FedAvg and RFA, but Semantic and Edge-case backdoors both evade FedAvg, NDC, RFA, and RSA. This phenomenon is more obvious in the SMP mode. Why does Trigger backdoor attack have a poor capability on attack success rate when the poisoned data proportion is the same as the other two backdoor attacks? We conjecture the reason is that it is harder for a model to learn the feature of the backdoor pattern from variant backgrounds than from a \ufb01xed background. Since Trigger backdoor attack poisons data by printing backdoor patterns on random data, but the backdoor patterns of 12 \fXMAM A PREPRINT 0 25 50 75 100 Iteration 0.135 0.140 0.145 0.150 0.155 0.160 Testing error rate 0 10 20 30 40 50 Iteration 0.25 0.50 0.75 Attack success rate FedAvg* FedAvg Krum Multi-Krum NDC RFA RSA XMAM Figure 6: The testing error rate of the global model under different aggregation methods when there is no attack. We can \ufb01nd that Krum is \ufb02uctuant and has a high testing error rate, RSA has a gap to the FedAvg, and other methods have comparable performance to FedAvg. Semantic and Edge-case backdoor attack are the data (containing special feature) itself, the success of Trigger backdoor attack needs more intensity and iterations. Viewing Fig. 9 vertically, we can observe the different performances on different attack modes. Visually, we can see that the attacks using hiding techniques successfully evade more aggregation methods than the attacks without using hiding techniques. Most notably, Multi-Krum is robust in black-box mode but fails in PGD and SMP mode (see Fig. 9\u2019s (b)(c)(e)(f)(h)). That means when backdoor attacks are under PGD mode and SMP mode, except Krum and our method, no existing aggregation methods can defend against them. About Krum, we will analyze it as follows. To illustrate why Multi-Krum fails but XMAM succeeds when the malicious local model updates are more concealing, we plot the distribution of 100 client local model updates using PCA dimension reduction technology. From Fig. 8, we can observe that with the scaling down magnitude increasing, the distance between malicious local model updates and benign local model updates are increasingly close, which makes them indistinguishable. In contrast, the SLOUs of malicious local model updates and benign local model updates still have a clear boundary when the malicious local model updates and benign ones are indistinguishable (Fig. 8(c)(f)). Since it collects n \u2212f local model updates that have a closer Euclidean distance to other local model updates, Multi-Krum fails when the scaling down magnitude is large enough. Fig. 10 shows the testing error rate of the global model on the main task. We can easily observe that whatever modes the backdoor attacks are under, our aggregation method ensures a low testing error rate, which is comparable to FedAvg*. However, other methods, especially RSA, have a higher testing error rate compared with XMAM. Now, we explain the two special phenomena in Fig. 9. The \ufb01rst one is that the attack success rate promotes about 30% in Semantic backdoor attacks even when there is no malicious client. Why does it happen? We surmise that the data on green cars are close to the data on birds in the output space of the clean global model. To verify it, we separately set the target label of poisoned data from class 0 to class 9 and run 100 iterations to observe the performance of the global model on different backdoor tasks. The results show that the data of green cars is more likely to be classi\ufb01ed as automobiles (class 1: 33.6%), birds (class 2: 29%), and frogs (class 6: 25.2%) in a clean global model. Detailed results are in Fig. 7. The second one is that Krum seems to have a more powerful capability than FedAvg* (FedAvg under no attacks) and XMAM in defending against Semantic attacks. However, as we explained in Figure 7, the global model has an innate accuracy of the backdoor task when there is no malicious client. Krum has a lower Attack Success Rate (ASR) because the global model trained by Krum has a lower innate accuracy of the backdoor task, which does not mean Krum is better than XMAM in defending against Semantic attacks. As in Krum, XMAM can also preclude malicious client models in each iteration with Semantic attacks. Therefore, we can conclude that Krum and XMAM both have the ability to evade Semantic attacks. However, XMAM will select more benign client models in each iteration when there are malicious clients so that XMAM has a good and similar performance to FedAvg* on the main task, but Krum is not good enough on the main task. Performance on adaptive attacks: The global model that uses Krum as the aggregation method has a worse convergence rate and a higher testing error rate, which is especially conspicuous when the data distribution is non-i.i.d. In this part, we further explain the weakness of Krum in defending against adaptive attack (Krum attack), which is designed by Fang et al. [2020] to challenge the robustness of Krum. We use Krum attack to test all aggregation methods on two datasets, CIFAR-10, and MNIST, Fig. 12 illustrates that Krum attack breakdowns the aggregation methods, Krum and Multi-Krum, and has a certain negative impact on the other methods except XMAM, which further demonstrates our method is more robust than Krum and Multi-Krum. 13 \fXMAM A PREPRINT 0 20 40 60 80 100 Iteration 0.0 0.1 0.2 0.3 Test accuracy class:0 class:1 class:2 class:3 class:4 class:5 class:6 class:7 class:8 class:9 Figure 7: The test accuracy of the global model on the backdoor testing dataset. We guess that the data of green cars are close to data of birds in the output space of the clean global model. So, we label the testing data as classes 0-9 to observe the test accuracy. The results show that the data of green cars are more likely to be classi\ufb01ed as automobile (class 1: 33.6%), bird (class 2: 29%), and frog (class 6: 25.2%), which veri\ufb01es our conjecture. (a) updates & no scaling (b) updates & \u03f5=5e-1 (c) updates & \u03f5=5e-2 (d) SLOUs & no scaling (e) SLOUs & \u03f5=5e-1 (f) SLOUs & \u03f5=5e-2 Figure 8: The dimensionality reduction graph of the 100 local model updates and the 100 corresponding SLOUs. The red dots (20%) are the malicious local model updates or SLOUs, and the blue dots (80%) are the benign local model updates or SLOUs. From left to right, we successively decrease the PGD parameter \u03f5 (i.e., we scale down the local model updates to a smaller norm) to observe the distribution of the 100 local model updates and the 100 corresponding SLOUs. Thus, we can \ufb01nd that the PGD hiding technique makes the malicious local model updates indistinguishable from the benign local model updates, but can not hide the corresponding malicious SLOUs from benign SLOUs. To further verify the robustness of XMAM, we design a new adaptive attack, XMAM attack, according to the same framework as the Krum attack. We \ufb01nd that even \u03bb is very small, like 1e\u221210, XMAM will not select the malicious local model updates. To \ufb01nd out the root cause, we plot the scatter diagrams using Principal Component Analysis (PCA) algorithm to explain why XMAM does not fail in XMAM-adaptive attack, but Krum fails in Krum-adaptive attack. From Fig. 11 we can see that when the reverse magnitude \u03bb becomes smaller and smaller, the malicious clients\u2019 model updates become closer and closer to the benign client\u2019s model updates so that Krum can select the malicious clients\u2019 model updates. Nevertheless, with the reverse magnitude \u03bb becoming smaller and smaller, the distance between the malicious clients\u2019 SLOU and the benign clients\u2019 SLOU has no signi\ufb01cant change. This experiment can illustrate XMAM\u2019s ability to defend against adaptive attacks. 3) Ef\ufb01ciency: Now we compare the screening complexity of the proposed aggregation methods. The server using the aggregation method FedAvg does not need to screen received local model updates, so its screening complexity is O(0). Krum and Multi-Krum compute the mutual distance of \u03c4 client local model updates. NDC and RSA clip and regularize \u03c4 local model updates, respectively. RFA \ufb01nds the geometric center by considering \u03c4 client local model updates until it satis\ufb01es the de\ufb01ned condition. In a word, all the above methods need to consider the \u03b6 parameters of the local model update. Unfortunately, the local model update in the deep learning model currently possesses millions of parameters (e.g., VGG-16 has 138M parameters). Instead of operating on the parameters of the local model update, our method considers the M probabilities of the Softmax layer\u2019s output. As far as we know, M is far less than \u03b6, which might be a millionfold. 14 \fXMAM A PREPRINT 0 20 40 60 80 100 Iteration 0.1 0.2 0.3 0.4 0.5 Attack success rate (a) Trigger & Black-box 0 20 40 60 80 100 Iteration 0.0 0.2 0.4 0.6 0.8 1.0 Attack success rate (b) Semantic & Black-box 0 20 40 60 80 100 Iteration 0.0 0.2 0.4 0.6 0.8 Attack success rate (c) Edge-case & Black-box 0 20 40 60 80 100 Iteration 0.10 0.11 0.12 0.13 0.14 0.15 Attack success rate (d) Trigger & PGD 0 20 40 60 80 100 Iteration 0.0 0.2 0.4 0.6 0.8 Attack success rate (e) Semantic & PGD 0 20 40 60 80 100 Iteration 0.0 0.2 0.4 0.6 Attack success rate (f) Edge-case & PGD 0 20 40 60 80 100 Iteration 0.1 0.2 0.3 0.4 Attack success rate (g) Trigger & SMP 0 20 40 60 80 100 Iteration 0.0 0.2 0.4 0.6 0.8 Attack success rate (h) Semantic & SMP 0 20 40 60 80 100 Iteration 0.0 0.2 0.4 0.6 0.8 Attack success rate (i) Edge-case & SMP 0 20 40 60 80 100 Iteration 0.00 0.25 0.50 0.75 Attack success rate FedAvg* FedAvg Krum Multi-Krum NDC RFA RSA XMAM Figure 9: The attack success rate of three backdoor attacks (Trigger attack, Semantic attack, and Edge-case attack) on the backdoor task under three modes (Black-box mode, PGD mode, and SMP mode). We compare the performance of the seven aggregation methods (FedAvg, NDC, RSA, RFA, Krum, and Multi-Krum) and the benchmark FedAvg*. We can see that some well-known robust methods are breached when the attack changed from the black-box mode to the PGD and SMP modes, and our method always maintains similar performance to FedAvg*. Table 2: The screening complexity for seven aggregation methods. \u03b6 denotes the number of local model update parameters, M means the number of label classes and \u03c4 is the number of collected local model updates in each iteration. The screening time is recorded by experiments on dataset CIFAR-10 and network VGG9. Method Screening complexity Screening time (second) FedAvg O(0) 0 Multi-Krum O(\u03c4 2\u03b6) 116.98 Krum O(\u03c4 2\u03b6) 109.19 RFA O(\u03c4\u03b6R\u2217) 39.93 RSA O(\u03c4\u03b6) 0.78 NDC O(\u03c4\u03b6) 0.34 XMAM O(\u03c4 2MR\u2217) 0.0079 For XMAM, the screening complexity covers two-parts. The \ufb01rst part is the time of training a random matrix to get the SLOUs. The second part is the time of clustering these SLOUs, which we set as O(\u03c4 2MR\u2217), where R\u2217is the rounds needed by the clustering algorithm to \ufb01nd the cluster centers. Note that the time of training a random matrix could be negligible. Therefore, the ultimate screening complexity of XMAM is O(\u03c4 2MR\u2217). Table 2 gives the concrete screening complexity for seven aggregation methods and the average screening time for ten arbitrary iterations. We run the procedure of screening on a computer with NVIDIA TITAN X GPU. The results show that compared with the same type of methods, Krum and Multi-Krum, XMAM improves the screening ef\ufb01ciency by tens of thousands of times. It is no surprise the XMAM reduces the dimension of the detection object from \u03b6 to M. 15 \fXMAM A PREPRINT 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 Testing error rate (a) Trigger & Black-box 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 Testing error rate (b) Semantic & Black-box 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 0.18 Testing error rate (c) Edge-case & Black-box 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 Testing error rate (d) Trigger & PGD 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 Testing error rate (e) Semantic & PGD 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 Testing error rate (f) Edge-case & PGD 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 Testing error rate (g) Trigger & SMP 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 Testing error rate (h) Semantic & SMP 0 20 40 60 80 100 Iteration 0.14 0.15 0.16 0.17 0.18 Testing error rate (i) Edge-case & SMP 0 20 40 60 80 100 Iteration 0.00 0.25 0.50 0.75 Attack success rate FedAvg* FedAvg Krum Multi-Krum NDC RFA RSA XMAM Figure 10: The attack success rate of three backdoor attacks (Trigger attack, Semantic attack, and Edge-case attack) on the main task under three modes (Black-box mode, PGD mode, and SMP mode). We compare the performance of the seven aggregation methods (FedAvg, NDC, RSA, RFA, Krum, and Multi-Krum) and the benchmark FedAvg*. We can see that in all backdoor attacks and modes, our aggregation method ensures a low testing error rate, which is comparable to FedAvg*. However, other methods, especially RSA, have a higher testing error rate than XMAM. 7 Discussions and limitations Discussion: Huang Huang et al. [2019] uses the interpretability technique to design a heatmap to explain the DNNs output, facilitating a more accurate detection for the poisoned model. However, their method requires a clean dataset encompassing all classes. Different from their method, our scheme only needs a generated random matrix, and it directly uses the probability distribution of the Softmax layer\u2019s output to judge the quality of a model. KolouriKolouri et al. [2020] feeds a group of Universal Litmus Patterns (ULPs) through a model and pools the logit layer\u2019s output to classify it as poisoned or clean. Nevertheless, optimizing the classi\ufb01er and ULPs requires hundreds of pre-trained clean and poisoned models, which is impractical in FL because the server has no dataset. HuangHuang et al. [2020] proposes One-Pixel Signature for backdoor detection. As in Kolouri et al. [2020], One-Pixel Signature also demands pre-trained clean and poisoned models. Limitations: As we know, the data distribution in FL is in a non-i.i.d. scenario, and our experimental datasets are also distributed in a non-i.i.d. way. In the early stages of training, the local model updates submitted by the benign clients are more heterogeneous than the local model updates in the middle and late stages of training. Therefore, our method is not guaranteed to be effective at the early stages of training. Note that in the middle and late stages of training, our method will occasionally fail to preclude all malicious local model updates in a certain round, but this does not mean that our defense is ineffective because backdoor attacks require multiple rounds to succeed. In the above experiments, we demonstrate that our method is the most robust aggregation method compared with the other six under a certain proportion (20%) of malicious clients. Then, we conduct an experiment to observe our method\u2019s tolerance in different proportions of malicious clients. The results (Fig. 13) show that our method can tolerate at least 45% of malicious clients in the black-box mode and about 30% of malicious clients in the PGD mode. However, we must admit that the results of our defense are still not enough, and we will \ufb01nd ways to improve XMAM\u2019s tolerance to a larger number of malicious clients in future work. 16 \fXMAM A PREPRINT (a) SLOU & \u03bb = 2\u221212 (b) SLOU & \u03bb = 2\u221214 (c) SLOU & \u03bb = 2\u221216 (d) SLOU & \u03bb = 2\u221218 (e) SLOU & \u03bb = 2\u221220 (f) update & \u03bb = 2\u221212 (g) update & \u03bb = 2\u221214 (h) update & \u03bb = 2\u221216 (i) update & \u03bb = 2\u221218 (j) update & \u03bb = 2\u221220 Figure 11: The PCA scatter diagram of 100 client models\u2019 SLOUs (top) and updates (bottom). We use CIFAR-10 and VGG9 in this experiment and malicious clients occupy 20 %. 0 10 20 30 40 50 Iteration 0.2 0.4 0.6 0.8 Attack success rate (a) CIFAR-10 0 10 20 30 40 50 Iteration 0.0 0.2 0.4 0.6 0.8 Attack success rate (b) MNIST 0 10 20 30 40 50 Iteration 0.25 0.50 0.75 Attack success rate FedAvg* FedAvg Krum Multi-Krum NDC RFA RSA XMAM Figure 12: The attack success rate of the Krum attack on two datasets (CIFAR-10 and MNIST). We can see from the results that Krum, and Multi-Krum seriously deteriorate and some other methods are slightly affected, but our method always has comparable performance to FedAvg*. 8 Related works Poisoning attacks refer to destroying the training result of machine learning. A typical poisoning attack is data poisoning attacksBiggio et al. [2012], Chen et al. [2017], Fang et al. [2018], Gu et al. [2017], Jagielski et al. [2018], Li et al. [2016], Mu\u00f1oz-Gonz\u00e1lez et al. [2017], Nelson et al. [2008], Rubinstein et al. [2009], Shafahi et al. [2018], Suciu et al. [2018], Wang and Gong [2019], Xiao et al. [2015], Yang et al. [2017], Fang et al. [2021]. Recently, multiple poisoning attacks Bagdasaryan et al. [2020], Xie et al. [2020], Wang et al. [2020] have threatened FL. In addition to data poisoning attacks, FL also suffers from local model poisoning attacksFang et al. [2020], Li et al. [2019], Baruch et al. [2019], Xie et al. [2019], He et al. [2020], Bagdasaryan et al. [2020], Wang et al. [2020], Bhagoji et al. [2019], which are more potent than the former. Furthermore, from the perspective of the attack\u2019s purpose, these poisoning attacks can be divided into untargeted attacksFang et al. [2020], Li et al. [2019], Baruch et al. [2019], Xie et al. [2019], He et al. [2020], which aim to deteriorate the global model, and backdoor (targeted) attacksBagdasaryan et al. [2020], Bhagoji et al. [2019], Xie et al. [2020], Wang et al. [2020], which aim to induce the global model to make some attacker-chosen mistake in certain input without deteriorating the global model. The latter is more threatening to FL owing to its hidden characteristic. In central learning, all data is used to train a model jointly, and it is assumed that the model is transparent to the attacker. Therefore, in the poisoning-based backdoor attacks Gu et al. [2017], Chen et al. [2017], Liao et al. [2018], Liu et al. [2017a], Zhao et al. [2020], Bagdasaryan et al. [2020], the attacker can optimize the trigger in the poisoned data according to the performance of the model on them. In FL, the attacker does not know what the global model of the current iteration will be, so it is impossible to optimize the trigger in the poisoned data based on the performance of the global model on them. In addition, apart from the poisoning-based backdoor attacks, when the data can not be tampered 17 \fXMAM A PREPRINT 0 5 10 15 20 25 30 35 40 45 50 Fraction of malicious clients (%) 0.0 0.2 0.4 0.6 0.8 1.0 Attack success rate Trigger Semantic Edge-case (a) Black-box 0 5 10 15 20 25 30 35 40 45 50 Fraction of malicious clients (%) 0.0 0.2 0.4 0.6 0.8 1.0 Attack success rate Trigger Semantic Edge-case (b) PGD Figure 13: The attack success rate of three backdoor attacks under different proportions of malicious clients. We can see that our defense can tolerate 45% malicious clients under black-box mode can about 30% malicious clients under PGD mode. with, the backdoor attacker in the central learning can also achieve the backdoor function by directly perturbing the model weight, which is the so-called Targeted Weight Perturbation (TWP) Dumford and Scheirer [2020]. Similarly, this is dif\ufb01cult to achieve in FL because the backdoor attacker in FL can only control its own model, and can not directly tamper with the global model. Currently, there are three types of backdoor attacks in FL: Trigger backdoor Gu et al. [2017], Semantic backdoor Li et al. [2020], and Edge-case backdoor Wang et al. [2020]. All of them belong to the poisoning-based backdoor attacks. There are three types of defense approaches proposed in centralized learning Li et al. [2020]: i) trigger-backdoor mismatch, ii) trigger elimination, and iii) bcakdoor elimination. The trigger-backdoor mismatch based defenses Liu et al. [2017b], Doan et al. [2020], Udeshi et al. [2019], Qiu et al. [2021] try to modify or reverse the trigger in the poisoned data so that the modi\ufb01ed poisoned data can not match the hidden backdoor in the model to prevent backdoor activation. The trigger elimination based defenses Gao et al. [2019], Subedar et al. [2019], Du et al. [2019], Javaheripi et al. [2020] attempt to distinguish between poisoned data and benign data, and preclude the poisoned data before training. Both of the above two types of defenses require direct contact with client data, which is impractical in FL. The bcakdoor elimination based defenses Wang et al. [2019], Chen et al. [2019], Huang et al. [2019], Xu et al. [2019], Huang et al. [2020] try to prune the neurons in the model that can be activated by the trigger, or refuse to deploy the infected model through detection. This type of defense requires the server having a large batch of data or plenty of models that have been labeled as infected or uninfected to train a meta-classi\ufb01er, which is dif\ufb01cult to obtain in FL. The existing defenses in FL can be roughly divided into three categories: (i) limiting the update, (ii) \ufb01nding the \u201ccenter\u201d, and (iii) detecting and precluding. The \ufb01rst category is limiting the update Sun et al. [2019], Li et al. [2019]. The core of this type of defense is to punish the local model update that has a large norm or regularize all the local model updates to a small norm. The second category is \ufb01nding the \u201ccenter\u201d Pillutla et al. [2019]. The core of this type of defense is to exploit the local model updates submitted by clients to \ufb01nd a compromised update, which is the \u201ccenter\u201d of the local model updates, to update the global model. The third category is detecting and precluding Blanchard et al. [2017]. The core of this type of defense is to detect the malicious local model update and preclude it. 9"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2209.15245v2",
+                    "title": "Fed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction",
+                    "abstract": "Due to limited communication capacities of edge devices, most existing\nfederated learning (FL) methods randomly select only a subset of devices to\nparticipate in training for each communication round. Compared with engaging\nall the available clients, the random-selection mechanism can lead to\nsignificant performance degradation on non-IID (independent and identically\ndistributed) data. In this paper, we show our key observation that the\nessential reason resulting in such performance degradation is the\nclass-imbalance of the grouped data from randomly selected clients. Based on\nour key observation, we design an efficient heterogeneity-aware client sampling\nmechanism, i.e., Federated Class-balanced Sampling (Fed-CBS), which can\neffectively reduce class-imbalance of the group dataset from the intentionally\nselected clients. In particular, we propose a measure of class-imbalance and\nthen employ homomorphic encryption to derive this measure in a\nprivacy-preserving way. Based on this measure, we also design a\ncomputation-efficient client sampling strategy, such that the actively selected\nclients will generate a more class-balanced grouped dataset with theoretical\nguarantees. Extensive experimental results demonstrate Fed-CBS outperforms the\nstatus quo approaches. Furthermore, it achieves comparable or even better\nperformance than the ideal setting where all the available clients participate\nin the FL training.",
+                    "authors": "Jianyi Zhang, Ang Li, Minxue Tang, Jingwei Sun, Xiang Chen, Fan Zhang, Changyou Chen, Yiran Chen, Hai Li",
+                    "published": "2022-09-30",
+                    "updated": "2023-06-06",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction With the booming of IoT devices, a considerable amount of data is generated at the network edge, providing valuable resources for learning insightful information and enabling intelligent applications such as self-driving, video analytics, anomaly detection, etc. The traditional wisdom is to train machine learning models by collecting data from devices and performing centralized training. Data migration usually raises serious privacy concerns. Federated learning (FL) (McMahan et al., 2017a) is a promising technique to mitigate such privacy concerns, enabling a large number of clients to learn a shared model collaboratively, and the learning process is orchestrated by a central server. In particular, the participating clients first download a global model from the central server and then compute local model updates using their local data. The clients then transmit the local updates to the server, where the local updates are aggregated and then the global model is updated accordingly. In practice, due to limited communication and computing capabilities, one usually can not engage all the available clients in FL training to fully utilize all the local data. Therefore, most FL methods only randomly select a subset of the available clients to participate in the training in each communication round. However, in practice, the data held by different clients are often typically non-IID (independent and identically distributed) due to various user preferences and usage patterns. This leads to a serious problem that the random client selection strategy often fails to learn a global model that can generalize well for most of the participating clients under non-IID settings (Goetz et al., 2019; Cho et al., 2020; Nishio & Yonetani, 2019; Yang et al., 2020). Several heuristic client selection mechanisms have been proposed to tackle the non-IID challenge. For example, in the method of (Goetz et al., 2019), the clients with larger local loss will have a higher probability to be selected to participate in the training. Power-of-Choice (Cho et al., 2020) selects several clients with the largest loss from a randomly sampled subset of all the available clients. However, selecting clients with a larger local loss may not guarantee that the final model can have a smaller global loss. Another limitation of previous research on client selection is 1 arXiv:2209.15245v2  [cs.LG]  6 Jun 2023 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction 0 25 50 75 100 125 150 175 200 Training Rounds 65 70 75 80 85 90 Test Accuracy (%) All Random Class-Balanced (a) Global Balanced & One-class 0 25 50 75 100 125 150 175 200 Training Rounds 70.0 72.5 75.0 77.5 80.0 82.5 85.0 87.5 90.0 Test Accuracy (%) All Random Class-Balanced (b) Global Balanced & Two-class 0 25 50 75 100 125 150 175 200 Training Rounds 40 50 60 70 80 90 Test Accuracy (%) All Random Class-Balanced (c) Global Imbalanced & One-class 0 25 50 75 100 125 150 175 200 Training Rounds 40 50 60 70 80 90 Test Accuracy (%) All Random Class-Balanced (d) Global Imbalanced & Two-class Figure 1. Three different FL client selection strategies on MNIST. All means engaging all the 100 clients in training. Random means randomly selecting 10 clients. Class Balanced means that we keep the class-balance by intentionally selecting 10 clients. In Figure 1a and 1b, the global dataset of all the 100 clients\u2019 training data is class-balanced. In Figure 1c and 1d, the global dataset is classimbalanced. Each client has only one class of data in (a) and (c) and each client has two classes of data in (b) and (d). The results show significant performance degradation with imbalanced data from random client selection. It is worth noting that when the global dataset is class-imbalanced, selecting all the clients leads to worse performance compared with the Class Balanced strategy, which suggests the importance of keeping class-balance for client selection. the missing comparison between their strategy and the ideal case, where all the available clients participate in the training. In general, existing works not only miss a vital criterion that can measure the performance of their methods, but also fail to investigate the essential reason why random client selection can lead to performance degradation on non-IID data compared with fully engaging all the available clients. In this paper, we focus on image classification tasks. First, we demonstrate our key observation for the essential reason why random client selection results in performance degradation on non-IID data, which is the class-imbalance of the grouped dataset from randomly selected clients. Based on our observation, we design an efficient heterogeneity-aware client sampling mechanism, i.e., Federated Class-Balanced Sampling (Fed-CBS), which effectively reduces the classimbalance in FL. Fed-CBS is orthogonal to numerous existing techniques to improve the performance of FL (Li et al., 2018; Wang et al., 2020b; Karimireddy et al., 2019; Chen et al., 2020; Reddi et al., 2020; Hao et al., 2021; Yang et al., 2021) on non-IID data, meaning Fed-CBS can be integrated with these methods to improve their performance further. Our major contributions are summarized as follows: \u2022 We reveal that the class-imbalance is the fundamental reason why random client selection leads to performance degradation on non-IID data in Section 2. \u2022 To effectively reduce the class-imbalance, we design an efficient heterogeneity-aware client sampling mechanism, i.e., Fed-CBS, based on our proposed classimbalance metric in Section 3. We provide theoretical analysis on the convergence of Fed-CBS in Section 4, as well as the analysis of the NP-hardness of this problem. \u2022 We empirically evaluate Fed-CBS on FL benchmark (non-IID datasets) in Section 5. The results demonstrate that Fed-CBS can improve the accuracy of FL models on CIFAR-10 by 2% \u223c7% and accelerate the convergence time by 1.3\u00d7 \u223c2.8\u00d7, compared with the state-of-the-art method (Yang et al., 2020) that also aims to reduce class-imbalance via client selection. Furthermore, our Fed-CBS achieves comparable or even better performance than the ideal setting where all the available devices are involved in the training. 2. Preliminary and Related Work We first clarify three definitions. The local dataset is the client\u2019s own locally-stored dataset, which is inaccessible to other clients and the server. Due to the heterogeneity of local data distribution, the phenomenon of class-imbalance frequently happens in most of the local datasets. The global dataset is the union of all the available client local datasets. It can be class-balanced or class-imbalanced, but it is often imbalanced. The grouped dataset is the union of several clients\u2019 local datasets which have been selected to participate in training for one communication round. It follows that the grouped dataset is a subset of the global dataset. 2.1. Pitfall of Class-Imbalance in Client Selection Some recent works (Yang et al., 2020; Wang et al., 2020b; Duan et al., 2019) have identified the issue of classimbalance in the grouped dataset by random selection under non-IID settings. Since class-imbalance degrades the classification accuracy on minority classes (Huang et al., 2016) and leads to low training efficiency, we are motivated to verify whether the class-imbalance of the randomly-selected grouped dataset is the essential reason accounting for the performance degradation. We conduct some experiments on MNIST to verify our proposition1. As shown in Figure 1a and Figure 1b, the random selection mechanism shows the worst performance when the global label distribution is class-balanced. If we keep the grouped dataset class-balanced by manually selecting the clients based on their local label distribution, we can 1Detailed experiment settings are listed in the Appendix (Section C.1) 2 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction obtain accuracy comparable to the case of fully engaging all the clients in training. Another natural corollary is that when the global dataset is inherently class-imbalanced, engaging all clients in training may lead to worse performance than manually keeping the grouped dataset class-balanced. The results in Figure 1c and Figure 1d prove our hypothesis and verify the importance of class-imbalance reduction. This also indicates that only keeping diversity in the data and fairness for clients is not enough, which was missed in the previous literature (Balakrishnan et al., 2021; Huang et al., 2021; Yang et al., 2020; Wang et al., 2020b; Shen et al., 2022; Wang et al., 2021). More experimental results on larger datasets will be provided to verify the importance of class-imbalance reduction (Section 5). 2.2. Related Work Some effort has been made to improve client selection for FL in previous literature. (Cho et al., 2020; Goetz et al., 2019) select clients with larger local loss, but this cannot guarantee that the final global model has a smaller global loss. Focusing on the diversity in client selection, the authors of (Balakrishnan et al., 2021) select clients by maximizing a submodular facility location function defined over gradient space. A fairness-guaranteed algorithm termed RBCS-F was proposed in (Huang et al., 2021), which models the fairness-guaranteed client selection as a Lyapunov optimization problem. Although diversity and fairness are important, the experimental results in Section 2.1 demonstrate that they are not enough for client selection if the classimbalance issue is not considered. The authors in (Ribero & Vikalo, 2020) model the progression of model weights by an Ornstein-Uhlenbeck process and design a sampling strategy for selecting clients with significant weight updates. However, the work only considers the identical data distribution setting. Following the existing works (Goetz et al., 2019; Cho et al., 2020), we only focus on the data heterogeneity caused by non-IID data across clients. Additionally, we included a comparison of our method with other clusteredbased client sampling algorithms in the appendix. To the best of our knowledge, (Duan et al., 2019) and (Yang et al., 2020) are the first two attempts to improve client selection by reducing class-imbalance. An extra virtual component called a mediator is introduced in Astraea of (Duan et al., 2019), which has access to the local label distributions of the clients. With these distributions, Astraea will conduct client selection in a greedy way. The method of (Yang et al., 2020) first estimates the local label distribution of each client based on the gradient of model parameters and adopts the same greedy way to select clients as Astraea. Since directly knowing the exact value of local label distributions of clients in Astraea will cause severe concerns on privacy leakage, we consider the method in (Yang et al., 2020) as the state-of-the-art method aiming to improve client selection through class-imbalance reduction. However, the solution presented by (Yang et al., 2020) has several limitations. First, their method requires a classbalanced auxiliary dataset that consists of all classes of data at the server. However, that is not always available in some large-scale FL systems since it requires the server to collect raw data from clients, which breaches privacy. Second, their estimations of the clients\u2019 local label distribution are not accurate as shown in Figure 2. Theorem 1 in (Yang et al., 2020) supports their estimations, but it cannot be generalized to multi-class classification tasks since it has only been proved in the original paper (Anand et al., 1993) for two-class classification problems. Finally, the performance of greedily conducting the client selection is not guaranteed due to the nature of the greedy algorithm. We provide an example in Figure 3 to show its weakness. Their method will select C1 as the first client since it is the most class-balanced one. Then C2 will be selected because the grouped dataset of C1 \u222aC2 is the most classbalanced among the choices C1 \u222aC2, C1 \u222aC3 and C1 \u222aC4. Similarly, it will choose C3 since the grouped dataset of C1 \u222aC2 \u222aC3 is more class-balanced than C1 \u222aC2 \u222aC4. Their method is deterministic and thus only one combination {C1, C2, C3} is obtained. However, this is clearly not the optimal solution since {C1, C3, C4} is more class-balanced than {C1, C2, C3}. The above weaknesses motivate us to design a more effective solution for this problem. 3. Methodology We first propose a metric to measure class-imbalance in Section 3.1. Then we derive the measure with privacypreserving techniques in Section 3.2. Based on this measure, we then design our client sampling mechanism and show its superiority in Section 3.3. 3.1. Class-Imbalance Measure Assume there are B classes of data in an image classification task, where B \u22652. In the k-th communication round, we assume there are Nk available clients and we select M clients from them. To make the presentation concise, we ignore the index \u201ck\u201d and assume the set of indices for the available clients is {1, 2, 3, ..., N} and the n-th available client has its own training dataset Dn. We adopt the following vector of size B to represent the local label distribution of Dn, where \u03b1(n,b) \u22650 and PB b=1 \u03b1(n,b) = 1, \u03b1n = \u0002 \u03b1(n,1), \u03b1(n,2), ..., \u03b1(n,b), ..., \u03b1(n,B) \u0003 . (1) We aim to find a subset M of {1, 2, 3, .., N} of size M, such that the following grouped dataset Dg M = S n\u2208M Dn is class-balanced. Assuming the n-th client\u2019s local dataset has 3 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction Figure 2. The exact local label distributions and the estimated ones of the first 5 clients in the experiment of (Yang et al., 2020). Label distribution quantifies the ratio between the number of data from 10 classes (C-1, C-2, ..., C-10) in each client\u2019s local dataset. (a) Greedy method (b) Our method Figure 3. An example demonstrating the weakness of greedy method to deal with class imbalance. Supposing we work on a 6-class classification task and aim to select 3 clients from 4 available clients C1, C2, C3, C4. Each of them has 30 images. The compositions of their local datasets are [5, 5, 5, 5, 5, 5], [6, 6, 6, 6, 6, 0], [0, 0, 0, 10, 10, 10] and [10, 10, 10, 0, 0, 0] respectively. The greedy method in (Yang et al., 2020) is deterministic. It can only derive one result {C1, C2, C3} instead of the optimal solution {C1, C3, C4} (see the text description). But our method is based on probability modeling, which directly models the distribution of the optimal solution {C1, C3, C4}. Thus when sampling from it, the optimal solution can be returned with high probability. qn training samples, the following vector \u03b1g M can represent the label distribution of the grouped dataset Dg M, \u03b1g M = P n\u2208M qn\u03b1n P n\u2208M qn = \u0014P n\u2208M qn\u03b1(n,1) P n\u2208M qn , ..., P n\u2208M qn\u03b1(n,b) P n\u2208M qn , ..., P n\u2208M qn\u03b1(n,B) P n\u2208M qn \u0015 . Instead of dealing with the Kullback-Leibler (KL) divergence as (Duan et al., 2019; Yang et al., 2020), which is complicated to analyze, we propose the following function to measure the magnitude of class-imbalance of M, which we call Quadratic Class-Imbalance Degree (QCID): QCID(M) \u225c B X b=1 ( P n\u2208M qn\u03b1(n,b) P n\u2208M qn \u22121 B )2. Essentially, QCID(M) reflects the L2 distance between the distribution of the grouped dataset Dg M and the ideally class-balanced dataset that has a uniform label distribution. Although there exist several more commonly-used probabilistic distances other than L2, it is easier to analyze QCID and more efficient to calculate while keeping privacy as shown in the next section. 3.2. Privacy-Preserving QCID Derivation Our privacy goal is to calculate the value of QCID while keeping clients\u2019 local distributions {\u03b1n} hidden from the server since it contains sensitive information. Unlike Kullback-Leibler (KL) divergence which is difficult to analyze, we can expand the expression of QCID to explore how the pairwise relationships of the clients\u2019 local label distributions {\u03b1m} affects the class-imbalance degree of M, where m \u2208M. Below we provide a theorem to show the feasibility of our method. Theorem 3.1. The QCID value is decided by the sum of inner products between each two vectors \u03b1m, \u03b1m\u2032 \u2208{\u03b1m} with m \u2208M, i.e., QCID(M) = P n\u2208M,n\u2032\u2208M qnq\u2032 n\u03b1n\u03b1T n\u2032 (P n\u2208M qn)2 \u22121 B Theorem 3.1 reveals the fact that there is no need to know the local label distribution of each client to calculate the QCID, as long as we have access to the inner products between each other. To derive the QCID for any subset M \u2286{1, 2, 3, .., N}, we only need to know the following N \u00d7 N matrix S with element sn,n\u2032 being \u03b1n\u03b1T n\u2032, which is the inner product between the local label distributions of the available clients n and n\u2032. S = \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 q1q1\u03b11\u03b1T 1 q1q2\u03b11\u03b1T 2 \u00b7 \u00b7 \u00b7 q1qN\u03b11\u03b1T N q2q1\u03b12\u03b1T 1 q2q2\u03b12\u03b1T 2 \u00b7 \u00b7 \u00b7 q2qN\u03b12\u03b1T N . . . . . . ... . . . qNq1\u03b1N\u03b1T 1 qNq2\u03b1N\u03b1T 2 \u00b7 \u00b7 \u00b7 qNqN\u03b1N\u03b1T N \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb Although it is possible to calculate QCID with S, another concern arises, can a malicious party infer the values of {\u03b1i} from S ? Then we have another theorem to provide privacy protection. Theorem 3.2. One can not derive the values of {\u03b1i} from the value of S. 4 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction Based on these two theorems, our privacy goal can be simplified as enabling the server to derive S without access to {\u03b1i}. There are several ways to achieve our goal. One option is to leverage the server-side trusted execution environments (TEEs), e.g., Intel SGX (Anati et al., 2013), which allows calculating S without leaking information of {\u03b1n}. Another potential solution is to adopt Fully Homomorphic Encryption (FHE) (Chen et al., 2017; Brakerski et al., 2014; Fan & Vercauteren, 2012; Halevi & Shoup, 2014; 2015) to enable the server to compute on encrypted data (i.e., {\u03b1i}) to derive S. We provide an example of the system skelon in Section A.2 to illustrate how to derive S without knowing the local label distributions {\u03b1i} using FHE. Since we focus on efficient algorithms to reduce class-imbalance instead of designing the fundamental infrastructure for computing (which is beyond our scope and not a contribution of this paper), we leave the detailed system design for future work. 3.3. A Client Sampling Mechanism To select the most class-balanced grouped dataset Dg M, we need to find the optimal subset M\u2217that has the lowest QCID value, which is defined as follows: M\u2217\u225c arg min M\u2286{1,2,3,..,N} P n\u2208M,n\u2032\u2208M qnqn\u2032\u03b1n\u03b1T n\u2032 (P n\u2208M qn)2 \u22121 B . The main challenge is computational complexity. To find the exact optimal M\u2217, we need to loop through all the possible cases and find the lowest QCID value. The computational complexity thereafter will be O \u0010\u0000N M \u0001 \u00d7 M 2\u0011 , which is unacceptable when N is extremely large. A probability approach To overcome the computational bottleneck, instead of treating M as a determined set, we consider it as a sequence of random variables, i.e. M = {C1, C2, ..., Cm, ..., CM} and assign it with some probability. Our expectation is that M should have higher probability to be sampled with if it is more class-balanced. This means P(C1 = c1, C2 = c2, ..., Cm = cm, ..., CM = cM) should be larger if M = {c1, c2, ..., cM} has a lower QCID value. Our sampling strategy generates the elements in M in a sequential manner, i.e., we first sample M1 = {c1} according to the probability of P(C1 = c1), then sample c2 to form M2 = {c1, c2} according to the conditional probability P(C2 = c2|C1 = c1). The same procedure applies for the following clients until we finally obtain M = {c1, c2, ..., cM}. In the following, we will design proper conditional probabilities such that the joint distribution of client selection satisfies our expectations. Let Tn denote the number of times that client n has been selected. Once client n has been selected in a communication round, Tn \u2192Tn+ 1, otherwise, Tn \u2192Tn. Inspired by combinatorial upper confidence bounds (CUCB) algorithm (Chen et al., 2013) and previous work in (Yang et al., 2020), in the k-th communication round, the first element is designed to be sampled with the following probability: P(C1 = c1) \u221d 1 [QCID(M1)]\u03b21 + \u03bb s 3 ln k 2Tc1 , \u03b21 > 0, where \u03bb above is the exploration factor to balance the trade-off between exploitation and exploration. The second term will add a higher probability to the clients that have never been sampled before in the following communication rounds. After sampling C1, the second client is defined to be sampled with probability P(C2 = c2|C1 = c1) \u221d 1 [QCID(M2)]\u03b22 1 [QCID(M1)]\u03b21 + \u03b1 q 3 ln k 2Tc1 , \u03b22 > 0. For the m-th client, where 2 < m \u2264M, we define P(Cm = cm|Cm\u22121 = cm\u22121, ..., C2 = c2, C1 = c1) \u221d[QCID(Mm\u22121)]\u03b2m\u22121 [QCID(Mm)]\u03b2m , \u03b2m\u22121, \u03b2m > 0. With the above sampling process, the final probability to sample M is P(C1 = c1, C2 = c1, ..., CM = cM) = P(C1 = c1) \u00d7 P(C2 = c2|C1 = c1) \u00b7 \u00b7 \u00b7 \u00d7 P(CM = cM|CM\u22121 = cM\u22121, ..., C2 = c2, C1 = c1) \u221d 1/[QCID(M)]\u03b2M . Since \u03b2M > 0, this matches our goal that the M with lower QCID value should have higher probability to be sampled with. Our mechanism, Fed-CBS, is summarized in Algorithm 1 . Algorithm 1 Fed-CBS Initialization: initial local model w(0), client index subset M = \u2205, K communication rounds, k = 0, Tn = 1 while k < K do // Client Selction: for n in {1, 2, ..., N} do if n \u2208M then Tn \u2192Tn + 1 else Tn \u2192Tn; end if end for Update M using our proposed sampling strategy in Section 3.3 // Local Updates: for n \u2208M do w(k+1) n \u2190Update(w(k)). end for // Global Aggregation: w(k+1) \u2190Aggregate(w(k+1) n ) for n \u2208M end while 5 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction Details and analysis For any 1 < m < M, we have P(C1 = c1, C2 = c1, ..., Cm = cm) \u221d 1 [QCID(Mm)]\u03b2m . This means when we generate the first m elements of M, we expect the Mm should be more class-balanced since the Mm with lower QCID value has a higher probability of being sampled. This is different from the algorithm in (Yang et al., 2020), which greedily chooses the cm from {1, 2, .., N}/Mm\u22121 that makes Mm the most classbalanced one. Unlike the greedy algorithm which has no guarantees on finding the optimal client set, our method can generate the globally optimal set of clients in the sense of probability. An example is provided in Figure 3 to demonstrate that our method can overcome the pitfall of the greedy method. After selecting the first two clients, {C1, C3} our method is less class-balanced than {C1, C2} chosen by the greedy method. However, after making the last choice, our method has the chance to derive a perfectly class-balanced set {C1, C3, C4}. In contrast, the greedy method can only get one result {C1, C2, C3}, which is less class-balanced. We require the distribution of P(C1 = c1, C2 = c1, ..., Cm = cm) to be more dispersed when m is small. This is because we expect our sampling strategy to explore more possible cases of client composition at the beginning. We require the distribution of P(C1 = c1, C2 = c1, ..., Cm = cm) to be less dispersed when m is large. This is because as we approach the end of our sampling process, we expect our sampling strategy can find the Mm that is more class-balanced. Especially when m = M, we hope the strategy to find the client cM which can make M the most class-balanced. Since P(C1 = c1, C2 = c1, ..., Cm = cm) \u221d 1 [QCID(Mm)]\u03b2m we can set 0 < \u03b21 < \u03b22 < ... < \u03b2M to satisfy the above requirements. Remark: We set a lower bound for QCID(Mm) as Lb since QCID(Mm) = 0 in some special cases will cause P(Cm = cm|Cm\u22121 = cm\u22121, ..., C1 = c1) \u2192\u221e. When viewing the conditional distribution as the likelihood in Bayesian inference, our probability can be interpreted as an estimate of the posterior distribution. This allows us to comprehend our algorithm through the lens of Bayesian sampling (Welling & Teh, 2011; Liu & Wang, 2019; Zhang et al., 2020a; 2019). In our future studies, we will further analyze the connection between them. Below we present two theorems to show the superiority of our proposed sampling strategy. Theorem 3.3 (Class-Imbalance Reduction). We denote the probability of selecting M with our strategy with \u03b2M as P\u03b2M and the probability of selecting M with the random selection as Prand. Our method can reduce the expectation of QCID compared to the random selection mechanism. In other words, we have EM\u223cP\u03b2M QCID(M) < EM\u223cPrandQCID(M). Furthermore, if increasing the value \u03b2M, the expectation of QCID can be further reduced, i.e., for \u03b2\u2032 M > \u03b2M, we have EM\u223cP\u03b2\u2032 M QCID(M) < EM\u223cP\u03b2M QCID(M). Theorem 3.4 (Computation Complexity Reduction). The computation complexity of our method is O \u0000N \u00d7 M 2\u0001 , which is much smaller than the exhaustive search of O \u0010\u0000N M \u0001 \u00d7 M 2\u0011 . Theorem 3.4 shows that the computation complexity of our method is independent of the number of classes. Since the dimension of neural networks is typically much larger than the class distribution vector \u03b1n, the additional communication cost is almost negligible. Besides, we also prove the NP-hardness of the problem formally in Section B.3 in the appendix. 4. Convergence Analysis To analyze the convergence of our method, we first define our objective functions and adopt some general assumptions. Our global objective function e F > 0 can be decomposed as e F = 1 B PB b=1 e Fb, where e Fb is the averaged loss function with respect to all the data of the b-th class in the global dataset. Similarly, the n-th client\u2019s local objective function Fn can be decomposed as Fn = PB b=1 \u03b1(n,b)Fn,b, where Fn,b is the averaged loss function with respect to all the data of the b-th class in the n-th client\u2019s local dataset, and \u03b1(n,b) is defined in Equation 1. Moreover, let w(k) denote the global model parameters at the k-th communication round and w(0) denote the initial global model parameters. If not stated explicitly, \u2207denotes \u2207w throughout the paper. Assumption 4.1 (Smoothness). The global objective function e F and each client\u2019s averaged loss function Fn,b are Lipschitz smooth, i.e. \r \r \r\u2207e F(w) \u2212\u2207e F(w\u2032) \r \r \r \u2264L e F \u2225w\u2212w\u2032\u2225and \u2225\u2207Fn,b(w) \u2212\u2207Fn,b(w\u2032)\u2225\u2264Ln,b\u2225w \u2212w\u2032\u2225, \u2200n, b, w, w\u2032. Assumption 4.2 (Unbiased Gradient and Bounded Variance). The stochastic gradient gn at each client is an unbiased estimator of the local gradient: E\u03be [gn(w | \u03be)] = \u2207Fn(w), with bounded variance E\u03be h \u2225gn(w | \u03be) \u2212\u2207Fn(w)\u22252i \u2264\u03c32, \u2200w, where \u03c32 \u22650. Assumption 4.3 (Bounded Dissimilarity). There exist two non-negative constants \u03b4 \u22651, \u03b32 \u22650 such that PB b=1 1 B \r \r \r\u2207e Fb(w) \r \r \r 2 \u2264\u03b4 \r \r \rPB b=1 1 B \u2207e Fb(w) \r \r \r 2 + \u03b32, \u2200w. Assumption 4.4 (Class-wise Similarity). For each class b, the discrepancy between the gradient of global averaged 6 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction all rand pow-d Fed-cucb Fed-CBS Communication Rounds \u03b1=0.1 757\u00b1155 951\u00b1202 1147\u00b1130 861\u00b1328 654\u00b196 \u03b1=0.2 746\u00b195 762\u00b1105 741\u00b1111 803\u00b1220 475\u00b1110 \u03b1=0.5 426\u00b167 537\u00b1115 579\u00b1140 1080\u00b1309 384\u00b174 E[QCID](10\u22122) \u03b1=0.1 1.01\u00b10.01 8.20\u00b10.21 12.36\u00b10.26 7.09\u00b12.27 0.62\u00b10.20 \u03b1=0.2 0.93\u00b10.03 7.54\u00b10.27 10.6\u00b10.48 5.93\u00b11.01 0.51\u00b10.12 \u03b1=0.5 0.72\u00b10.03 5.87\u00b10.24 7.36\u00b10.57 6.47\u00b10.77 0.36\u00b10.04 Table 1. The communication rounds required for targeted test accuracy and the averaged QCID values. The targeted test accuracy is 45% for \u03b1 = 0.1, 47% for \u03b1 = 0.2 and 50% for \u03b1 = 0.5. The results are the mean and the standard deviation over 4 different random seeds. 0 500 1000 1500 2000 2500 3000 Communication Round 30 35 40 45 50 55 Test accuracy(%) =0.1 0 500 1000 1500 2000 2500 3000 Communication Round 30 35 40 45 50 55 Test accuracy(%) =0.2 all rand pow-d Fed-cucb Fed-CBS (ours) 0 500 1000 1500 2000 2500 3000 Communication Round 30 35 40 45 50 55 Test accuracy(%) =0.5 Figure 4. Test accuracy on Cifar-10 under three heterogeneous settings. loss function and the local one is bounded by some constant in l2 norm. That means, for every n and b, we have \r \r \r\u2207e Fb(w) \u2212\u2207Fn,b(w) \r \r \r 2 \u2264\u03ba2 n,b, \u2200w. Assumptions 4.1, 4.2 and 4.3 have been widely adopted in previous literature on the theoretical analysis of FL (Li et al., 2019; Cho et al., 2020; Wang et al., 2020a). Assumption 4.4 is based on the similarity among the data from the same class. Similar to the standard setting (Wang et al., 2020a), the convergence of our algorithm is measured by the norm of the gradients, stated in Theorem 4.5. Theorem 4.5. Under Assumptions 4.1 to 4.4, if the total communication rounds K is pre-determined and the learning rate is set as \u03b7 = s 10L\u221a \u03c4(\u03c4\u22121)K , where s < 1, L = max{n,b} Ln,b and \u03c4 is the number of local update iterations, the minimal gradient norm of e F is bounded as: min k\u2264K \r \r \r\u2207e F \u0010 w(k)\u0011\r \r \r 2 \u22641 V [ \u03c32s2 25\u03c4K + sL e F \u03c32 10L p \u03c4(\u03c4 \u22121)K + 5\u03ba2 + 10L p \u03c4(\u03c4 \u22121) e F \u0000w(0)\u0001 s \u221a K + \u03b32E[QCID]], where V = 1 3 \u2212\u03b4BE[QCID] and \u03ba = max{n,b} \u03ban,b. If the class-imbalance in client selection is reduced, E[QCID] will decrease. Consequently, 1 V and E[QCID] V will also decrease, making the convergence bound on the right side tighter2. Therefore, Theorem 4.5 not only provides a convergence guarantee for Fed-CBS, but also proves 2Theorem 4.5 requires the \u03b2M in our method to be large enough to make E[QCID] < 1 3\u03b4B according to Theorem 3.3. How to the class-imbalance reduction in client selection could benefit FL, i.e., more class-balance leads to faster convergence. 5. Experiments We conduct thorough experiments on three public benchmark datasets, CIFAR-10 (Krizhevsky et al.), FashionMNIST (Xiao et al., 2017) and FEMNIST in the Leaf Benchmark (Caldas et al., 2018). In all the experiments, we simulate cross-device federated learning (CDFL), where the system runs with a large number of clients with only a fraction of them available in each communication round, and we make client selections on those available clients. The results show that our method can achieve faster and more stable convergence compared with four baselines: random selection (rand), Power-of-choice Selection Strategy (pow-d) (Cho et al., 2020), the method in Yang et al. (2020) (Fed-cucb), and the ideal setting where we select all the available clients (all). To compare them efficiently in the main text, we present the results from Cifar-10 where the whole dataset is divided to 200 (or 120) clients, since we need to engage all the clients for the ideal setting. To simulate more realistic settings where there are thousands of clients, we conduct our method on FEMNIST in the Leaf Benchmark with more then 3000 clients. Due to the space limit, we move the results of FEMNIST, Fashion-MNIST, and the ablation studies to Section C.5 & D in the Appendix. For Fashion-MNIST, we adopt FedNova (Wang explicitly derive a lower bound for \u03b2M is also very interesting and we leave it as a theoretical future work. 7 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction et al., 2020a) to show that our method can be organically integrated with existing orthogonal works which aim at improving FL. Experiment Setup We target cross-device settings where the devices are resource-constrained, i.e., most of the devices do not have sufficient computational power and memory to support the training of large models. Therefore, we adopt a compact model with two convolutional layers followed by three fully-connected layers and FedAvg (McMahan et al., 2017b) as the FL optimizer. The batch size is 50 for each client. In each communication round, all of them conduct the same number of local updates, which allows the client with the largest local dataset to conduct 5 local training epochs. In our method, we set the \u03b2m = m, \u03b3 = 10 and Lb = 10\u221220. The local optimizer is SGD with a weight decay of 0.0005. The learning rate is 0.01 initially and the decay factor is 0.9992. We terminate the FL training after 3000 communication rounds and then evaluate the model\u2019s performance on the test dataset of CIFAR-10. More details of the experiment setup are listed in Section C.2. 5.1. Results for Class-Balanced Global Datasets In this experiment, we set 200 clients in total with a classbalanced global dataset. The non-IID data partition among clients is based on a Dirichlet distribution parameterized by the concentration parameter \u03b1 in Hsu et al. (2019). Roughly speaking, as \u03b1 decreases, the data distribution will become more non-iid. In each communication round, we uniformly and randomly set 30% of them (i.e., 60 clients) available and select 10 clients from those 60 available ones to participate in the training. As shown in Table 4, our method can achieve the lowest QCID value compared with other client selection strategies. As a benefit of successfully reducing the class-imbalance, our method outperforms the other three baseline methods and achieves comparable performance to the ideal setting where all the available clients are engaged in training. As shown in Table 4 and Figure 4, our method can achieve faster and more stable convergence. The enhancement in stability can also be perceived as a reduction in gradient variance, a concept that has been explored in previous studies (Johnson & Zhang, 2013; Zhang et al., 2020b; Defazio et al., 2014; Zhao et al., 2018; Chatterji et al., 2018). It is also worth noting that due to the inaccurate distribution estimation and the limitations of the greedy method discussed in Section 2.2, the performance of Fed-cucb is much worse than ours. 5.2. Results for Class-Imbalanced Global Datasets In real-world settings, the global dataset of all the clients is not always class-balanced. Hence, we investigate two different cases to show the superiority of our method and provide more details of their settings in Section C.3. To simplify the construction of a class-imbalanced global dataset, each client only has one class of data with the same quantity. We report the best test accuracy in Table 2 and present the corresponding QCID values in Section C.4. 5.2.1. CASE 1: UNIFORM AVAILABILITY Settings. There are 120 clients in total, and the global dataset of these 120 clients is class-imbalanced. To measure the degree of class imbalance, we let the global dataset have the same amount of n1 data samples for five classes and the same amount of n2 data samples for the other five classes. The ratio r between n1 and n2 is respectively set to 3 : 1 and 5 : 1. In each communication round, we uniformly set 30% of them (i.e., 36 clients) available with replacement and select 10 clients to participate in the training. As shown in Table 2 and Figure 5, our method can achieve faster and more stable convergence, and it even achieves slightly better performance than the ideal setting where all the available clients are engaged. The performance of Fedcucb (Yang et al., 2020) is better than the results on the class-balanced global dataset, which is partly due to the simplicity of each client\u2019s local dataset composition in our experiments. The third line in Figure 2 indicates Fed-cucb can accurately estimate this simple type of label distribution. 0 500 1000 1500 2000 2500 3000 Communication Round 25 30 35 40 45 50 55 60 Test accuracy(%) 3:1 rand pow-d Fed-CBS(ours) Fed-cucb all 0 500 1000 1500 2000 2500 3000 Communication Round 25 30 35 40 45 50 55 60 Test accuracy(%) 5:1 Figure 5. Test accuracy on Cifar-10 with class-imbalanced global dataset in Case 1. 5.2.2. CASE 2: NON-UNIFORM AVAILABILITY Settings. There are 200 clients in total. In each communication round, 30% of them (i.e., 60 clients) are set available uniformly in each training round with replacement. By nonuniformly setting the availability, the global dataset of those 60 available clients is always class-imbalanced. To measure the degree of class imbalance, we make the global dataset have the same amount of n1 data samples for the five classes and have the same amount of n2 data samples for the other five classes. The ratio r between n1 and n2 is set to 3 : 1 and 5 : 1. We select 10 clients to participate in the training. As shown in Table 2 and Figure 5, our method consistently achieves higher test accuracy and more stable convergence, and it also outperforms the ideal setting where all the available clients are engaged. Since the global dataset of the available 60 clients in each communication round is always class-imbalanced, engaging all of them is not the optimal selection strategy in terms of test accuracy. 8 \fFed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction all rand pow-d Fed-cucb Fed-CBS Case 1 3:1 55.17\u00b10.94 50.99\u00b10.97 53.51\u00b10.34 55.11\u00b10.26 56.86\u00b10.34 5:1 50.93\u00b11.64 47.36\u00b12.34 52.73\u00b11.85 53.75\u00b10.58 54.94\u00b10.73 Case 2 3:1 54.01\u00b10.60 50.81\u00b12.03 53.98\u00b11.87 54.48\u00b11.31 57.71\u00b10.50 5:1 50.42\u00b11.27 48.33\u00b13.03 53.54\u00b11.18 53.38\u00b11.48 57.99\u00b10.46 Table 2. Best test accuracy for our method and other four baselines. 0 500 1000 1500 2000 2500 3000 Communication Round 10 20 30 40 50 60 Test accuracy(%) 3:1 all rand pow-d Fed-CBS (ours) Fed-cucb 0 500 1000 1500 2000 2500 3000 Communication Round 10 20 30 40 50 60 Test accuracy(%) 5:1 Figure 6. Test accuracy on Cifar-10 with class-imbalanced global dataset in Case 2. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2205.15202v1",
+                    "title": "A Small Leak Will Sink Many Ships: Vulnerabilities Related to Mini Programs Permissions",
+                    "abstract": "As a new format of mobile application, mini programs, which function within a\nlarger app and are built with HTML, CSS, and JavaScript web technology, have\nbecome the way to do almost everything in China. This paper presents our\nresearch on the permissions of mini programs. We conducted a systematic study\non 9 popular mobile app ecosystems, which host over 7 million mini programs,\nand tested over 2,580 APIs to understand these emerging systems better. We\nextracted a common abstracted model for mini programs permission control and\nrevealed six categories of potential security vulnerabilities in the permission\nenvironments. It is alarming that the current popular mobile app ecosystems\n(host apps) under study have at least one security vulnerability. We present\nthe corresponding attack methods to dissect these potential weaknesses further\nto exploit the discovered vulnerabilities. To prove that the revealed\nvulnerabilities may cause severe consequences in real-world use, we show three\nkinds of attacks related to the mini programs' permissions. We have responsibly\ndisclosed the newly discovered vulnerabilities, officially confirmed and\nrevised. Finally, we put forward systematic suggestions to strengthen the\nstandardization of mini programs.",
+                    "authors": "Jianyi Zhang, Leixin Yang, Yuyang Han, Zhi Sun, Zixiao Xiang",
+                    "published": "2022-05-30",
+                    "updated": "2022-05-30",
+                    "primary_cat": "cs.CR",
+                    "cats": [
+                        "cs.CR"
+                    ],
+                    "main_content": "Introduction Mini programs are light (commonly 2-4 MB) applications that run inside a speci\ufb01c mobile app (host app) [1]. As a new form of mobile application, leveraging web technologies like HTML, CSS, and JavaScript, mini programs are taking over the iOS and Android app ecosystems in China. The mini program technology enables the \"super app\" to bundle features and capabilities into a single mobile native APP, which lets the users never need to leave this native app. We call this a host app. Many host app vendors, such as Tencent (WeChat), ByteDance (TikTok) and Alibaba (Alipay), provide their own framework to support the mini programs [2]. There are various ways to launch a mini program in these host apps. Users can scan a QR code, directly search the name in the host app, share with a group or friend, launch it in a content article, or even link between mini programs. Figure 1a shows how to launch the Tesla mini program in WeChat as an example. With the clear interface and fast loading times, mini programs are very easy to use. More and more people use mini programs in their everyday life and need not worry about installing too many apps. There is a mini program for just about anything in China, whether to pay bills, play games, order a taxi, or book a doctor\u2019s appointment. As of 2021, there are over 1.26 billion monthly active users (MAU) of WeChat [3], 1 billion of TikTok [4], and over 658 million of Alipay [5]. The total number of the mini programs\u2019 users is close to that of Facebook, the most popular social network worldwide, which has about 2.89 billion MAU [6]. Since the widespread use of mini programs, any incorrect permission assignments or settings can result in serious security and privacy problems. However, there is not much research focus on this issue. That is not only because it is a new mobile application format, but also, more importantly, the permission structure of mini programs is entirely different from any other current permission-based security model. As we know, the mobile operating system (OS) is responsible for allowing or denying the use of speci\ufb01c resources at the app\u2019s run time [7]. The developers should declare a list of permissions that the user must accept before installing or running an application. Then the OS uses this security model to restrict every mobile app, native or hybrid, access to advanced or dangerous functionality on the arXiv:2205.15202v1  [cs.CR]  30 May 2022 \fRunning Title for Header (a) Two ways to access a WeChat mini program. (b) These mini programs, which operate as a separate mobile app, can run in the background and function within another larger app (host app). Figure 1: Tesla mini program in WeChat. Source: Screenshot. device [8]. Unlike the current access control models and methods, the permissions of mini programs are based on the host app authorization. That is, Android or iOS decides whether or not to allow the host apps to have some speci\ufb01c permissions, and the host apps authorize users to send pictures or other operations in the mini program. Therefore, from the view of OS, the permissions of mini programs and the host apps are the same, which means mini programs may apply for permissions from OS by using the reputation of host apps. It is impossible to control the permissions of mini programs directly. Once the host app does not authorize the correct permissions to the mini programs, it may cause security problems[9]. The permission issues in mobile applications have been fully studied in the past, and its permissions management mechanism is relatively complete and formal[10, 11, 12, 13, 14, 15, 16, 17, 18]. M-Perm[11] can identify normal, dangerous, and third-party permissions requests in applications to detect permission abuse. DroidNet[12] can provide advice on whether to accept or reject requests related to sensitive behaviors, helping users to implement low-risk resource access control on untrusted applications to protect user privacy. However, these studies cannot solve the relevant security issues in the permissions of the mini programs. In this paper, we present our systematic analysis of the current mini programs\u2019 permissions, where we dissect its framework, ecosystem, and potential vulnerabilities. We refer to the de\ufb01nition of sensitive permissions in Android and iOS and conduct a series of sensitive permission speci\ufb01cation tests on mini programs. Speci\ufb01cally, we systematically studied 9 popular mobile app ecosystems, which host more than 7 million mini programs, and established an abstract model for the unique permissions application process of existing mini programs. According to the host App and OS allow or reject permission application, we divide the application process into three situations: Host App Allow, OS Allow; Host App Reject, OS Allow; OS Reject. The majority of the vulnerabilities belong to Host App Reject, OS Allow. According to this abstract model, we investigated more than 2580 APIs and revealed six categories of potential security vulnerabilities that are common in most of the mini programs we studied. According to the types of leaked information, we present three kinds of proof-of-concept attacks to analyze these potential weaknesses further. To prove that the exposed vulnerabilities may cause serious consequences on the real-world systems, we described three interesting cases of APIs and illegal mini programs. To mitigate the threat of these vulnerabilities, we have listed recommendations for mini programs, developers, and users. Finally, to ensure that the mini programs\u2019 mechanism of different host apps had enough time to \ufb01x the vulnerabilities, we contacted them individually about the vulnerabilities several months before submitting this manuscript. This allowed several different host apps to \ufb01nish patching the reported vulnerabilities in writing. We also disclosed the vulnerabilities to various security response platforms, including Tencent Security Response Center. In summary, we have made the following contributions in this paper: \u2022 We deeply analyzed the current mini programs permission and presented a common abstracted model of the permission control of mini programs. To our best knowledge, we systematically studied the mini programs and the permissions for the \ufb01rst time. \u2022 We have detected more than 2,580 APIs. Through large-scale tracking and analysis of sensitive permission APIs, we have found six categories of potential security vulnerabilities in the process of processing sensitive permission applications by mini programs. 2 \fRunning Title for Header WebView WebView JSCore WebView App Service View Native Host App OS Mini Program request data API Figure 2: The common framework of mini programs \u2022 We conducted empirical research on the currently popular 9 host apps and revealed the security issues corresponding to the six types of potential security vulnerabilities we discovered in the real world. We also showed three real-life attacks on the mini programs permissions to prove that the revealed vulnerabilities may cause serious consequences in real-world use. \u2022 Following the practice of responsible disclosure, we have reported all the discovered design \ufb02aws, of\ufb01cially con\ufb01rmed and revised. \u2022 To mitigate these potential vulnerabilities, we put forward suggestions to strengthen the standardization of the entire mini programs\u2019 permission, thereby enhancing user privacy protection. 2 Background 2.1 Framework of Mini Programs Mini programs are a category of applications embedded in host apps without the need of downloading and installing [19]. The \ufb02ow of mini program framework consists of two components: View (the rendering layer) and App Service (the logic layer), which are respectively managed by two separate threads, as exhibited in Figure 2. The interface of View is rendered by WebView component, which handles page displaying and the user event interaction behavior, while the App Service employs JsCore threads to run JavaScript, for controlling the generation and processing of mini programs data. The communication between two threads is relayed by the Native app (refers to the client). The host app in the framework determines whether the mini program has the permission for acquiring the speci\ufb01c data through the corresponding API. That is, OS determines whether to allow host apps to have some speci\ufb01c permissions, and host apps then transmit authorized data to the mini program through the API. In other words, the permission of mini programs is inherited from the host app. Hence, if a host app does not properly manage the data and permission, data privacy and security issues will occur at its mini programs. In this paper, we will focus on the App Service component to study the authorization mechanism of the host app and the design of API for mini programs. 2.2 Difference to Other Apps As a new emerging paradigm, mini program has the substantial difference to the existing native app, hybrid app, progressive web app (PWA), and instant app. Yet, there is little research on mini programs, especially for their potential security and privacy issues. We brie\ufb02y summarize the difference between mini programs and other categories of apps here. A native app is a mobile application developed speci\ufb01cally for installing on device\u2019s OS like Android or iOS. A hybrid app looks like a native app, but at core, it is one kind of web app wrapped in a native container which loads the information on the page when a user navigates through the application. Although both hybrid apps and mini programs apply web technology, the hybrid app is still one type of native app that is under the management by OS. In contrast, as 3 \fRunning Title for Header Figure 3: Under the iOS system, the system permission pop-up prompt is displayed when Alipay requests runtime permission (left) and the pop-up prompt when a mini program in Alipay requests permission (right). a light version of the app, the mini program can only run on a native app\u2019s (host app\u2019s) interface and apply for the host app\u2019s permissions. Similar to the mini program, PWA also applies web technology, but differently, PWA is a type of webpage or website that runs in the browser and can be added to the home screen. Hence, the host environment of PWA is the browser and the OS manages the PWA\u2019s permission through the browser. Mini programs can be considered as one type of \u201cInstant\u201d app embedded in host apps for disposable interactions at a fraction of cost of an app. The host environment is a platform with extra capabilities that can support seamless service and access control for the user data. Google\u2019s instant app [20] is very similar to mini programs. Both of them allow users to access the application\u2019s content without additionally installing the application, which can save the application space on the device. In essence, Google\u2019s instant app is still a native app and under the OS\u2019s permission control while the mini program is under the host apps\u2019. 2.3 Authorization Permissions in mobile apps can be divided into two types, install-time permissions and runtime permissions[21]. Here, the runtime permission is also called the dangerous permission, which is related to users\u2019 privacy and can access users\u2019 private data, such as location information, contacts information, etc. These information are considered to be sensitive and should acquire user\u2019s authorization for access. When requesting the runtime permission, the system will display a prompt window, as shown in Figure 3. According to Section 2.1, the framework provides rich APIs to support the mini programs request a resource such as user information, location information, payment functions, etc. But the user does not authorize the API directly. In mini program development, the framework divided the dangerous APIs into multiple scope according to the scope of usage. The users can select scope to authorize. After a scope is authorized, all of its APIs can be used directly. 2.4 Permission Control Abstract Model We conclude an abstract model common to all mini programs\u2019 permission control, depicted as follows. Mini programs are \"sub-applications\" built on mature mobile applications which are then built on the OS. So, mini programs need to pass two layers of barriers when applying for sensitive permissions from users. That is, mini programs \ufb01rst apply to their host app where they are located and then the host App applies to the OS. As shown in Figure 4, according to the host app and OS allow or reject permission application scenarios, we categorize the application process into the following three cases: 4 \fRunning Title for Header Host App  OS \u2461 Allow or Reject ? Allow! Host App Reject, OS Allow OS Reject  Request A \u2462 Allow or Reject ? \u2461 Allow or Reject ? \u2463 Allow or Reject ? Reject! \u2461 Allow or Reject ? Reject! \u2460 Request A \u2460 Request B \u2462 Request B \u2460 Request C Allow! Mini Program  Allow! Obtained Permission A Obtained Permission A Host App Allow, OS Allow Obtained Permission B Not Obtained Permission B Not Obtained Permission C Not Obtained Permission C Figure 4: The sensitive permission application process of mini programs. Mini programs run in speci\ufb01c mobile applications (host apps), while mobile applications run in the OS. \u2022 Both Host App and OS Allow If a mini program applies for sensitive permission A, but the user has not applied this permission, the host app will pop up a window (as shown in the left of Figure 3) to ask whether the user is willing to grant the permission. If it agrees, the host app will continue to apply for sensitive permission A to the OS, and the OS will pop up a window (the right side of Figure 3) to ask if the user is willing to grant this permission to the mini program. The mini program will successfully obtain the requested sensitive permission once the user agrees. Later on, when the user uses the mini program again, he can directly call the interface. \u2022 Host App Reject but OS Allow In this case, the host app has obtained permission B given by the OS, but when its mini program wants to apply for permission B, the user chooses to refuse and then directly enters the interface fail callback. Hence, the mini program fails to obtain permission B. In our later analysis in Section 3, most vulnerabilities we discover belong to this category. \u2022 OS Reject If the host app does not get the permission C given by the OS, neither the host app nor the mini programs in it can get this permission. It is worth noting that when the developer calls the API requiring authorization, the authorization setting will appear in the mini programs\u2019 authorization setting page regardless of the user\u2019s permission or not. The user can modify the authorization, and the calling result will change accordingly until the user actively deletes the mini programs. But user authorization setting is not synchronized with the server. If the user changes the device or switches his account on the same device, the authorization will be prompted again. 3 Security Analysis After getting the background knowledge of mini programs, we start to perform the security analysis of mini programs\u2019 sensitive permission application process. According to the three scenarios as we have discussed in Section 2.4, we discover six categories of potentially vulnerable modes during mini programs\u2019 processing of sensitive permission applications, as presented in Figure 5. Based on these vulnerabilities, we have created three proof-of-concept attacks. In this section, we will \ufb01rst illustrate the principle for determining the sensitive permission in Section 3.1. Then, we elaborate the detailed vulnerability in Section 3.2 and present our proof-of-concept attacks in Section 3.3. 5 \fRunning Title for Header Proof-of-concept  Attacks Vulnerabilities Permission Application  Host App Allow, OS Allow Host App Reject, OS Allow OS Reject Security Risks Permission Encapsulation Leaked API Forgotten API Under Permission Control Parameter Settings Ignored Invalid Permission Setting Qualified Conditions Ignored Secretly Transfer Sensitive Data Reuse Cache Files Permission Management Issues Permissions Difference in Different Environments Contacts Attack Location Information Attack Clipboard Content Attack WebView Bypasses Permission Control Path1 Path2 Collect and use user privacy, personal information abuse. Figure 5: The vulnerabilities and attacks in the mini programs permissions. 3.1 Whether to Request Authorization According to 2.3, we know that the mini program framework puts the sensitive APIs into the scope. The framework determines whether the sensitive data can be obtained or processed by the mini program in the background as a rule for sensitive operations. For example, in the WeChat mini programs, wx.getLocation obtains the current geographical location in the background. If there is no user authorization, the current location information will be transmitted without the user\u2019s awareness. Therefore, the mini program framework puts such APIs into scope for uni\ufb01ed authorization and management. In contrast, the API of wx.chooseImage is to select pictures from local photo albums or take photos with cameras. Although the data involved are sensitive, the user needs to select and other interactive operations to send out the data. The framework will consider the user know and authorize this operation, and this data cannot be transmitted without the user\u2019s awareness. Hence, the mini program framework does not put this kind of API into scope. 3.2 Potential Vulnerabilities We conduct a large-scale analysis of APIs from different host apps and target APIs related to sensitive permission. A total of 2,580 APIs are detected by us. Through large-scale tracking and analysis, examining the documentation, available source codes, and demos, we identify the following six categories of potential security vulnerabilities in the process of sensitive permission applications by mini programs. 3.2.1 Reuse Cache Files When a user quits or deletes the mini programs, the cache \ufb01les in the corresponding path should also be deleted, for reducing the user\u2019s storage space and avoiding being reused. However, suppose the user closes the mini program in the host app and opens it again for operation. The previously cached temporary \ufb01le is still there without being deleted. So, an attacker has the chance to obtain the local temporary \ufb01le path of the previous \ufb01le without the user\u2019s awareness. 3.2.2 Permission Encapsulation in Leaked API Some APIs related to sensitive permissions do not encapsulate permissions well (we refer to these APIs collectively as PEL-API). This means that any mini program can directly call these APIs to obtain the corresponding permissions, ignoring the need to apply permission within the host app. The application process of sensitive permissions corresponding to such an issue is shown in Figure 6. When the host app obtains a dangerous permission B from the OS, the mini programs can get this permission without asking the user\u2019s willingness in a pop-up window. Even having the pop-up inquiry and the user rejects it, the mini programs can still get this permission. In particular, we divide these vulnerabilities into the following four categories. Quali\ufb01ed Conditions Ignored. According to the description in Section 3.1, when the user must manually operate the sensitive permissions involved to view, select, or transmit private data, the mini programs will default that the user has allowed the use of rights in self-operation so that no pop-up will be made. In other words, there is no scope (mentioned in 2.3) corresponding to these APIs, and as long as the OS opens the relevant permissions for the host app, the mini programs can call these APIs and get data at will. At this time, sensitive data can only be used when the user interacts 6 \fRunning Title for Header \u2461 Allow or Reject ? \u2460 Request B Allow! Obtained Permission B Obtained Permission B No Ask! \u2463 Allow or Reject ? Reject! \u2462 Request B Obtained Permission B Host App  OS Mini Program  Figure 6: When Host App Reject and OS Allow, mini programs illegally obtain the process of sensitive permissions. with mini program, which is equivalent to the user controlling his own sensitive information. Mini programs cannot steal the user\u2019s sensitive information through this type of API. However, since the API of mini programs will be updated irregularly, the newly-launched API that involves the same permissions may have the function of stealing users\u2019 private data. Many old APIs that involve the same permissions do not divide scope into corresponding ones, so these new APIs become a security risk. We believe that these APIs ignore quali\ufb01cation of sensitive permission (whether private data will be stolen without the user\u2019s awareness). Forgotten API Under Permission Control. The host apps handle whether the sensitive information can be passed to the mini program. For example, almost all host apps consider obtain \u201clocation information\u201d is the dangerous permission. Whenever the mini program wants to use the current user\u2019s location, it will \ufb01rst apply to the user to ask for his consent. If the user refuses, the mini program itself cannot locate the user\u2019s speci\ufb01c location, regardless of whether the host app has obtained the permission. However, the designer neglected that some APIs should ask for authorization before sending the location information to the mini programs. In other words, these neglected APIs do not belong to any scope (mentioned in 2.3) and can be used without user authorization. Parameter Settings Ignored. From the point of view of its functions, some APIs may not actively obtain the user\u2019s sensitive permissions. However, the designer neglects the parameter settings in API, which will also steal users\u2019 privacy. For example, my.chooseCity in the Alipay mini programs is an API to open the city selection list. The parameter showLocatedCity indicates whether to display the currently located city. If it is set to true, the user\u2019s current city will be directly located regardless of whether the host program opens the location permission for the mini programs. If the user does not do anything, the background cannot see any content about the location information. However, as long as the user selects the area located by the system, the background will return to the current city and latitude and the user\u2019s longitude, even if the location permission is closed at this time. Invalid Permission Setting. The permission settings of some APIs are inconsistent with the descriptions in their of\ufb01cial documents. To some extent, this shows that the API related to sensitive permissions in mini programs does not encapsulate permissions well. For example, wx.choosePoi in the WeChat mini programs realizes the function of opening the map and selecting the location. It is indicated in the document that the invocation of this API requires the authorization of scope.userLocation. However, in the actual test, we found that it is inconsistent with the of\ufb01cial document description the location can be selected without user authorization at all. When the user chooses precise positioning, the background will return the latitude and longitude data of the current user. 3.2.3 Stealthily Transferring Sensitive Data Vertical: The application process for this vulnerability is shown in Figure 6. When some mini programs developed by serious cooperative companies involve sensitive permissions like geographical location, their host apps will ignore applying for permissions from users. The method shown as in Figure 6 can bypass user authorization for stealthily transmitting user sensitive information. Horizontal: Different mini programs developed by the same company may share user information. All the user sensitive permissions acquired by the mini programs should be made public, and the user has the right to close the acquisition of sensitive permissions by the mini programs. Our empirical study found that some mini programs obtain and leverage the user information in their associated mini programs by default, including account information, shipping address, etc., but never request user\u2019s consent. This type of mini programs skips the permission application step, and 7 \fRunning Title for Header the authorized permission in the settings is empty, resulting that a user is unable to close authorization of relevant user information. Users cannot fully control the transmission way and use the scope of their personal information, which risks continuous opening of permissions and unknown use of personal information. 3.2.4 Permission Management Issues A mini program may continue to use sensitive permissions to collect users\u2019 privacy information even a user wants to turn off the permissions after using it. In particular, it can be divided into the following three situations. Permission Setting Disappears. The permission setting page provides convenience for users to manage permissions of mini programs. However, some mini programs may get permanent authorization after one-time authorization due to the lack of setting pages showing permissions. In this case, users cannot view what permissions they have granted to a mini program, and they cannot cancel the previously granted permission. As long as the host app is not uninstalled, the permission will remain open to the mini programs. Thus, a mini program can use the previous authorization to continuously gather and use the user\u2019s personal information, posing a security risk. Permission Cannot be Deleted. Regarding the validity period of authorization of mini programs, once a user explicitly agrees or rejects the authorization, such an authorization relationship will be recorded in the background until the user actively deletes the mini programs. However, the permissions some mini programs may not be able to delete, which can cause harmful consequences. These mini programs will be able to use the previous authorization to continue collecting and using users\u2019 personal information. Unable to Completely Remove Permissions. Although the authorization settings of some mini programs are deleted after a user actively removes the mini programs, the sensitive personal information (such as ID number) involved will be retained. And such information can be leveraged for queries and other operations. At this time, the setting options of related permissions will not be in the mini programs\u2019 settings page, and the user cannot completely remove the permission. Hence, although a user cancels the authorization, the related mini programs do not update the authorization in time, thus continuously collecting and using the user\u2019s personal information. This leads to a risk of using a user\u2019s personal information for abuse. 3.2.5 Webview Bypasses Permission Control The mini programs can use the web-view page bearing component to open the H5 page in the mini programs. In this process, the loaded H5 page needs to manually import JS \ufb01les (which is a web development toolkit based on the host apps for web developers) provided by each platform. In this way, developers can apply the capabilities of mobile phone system such as taking pictures, selecting pictures, and location with the help of host apps to provide users with a better web experience. After empirical study, we prove that webview can bypass the speci\ufb01ed API when the mini programs or host apps do not prompt for permission application or the user refuses after prompting. There are two scenarios to illegally obtain sensitive permissions for mini programs after using the web-view component. First, the mini programs may completely ignore the OS\u2019s permission control over host apps and host apps\u2019 permission control over mini programs. They can directly access any sensitive permissions without notifying a user. Even when OS rejects sensitive permissions from a host app, the mini programs can still obtain such permissions through the web-view page-bearing component. Second, only the OS\u2019s permission control over host apps is considered, while the host apps\u2019 permission control over mini programs is ignored. The speci\ufb01c process is shown in Figure 6. In this case, if the OS\u2019s permission control on host apps is turned off, the mini programs will not obtain the corresponding sensitive permissions. If the OS\u2019s permission control on the host app is turned on, a user allows host apps to use certain sensitive permission. Hence, any mini program in host apps can obtain sensitive permission. We would like to remark that both scenarios may result in the disclosure of user privacy. 3.2.6 Permission Issues in Different Environments The processing details for some APIs are not the same between different OS and versions. Since the mini program framework can not handle these APIs differently depending on the running environment, the same operation or program code will have different results. For example, apps can read the clipboard contents without the user\u2019s manually selecting \u201cPaste\u201d when the user copies something. This is by design. Nevertheless, if the user copies sensitive information and leaves it on the clipboard, all apps can capture it and maybe send it to a remote server. Copying private things from a clipboard is risky. Different versions of OS have different feedback on this. The Android or old version of iOS will not inform the user when an application reads the clipboard. Many host apps will also be silent when the mini program reads the clipboard. If OS or host app does not consider the clipboard permission is dangerous, mini programs with access to the clipboard can steal the clipboard information of users in the background. 8 \fRunning Title for Header Table 1: The list of collected 9 host apps. Company Host App Monthly Active Users Tencent WeChat 1.26 billion [3] Tencent QQ 595 million [22] Alibaba Alipay 658 million [5] ByteDance TouTiao 400 million [23] TouTiao speed Edition TikTok 1 billion [4] Baidu Baidu 607 million [24] Multi Vendor QuickAPP 130 million* [25] China UnionPay UnionPay 10.6 million [26] * Only contains Huawei\u2019s data. 3.3 Proof-of-concept Attacks The vulnerabilities mentioned above we found can result in users\u2019 sensitive information leakage. Here, we provide three proof-of-concept attacks to prove the seriousness of these vulnerabilities. 3.3.1 Location Information Attack An attacker can obtain the user\u2019s location information through the issues mentioned above discovered. For example, the second, third, and fourth issues discussed in Section 3.2.2 all involve location information leakage. An attacker may simply call an API with a bug in the design of host apps to obtain user\u2019s location information. Based on which, an attacker can infer personal information such as user\u2019s hobbies and sports patterns. More serious consequences can also be resulted, such as being tracked and personal attacks. If an attacker steals a massive amount of personal private information, the security and trust of the whole society will become an issue. 3.3.2 Contacts Attack Some mini programs do separate the permission of \u201ccontacts\u201d, which can result in serious consequence. For example, the WeChat mini programs do not divide the permissions of \u201ccontacts\u201d separately, so an attacker with ulterior motives can return and match similar mobile phone numbers in the background by calling wx.searchContacts. Once a user is enticed to click the button bound to the event, a malicious mini program can traverse the user\u2019s address book information for fraud. 3.3.3 Clipboard Content Attack Since the mini program framework does not restrict the apps to read the clipboard, developers only need a few lines of code to see what users have just copied. If users copy an online banking password to paste and leave this private information on the clipboard, a malicious mini program can read it in the background and see that data directly. The same goes for other sensitive data like names, addresses, credit card numbers, or even private photos. Mini programs can capture everything of user\u2019s clipboard and do whatever they want with it. The copied text could be sent to a remote server without user\u2019s awareness. 4 Empirical Study This section presents our empirical study for analyzing the current mini programs permissions. Our goal is twofold. First, we collect the current popular mini program platforms (Table 1), and then expose potential vulnerabilities as outlined in Section 3. We also exhibit the security issues exposed in the real world through detection. Second, we conduct case studies to show some real attacks on the permissions of mini programs, to prove that the revealed vulnerabilities may cause serious consequences in real-world use. 4.1 Mini Programs Ecosystem As of June 2021, the number of mini programs in the whole network exceeded 7 million. We have identi\ufb01ed 9 popular host apps developed by 6 companies, which are listed in Table 1. These 9 host apps are used by tremendous amounts of users. Each host app has its own development tools. We apply the respective development tools to test on different host apps. Through empirical study, we discuss their vulnerabilities and list them in Figure 7. In this \ufb01gure, the orange blocks indicate that the host app has corresponding vulnerabilities, green blocks indicate that the host app has \ufb01xed 9 \fRunning Title for Header WeChat QQ Alipay Baidu TikTok TouTiao TouTiao Speed Edition QuickAPP web-view bypasses Permission management Environmental differences Cache-related issues Existed Vulnerability Fixed Vulnerability after Our Report Unknown or No Discovery Yet Parameter Invalidation Secretly transfer Ignore Forgotten Non-existent UnionPay Cell format: Left Android Right iOS Figure 7: The vulnerabilities distributions in the collected 9 host apps, where the vertical axis lists the names of host apps and the horizontal axis lists the vulnerabilities that we have discussed in Section 3. Cell format: <analysis result on Android> | <analysis result on iOS> Table 2: The list of Collected PEL-APIs. Vulnerabilities Host App API Ignore WeChat wx.searchContacts Forgotten QQ MapContext.moveToLocation MapContext.getCenterLocation Parameter Alipay my.chooseCity Invalidation WeChart wx.choosePoi vulnerabilities, gray block indicates that the host app does not have such a vulnerability, and light yellow blocks indicate that it is uncertain if there exists such a vulnerability. 4.2 Vulnerability Analysis 4.2.1 Vulnerable Caching Mechanism Our study discovered that in WeChat, when a user closes the used mini programs (which have not been deleted from the recent use) and opens the mini programs again for operation, the previously cached temporary \ufb01les still exist. Because the local temporary \ufb01le path can be obtained in the background, the temporary \ufb01le can be reused without the user\u2019s consent before being recycled. In QQ and ByteDance, as long as the user exits the mini programs or reopens the previously used mini programs, the previous temporary \ufb01les will be automatically deleted. Hence, they don\u2019t have the vulnerable caching issue. This vulnerability has not been found temporarily in mini programs of other host apps. 4.2.2 Vulnerable Issues API As discussed in Section 3.2.2, this kind of issue comes from that the API related to sensitive permissions does not encapsulate permissions well. In order to \ufb01nd APIs with such vulnerabilities, we set two criteria for analysis: 1) whether Android and iOS treat the involved permissions as permissions that need to be granted by the user; 2) whether using permissions by these APIs involves the interaction between users and mini programs. If the user must manually operate to view, select or transmit private data, then the mini programs default that the user has allowed the use of the permissions without prompting, so these APIs are not within the scope of our research. Table 2 summarizes the APIs that we have found so far that are related to sensitive permissions but do not encapsulate permissions well. 10 \fRunning Title for Header 4.2.3 Vulnerable Permissions Transfer Vertical: \"Amap\" in the Alipay mini program can be opened directly to accurately locate the user, ignoring the application of the mini program to the user for location permission. Although Alipay and AMap have reached an in-depth cooperative relationship (both belong to Alibaba Group), this does not mean that their operations can bypass the user\u2019s willingness. No abnormality was found in the public test code of \u201cdemo\u201d provided by Alipay, but using this mini program alone can directly locate users accurately. This shows that there may be an inconsistency between the source code of \u201cdemo\u201d and the public test code provided. They may use other ways to bypass the user authorization to transmit the user\u2019s location secretly. Horizontal: Some companies may share user information among different mini programs. For example, after logging in the \u201cPinduodu\u201d mini program in WeChat, the same login information is displayed directly when the user \ufb01rst uses another mini program named \u201cPinduoduo Coupon\u201d, which is from the same company Pinduoduo Inc. This type of mini program skips the permission application step, and the authorized permission in the settings is empty. Hence, the user cannot close the authorization of relevant user information. Users cannot fully control the dissemination of their personal information and the scope of use, which will lead to the continuous opening of permission and the risk of personal information being used in unknown circumstances. 4.2.4 Insecure Permission Management The permission management of UnionPay mini programs is rather messy. Since the security of UnionPay is mainly related to user identity information and payment, it is not very strict to consider other sensitive permissions such as microphones, geographic locations, cameras, and photo albums. On the privacy settings page of the UnionPay APP, we can see the authorization information for mini programs. However, only the authorization of the phone number and identity information (name, ID number) is displayed. In contrast, other permissions such as location, access to mobile phone albums, and the camera are ignored, and there is no permission setting page inside mini programs. When a user grants mini programs sensitive permission such as a microphone, geographical location, camera, and photo album, as long as the UnionPay app is not uninstalled, the mini programs can always access these permissions. We found that, in ByteDance, the permission of mini programs cannot be deleted with the deletion of the mini programs. In the of\ufb01cial ByteDance document, there is no clear explanation on the validity period of the authorization and whether the authorization will remain open after deleting mini programs. Hence, mini programs under ByteDance may use previous authorizations to continuously access and collect user personal information, posing a security risk. Although the authorized information of Alipay mini programs can be deleted after a user actively deletes the mini program, the sensitive personal information involved before (such as ID number) will be retained by individual mini programs. We have canceled the authorization and removed the mini program, but the relevant information can still be consulted when entering the mini programs again after a certain period. At this time, the permission setting of the mini program becomes empty, and the user cannot set the permission on the setting page. This situation indicates that after the user cancels the authorization, the related mini programs do not update the authorization in time and may continue to collect and use the user\u2019s sensitive information after the user has canceled the authorization. 4.2.5 Insecure Webview Component Some mini programs can carry web pages through the webview component. The communication between the webpage and mini programs is realized by the interfaces provided by the webpage development kit based on host apps provided by each platform for web page developers. None of the communication outside the provided interface is supported. Neither QQ nor ByteDance found any interfaces involving dangerous permissions. In the WeChat host app under Android, it can be determined that the user\u2019s storage information can be obtained using the webview component. That is, the \ufb01rst scenario described in Section 3.2.5 can be used to bypass the normative permission control. The difference is that in WeChat mini programs under iOS, it is used to obtain the user\u2019s album permission, and the second scenario is used to obtain the user\u2019s camera permission. The mini programs in Alipay bypass the standard permission control through the second scenario described in Section 3.2.5, to obtain sensitive permissions such as the user\u2019s camera, photo album, location, etc., even if these permissions are turned off in the setting page of mini programs at this time. That is, when users reject these sensitive permissions, the sensitive information can still be obtained by using the webview component. The mini programs under Baidu, in the iOS environment, can use the second scenario to obtain the camera permission to take pictures. QuickApp in Huawei can obtain the users\u2019 stored information (pictures, audios, videos, documents, etc.) through it. This will cause users to disclose their precise location to the attacker without their awareness. If the user\u2019s mobile phone turns on the GPS when taking photos, then once the mini programs can access the camera and photo album, the malicious mini programs only need to \ufb01lter out which photo taken by the user\u2019s mobile phone to know its geographical location. 11 \fRunning Title for Header WeChat Baidu  TikTok  TouTiao  TouTiao Speed Edition  Alipay  QQ  1 2 3 4 5 6 Quick APP  (HUAWEI)  Figure 8: Examples of getting clipboard contents in Android. Only case 2 and case 3 will prompt the user when the mini programs get the contents of the user\u2019s clipboard. 4.2.6 Vulnerable Clipboard Mechanism After iOS 14 upgrade, there will be a pop-up to informs the user when an app reads the contents of the clipboard. Hence, there will be a noti\ufb01cation when the mini programs read the clipboard no matter which host apps they run inside. Android is more complex since there are multiple versions. Xiaomi MIUI12(MIUI is a third-party mobile phone OS deeply optimized, customized, and developed by Xiaomi based on Android) splits the permission of the clipboard independently, and users can monitor the reading and writing behavior of each application. However, in other Android phones (such as Huawei, Vivo, etc.), the permissions control of the clipboard will not inform the users. Under these OS versions, we test the clipboard permissions on different host apps in the Android environment, and check their corresponding prompts, with results shown in Figure 8. In order to exhibit that the mini programs have obtained the contents of a user\u2019s clipboard, we display a modal dialog box. We found that 6 out of the 8 host apps (cases 1, 4, 5, 6 in Figure 8) did not give any prompt to the user when obtaining the clipboard information, which may be vulnerable to the theft of clipboard content described in Section 3.3.3. Cases 2 and 3 in Figure 8 show a design example that the mini programs will prompt when obtaining the contents of a user\u2019s clipboard. However, case 2 only reminds the user that the mini programs have obtained the clipboard\u2019s contents through a bubble, but the user only knows that it cannot be blocked (same as the prompt in the iOS environment). In case 3, obtaining the clipboard\u2019s content is set as the user\u2019s permission to operate. The user must select \u201cAllow\u201d before the mini program can obtain the corresponding information. We consider these designs with the best security usability practice. It should be noted that, although it is set as the user\u2019s permission to operate in case 3, there is still an issue with Alipay. In version 10.2.26.8000, Alipay will pop up a window to ask the user \"Apply to obtain the contents of your clipboard\", and the contents of the clipboard have been pasted on the page before clicking \"Reject\" or \"Allow\" (this is equivalent to the mini program still being able to obtain the clipboard information after the user refuses authorization). However, in version 10.2.23.7100, users cannot get any content after clicking Reject. 12 \fRunning Title for Header API Dataset  WeChat ByteDance QQ Ali Baidu Triggering  Results report Automatic Code Comparison and Manual Analysis Source code analysis Test API in development tools Decompile Static  Analysis Sensitive Permission Detection System Source File \u2026\u2026 Detection in popular Mini Programs User behavior imitation module Status monitoring and recursive module Data acquisition and storage module Statistics Figure 9: Overview of measurement methodology 4.3 An Example WeChat is an application in China with more than 1 billion active users, and it is also one of \ufb01ve applications in the world that have surpassed this milestone. 78% of Chinese people aged 16-64 are using WeChat [27]. So, we take WeChat as an example and detect the API with permission issues in the popular mini programs and expose security issues hidden in the real world. MiniCrawler[28], an open source WeChat mini program crawler, can be used to automatically download, unpack and index the mini programs from WeChat. However, the tool has strict requirements on the WeChat version, and the latest WeChat version API has been replaced, so the tool cannot be used. Our measurement method is shown in Figure 9. 4.3.1 Implementation In order to ensure the smooth progress of the experiment, we did not choose the Android simulator (according to the previous test experience, running small programs with the Android simulator will be very stuck). Instead, we select an Android entity machine with ROOT permission to carry out our operation, as shown in Figure 9. Our experiment is mainly divided into the following three parts: Triggering We use Airtest, a UI automated testing tool based on image recognition and poco control recognition, to achieve automated clicks on mini programs to obtain the source \ufb01les (that is, downloading the source \ufb01le package of mini programs from the server to local). Notably, since mini programs can be used in parallel with the host apps after clicking, each mini program needs to be completely exited after the simulation click is over. Otherwise, too many open mini programs will cause the Android emulator slow. In the process of implementation, to achieve as far as possible that each page of mini programs can be loaded successfully to obtain all the source \ufb01les of mini programs, we adopted LSH image detection algorithm for near repetition to determine whether the current page is consistent with the previous page if it is consistent, and treat them as the same page. If it is inconsistent, it is considered as the next page. In order to illustrate our experiment more clearly, the simulation click part is separated into the following three sub-modules: User Behavior Imitation Module: The function of this module is to imitate the process of users clicking into mini programs. The program completely replaces the human operation: Automatically clicking the \u201csearch\u201d icon to enter the search bar through the simulated click technology, automatically input the name of mini programs stored in the sample library in the search bar through the \ufb01le retrieval technology, and identify the accurately searched mini programs based on the image recognition and poco control, to imitate the user behavior and enter mini programs. Status Monitoring and Recursive Module: The main function of this module is to monitor the current interface position of mini programs in real-time and recursively scan the page content. By de\ufb01ning a page scanner and combining LSH image detection algorithm, the depth of the current page is monitored in real-time to determine whether to enter the next interface or call back to the previous interface through recursive function, for scanning as many different interfaces of the same mini programs as possible and trying to get all the cache packets of mini programs. Data Acquisition and Storage Module: The main function of this module is to obtain the source \ufb01les of mini programs. Take WeChat mini programs as an example. In the scenarios of WeChat version 8.0.2, the source \ufb01les of a certain WeChat mini program are all in the path \u201c/data/data/com.tencent.mm/MicroMsg/.../appbrand/pkg\u201d. Then, we use the shell command to dump the source code of the mini program to the folder named after the WeChat mini program to facilitate the follow-up work. At the same time, it is also necessary to delete the code packages that have been transferred to other locations in a speci\ufb01c folder in time to prevent confusion with the code packages of the next WeChat mini program. 13 \fRunning Title for Header 0 0.05 0.1 0.15 0.2 0.25 0.3 Education Travel Traffic Finance Self-operated by merchants Catering Industry Medical Industry Realty Business Sports Public Welfare Serving People's Livelihood Express Delivery Figure 10: The proportion of wx.getClipboardData in different categories of WeChat mini programs. Decompile: We obtain the source code of the speci\ufb01ed mini programs through decompilation. We have added a graphical operation interface to wxappUnpacker [29] for the decompilation script and con\ufb01gured Nodejs, the running environment required for the decompilation script, which can decompile the mini program source \ufb01les obtained in the previous step to obtain their source code. Statistics: This step counts the usage of APIs with permission issues in popular mini programs in the world. Based on the issues found in section 3.2, we sorted out different mini programs platforms and listed APIs that have permission issues. Screen these APIs from the source code of the mini programs, and measure the usage of the problematic API in popular mini programs in the real world. 4.3.2 Results We refer to the \"Aladdin Index\" (an intelligent platform for data analysis of the entire network of mini programs), to select 578 popular mini programs for research, and get 977 mini programs packages after decompression. We take the permission of the clipboard as an example to detect the severity. Figure 10 shows the proportion of wx.getClipboardData in various WeChat mini programs. Almost every kind of WeChat mini program use the vulnerable API as a standard API, which has the potential risk that some mini programs apply these API to invade user privacy. 4.4 Case Studies In this section, we present our case study of some representative APIs and illegal mini programs. 4.4.1 Stealing Location Information In the QQ mini program development tool, the index.js \ufb01le used qq.createMapContext to create a MapContext object; then we used MapContext.moveToLocation to move the map center to the current location (at this time, the map component show-location is set to true); then use MapContext.getCenterLocation to get the latitude and longitude of the current map center. In the whole process, the user\u2019s geographic location, latitude, and longitude can be accurately obtained in the background without the user\u2019s authorization. When working with Tencent\u2019s location service to accurately obtain the location, the user\u2019s speci\ufb01c location can be obtained. We con\ufb01rmed that the QQ mini programs could obtain the coordinates of the center point of the current location without the user\u2019s authorization, regardless of the Android or iOS system, and successfully transfer the coordinate value, speci\ufb01c location, and other information to the mini program background. Without authorization, the mini programs can obtain the user\u2019s precise location. Once the mini programs associate the location information with the account information in the background, the user\u2019s personal information is completely exposed. This is equivalent to unauthorized access to the user\u2019s whereabouts (belonging to personally sensitive information), and the mini programs are suspected of obtaining personal information in violation of regulations. 14 \fRunning Title for Header 4.4.2 Stealing Contacts The API provided by the WeChat mini programs: wx.searchContacts is to \ufb01nd the contacts and match a similar mobile phone number. This API does not specify the number of calls within a period. Suppose the phoneNumber parameter (the number to be searched) in this function is written in a loop. In that case, most of the information in the user contacts can be obtained by traversing in sequence. Speci\ufb01cally, set a button in index.wxml to bind an event. Write the API wx.searchContacts into this event in the index.js \ufb01le, and click this button, the foreground will not respond, and the background will get the information of the user\u2019s contacts. At this time, if the host environment WeChat has already obtained the contacts\u2019 permission, the mini programs can obtain part of the contact information without the user\u2019s awareness. After testing, we have con\ufb01rmed that WeChat mini programs can use this method to obtain information of some users\u2019 contacts and successfully pass the obtained information into the mini programs\u2019 background, whether in Android or iOS, without the user\u2019s authorization and the user\u2019s awareness. Attackers can take advantage of this vulnerability and bind wx.searchContacts to an inductive button to entice users to click. Sensitive information in the user\u2019s address book may be read and uploaded to the background, resulting in information leakage. 4.4.3 Stealing Clipboard Information Take the WeChat mini programs as an example. During the test, we write wx.getClipboardData in the onload function in the JS code so that the clipboard content can be easily obtained without the user\u2019s awareness. Even if it is not written in the onload function, binding this event to the button control can also trigger it. In the real world, mini programs can write some inductive slogans on the button to induce the user to click and then obtain the contents of the user\u2019s clipboard. After a large number of tests, we have con\ufb01rmed that many host apps (WeChat, ByteDance, Baidu, QuickAPP) can obtain user clipboard information and successfully transfer the obtained information into the mini programs\u2019 background without the user\u2019s authorization and awareness. Suppose the copied content is not destroyed after the user pastes it into an application. In that case, the content can still be obtained when the user opens a mini program, thus causing the leakage of sensitive information of the user. For example, when a user copies the name of a certain product, after opening a shopping mini program, it can read the content of the user\u2019s clipboard and upload the private information in the background. The vendor of this mini program will know the user may want to buy this product and push similar commodities or analyze the user\u2019s behavior to push advertisements accurately. 4.5 Responsible Disclosure We reported the results of our investigation to the Tencent Security Response Center on May 6, 2021, and August 27, 2021, and received their vulnerability con\ufb01rmation. We also reported vulnerabilities regarding location and contacts permissions to CVE, received their con\ufb01rmation of the vulnerability on May 18, 2021, and August 29, 2021, and obtained CVE numbers CVE-2021-33057 and CVE-2021-40180. 5 Discussion 5.1 Limitations and Future Works 1) We conducted a series of mini programs\u2019 permissions tests based on personal accounts. In other words, the tested APIs are all for individuals. According to our statistics, the number of APIs open to non-individual developers and certi\ufb01ed mini programs is 4.6% of the total API numbers. This type of interface usually includes obtaining the user\u2019s mobile phone number, sports data, etc. Different host apps have different attitudes towards such interfaces. For example, the WeChat mini programs allow individual developers to obtain user sports data, while in the Alipay mini programs, this interface is only open to corporate users. Generally, those interfaces that are only open to enterprise users will be more strictly managed by host apps since these can obtain more data. It is very dif\ufb01cult for us to test the APIs used by the enterprise accounts. In the future, we will cooperate with enterprises to test and research these APIs. 2) In the 4.3, we take the permissions of the clipboard in the WeChat mini programs as an example to test it in the mini programs that are currently online. The percentage of detected problems is not large. However, it does not mean that the vulnerabilities we put forward do not appear in mini programs in other host apps. As more global companies join the mini programs related discussion, the number of emerging host apps supporting mini programs (such as UnionPay, NetEase) increases. They just appeared, and the vulnerability proposed in this article may not exist. When host apps gradually add new mini programs\u2019 functions or APIs, they may also make the vulnerabilities we found in this article, so the host apps still need to pay more attention to the permissions in the future. 15 \fRunning Title for Header 5.2 Mitigation Measures The purpose of our empirical analysis was to draw people\u2019s attention to the neglected security issue of the improper use of sensitive permissions in mini programs. In the face of the rapid development of mini programs and the sharp rise in the demand for personal privacy protection of mini programs, it is necessary to strengthen the coordination of host apps, apps vendors, and users to form a management system for personal privacy protection of mini programs. The responsibility of host apps is the biggest. In essence, mini programs are still software that provides various services for mobile users. Mini programs are put on shelves through the host apps, and host apps should strictly control the management of permissions, actively rectify APIs related to users\u2019 sensitive permissions, and protect user privacy with the most signi\ufb01cant effort. For the apps vendors, it is necessary to consider how the existing OS handles speci\ufb01c APIs and the results feedback. For example, face the clipboard issues, the developer can highlight any word in the app and select \u201cCopy\u201d or automatically clear the clipboard after a while. Users also need to enhance the security awareness to protect personal information when using mini programs, be vigilant against mini programs of unknown origin, and do not quickly authorize their private information to the mini programs to avoid being illegally collected and leaked, causing unnecessary losses. At the same time, users should be encouraged to report illegal activities actively. 6 Related Work Reason Exploration. Almomani et al.[30] demonstrated, discussed and compared the latest technologies in the \ufb01eld of Android permissions, and conducted the latest research on Android permissions, revealing that the Android permissions faces various security issues. Zheran Fang et al. [31] investigate the arising issues in Android security, including coarse granularity of permissions, incompetent permission administration, insuf\ufb01cient permission documentation, over-claim of permissions, permission escalation attack, and TOCTOU (Time of Check to Time of Use) attack and put forward several methods to further reduce Android security risks. Joel Reardon et al.[13] searched for sensitive data being sent over the network for which the sending app did not have permissions to access it by mechanisms to monitor the application\u2019s runtime behavior and network traf\ufb01c. They found that apps can circumvent the permission model and gain access to protected data without user consent by using both covert and side channels and determined how this unauthorized access occurs. Mujahid [32] implement a technique in a tool called PERMLYZER, which automatically detects permission issues from apps APK. Detection and Protection. In the Android, the most common is the permission management mechanism of Android[33, 34, 35, 36]. DroidNet[12] is an Android permission control and recommendation system, which is an Android permission control framework based on crowdsourcing. It provides recommendations on whether to accept or reject the permission requests based on decisions from peer expert users, which can help users implement low-risk resource access control for untrusted applications and protect users\u2019 privacy. HybridGuard[37], a framework based on the subject authority and \ufb01ne-grained policy execution for web mobile applications, can accurately monitor all web codes to ensure the security of mobile applications, in which an interception and policy code is implemented in a single JavaScript \ufb01le, and whether to intercept them is determined by wrapping API about device resource access and DOM operation and checking the policy. M-Perm[11] is a detection tool that combines string analysis and static analysis to identify normal, dangerous, and third-party permission requests in applications to detect permission abuse. Cusper[38] is a new modular design in the Android permission model, which separates the management of system permissions from custom permissions declared by untrusted third-party applications. It introduces backwards compatible naming conventions for custom permissions to systematically eliminate and prevent the loopholes of custom permissions. The mainstream approach for enhancing the Android permission mechanism is to identify over-declared permissions requested by an app [39, 40, 41, 42], and recommend appropriate permissions for an app [43, 44]. TERMINATOR [16] provides a safe, reliable, yet non-disruptive approach to protect mobile users against permission misuses.Bin Liu et al.[14] proposed a Personalized Privacy Assistant (PPA) for mobile applications, which can manage mobile permissions of mobile applications and predict the privacy settings that users want by asking some questions, and proposed a method to learn the privacy pro\ufb01le of permission settings. Bao [45] also propose two novel approaches to realize permission recommendation. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2205.15170v1",
+                    "title": "GAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework",
+                    "abstract": "Using generative adversarial network (GAN)\\cite{RN90} for data enhancement of\nmedical images is significantly helpful for many computer-aided diagnosis (CAD)\ntasks. A new attack called CT-GAN has emerged. It can inject or remove lung\ncancer lesions to CT scans. Because the tampering region may even account for\nless than 1\\% of the original image, even state-of-the-art methods are\nchallenging to detect the traces of such tampering.\n  This paper proposes a cascade framework to detect GAN-based medical image\nsmall region forgery like CT-GAN. In the local detection stage, we train the\ndetector network with small sub-images so that interference information in\nauthentic regions will not affect the detector. We use depthwise separable\nconvolution and residual to prevent the detector from over-fitting and enhance\nthe ability to find forged regions through the attention mechanism. The\ndetection results of all sub-images in the same image will be combined into a\nheatmap. In the global classification stage, using gray level co-occurrence\nmatrix (GLCM) can better extract features of the heatmap. Because the shape and\nsize of the tampered area are uncertain, we train PCA and SVM methods for\nclassification. Our method can classify whether a CT image has been tampered\nand locate the tampered position. Sufficient experiments show that our method\ncan achieve excellent performance.",
+                    "authors": "Jianyi Zhang, Xuanxi Huang, Yaqi Liu, Yuyang Han, Zixiao Xiang",
+                    "published": "2022-05-30",
+                    "updated": "2022-05-30",
+                    "primary_cat": "eess.IV",
+                    "cats": [
+                        "eess.IV",
+                        "cs.CR",
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Due to the privacy of medical images, the lack of data has always been a signi\ufb01cant problem for machine learning tasks related to medical images. One way to solve this problem is the GAN [1], which can generate images that are highly similar to real images. GAN has been widely concerned in the medical image \ufb01eld. Several studies have used GAN to generate medical images for data enhancement and achieved gratifying performance. The image quality generated by GAN is enough to confuse radiologists. Therefore, once this technology is used for malicious attacks, it will lead to serious consequences. Using the deep convolution neural network can detect the GAN-generated image [3\u20135]. Moreover, the detection accuracy can be improved through feature engineering [6\u201311]. To the knowledge of this paper, there is no detection method for GAN forged medical images. Although there is no speci\ufb01c solution to detect the medical images generated by GAN, there are some domain generic methods. For example, Frank et al. used discrete cosine transform (DCT) to detect GAN generated images [10]. Marra et al., through incremental learning, can detect new GAN-generated images with only a small number of samples [12]. Cozzolino et al. learn feature extraction through auto-encoders and generalize the model through a small number of samples [13]. CT-GAN [2], a GAN dif\ufb01cult to detect even with state-of-the-art methods, emerged. It can inject or remove large lung nodules from CT images. Examples of CT-GAN inject/remove tampering of lung nodules are shown in Fig. 2. The number of large lung nodules is a signi\ufb01cant marker of lung cancer. Therefore, CT-GAN can make doctors misjudge \u2217Citation: Authors. Title. Pages.... DOI:000000/11111. arXiv:2205.15170v1  [eess.IV]  30 May 2022 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework CT slice image Sub-images Detector network Combine the  detect results GLCM \u00d74 Flatten PCA SVM Fake or not? Step 1: Local detection Step 2: Global classification Heatmap  Features(256) Features(40,000) r1 \u22ef \ud835\udc5f \ud835\udc64 \u22ee \u22f1 \u22ee \ud835\udc5f\u210e \u22ef \ud835\udc5f\ud835\udc64\u00d7\u210e Figure 1: Overview of our method. We cut out small sub-images from CT slices to train the local detection neural network. Each sub-image will be detected and output a tampered probability. The detection results were combined according to the position to generate a heatmap. Then we use GLCM to extract the features from the heatmap, which are used for PCA and SVM model training. We use the trained model for global classi\ufb01cation. the patient\u2019s condition, seriously threatening the patient\u2019s life safety. In addition, this attack may also be used to defraud medical insurance and maliciously discredit competitors. origin tampered Figure 2: CT-GAN tampered samples. The \ufb01rst row shows the removal tampering of CT-GAN. A lung nodule is removed from the CT slice image by CT-GAN. The second row shows the injection tampering of CT-GAN. A small nodule was tampered with as a large nodule by CT-GAN. However, new attacks like CT-GAN challenge the current detection methods. CT-GAN generates only a minimal area, and the surrounding area is used as a constraint condition to train a conditional generative adversarial network (CGAN). In that case, the generated image will be closer to the real image. We call the attack using CGAN to forge a very small region in an image as GAN-based small region forgery attack. At present, there is no solution that can effectively detect the GAN-based small region forgery attack in medical image. The characteristic of the attack is that the ratio of the region generated by GAN is very small. Although some methods, such as R\u00f6ssler et al. [4], can detect partial generation, such as face manipulation. However, because medical images\u2019 style, content, and storage format are very 2 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework different from the normal images and the tampered region is too small, even state-of-the-art detection methods can not effectively detect GAN-based small region forgery attacks in medical image. It is conceivable that our medical image security is facing a considerable threat. In order to solve above-mentioned problem, we propose a novel cascade framework based on a local detection network and a global classi\ufb01cation method that can detect GAN-based small region forgery attacks in medical images. The \ufb01rst stage is local detection. We crop a small sub-image from the CT slice image to train the detector network. The sub-image size is small enough so that interference information in authentic regions will not affect the detector. Because the training data only has a single channel and the training data size is small, it is easy to over-\ufb01t. Therefore, we design a lightweight neural network with fewer parameters and use early stopping to prevent over-\ufb01tting. After training, the detector can detect the tampered region effectively. Then, we traverse the entire CT slice image by sub-image. The detection result of all sub-image will be combined and output as a heatmap. It can indicate which area may be tampered. The second stage is global classi\ufb01cation. We use the gray level co-occurrence matrix (GLCM) to extract features from the heatmap. Then train the PCA and SVM model with the GLCM features. Since CT-GAN can adjust the size of the tampered area to a certain extent, we use GLCM features to train PCA and SVM models for global classi\ufb01cation. Compared with the method that uses the whole image as input, this method can locate the tampered coordinate and requires less training data but has faster training speed and higher accuracy. The main contributions are as follows: \u2022 A novel cascade framework based on local detection and a global classi\ufb01cation is proposed to detect and locate the tampering regions caused by CT-GAN attacks including both injection and removal. \u2022 A local detection network with channel attention, spatial attention, depthwise separable convolution, and residual, which can better \ufb01nd the information of small areas in the image and prevent over-\ufb01tting. \u2022 We design a global classi\ufb01cation method base on PCA and SVM with the gray level co-occurrence matrix (GLCM) as input features, which can effectively cooperate with local detection to classify medical images. \u2022 Experiments show that for GAN-based small region forgery attacks in medical image like CT-GAN, our method can achieve excellent performance. The rest of this paper is organized as follows. In Section 2, we discussed the background and related work of the detection of GAN generated images in recent years. Moreover, in Section 3, we explained our method in detail. Furthermore, in Section 4, we describe our experimental results. In Section 5, we discuss our method, and in Section 6 we draw our conclusions. 2 Background and related works 2.1 Medical image The medical image uses some particular medium to interact with the human body to show the structure of the internal tissues or organs of the human. Digital imaging and communications in medicine (DICOM) is an international standard for medical images and their related information. It is widely used in various radiological diagnostic equipment (X-ray, CT, MR, ultrasound, etc.). All medical images of patients are stored in DICOM \ufb01le format. The data used in this paper are mainly CT images with DICOM format. CT equipment scans slices one after another around a certain part of the patient\u2019s body. Fig. 3 shows an example of a CT scan. The scanned image is multi-layered. A three-dimensional image can be formed by stacking layers of slice images on the z-axis. The de\ufb01nition of medical images such as CT is positively correlated with radiation dose. In contrast, high-dose radiation may damage patient\u2019s health, so it is dif\ufb01cult to improve the de\ufb01nition of medical images. Besides, medical images have only one channel. So GAN can easily \ufb01t the distribution of medical images than normal three-channel color images. 2.2 Generative adversarial network Since GAN was proposed by Goodfellow et al., it has been one of the hot spots in the CV \ufb01eld. The GAN model is different from the traditional neural network structure. GAN includes a generative model G and a discriminative model D. G generates a new sample from random noise, and D distinguishes whether the input sample is a real sample. The task of G is to generate images that D cannot distinguish. At the same time, the task of D is to distinguish between the images generated by G and the real images. The two networks compete against each other during training through this min-max game. In this way, G can learn the data distribution of the real sample. Up to now, GAN has derived a large number of variants, such as WGAN [14], PGGAN [15], StyleGAN [16] and so on. These variants are widely used in various CV tasks. 3 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework (a) (b) Figure 3: CT scan schematic. (a) is the cartoon schematic of CT scanning operation. (b) is a part of a CT scan. A complete CT scan may include about 300 slice images, and only a portion is selected here. 2.3 Application of GAN in medical image Medical images are different from normal images and have robust privacy. Even though there are many public data sets such as LIDC-IDRI, DDSM MIAS, OASIS, etc., the medical data sets are still insuf\ufb01cient. Because GAN can effectively alleviate the lack of training data, there are also a large number of researches of medical imaging using GAN. In recent years, the more frequently used GAN variants in medical imaging are pix2pix [17] and CycleGAN [18]. GAN is widely used in image synthesis [19\u201323], noise reduction [24\u201327], cross-modality [28\u201333], and many other aspects, providing signi\ufb01cant help for CAD. 2.4 Detect the GAN-generated image Because of the high performance of GAN, it has gradually become a trend to use deep learning to distinguish whether an image is generated by GAN. Due to the excellent performance of convolutional neural networks (CNN) in CV tasks, CNNs, such as ResNet [34], XceptionNet [35], and Ef\ufb01cientNet [36], are widely used in various CV \ufb01elds, including digital image forensics [3,5,7]. Besides, Andreas et al. [4] prove the superior performance of XceptionNet in image source detection. Using some features can make the network perform better. A way to distinguish whether an image is generated by GAN is to use GAN \ufb01ngerprint. [8]. GAN will leave special \ufb01ngerprints in the generated image due to its structure. Through deep learning, learn those \ufb01ngerprints as a feature. Then it can be used to distinguish the source of the image. Some people use the shortcomings of GAN to \ufb01nd some special features to better distinguish whether a image is generated by GAN. For example, McCloskey and Albright \ufb01nd that the saturated or underexposed pixels of image will be suppressed by the normalization operation of the GAN generator [6], and use this feature to distinguish the real camera images and GAN images. Because the statistical characteristics of GAN images are different from real images, some people use three co-occurrence matrices on RGB channels as features to distinguish the source of the image [9]. Zhang et al. suppress the image content information by converting the image to the YCrCb color space and then use the Scharr operator and the gray-level co-occurrence matrix(GLCM) to obtain edge features, allowing them to simultaneously detect GAN images and copy-move images [7]. In addition, someone distinguishes the source of the image from the defects of up-sampling operations in GAN. Frank et al. found that the up-sampling in GAN will cause grid-like artifacts in the generated images after DCT operation [10], which can be used to distinguish the source of the image. Durall et al. found that the images generated by GAN cannot reproduce the actual spectral distribution [11], which is also due to the upsampling operation. Therefore, after using azimuthal integration to extract the spectral features, using SVM or K-Means can distinguish the source of the image without the need to train a deep CNN. 2.5 Challenge The CT-GAN paper also proposed some detection methods that may be useful. Unfortunately, these methods are not suitable for GAN-based small region forgery attack in medical image like CT-GAN. The reasons are as follows. 4 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework On the one hand, there is a huge difference between medical images and normal images. Medical images show the structure and density of human internal tissues or organs, so they have unique content and style. Medical images, such as CT, MR, X-ray, etc., are all taken with special equipment different from general photographing equipment and are saved according to the DICOM standard. Medical images are all single-channel in terms of the image data format, and the pixel values range of the medical image is about 4096. Compared with the normal gray-scale image ranging from 0 to 255, the range of pixel values of medical images is 16 times larger. Therefore, the pre-training model of normal images only has little effect on medical images. In addition, methods that need to extract features from three channels of an image, such as [7] that needs to compare three features extract from different channels, are not appropriate. The co-occurrence matrix is one of the most effective features to distinguish whether an image is generated by GAN. However, due to the expansion of the pixel value range, the cost of calculating the co-occurrence matrix will increase to unacceptable. So the method using the co-occurrence matrix [9] cannot work too. (a) (b) Figure 4: (a) is a tampered CT image. The nodules in the red box are injected by CT-GAN. (b) is the heatmap corresponding to (a), in which the bright red spot corresponds to the injected nodule. Because the preset sliding window stride is greater than 1, the size of the heatmap is smaller than the original CT slice image. In order to facilitate observation, we enlarged the thermal map, superimposed it on the CT image, and adjusted their colors. On the other hand, GAN-based small region forgery attacks are more dif\ufb01cult to detect. Take CT-GAN as an example. CT-GAN is a 3D CGAN. It designs a 3D network that references the pix2pix structure. The generator of CT-GAN is a 3D UNet [37] structure. It cuts out a small cuboid from a series of CT slices of the patient, then scales it into a small cube of 323 pixels and masks the 163 pixels in the center of the cube to zero. This cube with a masked center is input into the generator as a condition. CT-GAN trained two models. Those models can generate large or small nodules in the cube\u2019s center. It is worth mentioning that the size of a CT image is (512 \u00d7 512) pixels, in which the region modi\ufb01ed by CT-GAN is less than (32 \u00d7 32). That means the minimum number of pixels that have been tampered with is only 1/1024 of the total. Fig. 4 shows a CT image injected into a lung nodule. Hence, as we can see from Fig. 2 and Fig. 4, Each tampering operation by CT-GAN will modify 1/1024 to 1/256 of the pixels of the image. A CT image has been tampered with at four different locations, and the tampered pixels only account for about 1%. Generally, a CT image does not need to be operated so frequently. In other words, 99% of an image is interference information. The untampered part is equivalent to the \"cover\" of the tampered part, which seriously hinders the model from learning the difference between positive and negative images. That is why even state-of-the-art methods are challenging to detect the tampering trace of CT-GAN. Because of this, many methods based on statistical characteristics, such as [6,10,11], are ineffective. More serious is that CNN is not sensitive enough to small tampered regions, so it is dif\ufb01cult to detect such attacks accurately. Unfortunately, almost all current detection methods are based on deep CNN, so it is challenging to detect directly. The current methods for distinguishing whether an image is generated by GAN aim at the images wholly or mostly generated by GAN. There is no special detection model for GAN-based small region forgery attack in medical image like CT-GAN for the time being. In Section 4.4, We have tried to use the whole CT image as input to train the state-of-the-art network. Unfortunately, the result is inferior. 5 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework 3 Our method 3.1 Motivation Medical images are critical private information and are vitally important to the patient\u2019s life. At present, the integrity of medical images faces the threat of GAN-based small region forgery attacks. However, there is no practical method to detect GAN-based small region forgery attacks in medical images. There are two main reasons why attacks like CT-GAN are dif\ufb01cult to detect. First, medical images\u2019 style, content, and storage format are very different from the normal images. If we convert these DICOM images to any other image format, it will lose information like pixels or meta-data. Therefore, the model training by normal image can not be well generalized to CT images, making the pre-training model unusable. Training a new model will need much more data. Unfortunately, the sample data of medical images is very limited due to the restrictions on the use of data concerning health under the privacy regulations like GDPR or CCPA. Hence, we cannot try to solve this attack from the perspective of training data. Moreover, what makes the detection task more challenging is that the tampered region is very small while the entire CT image is large. As mentioned above, the tampered region is less than (32 \u00d7 32) when the entire CT is (512 \u00d7 512). This means the ratio of a single tampered region in the original image may be less than 0.4%. This greatly reduces the sensitivity of general CNN detection methods since the loss of spatial information limits the learning ability of CNN. Hence, directly detecting the whole image will result in very low accuracy. Based on the above, even state-of-the-art methods are challenging to detect CT-GAN attacks. Although no speci\ufb01c method can be implemented directly to detect the GAN-based small region forgery attack in medical images, some research works can still inspire us to design an effective method. Andreas et al. [4] used a face tracking method to extract the face area of the image. They found that if the extracted facial information is used as the input of the detector, it will be more accurate than directly using the entire image as input. It means that the neural network can achieve a better performance if the classi\ufb01er focuses on more precise regions. Following their idea, we refer to the common preprocessing method of copy-move forgery detection, making the detector pay more attention to the local part of the image through a sliding window. Speci\ufb01cally, we split the target CT image into many small sub-images to train a local detector with a corresponding method for using local classi\ufb01cation results to determine global classi\ufb01cation results. Christian et al. [35] replaced the Inception modules with depthwise separable convolutions and proposed their method named XceptionNet for computer vision. Since XceptionNet makes more ef\ufb01cient use of model parameters, compared to Inception V3 [38], it shows better runtime performance and higher accuracy on large-scale datasets like ImageNet while having fewer parameters than general deep CNN. This architecture can effectively reduce over\ufb01tting when we cannot collect more data. Therefore, considering these features, we designed our method inspired by the XceptionNet to detect the GAN-based local tampering attacks. 3.2 Overview Our detection method is mainly divided into two stages: local detection and global judgment. The method we propose is outlined below. In the local detection stage, small sub-images are cut out from CT slices in a planned way to train the local detector neural network. The size of the cropped sub-image is small enough for the tampered area. In this way, the authentic regions are not enough to hinder the detector. The detector can focus on learning the difference between the real image and the GAN-generated image. The tampered area may be hidden in the original image as the background when testing. Therefore, to minimize the missed judgment, our method detects each sub-image divided by the sliding window and predicts the tampered probability of each sub-image. When all sub-images were detected, the results were combined according to the position to generate a heatmap. This heatmap can intuitively re\ufb02ect which area in the original image may have been tampered with by GAN. In the global classi\ufb01cation stage, we use GLCM to extract the features from the heatmap, which are used for PCA and SVM model training. GLCM can make the features of the heatmap more prominent. We use the trained model for global classi\ufb01cation. Intuitively, our method allows the neural network to observe the details of the image more carefully instead of looking at the overall situation. Thus it has a better performance when facing GAN-based small region forgery attack. 3.3 Local detection network architecture Because our training data is insuf\ufb01cient and the hardware is not powerful, we tend to use lightweight networks as the local detector. Using depthwise separable convolution can reduce a large number of required training parameters while maintaining a good training effect. For example, XceptionNet [35] and MobileNet [39] both construct the primary 6 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework Input  Conv 32 3\u00d73 Conv 128 3\u00d73 SeparableConv 256 3\u00d73 MaxPooling 3\u00d73 Strdie2 Spatial Attention MaxPooling 3\u00d73 Strdie2 SeparableConv 512 3\u00d73 SeparableConv 1024 3\u00d73 SeparableConv 1024 3\u00d73 SeparableConv 1024 3\u00d73 SeparableConv 1024 3\u00d73 SeparableConv 1024 3\u00d73 SeparableConv 1024 3\u00d73 Global Average Pooling FC 2048 SELU Output Residual Conv 256 1\u00d71  Strdie 2 Residual Conv 512 1\u00d71 Strdie 2 Residual Conv 1024 1\u00d71 Strdie 2 Residual Conv 1024 1\u00d71 Stride 2 FC 2 Softmax Channel Attention Channel Attention B1 B2 B3 B4 B5 Figure 5: The network architecture. If there is no description, the default stride of the convolution operation is 1, the padding operation defaults to \"SAME\", the activation function defaults to Relu, and each convolution and depthwise separable convolution layer are followed by batch normalization by default. part of the network with depthwise separable convolution, and they perform well in image classi\ufb01cation. But our classi\ufb01cation task does not need the too deep network. Because the training data structure is too simple, using a network like XceptionNet or MobileNet will waste many computing resources and may lead to network degradation or over\ufb01tting. Therefore, we designed a shallower network as our sub-image classi\ufb01er based on the depthwise separable convolution. Our network structure is shown in Fig. 5. The network\u2019s input is a (32 \u00d7 32) image matrix, and the features of the image are extracted through a small number of traditional convolutions and a large amount of depthwise separable convolutions. The attention mechanism can effectively improve the performance of the deep learning model. The attention mechanism is often used by copy-move detection and other detail-oriented tamper detection methods. Inspired by Woo et al. [40], we designs a simple attention mechanism for our network. Our attention mechanism is shown in Fig. 6. The spatial attention and channel attention can be computed by Eq. (1) and (2). This network sets the channel attention module after the convolution blocks with the largest number of channels. It is more signi\ufb01cant to use channel attention here. Similarly, because the pooling layer will further reduce the size of the feature image, we set the spatial attention module before the convolution block containing the pooling operation, where the feature image size is the largest. After adding the attention module, the training cost does not increase much, but it can signi\ufb01cantly improve the global classi\ufb01cation performance. As(F F F) = \u03c3(Conv([MAXs(F F F); AV Gs(F F F)])) (1) 7 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework Input feature MaxPool 7*7 Convolution AveragePool AveragePool MaxPool Share FC layer Sigmoid according to  spatial position according to  channel Output  Spatial Attention Channel Attention \ud835\udf0e Figure 6: The attention module of our network. The number of nerve cells in the three FC layers of channel attention is C, C/4, C, where C means the number of channels. The stride of convolution is 1, using the SAME Padding, and the activation function is Sigmoid. Ac(F F F) = \u03c3(FC(MAXc(F F F)) + FC(AV Gc(F F F))) (2) The design of residual block refers to ResNet [34]. The number of convolution kernels channels is unchanged when the input feature image size is the same as the output. The number of convolution kernels and channels is doubled when the input feature image size is different from the output (through the pooling layer). At the end of our network. Selu [41] is used as the activation function in the full connection layer of the network. The Selu function is given by Eq. (3), where \u03bb and \u03b1 are two meticulously designed numbers. It has better performance than Relu in the full connection layer. Our network can save computing resources through the above network structure while maintaining high accuracy. Selu(x) = \u03bb \u001a x (x > 0) \u03b1ex \u2212\u03b1 (x \u22640) (3) 3.4 Global classi\ufb01cation method In our method, the window size is \ufb01xed. This is slightly different from the sliding window commonly used in target detection tasks. In the task of target detection, if the window only covers a part of the target, the model may not be able to classify the target correctly, so it is necessary to adjust the window size and traverse the image multiple times. However, in our task, even if the window is only a part of the GAN generation region, the model can determine whether it is generated by GAN with high accuracy. Therefore, we only need a smaller window size to avoid the in\ufb02uence of authentic regions. First of all, calculate a series of coordinates as the center coordinates of sub-images. Then, crop a sub-image with a size of (32 \u00d7 32) according to these center coordinates. The crop operation is because the practical part of the CT image is only the inside of a circle tangent to the square frame. Furthermore, most of the tampering occurs in this area. It will waste a lot of time and space if the extra part is included in the calculation. The size of the CT image is marked as CTsize, the sub-image size is marked as imgsize, and the stride marked as s. We calculate the longitudinal coordinates of all rows as follow: y = {imgsize 2 + i \u00d7 s} (4) Where i = 0, 1, 2, ..., \u230aCTsize \u2212imgsize s \u230b (5) Then, for a row with y = h, we calculate the horizontal ordinates as follow: x = {ctsize/2 + j \u00d7 s} (6) 8 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework Where j = 0, 1, 2, ..., \u230aw\u2212imgsize 2s \u230b w = \u230a p (ctsize/2)2 \u2212h2\u230b (7) Our method can record the output result of each sub-image. Then classify whether an image has been tampered with according to these results. In addition to output a \ufb01nal prediction result, our model can also generate a heatmap based on the results of each sub-image (see Fig. 4). For the area not counted by the formulas above, we default it to have tampered with the probability of 0. Generally speaking, attackers usually set the tampered region square or round for GAN-based small region forgery attacks. Therefore, after local detection of the tampered slice, a series of sub-images will be judged as positive in theory. Therefore, the tampering trace of CT-GAN can be detected through some \ufb01xed modes, such as the vertical and horizontal continuous n sub-images are judged to be false. However, this detection method is not perfect. The size of the tampered area of GAN is not \ufb01xed. The attacker can set the size of the CT-GAN tampered area to a certain extent. Moreover, many studies use GAN to generate more extensive and higher resolution images. Therefore, we need a \ufb02exible global classi\ufb01cation method. Firstly, to make the features more prominent, GLCM is used to extract the texture features in the thermal map. After rounding the local detection results (heatmap) \u00d7 100, calculate the GLCM with a distance of 1 at four angles of 0 \u00b0, 45 \u00b0, 90 \u00b0 and 135 \u00b0, respectively, then get the feature matrix of (100 \u00d7 100 \u00d7 4). Secondly, a PCA model is trained to reduce the feature to 256 dimensions. Thirdly, the feature data after dimensionality reduction are used to train an SVM model, and the best parameters of the model are found by grid search. Finally, we use the trained PCA + SVM to make the global classi\ufb01cation. This method can adapt to different GAN tampering area sizes. Algorithm 1 Generate the GLCM of heatmap Input: heatmap\u2014The heatmap matrix with size H \u00d7 W. a, b\u2014Two constants determined by angle and distance. Output: GLCM\u2014A matrix with size g \u00d7 g, g is the gray levels number. heatmap = heatmap \u00d7 100, then round heatmap to integer, Initialize GLCM to 0 matrix x = 0, y = 0 while x < W do while y < H do if 0 < (x + a) < W and 0 < (y + b) < W then g1 = heatmap(x, y) g2 = heatmap(x + a, y + b) GLCM[g1, g2] = GLCM[g1, g2] + 1 end if end while end while 4 Experiments 4.1 Implementation Details 1) Data set We use the source code of CT-GAN, train the inject and remove models with the LUNA16 data set [42], and use the trained models to generate 3540 different CT scan samples. Among them, 1776 scans were injected lung cancer lesions (Equivalent to a large-diameter lung nodule), and 1764 scans were removed lung cancer lesions. For each fake sample, we select the tampered point and two slices before and after it, \ufb01ve CT slices, and the corresponding \ufb01ve slices before tampering. In the end, 35400 CT slice images were obtained. The tampering points are the CT slices with lung nodules. Therefore, we randomly selected about half of the real CT slice images (about 8850) and replaced them with slices at random locations. Among the 35400 CT slice images, 1200 images are randomly selected as the test set, 4800 images are randomly selected as the training set of global classi\ufb01cation, 2000 images are randomly selected as the veri\ufb01cation set of local detection, and the remaining 27400 slices are used as the training set of local detection. We mark the test set described in the previous paragraph as the test set CTGAN-ALL. Besides, we divide the test set CTGAN-ALL into two parts according to inject or remove tampering. The large nodule injected CT slice images, and the real large nodule images were marked as CTGAN-INJ. The large nodule removed CT images, and the real small nodule images were marked as CTGAN-REM. 9 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework In addition, eight CT scans different from the above data sets were retained. Two of them were real lung CT scans. One of them had malignant lung cancer lesions, and the other did not. These two scans were marked as MAL and BEN. In addition, one, two, and three large nodules were injected into three scans, respectively. The three scans were marked as INJ1, INJ2, and INJ3. Similarly, one, two, and three large nodules were removed from the remaining three scans marked as REM1, REM2, and REM3. Furthermore, we add impulse noise and gaussian noise to 5000 CT slice images, then use CycleGAN to reduce noise. In the end, 5000 images modi\ufb01ed by CycleGAN were obtained. The images without noise and the image denoised by CycleGAN, these 10,000 slice images are marked as the data set CycleGAN. We mark the images denoised by CycleGAN as the positive class and the images without noise as the negative class. Among them, 8000 images are used as the training set, 1000 images are used as the veri\ufb01cation set, and 1000 images are used as the test set. For each slice image in the test set, we use a (32 \u00d7 32) window to traverse the whole CT image (with the size of (512 \u00d7 512)) with 4 pixels stride. Our method use Eq. (4) and (6) to traverse the image. For each slice image of the training set and the cross-validation set, we use the method of shifting by one pixel for data enhancement so that each slice image in the training set can generate 25 sub-image images. For the fake image (positive class), we mark the coordinates of the injection center point as (0,0), take 25 coordinate points in the rectangle from (-2,-2) to (2,2). Then use these coordinate points as the center point, cut out 25 sub-images with the size of (32 \u00d7 32). For the real image (negative class), we take 10 coordinate points in the rectangle from (-2,-2) to (-1,2), and then randomly select 20 different coordinates from the coordinates calculated by Eq. (4) and (6). Taking these coordinate points as the center, and cut out 25 sub-images with the size of (32 \u00d7 32). By adding n negative samples corresponding to the positive samples, the model can better learn the difference between the images before and after tampering. We found that when n = 10, the model\u2019s performance is better. 2) Setup All experiments were implemented using the Tensor\ufb02ow 1.13 framework and were trained on a single NVIDIA GTX2080TI GPU. The parameters of the training phase are as follows. We set the initial learning rate to 0.0005 and use exponential decay, which decays every 600 steps and with a decay rate of 0.85. The mini-batch size is 56, the batch normalization decay parameter is 0.95, and the L2 regularization weight decay parameter is 0.0001. We use Adam optimizer to minimize cross-entropy loss. Except for the learning rate, the default parameters of the Adam optimizer are used, namely \u03b21 = 0.9, \u03b22 = 0.999, \u03f5 = 1 \u00d7 10\u22128. The early stopping is set to stop training when the accuracy of the validation set no longer increases for three consecutive epochs. If the early stop is not triggered, the training will stop after 30 epochs. 3) Evaluation We regard the tampered slice image as a positive example and the real slice image as a negative example. The number of positive and negative samples in the actual scene may differ. Therefore, in addition to accuracy(ACC), we also use precision(P), recall(R) and F1-score(F1) to evaluate the model\u2019s performance. The tampering operation of CT-GAN is aimed at 3D medical images. The number of slices involved in a tampering operation can easily reach more than 30. Besides, the number of tampered slices is more if the slice interval is small. However, if the same area of 10 consecutive slices is predicted as the tampering area, we can judge that this position has been tampered with easily. However, if the above indicators are calculated in the unit of 2D slice image, the value will be de\ufb01cient, which is unreasonable. Therefore, when detecting the complete CT scan, this paper also takes the 3D tampered area (a series of slice images) as the unit and counts the indicators in the following way. For a tampered area, when 9 or more of the 10 consecutive slices, which including the tampered central slice (these slices must be positive examples in this experiment), are judged as positive examples, we consider that the tampering trace is accurately found and marked as a true positive example. Otherwise, it is regarded as a missing report and marked as a false negative example. Similarly, for a real area, suppose 9 or more consecutive slices in the real slice are judged as positive examples, or 9 of the 10 consecutive slices are judged as positive examples. In that case, it will be regarded as false positives. Finally, the precision, recall and F1-score are calculated in the above way. 4.2 Ablation Study In order to verify the effectiveness of each module in our method, we conducted ablation studies. Four experiments were used to verify the effectiveness of local detection, attention mechanism, Selu activation function, and GLCM feature extraction. In each experiment, we ablate a module from our method. In these experiments, ablate local detection is to input the complete slice image (512 \u00d7 512) and use our network to train and predict directly. The experimental results are shown in Table 1. 10 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework Table 1: The Ablation Study Result of Our Method. \"-SW\" means that sliding windows are not used, the whole image is classi\ufb01ed directly without local detection. \"-Attention\" means that attention mechanism are not used. \"-Selu\" means that uses Relu instead of Selu in our network. \"-GLCM\" means that PCA and SVM are directly used to classify the heatmap without GLCM to extract features. Ablated module ACC P R F1 Ours-SW 0.6583 0.6554 0.6660 0.6607 Ours-Attention 0.9158 0.9561 0.8717 0.9119 Ours-Selu 0.9192 0.9499 0.8850 0.9163 Ours-GLCM 0.8717 0.8729 0.8700 0.8715 Ours 0.9350 0.9350 0.9350 0.9628 0.9628 0.9628 0.9050 0.9050 0.9050 0.9330 0.9330 0.9330 The experimental results show that using the sliding window to divide sub-images for local detection is very helpful to detect GAN-based small region forgery attacks like CT-GAN, which signi\ufb01cantly improves the performance of detection. Selu activation function and attention mechanism can be slightly helpful to the performance of our method. Moreover, the performance of our method can be signi\ufb01cantly improved by using GLCM. When the above modules are used together, the improvement effect is better. 4.3 Detection of CT-GAN inject or remove attack For general GAN-based small region forgery attack, it may not use the same GAN structure to train two different models like CT-GAN. Therefore, we divided the training set into two parts according to the same way as the test set CTGAN-INJ and CTGAN-REM, then trained detector models and tested the inject and remove models of CT-GAN separately. After the training, we got two detectors for different tampering models. After that, we tested on the two kinds of tampering respectively. The test results are shown in Table 2. Table 2: The Detection Results of CT-GAN Inject and Remove Attacks. The training and testing of the two are carried out separately. Test set ACC P R F1 CTGAN-INJ 0.8999 0.9762 0.8200 0.8913 CTGAN-REM 0.9670 0.9937 0.9400 0.9661 The experimental results show that although training data is reduced, our model can still detect CT-GAN\u2019s inject or remove model with a high F1-score. The detection accuracy and f1-score of the inject tampering model are about 90%, while the detection accuracy and f1-score of the remove tampering model are about 97%. The above results show that our method can still effectively detect the traces of tampering in the face of a single tampering model, but our method is more sensitive to the traces of removing tampering. 4.4 Compare to state-of-the-art methods 1) Detect CT slices Because other feature extraction methods for detecting GAN-generated images are not suitable for CT-GAN, we use XceptionNet and ResNet50, both of which are the most advanced DCNNs, as the baseline. Some existing studies have shown that XceptionNet has an excellent performance in GAN forged image detection, and its detection accuracy can be comparable with the most advanced detection methods. Therefore, we choose Xceptionnet as the baseline. In addition, because both XceptionNet and the local detection network take depthwise separable convolution as the main structure, we also select resnet50 as another baseline. Moreover, inspired by [10] we also tried to use the DCT of the sub-image as a feature(Ours-DCT). This experiment is divided into two kinds. One is to use the sliding window, the network as the local detector to predict the (32 \u00d7 32) sub-image. The other is to train and test in a general way. The input of the network is (512 \u00d7 512) 11 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework 0 20 40 60 80 epoch 0.3 0.4 0.5 0.6 0.7 accuracy Xception ResNet50 Our-whole (a) 0 15000 30000 45000 iter 0.6 0.7 0.8 0.9 1.0 accuracy Xception ResNet50 Ours-DCT Ours (b) Figure 7: The training accuracy curve. (a) Input with whole image (512 \u00d7 512). (b)Input with sub-image (32 \u00d7 32). complete slice images. The training set and test set are as described in Section 3. The training information is shown in Fig. 7, and the test results are shown in Table 3. Table 3: The Detect Result of CT-GAN with the State-of-the-art Methods and Ours. Where \"-W\" means whole slice image input. \"-DCT\" means the local detection network is trained with the DCT features extracted from the sub-image. Method ACC P R F1 XceptionNet-W 0.5912 0.5738 0.7064 0.6332 XceptionNet 0.7125 0.7986 0.5683 0.6641 ResNet50-W 0.5600 0.5488 0.6690 0.6030 ResNet50 0.6850 0.7643 0.5350 0.6294 Ours-W 0.6583 0.6554 0.6660 0.6607 Ours-DCT 0.8383 0.8512 0.8200 0.8353 Ours 0.9350 0.9350 0.9350 0.9628 0.9628 0.9628 0.9050 0.9050 0.9050 0.9330 0.9330 0.9330 Experimental results show that when using the current data set, even the most advanced DCNNs such as XceptionNet and ResNet50, the test accuracy and F1-score are only about 65%, which means that it is dif\ufb01cult for them to distinguish CT-GAN tampered images and real images. Their performance is improved when used as local detectors, but the performance is still unsatisfactory, which may be due to over\ufb01tting and network degradation. Our method will seriously over\ufb01t without the sliding window. However, using the sliding window, the model converges faster, and the accuracy and f1-score of our method are increased to 93%, an increase by a percentage of 28. However, all indicators have declined when using DCT as a feature, and the accuracy and f1-score are only about 86%. 2) Detect CT scans In order to test the performance of our method more comprehensively, we compared our method with the latest method on complete CT scans. The model used is still trained under the mixed condition of injecting and removing. The test set is the eight scans mentioned above. Fig. 8 shows several continuous CT slice images near the tampering center point and the corresponding heatmap. Table 4 shows the results of our experiment. The experimental results show that our model can effectively \ufb01nd the traces of CT-GAN tampering and is more stable than other methods. Our method can determine whether a scan has been tampered with automatically through a simple strategy. For example, when any n of m consecutive images are classi\ufb01ed to be positive, it is considered that this scan has been tampered with by CT-GAN. Therefore, even CT scan is three-dimensional, while our model is two-dimensional, our model can effectively assist in distinguishing whether a CT scan has been tampered with. In addition, for CT scans with smaller slice spacing, such as INJ1, INJ3, REM1 and REM2, our method can detect more consecutive positive samples (more than 15). However, when the slice spacing is larger, the continuous positive samples that the model can detect are fewer. Furthermore, many misjudgments will occur in places that are unrelated to lung nodules, such as folds of clothes and calci\ufb01ed muscle tissue, which doctors can easily identify. 12 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework Table 4: The Detection Results of Complete CT Scans. The \"spacing\" means the spacing between two adjacent slice images. The \"2D\" means the indicators are calculated in the unit of a 2D slice image. The \"3D\" means the indicators are calculated in the unit of a 3D tampering area. There is no difference between the \"2D\" and \"3D\" methods in detection but in evaluation. Test set Spacing (mm) Method TP TN FP FN Accuracy Precision Recall F1-score BEN 2.5 Xception 0 162 16 0 0.9101 Resnet50 0 166 12 0 0.9326 Ours-2D 0 0 0 178 178 178 0 0 0 0 0 0 1.0 1.0 1.0 MAL 0.625 Xception 0 433 47 0 0.9101 Resnet50 0 413 67 0 0.9326 Ours-2D 0 0 0 479 479 479 1 1 1 0 0 0 0.9979 0.9979 0.9979 INJ1 1.0 Xception 23 161 30 53 0.6891 0.4340 0.3026 0.3566 Resnet50 12 176 15 64 0.7041 0.4444 0.1579 0.2330 Ours-2D 43 43 43 182 182 182 9 9 9 33 33 33 0.8427 0.8427 0.8427 0.8269 0.8269 0.8269 0.5658 0.5658 0.5658 0.6719 0.6719 0.6719 Ours-3D 1 0 0 1.0 1.0 1.0 INJ2 2.5 Xception 25 25 25 68 68 68 1 1 1 37 37 37 0.7099 0.7099 0.7099 0.9615 0.9615 0.9615 0.4032 0.4032 0.4032 0.5682 0.5682 0.5682 Resnet50 21 68 1 41 0.6794 0.9545 0.3387 0.50 Ours-2D 15 69 0 47 0.6412 1.0 0.2419 0.3896 Ours-3D 2 0 0 1.0 1.0 1.0 INJ3 0.625 Xception 39 290 14 137 0.6854 0.7358 0.2216 0.3406 Resnet50 32 291 13 144 0.6729 0.7111 0.1818 0.2896 Ours-2D 79 79 79 304 304 304 0 0 0 97 97 97 0.7979 0.7979 0.7979 1.0 1.0 1.0 0.4489 0.4489 0.4489 0.6196 0.6196 0.6196 Ours-3D 2 0 0 1.0 1.0 1.0 REM1 1.8 Xception 11 88 2 55 0.6346 0.8462 0.1667 0.2785 Resnet50 7 89 1 59 0.6154 0.8750 0.1061 0.1892 Ours-2D 42 42 42 90 90 90 0 0 0 24 24 24 0.8462 0.8462 0.8462 1.0 1.0 1.0 0.6364 0.6364 0.6364 0.7778 0.7778 0.7778 Ours-3D 1 0 0 1.0 1.0 1.0 REM2 1.8 Xception 41 88 0 64 0.6346 0.8462 0.1667 0.2785 Resnet50 30 86 2 75 0.6010 0.9375 0.2857 0.4380 Ours-2D 80 80 80 88 88 88 0 0 0 25 25 25 0.8705 0.8705 0.8705 1.0 1.0 1.0 0.7619 0.7619 0.7619 0.8649 0.8649 0.8649 Ours-3D 2 0 0 1.0 1.0 1.0 REM3 2.5 Xception 20 50 1 61 0.5303 0.9524 0.2469 0.3922 Resnet50 19 50 1 62 0.5227 0.950 0.2346 0.3762 Ours-2D 59 59 59 51 51 51 0 0 0 22 22 22 0.8333 0.8333 0.8333 1.0 1.0 1.0 0.7284 0.7284 0.7284 0.8429 0.8429 0.8429 Ours-3D 3 0 0 1.0 1.0 1.0 5 Discussion and limitations 1) The correlation of sub-images In our method, we directly detect each sub-image without considering the correlation of sub-images. Although the current experimental results meet the requirements of the detection task, we believe that introducing this correlation into the detection task will probably help improve the detection accuracy, ef\ufb01ciency, or generalization. Hence, in the future, we plan to conduct more extensive experiments to \ufb01nd the correlation between these sub-images. For example, whether there is potentially hidden information between adjacent sub-images in the tampered region and how this hidden information helps to improve the performance. 2) Ef\ufb01ciency Since our method divides the medical images into multiple sub-images through a sliding window, the increased number of targets would reduce the detection ef\ufb01ciency. With the help of a high processing speed for a single sub-image, the overall speed to detect a complete scan of the lungs is acceptable. As mentioned before, the main reason for using this sliding window is that the tampered region only occupies a small ratio of a normal image, which causes 13 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework (a) (b) Figure 8: A part of CT slice images and corresponding heatmaps of scans. (a) scan INJ1. (b) scan REM1. The \ufb01rst column of each scan is the tampered CT slice image. The second column is the heatmaps output when XceptionNet is used as the local detector. The third column is the heatmaps output when ResNet50 is used as the local detector. The fourth column is the heatmaps output of our method. the existing detection methods that treat the target image as a whole to fail. Hence, the proposed method is the better choice from the detection point of view as it is the only detection method for the CT-GAN attack. When the forgery attack is applied to the whole image, although we notice that our method can also detect the tampered region with high accuracy, the ef\ufb01ciency still can not catch up with those methods that treat the target as a single unit, such as [7]. If we have prior knowledge of the size of the tampered region, it will help us select the proper processing method. 3) Generalization The two most commonly used GAN structures in the medical image \ufb01eld are pix2pix and CycleGAN. CT-GAN uses the pix2pix structure. The works such as [28\u201333] are based on the CycleGAN structure. Therefore, we chose CycleGAN to construct another data set to exam the performance of our method. The CycleGAN data set is described in Section 4.1. The medical images denoised by CycleGAN can be regarded as entirely generated by CycleGAN. The test results show that our method can classify CycleGAN tampered medical images and real medical images with 99.8% accuracy. In addition, if we do not use machine learning but use some \ufb01xed pattern for global classi\ufb01cation, it is challenging to classify GAN-based small region forgery and the image generated by GAN wholly. CycleGAN and pix2pix are the two most commonly used GAN structures in medical image synthesis. Our method can effectively detect the images generated by CycleGAN and pix2pix. Although our method can detect the images generated by the same or similar GAN, the detection effect of other GAN models not in the training set is not as good as the former. Many studies [3,6,7,9\u201311,43] want to improve the generalization ability of GAN detection methods. This may be achieved by studying the common defects of CNN or GAN. For example, Wang et al. [43] tested multiple latest image generation models and found that the images generated by CNN today have certain common defects. Chai et al. [3] summarizes which parts are likely to cause the face images generated by GAN to be recognized. However, the above studies did not take GAN-based small region forgery attack into consideration. How to combine these studies with GAN-based small region forgery attack is still a problem. We plan to study this in the future. 14 \fGAN-based Medical Image Small Region Forgery Detection via a Two-Stage Cascade Framework 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2102.13446v1",
+                    "title": "Safe Distributional Reinforcement Learning",
+                    "abstract": "Safety in reinforcement learning (RL) is a key property in both training and\nexecution in many domains such as autonomous driving or finance. In this paper,\nwe formalize it with a constrained RL formulation in the distributional RL\nsetting. Our general model accepts various definitions of safety(e.g., bounds\non expected performance, CVaR, variance, or probability of reaching bad\nstates). To ensure safety during learning, we extend a safe policy optimization\nmethod to solve our problem. The distributional RL perspective leads to a more\nefficient algorithm while additionally catering for natural safe constraints.\nWe empirically validate our propositions on artificial and real domains against\nappropriate state-of-the-art safe RL algorithms.",
+                    "authors": "Jianyi Zhang, Paul Weng",
+                    "published": "2021-02-26",
+                    "updated": "2021-02-26",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Reinforcement learning (RL) has shown great promise in various applications [Silver et al., 2017, Jin, 2017]. As such techniques start to be deployed in real applications, safety in RL [Garcia and Fernandez, 2015] starts to be recognized as a key consideration both during learning, but also during execution after training. Indeed, in many domains from medical applications to autonomous driving to \ufb01nance, the actions chosen by an RL agent can have disastrous consequences and therefore the corresponding risks need to be controlled both during training, but also during execution. While traditional RL does not take safety into account, recent work has started to studied it more actively. Safety takes various de\ufb01nitions in the literature. In its simplest sense, it means avoiding bad states [Geibel and Wysotzky, 2005], but it can take more general meaning such as decision-theoretic risk aversion [Borkar, 2010], or risk constraints [Prashanth and Ghavamzadeh, 2016], satisfaction of logic speci\ufb01cations [Alshiekh et al., 2018], but also simple bounds on expected cumulated costs [Yu et al., 2019]. For a given de\ufb01nition of safety, one may want to learn a policy that satis\ufb01es it, without constraining the training process. Such approach would provide a safe policy for deployment after training. In contrast, recent work in safe RL aims at enforcing safety during learning as well, which is a dif\ufb01cult task as the RL agent needs to explore. This paper follows this latter trend and safety is formulated as the satisfaction of a set of general constraints on distributions of costs or rewards. Thus, a safe policy is de\ufb01ned as a policy that respects some constraints in expectation or in probability. Our goal is to learn among safe policies one that optimizes the usual expected discounted sum of rewards. Furthermore, we also require safe learning, i.e., the safety constraints shall be satis\ufb01ed during training as well. To that aim, we \ufb01rst propose a general framework that accepts various safety formulations from bounds on CVaR to variance, to probability of reaching bad states. This general framework is made amenable by formulating the problem in the distributional RL setting, where distributions of returns are learned in contrast to their expectations. Based on this general distributional formulation, we extend an existing safe RL algorithm, Interior-Point Policy Optimization (IPO) [Liu et al., 2020], to the distributional setting, for which we formulate a performance bound. Contributions Our contributions are threefold: (1) We propose a general framework for safe RL where safety is expressed as the satisfaction of risk constraints, which is enforced during and after training. A risk constraint can be expressed as any (sub)differentiable function of a random variable representing a cumulative reward or cost. (2) In order to obtain a practical algorithm, we formulate our problem and solution method in the distributional RL setting. (3) Our proposition, called SDPO, is empirically validated on multiple domains with various risk constraints against relevant state-of-the-art algorithms. Submitted to the 37th Conference on Uncertainty in Arti\ufb01cial Intelligence (UAI 2021). arXiv:2102.13446v1  [cs.LG]  26 Feb 2021 \fAlgorithm PD CPO IPO PCPO SDPO Expectation \u0013 \u0013 \u0013 \u0013 \u0013 Variance \u0013 \u0017 \u0017 \u0017 \u0013 CVaR \u0013 \u0017 \u0017 \u0017 \u0013 (Sub)differentiable fun. \u0017 \u0017 \u0017 \u0017 \u0013 Safe learning \u0017 \u0013 \u0013 \u0013 \u0013 Safe execution \u0013 \u0013 \u0013 \u0013 \u0013 Table 1: Summary of related algorithms: which constraints are accepted, whether safety is guaranteed during learning/execution. PD (primal-dual) actually corresponds to several algorithms. 2 RELATED WORK Safe RL is becoming an important research direction in RL [Garcia and Fernandez, 2015]. In this paper, we distinguish three main non-exclusive aspects for safe RL: policy safety, algorithmic safety, and exploration safety in exploration. Policy safety corresponds to the goal of learning a safe policy such that its execution would avoid/limit the occurrence of bad outcomes (e.g., probability of reaching bad states or bound on performance). Safety can be modeled as additional constraints or penalization. In that sense, safe RL is related to risk-sensitive RL [Borkar, 2010, Chow and Ghavamzadeh, 2014, Chow et al., 2015] where the goal is to learn a policy that optimizes a risk-sensitive objective function, constrained RL [Achiam et al., 2017, Tessler et al., 2019, Miryoose\ufb01et al., 2019, Liu et al., 2020] where the goal is to learn a policy that optimizes some constraints, and risk-constrained RL [Geibel and Wysotzky, 2005, Borkar and Jain, 2014, Prashanth and Ghavamzadeh, 2016, Chow et al., 2017, Brazdil et al., 2020], which in some sense combines the previous settings. The works in those three areas, with a few exceptions (CPO [Achiam et al., 2017], IPO [Liu et al., 2020], PCPO [Yang et al., 2020]), do not provide any safety guarantee during learning. They are based on a primal-dual approach (PD). For the exceptions, they can only accept simple constraints on expected discounted total costs. Notably, our algorithm, called Safe Distributional Policy Optimization (SDPO), builds on IPO [Liu et al., 2020] and extends it to the distributional RL setting, which then allows the formulation of sophisticated constraints. See Table 1 for a summary. Algorithmic safety corresponds to the idea that running a safe RL algorithm should also guarantee some safety property, e.g., continuous policy improvement [Pirotta et al., 2013], convergence to stationary point [Yu et al., 2019], satisfaction of logic speci\ufb01cations [Alshiekh et al., 2018], satisfaction of constraints [Achiam et al., 2017, Yang et al., 2020] during learning. However, none of those propositions can take into account sophisticated safety constraints (e.g., on risk measure). Exploration safety focuses on an important aspect of safe RL: the exploration problem during learning in order to limit/avoid selecting dangerous actions.. In this context, safety is generally modeled as avoiding bad states. One main line of work [Turchetta et al., 2016, Berkenkamp et al., 2017, Wachi et al., 2018, Cheng et al., 2019] tries to prevent the choice of a bad action by learning a model. Other directions have been explored, for instance, by using a veri\ufb01cation method [Fulton and Platzer, 2018] or by correcting a chosen action [Dalal et al., 2018]. However, this type of approaches requires the assumption that the environment is deterministic. Although research has been active in safe RL, to the best of our knowledge, no ef\ufb01cient algorithm has been proposed for the general framework that we propose. In particular, our proposition can learn a risk-constrained policy while ensuring the satisfaction of the risk constraint during learning. Our proposition is based on distributional RL [Bellemare et al., 2017], which has demonstrated that estimating distributions of returns instead of their expectations can ensure better overall performance of RL algorithms. Most work [Dabney et al., 2018, Yang et al., 2019] in this area focuses on value-based methods, extending mostly the DQN algorithm [Mnih et al., 2015]. However, one recent work has also investigated the extension of the distributional setting to policy optimization [Barth-Maron et al., 2018]. Our work is based on the IQN algorithm [Dabney et al., 2018] instead of more recent propositions (e.g., [Yang et al., 2019]) because of its simplicity and because it perfectly \ufb01ts our purposes. Note that in IQN, the authors consider optimizing a risk-sensitive objective function, but they do not consider constraints, as we do. 3 BACKGROUND In this section, we present the notations, recall the de\ufb01nition of a Markov Decision Process (MDP) as well as its extension to Constrained Markov Decision Process (CMDP), and review the notions (e.g., CVaR) and the related deep RL algorithms, which we use to formulate our method. Notations For any set X, \u2206(X) denotes the set of probability distributions (or densities if X is continuous) over X. For any function f : Y \u2192\u2206(X) and any (x,y) \u2208X \u00d7Y, f(x | y) denotes the probability (or density value if X is continuous) of obtaining x according to f(y). For any n \u2208N, [n] denotes {1,2,...,n}. Vectors (resp. matrix) will be denoted in bold lowercase (resp. uppercase) with their components in normal font face with indices. For instance, v v v = (v1,...,vn) \u2208Rn or M M M = (mij)i\u2208[n], j\u2208[m] \u2208Rn\u00d7m. MDP Model A Markov Decision Process (MDP) [Sutton and Barto, 2018] is de\ufb01ned as a tuple (S ,A ,P,r,\u00b5 \u00b5 \u00b5,\u03b3), where S is a set of states, A is a set of actions, P : 2 \fS \u00d7A \u2192\u2206(S ) is a transition function, r : S \u00d7A \u2192R is a reward function, \u00b5 \u00b5 \u00b5 \u2208\u2206(S ) is a distribution over initial states, and \u03b3 \u2208[0,1) is a discount factor. In this model, a policy \u03c0 : S \u2192\u2206(A ) is de\ufb01ned as a mapping from states to distributions over actions. We also use notation \u03c0\u03b8 \u03b8 \u03b8 to emphasize that the policy is parameterized by \u03b8 \u03b8 \u03b8 (e.g., parameters of neural network). In the remaining, we identify \u03c0\u03b8 \u03b8 \u03b8 to its parameter \u03b8 \u03b8 \u03b8 for ease of notation. The usual goal in an MDP is to search for a policy that maximizes the expected discounted total reward: J(\u03b8 \u03b8 \u03b8) = E\u00b5 \u00b5 \u00b5,P,\u03c0\u03b8 \u03b8 \u03b8 [\u2211\u221e t=0 \u03b3tr(st,at)] (1) where E\u00b5 \u00b5 \u00b5,P,\u03c0\u03b8 \u03b8 \u03b8 is the expectation with respect to the distribution \u00b5 \u00b5 \u00b5, the transition function P, and \u03c0\u03b8 \u03b8 \u03b8. We de\ufb01ne the (state) value function of a policy \u03c0\u03b8 \u03b8 \u03b8 for state s as: V \u03b8 \u03b8 \u03b8(s) = EP,\u03c0\u03b8 \u03b8 \u03b8 [\u2211\u221e t=0 \u03b3tr(st,at)|s0 = s] (2) where EP,\u03c0\u03b8 \u03b8 \u03b8 is the expectation with respect to the transition function P and \u03c0\u03b8 \u03b8 \u03b8. The (action) value function is de\ufb01ned as follows: Q\u03b8 \u03b8 \u03b8(s,a) = EP,\u03c0\u03b8 \u03b8 \u03b8 [\u2211\u221e t=0 \u03b3tr(st,at)|s0 = s,a0 = a] (3) and the advantage function is de\ufb01ned as: A\u03b8 \u03b8 \u03b8(s,a) = Q\u03b8 \u03b8 \u03b8(s,a)\u2212V \u03b8 \u03b8 \u03b8(s). As there is no risk of ambiguity, to avoid clutter we drop \u00b5 \u00b5 \u00b5 and P in the notation of the expectation from now on. Reinforcement learning (RL) is based on MDP, but in RL, the transition and reward functions are not assumed to be known. Thus, in (online) RL, an optimal policy needs to be learned by trial and error. CMDP Model The MDP model can be extended to the Constrained MDP (CMDP) setting [Altman, 1999] in order to handle constraints. In a CMDP, m cost functions ci : S \u00d7 A \u2192R for i \u2208[m] are introduced in addition to the original rewards. For each cost function ci, the corresponding value functions can be de\ufb01ned. They are denoted with a subscript, e.g., Jci, Vci, or Qci. For a CMDP, the goal is to \ufb01nd a policy that maximizes the expected discounted total reward while satisfying constraints on the expected costs Jci(\u03b8 \u03b8 \u03b8): max \u03b8 \u03b8 \u03b8 J(\u03b8 \u03b8 \u03b8) s.t. Jci(\u03b8 \u03b8 \u03b8) \u2264di \u2200i \u2208[m], (4) where d d d = (d)i\u2208[m] \u2208Rm is a \ufb01xed vector constraint bound. Proximal Policy Optimization The family of policy gradient methods constitutes the standard approach for tackling an RL problem when considering parametrized policies. Such a method iteratively updates a policy parameter in the direction of a gradient given by [Sutton and Barto, 2018]: \u2207\u03b8 \u03b8 \u03b8J(\u03b8 \u03b8 \u03b8) = E(s,a)\u223cd d d\u03c0\u03b8 \u03b8 \u03b8 [A\u03b8 \u03b8 \u03b8(s,a)\u2207\u03b8 \u03b8 \u03b8 log\u03c0\u03b8 \u03b8 \u03b8(a | s)] where the expectation is taken with the respect to the stateaction visitation distribution of \u03c0\u03b8 \u03b8 \u03b8. One issue in applying a policy gradient method is the dif\ufb01culty of estimating A\u03b8 \u03b8 \u03b8 online. This issue motivates the use of an actor-critic scheme where an actor (\u03c0\u03b8 \u03b8 \u03b8) and a critic (e.g., A\u03b8 \u03b8 \u03b8 or V \u03b8 \u03b8 \u03b8 depending on the speci\ufb01c algorithm) are simultaneously learned. Learning the value function can help the policy update, such as reducing the gradient variance. Proximal Policy Optimization (PPO) [Schulman et al., 2017] is a state-of-the-art actor-critic algorithm, which optimizes instead a clipped surrogate objective function JPPO(\u03b8 \u03b8 \u03b8) de\ufb01ned by: \u2211\u221e t=0 min(\u03c9t(\u03b8 \u03b8 \u03b8)A \u00af \u03b8 \u03b8 \u03b8(st,at),clip(\u03c9t(\u03b8 \u03b8 \u03b8),\u03b5)A \u00af \u03b8 \u03b8 \u03b8(st,at)), (5) where \u00af \u03b8 \u03b8 \u03b8 is the current policy parameter, \u03c9t(\u03b8 \u03b8 \u03b8) = \u03c0\u03b8 \u03b8 \u03b8 (at|st) \u03c0 \u00af \u03b8 \u03b8 \u03b8 (at|st), and clip(\u00b7,\u03b5) is the function to clip between [1 \u2212\u03b5,1 + \u03b5]. This surrogate function was motivated as an approximation of that used in TRPO [Schulman et al., 2015], which was introduced to ensure monotonic improvement after a policy parameter update. Although PPO is more heuristic than TRPO, its advantages are its simplicity and lower sample complexity. Distributional Reinforcement Learning The key idea in distributional RL [Bellemare et al., 2017] is to learn a random variable to represent the discounted return Z Z Z\u03b8 \u03b8 \u03b8(s,a) = \u2211\u221e t=0 \u03b3tr r rt where r r rt is the random variable representing the immediate reward received at time step t when applying action a in state s and policy \u03c0\u03b8 \u03b8 \u03b8 thereafter. In contrast, standard RL algorithms directly estimate the expectation of Z Z Z\u03b8 \u03b8 \u03b8(s,a), since Q\u03b8 \u03b8 \u03b8(s,a) = EZ Z Z\u03b8 \u03b8 \u03b8 [Z Z Z\u03b8 \u03b8 \u03b8(s,a)] where the expectation is with respect to the distribution of Z Z Z\u03b8 \u03b8 \u03b8(s,a). Recall that any real random variable Z can be represented by its cumulative distribution denoted FZ(z) = P(Z \u2264z) \u2208[0,1], or equivalently by its quantile function (inverse cumulative distribution) denoted F\u22121 Z (p) = inf{z \u2208R | p \u2264FZ(z)} for any p \u2208[0,1]. For ease of notation, Zp denotes the pquantile F\u22121 Z (p). In the Implicit Quantile Network (IQN) algorithm, Dabney et al. [2018] proposed to approximate the quantile function of Z Z Z(s,a) with a neural network and to learn it using quantile regression [Koenker, 2005]. Concretely, the quantile function of Z Z Z(s,a) can be learned as follows. Denote \u02c6 Z Z Z(s,a) the approximated random variable whose quantile function is given by a neural network \u03a8(s,\u03c4), which takes as input a state s and a probability \u03c4 \u2208[0,1] and returns the corresponding \u03c4-quantile \u02c6 Z Z Z\u03c4(s,a) for each action a. After observing a transition (s,a,r,s\u2032), \u03a8 can be trained by sampling 2N values \u03c4 \u03c4 \u03c4 = (\u03c41,...,\u03c4N) and \u03c4 \u03c4 \u03c4\u2032 = (\u03c4\u2032 1,...,\u03c4\u2032 N) with the uniform distribution on [0,1]. By inverse transform sampling, sampling \u03c4 \u03c4 \u03c4 amount to sampling N values from \u02c6 Z Z Z(s,a) corresponding to \u02c6 Z Z Z\u03c41(s,a),..., \u02c6 Z Z Z\u03c4N(s,a)), and similarly for \u03c4 \u03c4 \u03c4\u2032 and sampling from \u02c6 Z Z Z(s\u2032,\u03c0(s\u2032)) where \u03c0 is the current policy. Those samples de\ufb01ne N2 TD errors in the distributional setting: \u03b4ij = r +\u03b3 \u02c6 Z Z Z\u03c4\u2032 j(s\u2032,\u03c0(s\u2032))\u2212\u02c6 Z Z Z\u03c4i(s,a) (6) 3 \fFollowing quantile regression, the following loss function for training the neural network \u03a8 in (s,a,r,s\u2032) is given by: LIQN = 1 N \u2211i\u2208[N] \u2211j\u2208[N] \u03be \u03ba \u03c4i (\u03b4i j) (7) where for any \u03c4 \u2208(0,1], \u03be \u03ba \u03c4 (\u03b4i j) = |\u03c4 \u2212I(\u03b4i j < 0)| L\u03ba(\u03b4ij) \u03ba is the quantile Huber loss with threshold \u03ba with L\u03ba(\u03b4) = 1 2\u03b4 2 for |\u03b4| \u2264\u03ba or \u03ba(|\u03b4|\u22121 2\u03ba) otherwise. Interior-Point Policy Optimization In the CMDP setting, Interior-point Policy Optimization (IPO) [Liu et al., 2020] is a recent RL algorithm to maximize an expected discounted total rewards while satisfying constraints on some expected discounted total costs. To deal with a constraint, IPO augments PPO\u2019s objective function with a logarithmic barrier function applied to it, which provides a smooth approximation of the indicator function. The constrained problem then becomes an unconstrained one with an augmented objective function: max \u03b8 \u03b8 \u03b8 JIPO(\u03b8 \u03b8 \u03b8) = JPPO(\u03b8 \u03b8 \u03b8)+\u2211i\u2208[m] ln(di\u2212Jci(\u03b8 \u03b8 \u03b8)) \u03b7 , (8) where \u03b7 is a hyper-parameter. As \u03b7 tends to \u221e, the solution of (8) tends to that of the original constrained problem. The objective JIPO is differentiable, therefore, we can apply a gradient-based optimization method to update the policy. 4 PROBLEM FORMULATION Let \u2206(R) denote the set of real random variables. Therefore, Z Z Z \u2208\u2206(R)S denotes a function from states to random variables. Given an (unknown) CMDP, the problem tackled in this paper can be expressed as a constrained optimization problem formulated in the distributional RL setting: max \u03b8 \u03b8 \u03b8 Es0\u223c\u00b5 \u00b5 \u00b5,Z Z Z\u03b8 \u03b8 \u03b8 [Z Z Z\u03b8 \u03b8 \u03b8(s0)] (9) s.t. \u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i ) \u2264di \u2200i \u2208[m] (10) where Z Z Z\u03b8 \u03b8 \u03b8(s) corresponds to the return distribution generated by policy \u03c0\u03b8 \u03b8 \u03b8 from the reward function, for all i \u2208[m], Y Y Y \u03b8 \u03b8 \u03b8 i (s) corresponds to the cumulated cost distribution from cost function ci, and \u03c1i : \u2206(R)S \u2192R is a (sub)differentiable function. Note that this formulation is strictly more general than problem (4) thanks to the possibly non-linear functions \u03c1i\u2019s. We recall a few common cases for \u03c1i in Table 2. The expectation is a simple example. For episodic MDPs with absorbing bad states, another simple example is the probability of bad states, which is de\ufb01ned like the expectation, but applied to a undiscounted cost equal to 1 for a bad state and 0 otherwise. CVaR is a widely-used risk measure in \ufb01nance. In this context, the \u03b1-CVaR of a portfolio is intuitively its expected return in the worst \u03b1 \u00d7100% cases. Here, \u03c1i De\ufb01nition Expectation Es0\u223c\u00b5 \u00b5 \u00b5,Y Y Y [Y Y Y(s0)] Prob. of bad states Es0\u223c\u00b5 \u00b5 \u00b5,Y Y Y [Y Y Y(s0)] \u03b1-CVaR of rewards Es0\u223c\u00b5 \u00b5 \u00b5[ 1 \u03b1 R \u03b1 0 Y\u03b6 (s0)d\u03b6] Variance Es0\u223c\u00b5 \u00b5 \u00b5[EY Y Y [Y Y Y(s0)2]\u2212EY Y Y [Y Y Y(s0)]2] Table 2: Common examples for \u03c1i. we adapted the de\ufb01nition to rewards (instead of costs). Naturally, a CVaR of an additional cost would also be possible. In contrast to previous methods, our framework can accept any (sub)differentiable de\ufb01nitions for \u03c1i (e.g., coherent risk measures). Note that we chose to take the mean (over initial states) of the CVaRs instead of the CVaR of the mean. The latter would have been possible as well, but because CVaR is a convex risk measure, our de\ufb01nition is an upperbound of the CVaR of the mean, which means that our formulation is more conservative and in that sense, safer. The same trick applies if \u03c1i were de\ufb01ned based on any other coherent risk measure, of which CVaR is only one instance. Similarly, for the variance, we use the mean (over initial states) of variances instead of the other way around. Since the initial states are sampled in an independent way, the Y Y Y(s0)\u2019s are independent. This means that our de\ufb01nition upperbounds the variance of the mean of the Y Y Y(s0)\u2019s, leading to a more cautious formulation, which is more desirable for safe RL. In this paper, we de\ufb01ne a safe policy as a policy satisfying constraints (10). Our goal is to learn a policy maximizing an expected discounted total rewards (9) among all safe policies (i.e., safe execution). Besides, we require that any policy used during learning be safe (i.e., safe learning). The formulation of (9)-(10) in the distributional RL setting serves two purposes. First, as observed in distributional RL, estimating the distributions of the cumulated rewards improves the overall performance. Second, many safety constraints (10), such as CVaR, become natural and simple to express in the distributional setting. 5 PROPOSED METHOD To solve problem (9)-(10) in the safe RL setting, we extend IPO to the distributional RL setting and combine it with an adaptation of IQN. Next, we explain the general principle of our approach, and then discuss some techniques to obtain a concrete ef\ufb01cient implementation. 4 \f5.1 GENERAL PRINCIPLE To adapt IPO, we rewrite the surrogate objective function used in PPO in the distributional setting: JPPO(\u03b8 \u03b8 \u03b8) = \u221e \u2211 t=0 min(\u03c9t(\u03b8 \u03b8 \u03b8)E\u03b8 \u03b8 \u03b8[Z Z Z \u00af \u03b8 \u03b8 \u03b8(st,at)\u2212Z Z Z \u00af \u03b8 \u03b8 \u03b8(st)], clip(\u03c9t(\u03b8 \u03b8 \u03b8),\u03b5)E\u03b8 \u03b8 \u03b8[Z Z Z \u00af \u03b8 \u03b8 \u03b8(st,at)\u2212Z Z Z \u00af \u03b8 \u03b8 \u03b8(st)]). (11) Problem (9)-(10) can then be tackled by iteratively solving the following problem with this surrogate function: max \u03b8 \u03b8 \u03b8 JPPO(\u03b8 \u03b8 \u03b8) s.t. \u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i ) \u2264di \u2200i \u2208[m]. (12) Now, following IPO, using the log barrier function, we reformulate problem (12) as an unconstrained problem: max \u03b8 \u03b8 \u03b8 JPPO(\u03b8 \u03b8 \u03b8)+ \u2211 i\u2208[m] ln(di \u2212\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i )) \u03b7i . (13) In contrast to convex optimization [Boyd and Vandenberghe, 2004], we introduce a constraint-dependent hyperparameter \u03b7i to better control the satisfaction of each constraint separately. Finally, we propose to solve problem (13) with an actorcritic architecture where both the actor and the critic are approximated with neural networks. For the critic, we adapt the approach proposed for IQN [Dabney et al., 2018] to learn random returns Z Z Z and random cumulated costs Y Y Y i\u2019s. For the actor, parameter \u03b8 \u03b8 \u03b8 of policy \u03c0\u03b8 \u03b8 \u03b8(a|s) is updated in the direction of the gradient of the objective function de\ufb01ned in (13): \u2207\u03b8 \u03b8 \u03b8JPPO(\u03b8 \u03b8 \u03b8)\u2212\u2211 i\u2208[m] 1 \u03b7i \u2207\u03b8 \u03b8 \u03b8\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i ) di \u2212\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i ) . (14) This gradient raises one dif\ufb01culty regarding the computation of \u2207\u03b8 \u03b8 \u03b8\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i ), which corresponds to the gradient of a critic with respect to the parameters of the actor. When \u03c1i is linear (i.e., for expectation constraints), the policy gradient theorem [Sutton et al., 2000] applies and speci\ufb01es how to compute \u2207\u03b8 \u03b8 \u03b8\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i ). However, when \u03c1i is non-linear (i.e., for more sophisticated risk constraints), the gradient in (14) cannot be obtained easily. To solve this issue, we propose a simple and generic solution, which consists in connecting the actor network to any critic network with a non-linear \u03c1i (see Figure 1 for an illustration where only one critic corresponding to non-linear \u03c1i is displayed). Using this construct, the exact gradient of \u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i ) can be computed by automatic differentiation if \u03c1i is (sub)differentiable and Y Y Y \u03b8 \u03b8 \u03b8 i is approximated with a neural network, as we assume. Note that in previous work, Dabney et al. [2018] who proposed to optimize a risk measure in IQN did not face this gradient issue because their algorithm is based on DQN [Mnih et al., 2015] and therefore does not have an actor network. As a side note, Figure 1: Architecture of SDPO where critic \u03a8 corresponds to the objective function and critic \u03a8i corresponds to constraint i. Both critics outputs a distribution. this construct could be used to deal with a more general problem than (9)-(10) where a non-linear transformation is also applied on the objective function. For instance, one may want to optimize the CVaR of some rewards subject to some other risk constraints, which is as far as we know a completely novel problem. We leave this investigation to future work. Like any interior point method, an initial feasible (i.e., safe) solution is needed. This requirement is actually not as strong as it seems. In many MDPs (or CMDPs), there is a known safe action for every state. For instance, in navigation problem, the action of not moving is safe if the current state is safe. In \ufb01nance, investing in cash or a risk-free asset is safe. For many problems, a dummy action that does not have any effect can be added to de\ufb01ne an initial safe action. More generally, when such a simple safe policy cannot be de\ufb01ned, an expert could possibly provide this initial safe policy or it could be obtained by pretraining with an imperfect simulator. 5.2 TECHNIQUES FOR EFFICIENT IMPLEMENTATION In this section, to simplify notations, we do not write the superscript \u03b8 \u03b8 \u03b8 for the random variables Z Z Z and Y Y Y i\u2019s. To make our \ufb01nal algorithm more ef\ufb01cient, we propose to learn Z Z Z(s) only, instead of Z Z Z(s,a) as it is the usual practice in distributional RL. This serves two purposes: (1) a state-dependent distribution is easier to learn, and (2) the advantage function can be easily estimated from a state value function alone. Note that for the constraints only Y Y Y i(s) is needed for any i \u2208[m]. Recall that the two random variables Z Z Z(s) and Z Z Z(s,a) are related by the following equation: Z Z Z(s,a) = R(s,a)+\u03b3Es\u2032\u223cP(\u00b7|s,a)[Z Z Z(s\u2032)] (15) Following IQN, random variable Z Z Z(s) is approximated by a random variable \u02c6 Z Z Z, which is represented by a neural network. The expectation of Z Z Z(s) can then be approximated by that 5 \fAlgorithm 1 SDPO Require: Constraint bound d d d, Initial policy network \u03c0\u03b8 \u03b8 \u03b8 0, Initial IQN network \u03a80, Hyperparameters \u03b5 for PPO clip rate and \u03b7i for each logarithmic barrier function. 1: for k = 0,1,... do 2: B \u2190run policy \u03c0\u03b8 \u03b8 \u03b8 for N trajectories 3: # update the IQN network 4: Sample \u03c41 < ... < \u03c4N from U [0,1] 5: # quantile regression 6: for i, j \u2208[N] do 7: \u03b4ij = r +\u03b3 \u02c6 Z Z Z\u03c4\u2032 j(s\u2032,\u03c0(s\u2032))\u2212\u02c6 Z Z Z\u03c4i(s,a) 8: end for 9: Update \u03a8k+1 with \u2207LIQN (see (7)) using B 10: Update \u03b8 \u03b8 \u03b8k+1 with \u2207J(\u03b8 \u03b8 \u03b8k) de\ufb01ned in (14) using B 11: end for of \u02c6 Z Z Z(s) with \u03c4 \u03c4 \u03c4 randomly uniformly sampled in [0,1]: E\u03b8 \u03b8 \u03b8[Z Z Z(s)] \u2248 N \u2211 i=1 (\u03c4i \u2212\u03c4i\u22121) \u02c6 Z Z Z\u03c4i(s). (16) setting \u03c40 = 0 by convention and assuming 0 < \u03c41 < \u03c42 < ... < \u03c4N < 1. The exact handling of the constraints depend on the de\ufb01nition of \u03c1i. As illustrative examples, we explain how they can be computed for some concrete cases. If \u03c1i is simply de\ufb01ned as an expectation, it can be dealt with like the objective function. For CVaR, it can be estimated as follows for a random variable Y Y Y(s0): c\u03b1(Y Y Y) \u2248c\u03b1( \u02c6 Y Y Y) = 1 \u03b1 \u2211 i|\u03c4i\u2264\u03b1 (\u03c4i \u2212\u03c4i\u22121) \u02c6 Y Y Y \u03c4i(s0) (17) Here, in contrast to the standard expectation (e.g., (16)), an implementation trick consists in sampling \u03c4 \u03c4 \u03c4 in [0,\u03b1] such as \u03c41 < \u03c42 < ... < \u03c4N = \u03b1 since (17) corresponds to the expectation conditioned on event \u201c \u02c6 Y Y Y \u2264\u02c6 Y Y Y \u03b1\u201d. For the variance, \u03c1i(Y Y Y) can be estimated by: N \u2211 i=1 (\u03c4i \u2212\u03c4i\u22121) \u02c6 Y Y Y \u03c4i(s0)2 \u2212   N \u2211 i=1 (\u03c4i \u2212\u03c4i\u22121) \u02c6 Y Y Y \u03c4i(s0) !2 (18) The pseudo code of our method is shown in Algorithm 1. 5.3 PERFORMANCE GUARANTEE BOUND For \ufb01xed \u03b7 \u03b7 \u03b7, solving (13) instead of (12) may incur a performance loss, which can be bounded under natural conditions, which we discuss below. Since this result uses weak Lagrange duality, we \ufb01rst recall the de\ufb01nition of the Lagrangian of (12): L (\u03b8 \u03b8 \u03b8,\u03bb \u03bb \u03bb) = JPPO(\u03b8 \u03b8 \u03b8)+ \u2211 i\u2208[m] \u03bbi(di \u2212\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8 i )) and its dual function: g(\u03bb \u03bb \u03bb) = max\u03b8 \u03b8 \u03b8 L (\u03b8 \u03b8 \u03b8,\u03bb \u03bb \u03bb). The following bound can be proven: Theorem 1. If \u03b8 \u03b8 \u03b8 \u2217 1 is an optimal solution of (12), \u03b8 \u03b8 \u03b8 \u2217 2 is the strictly feasible optimal solution of (13) and the unique stationary point of L (\u00b7,\u03bb \u03bb \u03bb \u2217) with \u03bb \u2217 i = 1 \u03b7i(di\u2212\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8\u2217 2 i )) then: JPPO(\u03b8 \u03b8 \u03b8 \u2217 1)\u2212JPPO(\u03b8 \u03b8 \u03b8 \u2217 2) \u2264\u2211 i\u2208[d] 1 \u03b7i (19) Proof. This result generalizes Theorem 1 of [Liu et al., 2020], whose proof implicitly uses convexity (which does not hold in deep RL) and follows from the discussion in page 566 of [Boyd and Vandenberghe, 2004]. We adapt the proof to our more general setting. We have: JPPO(\u03b8 \u03b8 \u03b8 \u2217 1) \u2264g(\u03bb \u03bb \u03bb \u2217) (20) = JPPO(\u03b8 \u03b8 \u03b8 \u2217 2)+ \u2211 i\u2208[m] \u03bb \u2217 i (di \u2212\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8\u2217 2 i )) (21) = JPPO(\u03b8 \u03b8 \u03b8 \u2217 2)+ \u2211 i\u2208[m] 1 \u03b7i (22) Step (20) holds by weak duality because \u03bb \u2217 i \u22650 for all i \u2208[m] (since \u03b8 \u03b8 \u03b8 \u2217 2 is strictly feasible). Step (21) holds because we have by de\ufb01nition of \u03b8 \u03b8 \u03b8 \u2217 2: \u2207\u03b8 \u03b8 \u03b8JPPO(\u03b8 \u03b8 \u03b8 \u2217 2)\u2212\u2211 i\u2208[m] \u2207\u03b8 \u03b8 \u03b8\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8\u2217 2 i ) \u03b7i(di \u2212\u03c1i(Y Y Y \u03b8 \u03b8 \u03b8\u2217 2 i )) = 0 (23) which implies that \u03b8 \u03b8 \u03b8 \u2217 2 maximizes L (\u00b7,\u03bb \u03bb \u03bb \u2217) since \u03b8 \u03b8 \u03b8 \u2217 2 is assumed to be its unique stationary point. Step (22) holds by de\ufb01nition of \u03bb \u03bb \u03bb \u2217. The conditions in this theorem are natural. In order to apply an interior point method, the constrained problem needs to be strictly feasible. The condition on the stationarity of \u03b8 \u03b8 \u03b8 \u2217 2 is reasonable and can be controlled by setting \u03b5 (used in the clipping function of JPPO) small enough. As a direct corollary, this result implies that if (13) could be solved exactly, the error made by algorithm SDPO is controllable via setting appropriate \u03b7i\u2019s. Naturally, in the online RL setting, this assumption does not hold perfectly, but this result still provides some theoretical foundation to our proposition. In the next section, we validate the algorithm using various experimental settings. 6 EXPERIMENTAL RESULTS The experiments are carried out in three different domains to validate our algorithm: random CMDPs, safety gym, as well as \ufb01nancial investment. See Appendix A for details about hyperparameter settings. Random CMDPs are CMDPs with N states and M actions, where transition probabilities P(s\u2032 | s,a) are randomly assigned with \u2308lnN\u2309positive values for each pair of stateaction, and rewards are sampled from a uniform distribution, 6 \f(a) rewards (b) constraint Figure 2: 2(a): Average performance over 10 runs of PPO, SDPO and PD-CVaR under the random CMDP for N = 1000. 2(b): 0.1-CVaR bounded by 51. Both SDPO and PDCVaR converge to the level indicated by the dashed line. i.e., r(s,a) \u223cU [0,1]. In the experiments, we set N = 1000 and M = 10. We consider two cases: a bound over the variance or a bound over the CVaR, both over the distribution of discounted total rewards. Safety gym [Ray et al., 2019] includes a set of environments designed for evaluating safe exploration in RL. They all correspond to navigation problems where an agent (i.e., Point, Car, Doggo) moves to some random goal positions to perform some tasks (i.e., Goal, Button, Push) while avoiding entering dangerous hazards or bumping into fragile vases. Each task has two dif\ufb01culty levels (i.e., 1 or 2). See Appendix A for more details. For space reasons, we only present a selection of results in this domain in the main paper. More experimental results in these Mujoco environments are shown in Appendix B. The third domain is the \ufb01nancial stock market. The RL agent can observe the close prices of the stocks in one day, i.e., the observation ot = p p pt = (1, p1,t,..., pN,t) for N selected stocks where the \ufb01rst component corresponds to cash. We further assume that all transactions are dealt at these prices. The action of the agent is de\ufb01ned by a portfolio vector, which corresponds to allocation weights over cash and stocks, i.e., at = w w wt+1 = (w0,t+1,...,wN,t+1), w0 (resp. wi for i \u2208[N]) is the weight for cash (resp. stock i) and \u2211N i=0 wi,t = 1. Naturally, for each stock, we want to maximize the pro\ufb01t. Thus, with reward function rt = ln\u2211N i=0 wi,t pi,t pi,t\u22121 , optimizing the undiscounted cumulative rewards can maximize the pro\ufb01t. We set the CVaR boundary d1 = 0 to avoid any possible loss. Detailed settings of the experiment are listed in Appendix A. In all our experiments, all the agents are initialized so that they are in a feasible region at the beginning. In practice, an initial safe policy can be de\ufb01ned using domain knowledge or by an expert, e.g., in Mujoco domain, the agent can be initialized to stay and doing nothing. For fairness, the PPO agent is also initialized with the same safe policy as all other agents. Two policy gradient algorithms with CVaR and variance constraints respectively, PD-CVaR [Chow and Ghavamzadeh, 2014] and PD-VAR, which is modi\ufb01ed from (a) rewards (b) constraint Figure 3: 3(a): Average performance over 10 runs of PPO, SDPO and PD-VAR under the random CMDP for N = 1000. 3(b): Variance bounded by 2. Both SDPO and PD-VAR converge to the level indicated by the dashed line. Algorithm 2 in [Prashanth and Ghavamzadeh, 2016] are used as baselines in the \ufb01rst domain. SDPO is compared with CPO [Achiam et al., 2017], PCPO [Yang et al., 2020], and IPO [Liu et al., 2020] in the second domain. PPO [Schulman et al., 2017] is evaluated on all domains to serve as a non-safe RL method. Note that in contrast to our architecture SDPO, none of those algorithms can tackle the problem de\ufb01ned in (9)-(10) in its most general form. The experiments are designed to evaluate SDPO in a variety of domains with various risk constraints and to answer the following questions: (A) How does SDPO compare with methods based on Lagrangian relaxation? (B) How does SDPO compare with other safe RL algorithms? Does the distributional formulation of SDPO help compared to IPO? (C) How does SDPO perform with multiple constraints (cumulative cost and probability of reaching a bad states)? (D) How does SDPO perform on a real domain? How does the constraint stringency impact the performance of SDPO? Question (A) To answer (A), we perform some experiments on the \ufb01rst domain, random CMDPs, with either a constraint on CVaR or a constraint on variance. Both are based on the rewards. Therefore, the \ufb01rst needs to be lowerbounded, while the second needs to be upper-bounded. The con\ufb01dence level is \ufb01xed to \u03b1 = 0.1 and the bound for CVaR is set to 51 and that for the variance is set to 2. The bounds were chosen so that they are not too restrictive. From the results in Figure 2, as expected, PPO without constraint achieves the best total rewards and converges faster than the constrained ones. When the CVaR value is bounded, PD-CVaR and SDPO both converge to a slightly worse but safe policy, however SDPO converges faster. From the results in Figure 3, similar observations can be drawn for PPO, PD-VAR, and SDPO. With regards to safety, we can again conclude than SDPO is superior. Question (B) To answer (B), we perform some experiments on the second domain, Safety gym, which is a much more dif\ufb01cult domain than random CMDPs. For this domain, 7 \f(a) rewards (b) constraint Figure 4: Average performance over 10 runs of PPO, SDPO, CPO, PCPO and IPO under Point-Goal1. They are bounded by the dashed line d1 = 25 in 4(b). we did not evaluate the methods based on Lagrangian relaxation: since they do not use a critic, they would not be competitive. In Safety gym, the agent is penalized by receiving a cost c1 = 1 when touching a fragile vase. With a constraint on expected total cost \u03c11(Y Y Y 1) = Es0\u223c\u00b5 \u00b5 \u00b5,Y Y Y 1[Y Y Y 1(s0)] \u2264d1, we are able to compare SDPO with other safe RL algorithms like CPO, PCPO and IPO. We only show the results for Point-Goal1 in Figure 4. For other tasks, please refer to Appendix B. According to Figure 4, SDPO, PCPO and IPO can explore safely, while CPO cannot satisfy the constraint well. This latter observation regarding CPO may be surprising since CPO was designed to solve CMDPs, but similar results were also reported in previous work [Ray et al., 2019]. Among these three latter algorithms, SDPO and IPO performs the best. In Figure 4 and in all the Safety-gym environments (see Appendix B), SDPO dominates IPO in terms of either returns or convergence rates (and sometimes both), which con\ufb01rms the positive contribution of the distributional critics. Question (C) To demonstrate that SDPO can satisfy multiple constraints, the safety gym environment is used again, but with a variation. We modify the hazard area to be end states where an agent receives a cost c2 = 1, and the episode is terminated. In addition to the previous constraint, another one is enforced: \u03c12(Y Y Y 2) = Es0\u223c\u00b5 \u00b5 \u00b5,Y Y Y 2[Y Y Y 2(s0)] \u2264d2, where Y Y Y 2 is the undiscounted cumulative cost distribution from cost function c2. Here, we set the bounds: d1 = 10 and d2 = 0.1. From Figures 5(a) and 5(b), PPO without constraints achieves much more goals, but at the cost of violating all the constraints. For constraint \u03c12, both SDPO and IPO agents can avoid entering into hazards during training. For constraint \u03c11, SDPO converges faster than IPO because of the adaption to distributional RL. Question (D) To answer (D), we switch to the \ufb01nance domain, where the stock market data of year 2019 is used. We run SDPO with a constraint on CVaR de\ufb01ned over rewards using different con\ufb01dence levels \u03b1. Note that since the CVaR is de\ufb01ned over rewards, it needs to be lower(a) rewards (b) constraints Figure 5: 5(a): Average performance over 5 runs of PPO, SDPO and IPO under Point-Goal2. 5(b): Average costs of PPO, SDPO and IPO under Point-Goal2. Figure 6: Average performance over 10 runs of PPO and SDPO with con\ufb01dence level \u03b1 = 0.01,0.05,0.1. bounded. We also run PPO as a baseline to show the performance without any constraints. From Figure 6, all agents manage to make pro\ufb01ts. With tighter constraint on risk (smaller \u03b1), the SDPO agent makes less pro\ufb01t. While PPO does not satisfy the constraint as expected, the curves for the constraint satisfaction of all SDPO agents are all similar. We therefore plot their average directly in Figure 6. PPO without constraint cannot avoid risk and thus suffers from \ufb02uctuation and loss at some time point. Interestingly, all the SDPO agents eventually perform better than PPO, which demonstrates that enforcing safety does not necessarily prevent good performance. Finally, SDPO with \u03b1 = 0.1 performs best. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1811.08052v1",
+                    "title": "Variance Reduction in Stochastic Particle-Optimization Sampling",
+                    "abstract": "Stochastic particle-optimization sampling (SPOS) is a recently-developed\nscalable Bayesian sampling framework that unifies stochastic gradient MCMC\n(SG-MCMC) and Stein variational gradient descent (SVGD) algorithms based on\nWasserstein gradient flows. With a rigorous non-asymptotic convergence theory\ndeveloped recently, SPOS avoids the particle-collapsing pitfall of SVGD.\nNevertheless, variance reduction in SPOS has never been studied. In this paper,\nwe bridge the gap by presenting several variance-reduction techniques for SPOS.\nSpecifically, we propose three variants of variance-reduced SPOS, called SAGA\nparticle-optimization sampling (SAGA-POS), SVRG particle-optimization sampling\n(SVRG-POS) and a variant of SVRG-POS which avoids full gradient computations,\ndenoted as SVRG-POS$^+$. Importantly, we provide non-asymptotic convergence\nguarantees for these algorithms in terms of 2-Wasserstein metric and analyze\ntheir complexities. Remarkably, the results show our algorithms yield better\nconvergence rates than existing variance-reduced variants of stochastic\nLangevin dynamics, even though more space is required to store the particles in\ntraining. Our theory well aligns with experimental results on both synthetic\nand real datasets.",
+                    "authors": "Jianyi Zhang, Yang Zhao, Changyou Chen",
+                    "published": "2018-11-20",
+                    "updated": "2018-11-20",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction 2 2 Preliminaries 4 2.1 Stochastic gradient MCMC . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Stein variational gradient descent . . . . . . . . . . . . . . . . . . . . 5 2.3 Stochastic particle-optimization sampling . . . . . . . . . . . . . . . . 5 3 Variance Reduction in SPOS 6 3.1 SAGA-POS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 SVRG-POS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 SVRG-POS+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Convergence Analysis 9 5 Experiments 12 5.1 Synthetic log-normal distribution . . . . . . . . . . . . . . . . . . . . 12 5.2 Bayesian logistic regression . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2.1 Variance-reduced SPOS versus SPOS . . . . . . . . . . . . . . 14 5.2.2 Variance-reduced SPOS versus variance-reduced SGLD . . . . 14 5.2.3 Impact of number of particles . . . . . . . . . . . . . . . . . . 14 6 Conclusion 14 A More details about the notations 19 B Convergence guarantees for SAGA-LD, SVRG-LD and SVRG-LD+ 20 C Proof of the theorems in Section 4 21 D Extra theoretical discussion for SAGA-POS, SVRG-POS and SVRG-POS+ 28 E Comparison between SPOS and its variance-reduction counterpart 30 F More experiments results 30 1 Introduction Sampling has been an e\ufb00ective tool for approximate Bayesian inference, which becomes increasingly important in modern machine learning. In the setting of big data, recent research has developed scalable Bayesian sampling algorithms such as stochastic gradient Markov Chain Monte Carlo (SG-MCMC) [WT11] and Stein \fZhang & Zhao & Chen 3 variational gradient descent (SVGD) [LW16]. These methods have facilitated important real-world applications and achieved impressive results, such as topic modeling [GCH+15, LZS16], matrix factorization [CFG14, DFB+14, cBCR16], di\ufb00erential privacy [WFS15, LCLC17], Bayesian optimization [SKFH16] and deep neural networks [LCCC16]. Generally speaking, these methods use gradient information of a target distribution to generate samples, leading to more e\ufb00ective algorithms compared to traditional sampling methods. Recently, [CZW+18] proposed a particle-optimization Bayesian sampling framework based on Wasserstein gradient \ufb02ows, which uni\ufb01ed SG-MCMC and SVGD in a new sampling framework called particle-optimization sampling (POS). Very recently, [ZZC18] discovered that SVGD endows some unintended pitfall, i.e. particles tend to collapse under some conditions. As a result, a remedy was proposed to inject random noise into SVGD update equations in the POS framework, leading to stochastic particle-optimization sampling (SPOS) algorithms [ZZC18]. Remarkably, for the \ufb01rst time, non-asymptotic convergence theory was developed for SPOS (SVGD-type algorithms) in [ZZC18]. In another aspect, in order to deal with large-scale datasets, many gradient-based methods for optimization and sampling use stochastic gradients calculated on a mini-batch of a dataset for computational feasibility. Unfortunately, extra variance is introduced into the algorithms, which would potentially degrade their performance. Consequently, variance control has been an important and interesting work for research. Some e\ufb03cient solutions such as SAGA [DBLJ14] and SVRG [JZ13] were proposed to reduce variance in stochastic optimization. Subsequently, [DRP+16] introduced these techniques in SG-MCMC for Bayesian sampling, which also has achieved great success in practice. Since SPOS has enjoyed the best of both worlds by combining SG-MCMC and SVGD, it will be of greater value to further reduce its gradient variance. While both algorithm and theory have been developed for SPOS, no work has been done to investigate its variance-reduction techniques. Compared with the research on SG-MCMC where variance reduction has been well explored by recent work such as [DRP+16, CFM+18, ZXG18], it is much more challenging for SPOS to control the variance of stochastic gradients. This is because from a theoretical perspective, SPOS corresponds to nonlinear stochastic di\ufb00erential equations (SDE), where fewer existing mathematical tools can be applied for theoretical analysis. Furthermore, the fact that many particles are used in an algorithm makes it di\ufb03cult to improve its performance by adding modi\ufb01cations to the way they interact with each other. In this paper, we take the \ufb01rst attempt to study variance-reduction techniques in SPOS and develop corresponding convergence theory. We adopt recent ideas on variance reduction in SG-MCMC and stochastic-optimization algorithms, and propose three variance-reduced SPOS algorithms, denoted as SAGA particle-optimization sampling (SAGA-POS), SVRG particle-optimization sampling (SVRG-POS) and a variant of SVRG-POS without full-gradient computations, denoted as SVRG-POS+. For all these variants, we prove rigorous theoretical results on their non-asymptotic convergence rates in terms of 2-Wasserstein metrics. Importantly, our theoretical \f4 Variance Reduced Stochastic Particle-Optimization Sampling results demonstrate signi\ufb01cant improvements of convergence rates over standard SPOS. Remarkably, when comparing our convergence rates with those of variancereduced stochastic gradient Langevin dynamics (SGLD), our theory indicates faster convergence rates of variance-reduced SPOS when the number of particles is large enough. Our theoretical results are veri\ufb01ed by a number of experiments on both synthetic and real datasets. 2 Preliminaries 2.1 Stochastic gradient MCMC In Bayesian sampling, one aims at sampling from a posterior distribution p(\u03b8| X) \u221d p(X |\u03b8)p(\u03b8), where \u03b8 \u2208Rd represents the model parameter, and X \u225c{xj}N j=1 is the dataset. Let p(\u03b8| X) = (1/Z) exp(\u2212U(\u03b8)), where U(\u03b8) = \u2212log p(X |\u03b8) \u2212log p(\u03b8) \u225c\u2212 N X j=1 log p(xi |\u03b8) \u2212log p(\u03b8) is referred to as the potential energy function, and Z is the normalizing constant. We further de\ufb01ne the full gradient F and individual gradient Fj used in our paper: Fj(\u03b8) \u225c\u2212\u2207\u03b8 log p(xj |\u03b8) \u22121 N \u2207\u03b8 log p(\u03b8) = 1 N \u2207\u03b8U(\u03b8) F(\u03b8) \u225c\u2207\u03b8U(\u03b8) = N X j=1 Fj(\u03b8) We can de\ufb01ne a stochastic di\ufb00erential equation, an instance of It\u00f3 di\ufb00usion whose stationary distribution equals to the target posterior distribution p(\u03b8| X). For example, consider the following 1st-order Langevin dynamic: d\u03b8t = \u2212\u03b2\u22121F(\u03b8t)dt + p 2\u03b2\u22121dWt , (1) where t is the time index; Wt \u2208Rd is d-dimensional Brownian motion, and \u03b2 a scaling factor. By the Fokker-Planck equation [Kol31, Ris89], the stationary distribution of (1) equals to p(\u03b8| X). SG-MCMC algorithms are discretized numerical approximations of It\u00f3 di\ufb00usions (1). To make algorithms e\ufb03cient in a big-data setting, the computationally-expensive term F is replaced with its unbiased stochastic approximations with a random subset of the dataset in each interation, e.g. F can be approximated by a stochastic gradient: Gk \u225cN B X j\u2208Ik Fj(\u03b8k) = \u2212\u2207log p(\u03b8k) \u2212N B X j\u2208Ik \u2207\u03b8k log p(xj |\u03b8k) where Ik is a random subset of {1, 2, \u00b7 \u00b7 \u00b7 , N} with size B. The above de\ufb01nition of Gk re\ufb02ects the fact that we only have information from B \u226aN data points \fZhang & Zhao & Chen 5 in each iteration. This is the resource where the variance we try to reduce comes from. We should notice that Gk is also used in standard SVGD and SPOS. As an example, SGLD is a numerical solution of (1), with an update equation: \u03b8k+1 = \u03b8k \u2212\u03b2\u22121Gkh + p 2\u03b2\u22121h\u03bek, where h means the step size and \u03bek \u223cN(0, I). 2.2 Stein variational gradient descent Di\ufb00erent from SG-MCMC, SVGD initializes a set of particles, which are iteratively updated to approximate a posterior distribution. Speci\ufb01cally, we consider a set of particles {\u03b8(i)}M i=1 drawn from some distribution q. SVGD tries to update these particles by doing gradient descent on the interactive particle system via \u03b8(i) \u2190\u03b8(i) + h\u03c6(\u03b8(i)), \u03c6 = arg max \u03c6\u2208F { \u2202 \u2202hKL(q[h\u03c6]||p)} where \u03c6 is a function perturbation direction chosen to minimize the KL divergence between the updated density q[h\u03c6] induced by the particles and the posterior p(\u03b8| X). The standard SVGD algorithm considers F as the unit ball of a vector-valued reproducing kernel Hilbert space (RKHS) H associated with a kernel \u03ba(\u03b8, \u03b8\u2032). In such a setting, [LW16] shows that \u03c6(\u03b8) = E\u03b8\u2032\u223cq[\u03ba(\u03b8, \u03b8\u2032)F(\u03b8\u2032) + \u2207\u03b8\u2032\u03ba(\u03b8, \u03b8\u2032)]. (2) When approximating the expectation E\u03b8\u2032\u223cq[\u00b7] with an empirical distribution formed by a set of particles {\u03b8(i)}M i=1 and adopting stochastic gradients G(i) k \u225cN B P j\u2208Ik Fj(\u03b8(i) k ), we arrive at the following update for the particles: \u03b8(i) k+1 = \u03b8(i) k + h M M X q=1 h \u03ba(\u03b8(q) k , \u03b8(i) k )G(i) k + \u2207\u03b8(q) k \u03ba(\u03b8(q) k , \u03b8(i) k ) i (3) SVGD then applies (3) repeatedly for all the particles. 2.3 Stochastic particle-optimization sampling In this paper, we focus on RBF kernel \u03ba(\u03b8, \u03b8\u2032) = exp(\u2212\u2225\u03b8\u2212\u03b8\u2032\u22252 2\u03b72 ) due to its wide use in both theoretical analysis and practical applications. Hence, we can use a function K(\u03b8) = exp(\u2212\u2225\u03b8\u22252 2\u03b72 ) to denote the kernel \u03ba(\u03b8, \u03b8\u2032). According to the work of [CZW+18, ZZC18], the stationary distribution of the \u03c1t in the following partial di\ufb00erential equation equals to p(\u03b8| X). \u2202t\u03c1t =\u2207\u03b8 \u00b7 (\u03c1t\u03b2\u22121F(\u03b8) + \u03c1tEY \u223c\u03c1tK(\u03b8 \u2212Y )F(Y ) \u2212\u03c1t(\u2207K \u2217\u03c1t) + \u03b2\u22121\u2207\u03b8\u03c1t) . (4) When approximating the \u03c1t in Eq.(4) with an empirical distribution formed by a set of particles {\u03b8(i)}M i=1, [ZZC18] derive the following di\ufb00usion process characterizing \f6 Variance Reduced Stochastic Particle-Optimization Sampling the SPOS algorithm. d\u03b8(i) t = \u2212\u03b2\u22121F(\u03b8(i) t )dt \u22121 M M X q=1 K(\u03b8(i) t \u2212\u03b8(q) t )F(\u03b8(q) t )dt + 1 M M X q=1 \u2207K(\u03b8(i) t \u2212\u03b8(q) t )dt + p 2\u03b2\u22121dW(i) t \u2200i (5) It is worth noting that if we set the initial distribution of all the particles \u03b8(i) 0 to be the same, the system of these M particles is exchangeable. So the distributions of all the \u03b8(i) t are identical and can be denoted as \u03c1t. When solving the above di\ufb00usion process with a numerical method and adopting stochastic gradients G(i) k , one arrives at the SPOS algorithm of [ZZC18] with the following update equation: \u03b8(i) k+1 = \u03b8(i) k \u2212h\u03b2\u22121G(i) k \u2212h M M X j=1 K(\u03b8(i) k \u2212\u03b8(j) k )G(j) k + h M M X j=1 \u2207K(\u03b8(i) k \u2212\u03b8(j) k ) + p 2\u03b2\u22121h\u03be(i) k (6) where \u03be(i) k \u223cN(0, I). And SPOS will apply update (6) repeatedly for all the particles \u03b8(i) k . Detailed theoretical results for SPOS are reviewed in the Supplementary Material (SM). 3 Variance Reduction in SPOS In standard SPOS, each particle is updated by adopting G(i) k \u225cN B P j\u2208Ik Fj(\u03b8(i) k ). Due to the fact that one can only access B \u226aN data points in each update, the increased variance of the \u201cnoisy gradient\u201d G(i) k would cause a slower convergence rate. A simple way to alleviate this is to increase B by using larger minibatches. Unfortunately, this would bring more computational costs, an undesired side e\ufb00ect. Thus more e\ufb00ective variance-reduction methods are needed for SPOS. Inspired by recent work on variance reduction in SGLD, e.g., [DRP+16, CFM+18, ZXG18], we develop three di\ufb00erent variance-reduction algorithms for SPOS based on SAGA [DBLJ14] and SVRG [JZ13] in stochastic optimization. 3.1 SAGA-POS SAGA-POS generalizes the idea of SAGA [DBLJ14] to an interactive particleoptimization system. For each particle \u03b8(i) k , we use {g(i) k,j}N j=1 as an approximation for each individual gradient Fj(\u03b8(i) k ). An unbiased estimate of the full gradient F(\u03b8(i) k ) is \fZhang & Zhao & Chen 7 calculated as: G(i) k = N X j=1 g(i) k,j + N B X j\u2208Ik (Fj(\u03b8(i) k ) \u2212g(i) k,j), \u2200i (7) In each iteration, {g(i) k,j}N j=1 will be partially updated under the following rule: g(i) k+1,j = Fj(\u03b8(i) k ) if j \u2208Ik, and g(i) k+1,j = g(i) k,j otherwise. The algorithm is described in Algorithm 3.1. Compared with standard SPOS, SAGA-POS also enjoys highly computational e\ufb03ciency, as it does not require calculation of each Fj(\u03b8(i) k ) to get the full gradient F(\u03b8(i) k ) in each iteration. Hence, the computational time of SAGA-POS is almost the same as that of POS. However, our analysis in Section 4 shows that SAGA-POS endows a better convergence rate. From another aspect, SAGA-POS has the same drawback of SAGA-based algorithms, which requires memory scaling at a rate of O(MNd) in the worst case. For each particle \u03b8(i) k , one needs to store N gradient approximations {g(i) k,j}N j=1. Fortunately, as pointed out by [DRP+16, CFM+18], in some applications, the memory cost scales only as O(N) for SAGA-LD, which corresponds to O(MN) for SAGA-POS as M particles are used. Algorithm 1 SAGA-POS Input: A set of initial particles {\u03b8(i) 0 }M i=1, each \u03b8(i) 0 \u2208Rd, step size hk, batch size B. Initialize {g(i) 0,j}N j=1 = {Fj(\u03b8(i) 0 )}N j=1 for all i \u2208 {1, ..., M}; 1: for iteration k= 0,1,...,T do 2: Uniformly sample Ik from {1, 2, ..., N} randomly with replacement such that |Ik| = B; 3: Sample \u03be(i) k \u223cN(0, Id\u00d7d), \u2200i ; 4: Update G(i) k \u2190 N P j=1 g(i) k,j + N B P j\u2208Ik (Fj(\u03b8(i) k ) \u2212g(i) k,j), \u2200i ; 5: Update each \u03b8(i) k with Eq.(6); 6: Update {g(i) k,j}N j=1, \u2200i : if j \u2208Ik, set g(i) k+1,j \u2190Fj(\u03b8(i) k ); else, set g(i) k+1,j \u2190g(i) k,j 7: end for Output:{\u03b8(i) T }M i=1 Remark When compared with SAGA-LD, it is worth noting that M particles are used in both SPOS and SAGA-POS. This makes the memory complexity M times worse than SAGA-LD in training, thus SAGA-POS does not seem to bring any advantages over SAGA-LD. However, this intuition is not correct. As indicated by our theoretical results in Section 4, when the number of particles M is large enough, the convergence rates of our algorithms are actually better than those of variance-reduced SGLD counterparts. \f8 Variance Reduced Stochastic Particle-Optimization Sampling 3.2 SVRG-POS Under limited memory, we propose SVRG-POS, which is based on the SVRG method of [JZ13]. For each particle \u03b8(i) k , ones needs to store a stale parameter e \u03b8(i), and update it occasionally for every \u03c4 iterations. At each update, we need to further conduct a global evaluation of full gradients at e \u03b8(i), i.e., e G(i) \u2190F(\u03b8(i) k ) = F(e \u03b8(i)). An unbiased gradient estimate is then calculated by leveraging both e G(i) and e \u03b8(i) as: G(i) k \u2190e G(i) + N B X j\u2208Ik [Fj(\u03b8(i) k ) \u2212Fj(e \u03b8(i))] (8) The algorithm is depicted in Algorithm 3.2, where one only needs to store e \u03b8(i) and e G(i), instead of gradient estimates of all the individual Fj. Hence the memory cost scales as O(Md), almost the same as that of standard SPOS. We note that although SVRG-POS alleviates the storage requirement of SAGAPOS remarkably, it also endows downside that full gradients, F(e \u03b8(i)) = PN j=1 F(e \u03b8(i)), are needed to be re-computed every \u03c4 iterations, leading to high computational cost in a big-data scenario. Algorithm 2 SVRG-POS Input: A set of initial particles {\u03b8(i) 0 }M i=1, each \u03b8(i) 0 \u2208Rd, step size h, epoch length \u03c4, batch size B. Initialize {e \u03b8(i)} \u2190{\u03b8(i) 0 }, e G(i) \u2190F(\u03b8(i) 0 ), \u2200i ; 1: for iteration k= 0,1,...,T do 2: if k mod \u03c4 =0 then 3: Option I i)Sample l \u223cunif(0, 1, .., \u03c4 \u22121) ii)Update e \u03b8(i) \u2190\u03b8(i) k\u2212l Update \u03b8(i) k \u2190e \u03b8(i), \u2200i iii)Update e G(i) \u2190F(\u03b8(i) k ), \u2200i 4: Option II i) Update e \u03b8(i) \u2190\u03b8(i) k ii)Update e G(i) \u2190F(\u03b8(i) k ), \u2200i 5: end if 6: Uniformly sample Ik from {1, 2, ..., N} randomly with replacement such that |Ik| = B; 7: Sample \u03be(i) k \u223cN(0, Id\u00d7d), \u2200i ; 8: Update G(i) k \u2190e G(i) + N B P j\u2208Ik [Fj(\u03b8(i) k ) \u2212Fj(e \u03b8(i))], \u2200i ; 9: Update each \u03b8(i) k with Eq.(6) 10: end for Output:{\u03b8(i) T }M i=1 Remark i) Similar to SAGA-POS, according to our theory in Section 4, SVRG-POS enjoys a faster convergence rate than SVRD-LD \u2013 its SGLD counterpart, although \fZhang & Zhao & Chen 9 M times more space are required for the particles. This provides a trade-o\ufb00between convergence rates and space complexity. ii) Previous work has shown that SAGA typically outperforms SVRG [DRP+16, CFM+18] in terms of convergence speed. The conclusion applies to our case, which will be veri\ufb01ed both by theoretical analysis in Section 4 and experiments in Section 5. 3.3 SVRG-POS+ The need of full gradient computation in SVRG-POS motives the development of SVRG-POS+. Our algorithm is also inspired by the recent work of SVRG-LD+ on reducing the computational cost in SVRG-LD [ZXG18]. The main idea in SVRGPOS+ is to replace the full gradient computation every \u03c4 iterations with a subsampled gradient, i.e., to uniformly sample |Jk| = b data points where Jk are random samples from {1, 2, ..., N} with replacement. Given the sub-sampled data, e \u03b8(i) and e G(i) are updated as: e \u03b8(i) = \u03b8(i) k , e G(i) = N b P j\u2208Jk Fj(\u03b8(i) k ). The full algorithm is shown in Algorithm 3.3. Algorithm 3 SVRG-POS+ Input : A set of initial particles {\u03b8(i) 0 }M i=1, each \u03b8(i) 0 \u2208Rd, step size h, epoch length \u03c4, batch size B. Initialize {e \u03b8(i)} \u2190{\u03b8(i) 0 }, e G(i) \u2190F(\u03b8(i) 0 ), \u2200i ; 1: for iteration k= 0,1,...,T do 2: if k mod \u03c4 =0 then 3: i) Uniformly sample Jk from {1, 2, ..., N} with replacement such that |Jk| = b; ii) Update e \u03b8(i) \u2190\u03b8(i) k e G(i) \u2190N b P j\u2208Jk Fj(\u03b8(i) k ), \u2200i ; 4: end if 5: Uniformly sample Ik from {1, 2, ..., N} with replacement such that |Ik| = B; 6: Sample \u03be(i) k \u223cN(0, Id\u00d7d), \u2200i ; 7: Update G(i) k \u2190e G(i) + N B P j\u2208Ik [Fj(\u03b8(i) k ) \u2212Fj(e \u03b8(i))], \u2200i ; 8: Update each \u03b8(i) k with Eq.(6) 9: end for Output:{\u03b8(i) T }M i=1 4 Convergence Analysis In this section, we prove non-asymptotic convergence rates for the SAGA-POS, SVRG-POS and SVRG-POS+ algorithms under the 2-Wasserstein metric, de\ufb01ned as W2(\u00b5, \u03bd) = \u0012 inf \u03b6\u2208\u0393(\u00b5,\u03bd) Z Rd\u00d7Rd \u2225X\u00b5 \u2212X\u03bd\u22252d\u03b6(X\u00b5, X\u03bd) \u0013 1 2 \f10 Variance Reduced Stochastic Particle-Optimization Sampling where \u0393(\u00b5, \u03bd) is the set of joint distributions on Rd \u00d7Rd with marginal distribution \u00b5 and \u03bd. Let \u00b5\u2217denote our target distribution, and \u00b5T the distribution of 1 M PM i=1 \u03b8(i) T derived via (5) after T iterations. Our analysis aims at bounding W2(\u00b5T, \u00b5\u2217). We \ufb01rst introduce our assumptions. Assumption 1 F and K satisfy the following conditions: \u2022 There exist two positive constants mF and mW, such that \u27e8F(\u03b8) \u2212F(\u03b8\u2032), \u03b8 \u2212 \u03b8\u2032\u27e9\u2265mF\u2225\u03b8 \u2212\u03b8\u2032\u22252 and \u27e8\u2207K(\u03b8) \u2212\u2207K(\u03b8\u2032), \u03b8 \u2212\u03b8\u2032\u27e9\u2264\u2212mK\u2225\u03b8 \u2212\u03b8\u2032\u22252. \u2022 F is bounded and LF-Lipschitz continuous with LF i.e. \u2225F(\u03b8)\u2225\u2264HF and \u2225F(\u03b8) \u2212F(\u03b8\u2032)\u2225\u2264LF\u2225\u03b8 \u2212\u03b8\u2032\u2225; K is LK-Lipschitz continuous for some LK \u22650 and bounded by some constant HK > 0. \u2022 K is an even function, i.e., K(\u2212\u03b8) = K(\u03b8). Assumption 2 There exists a constant DF > 0 such that \u2225\u2207F(\u03b8) \u2212\u2207F(\u03b8\u2032)\u2225\u2264 DF\u2225\u03b8 \u2212\u03b8\u2032\u2225. Assumption 3 There exits a constant \u03c3 such that for all j \u2208{1, 2, ..., N}, E[\u2225Fj(\u03b8) \u22121 N N X j=1 Fj(\u03b8)\u22252] \u2264d\u03c32/N 2 Remark i) Assumption 1 is adopted from [ZZC18] which analyzes the convergence property of SPOS. The \ufb01rst bullet of Assumption 1 suggests U(\u00b7) is a strongly convex function, which is the general assumption in analyzing SGLD [DK17, DM16] and its variance-reduced variants [ZXG18, CFM+18]. It is worth noting that although some work has been done to investigate the non-convex case, it still has signi\ufb01cant value to analysis the convex case, which are more instructive and meaningful to address the practical issues [DK17, DM16, ZXG18, CFM+18]. ii) All of the mf, LF, HF and DF could scale linearly with N. iii) K(\u03b8) = exp(\u2212\u2225\u03b8\u22252 2\u03b72 ) can satisfy the above assumptions by setting the bandwidth large enough, since we mainly focus on some bounded space in practice. Consequently, \u2207K can also be L\u2207K-Lipschitz continuous and bounded by H\u2207K; K can also be Hessian Lipschitz with some positive constant D\u22072K For the sake of clarity, we de\ufb01ne some constants which will be used in our theorems. C1 = H\u2207K + HF \u221a 2(\u03b2\u22121 \u22123HFLK \u22122LF) C2 = p 2(\u03b2\u22121LF + 2LKHF + HKLF + L\u2207K)2 + 2 C3 = \u03b2\u22121mF \u22122LF \u22123HFLK C4 = \u03b2\u22121DF + 4D\u22072K + 4HFL\u2207K + 2LFH\u2207K + 2HFLK + LFHK C5 = 2\u03b2\u22121\u03c32 + 2H2 K\u03c32 \fZhang & Zhao & Chen 11 Now we present convergence analysis for our algorithms, where \u03b1 is some positive constant independent of T. Theorem 1 Let \u00b5T denote the distribution of the particles after T iterations with SAGA-POS. Under Assumption 1 and 2, let the step size h < B 8C2N and the batch size B \u22659, the convergence rate of SAGA-POS is bounded as W2(\u00b5T,\u00b5\u2217) \u2264C1 \u221a M + 5 exp(\u2212C3h 4 T)W2(\u00b50, \u00b5\u2217) + 2hC4dM 1/2\u2212\u03b1 C3 + 2hC2 3 2\u221a d C3M \u03b1 + 24hC2 \u221a dN M \u03b1\u221aC3B (9) Theorem 2 Let \u00b5T denote the distribution of the particles after T iterations with SVRG-POS in Algorithm 3.2. Under Assumption 1 and 2, if we choose Option I and set the step size h < 1 8C2, the batch size B \u22652 and the epoch length \u03c4 = 4 hC3(1\u22122hC2(1+2/B)), the convergence rate of SVRG-POS is bounded for all T, which mod \u03c4 = 0, as W2(\u00b5T , \u00b5\u2217) \u2264C1 \u221a M + exp(\u2212C3h 56 T) \u221aC2 \u221aC3 W2(\u00b50, \u00b5\u2217) + 2hC4dM1/2\u2212\u03b1 C3 + 2hC2 3 2 \u221a d C3M\u03b1 + 64C2 3 2 \u221a hd M\u03b1\u221a BC3 (10) If we choose Option II and set the step size h < \u221a B 4\u03c4C2, the convergence rate of SVRG-POS is bounded for all T as W2(\u00b5T, \u00b5\u2217) \u2264C1 \u221a M + exp(\u2212C3h 4 T)W2(\u00b50, \u00b5\u2217) + \u221a 2hC4dM 1/2\u2212\u03b1 C3 + 5hC2 3 2\u221a d C3M \u03b1 + 9hC2\u03c4 \u221a d M \u03b1\u221aBC3 (11) Theorem 3 Let \u00b5T denote the distribution of particles after T iterations with SVRG-POS+. Under Assumption 1, 2 and 3, if we set the step size h \u2264 min{( BC3 24C24\u03c4 2) 1 3, 1 6\u03c4(C52/b+C2)}, then the convergence rate of SVRG-POS+ is bounded for all T as W2(\u00b5T,\u00b5\u2217) \u2264C1 \u221a M + (1 \u2212hC3/4)TW2(\u00b50, \u00b5\u2217) + 3C5d1/2 M \u03b1C3b1/21(b \u2264N) + 2h(C4dM 1/2\u2212\u03b1) C3 + 2hC3/2 2 d1/2 C3M \u03b1 + 4hC2(\u03c4d)1/2 \u22273h1/2d1/2C5 M \u03b1\u221aBC3 (12) Since the complexity has been discussed in the Section 3, we mainly focus on discussing the convergence rates here. Due to space limit, we move the comparison between \f12 Variance Reduced Stochastic Particle-Optimization Sampling convergence rates of the standard SPOS and its variance-reduced counterparts such as SAGA-POS into the SM. Speci\ufb01cally, adopting the standard framework of comparing di\ufb00erent variance-reduction techniques in SGLD [DRP+16, CFM+18, ZXG18], we focus on the scenario where mf, LF, HF and DF all scale linearly with N with N \u226bd. In this case, the dominating term in Theorem 1 for SAGA-POS is the last term, O( hC2 \u221a d M\u03b1B ) \u2248O( hN \u221a d M\u03b1B ). Thus to achieve an accuracy of \u03b5, we would need the stepsize hag = O( \u03b5M\u03b1B N \u221a d ). For SVRG-POS, the dominating term in Theorem 2 is O( \u221a hNd M\u03b1\u221a B) for Option I and O( \u03c4hN \u221a d M\u03b1\u221a B ) for Option II. Hence, for an accuracy of \u03b5, the corresponding step sizes are hvr1 = O( \u03b52M2\u03b1B Nd ) and hvr2 = O( \u03b5M\u03b1\u221a B \u03c4N \u221a d ), respectively. Due to the fact that the mixing time T for these methods is roughly proportional to the reciprocal of step size [CFM+18], it is seen that when \u03b5 is small enough, one can have hvr1 \u226ahag, which causes SAGA-POS converges faster than SVRG-POS (Option I). Similar results hold for Option II since the factor 1 \u221a B\u03c4 in hvr2 would make the step size even smaller. More theoretical results are given in the SM. Remark We have provided theoretical analysis to support the statement of i) in Remark 3.2 . Moreover, we should also notice in SAGA-POS, stepsize hag = O( \u03b5M\u03b1B N \u221a d ) has an extra factor, M \u03b1, compared with the step size O( \u03b5B N \u221a d) used in SAGA-LD [CFM+18]\u2217. This means SAGA-POS with more particles (M is large) would outperform SAGA-LD. SVRG-POS and SVRG-POS+ have similar conclusions. This theoretically supports the statements of Remark 3.1 and i) in Remark 3.2. Furthermore, an interesting result from the above discussion is that when hvr1 = O( \u03b52M2\u03b1B Nd ) in SVRG-POS, there is an extra factor M compared to the stepsize O( \u03b52B Nd ) in SVRG-LD [CFM+18]. Since the order of M 2\u03b1 is higher than M \u03b1, one expects that the improvement of SVRG-POS over SVRG-LD is much more signi\ufb01cant than that of SAGA-POS over SAGA-LD. This conclusion will be veri\ufb01ed in our experiments. 5 Experiments We conduct experiments to verify our theory, and compare SAGA-POS, SVRGPOS and SVRG-POS+ with existing representative Bayesian sampling methods with/without variance-reduction techniques, e.g. SGLD and SPOS without variance reduction; SAGA-LD, SVRG-LD and SVRG-LD+ with variance reduction. For SVRG-POS, we focus on Option I in Algorithm 3.2 to verify our theory. 5.1 Synthetic log-normal distribution We \ufb01rst evaluate our proposed algorithms on a log-normal synthetic data, de\ufb01ned as p(x|\u00b5) = 1 x \u221a 2\u03c0 exp(\u2212(ln x\u2212\u00b5)2 2 ) where x, \u00b5 \u2208R10. We calculate log-MSE of the sampled \u201cmean\u201d w.r.t. the true value, and plot the log-MSE versus number of passes \u2217For fair comparisons with our algorithms, we consider variance-reduced versions of SGLD with M independent chains. \fZhang & Zhao & Chen 13 Figure 1: Log-MSE of the mean parameter versus the number of dataset pass. through data [CFM+18], like other variance-reduction algorithms in Figure 1, which shows that SAGA-POS and SVRG-POS converge the fastest among other algorithms. It is also interesting to see SPOS even outperforms both SAGA-LD and SVRG-LD. 5.2 Bayesian logistic regression Following related work such as [DRP+16], we test the proposed algorithms for Bayesian-logistic-regression (BLR) on four publicly available datasets from the UCI machine learning repository: Australian (690-14), Pima (768-8), Diabetic (115120) and Susy (100000-18), where (N \u2212d) means a dataset of N data points with dimensionality d. The \ufb01rst three datasets are relatively small, and the last one is a large dataset suitable to evaluate scalable Bayesian sampling algorithms. Speci\ufb01cally, consider a dataset {Xi, yi}N i=1 with N samples, where Xi \u2208Rd and yi \u2208{0, 1}. The likelihood of a BLR model is written as p(yi = 1|Xi, \u03b1) = sigmoid(\u03b1TXi) with regression coe\ufb03cient \u03b1 \u2208Rd, which is assumed to be sampled from a standard multivariate Gaussian prior N(0, I) for simplicity. The datasets are split into 80% training data and 20% testing data. Optimized constant stepsizes are applied for each algorithm via grid search. Following existing work, we report testing accuracy and log-likelihood versus the number of data passes for each dataset, averaging over 10 runs with 50 particles. The minibatch size is set to 15 for all experiments. \f14 Variance Reduced Stochastic Particle-Optimization Sampling 5.2.1 Variance-reduced SPOS versus SPOS We \ufb01rst compare SAGA-POS, SVRG-POS and SVRG-POS+ with SPOS without variance reduction proposed in [ZZC18]. The testing accuracies and log-likelihoods versus number of passes through data on the four datasets are plotted in Figure 2. It is observed that SAGA-POS converges faster than both SVRG-POS and SVRGPOS+, all of which outperform SPOS signi\ufb01cantly. On the largest dataset SUSY, SAGA-POS starts only after one pass through data, which then converges quickly, outperforming other algorithms. And SVRG-POS+ outperforms SVRG-POS due to the dataset SUSY is so large. All of these phenomena are consistent with our theory. 5.2.2 Variance-reduced SPOS versus variance-reduced SGLD Next we compare the three variance-reduced SPOS algorithms with its SGLD counterparts, i.e., SAGA-LD, SVRG-LD and SVRG-LD+. The results are plotted in Figure 3. Similar phenomena are observed, where both SAGA-POS and SVRG-POS outperform SAGA-LD and SVRG-LD, respectively, consistent with our theoretical results discussed in Remark 3.1 and 3.2. Interestingly, in the PIMA dataset case, SVRG-LD is observed to perform even worse (converge slower) than standard SGLD. Furthermore, as discussed in Remark 4, our theory indicates that the improvement of SVRG-POS over SVRG-LD is more signi\ufb01cant than that of SAGA-POS over SAGA-LD. This is indeed true by inspecting the plots in Figure 3. 5.2.3 Impact of number of particles Finally we examine the impact of number of particles to the convergence rates. As indicated by Theorems 1-3, for a \ufb01xed number of iterations T, the convergence error in terms of 2-Wasserstein distance decreases with increasing number of particles. To verify this, we run SAGA-POS and SVRG-POS for BLR with the number of particles ranging between {1, 2, 4, 8, 16}. The test log-likelihoods versus iteration numbers are plotted in Figure 4, demonstrating consistency with our theory. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1809.01293v5",
+                    "title": "Stochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory",
+                    "abstract": "Particle-optimization-based sampling (POS) is a recently developed effective\nsampling technique that interactively updates a set of particles. A\nrepresentative algorithm is the Stein variational gradient descent (SVGD). We\nprove, under certain conditions, SVGD experiences a theoretical pitfall, {\\it\ni.e.}, particles tend to collapse. As a remedy, we generalize POS to a\nstochastic setting by injecting random noise into particle updates, thus\nyielding particle-optimization sampling (SPOS). Notably, for the first time, we\ndevelop {\\em non-asymptotic convergence theory} for the SPOS framework (related\nto SVGD), characterizing algorithm convergence in terms of the 1-Wasserstein\ndistance w.r.t.\\! the numbers of particles and iterations. Somewhat\nsurprisingly, with the same number of updates (not too large) for each\nparticle, our theory suggests adopting more particles does not necessarily lead\nto a better approximation of a target distribution, due to limited\ncomputational budget and numerical errors. This phenomenon is also observed in\nSVGD and verified via an experiment on synthetic data. Extensive experimental\nresults verify our theory and demonstrate the effectiveness of our proposed\nframework.",
+                    "authors": "Jianyi Zhang, Ruiyi Zhang, Lawrence Carin, Changyou Chen",
+                    "published": "2018-09-05",
+                    "updated": "2020-03-29",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Recently there has been extensive development of scalable Bayesian sampling algorithms, such as stochastic gradient MCMC (SG-MCMC) [Welling and Teh, 2011, Chen et al., 2014, Ding et al., 2014, Chen et al., Proceedings of the 23rdInternational Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2020, Palermo, Italy. PMLR: Volume 108. Copyright 2020 by the author(s). 2015] and Stein variational gradient descent (SVGD) [Liu and Wang, 2016]. SG-MCMC is a family of scalable Bayesian sampling algorithms built on It\u00f3 diffusions, stochastic di\ufb00erential equations (SDEs) with appropriately designed coe\ufb03cients whose stationary distributions match the target distributions. One potential issue of SG-MCMC is that samples may be highly correlated partially due to the nature of Markov chains, leading to undesired low sample e\ufb03ciency. SVGD, on the other hand, belongs to the family of particleoptimization-based sampling methods that optimize a set of interacting particles to minimize some distance metric (e.g., KL-divergence) between the target distribution and the particle-induced approximate distribution. By optimization, one seeks to maintain an optimal set of particles. Recent development of SVGD has shown that the underlying mathematical principle is based on a family of nonlinear partial di\ufb00erential equations (PDEs) [Liu, 2017]. Although achieving signi\ufb01cant practical successes [Liu and Wang, 2016, Feng et al., 2017, Liu et al., 2017, Haarnoja et al., 2017, Zhang et al., 2018a, Zhang et al., 2019, Liu and Zhu, 2018], little theory is available to fully understand its non-asymptotic convergence properties. A recent theoretical development has interpreted SVGD as a special type of gradient \ufb02ows, and developed theory to disclose its asymptotic convergence behavior [Liu, 2017]. The asymptotic theory is also studied in [Lu et al., 2018]. A very recent work [Liu and Wang, 2018] investigated non-asymptotic properties of SVGD, limited to the region of \ufb01nite particles and in\ufb01nite time with restricted conditions. In [\u015eim\u015fekli et al., 2018] the convergence property of the sliced-Wasserstein \ufb02ow are only considered under an in\ufb01nite-particle setting. Recently, [Chen et al., 2018] uni\ufb01ed SG-MCMC and SVGD by proposing a particle-optimization-sampling (POS) framework to interpret both as Wasserstein gradient \ufb02ows (WGFs). Generally, a WGF is a PDE de\ufb01ned on the space of probability measures, describing the evolution of a density over time. [Chen et al., 2018] de\ufb01ned a WGF by combining the corresponding PDEs for both SG-MCMC and SVGD, and solved it with deterministic particle approximations. However, arXiv:1809.01293v5  [stat.ML]  29 Mar 2020 \fStochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory due to its di\ufb00usion nature, deterministic-particle approximation leads to a hard-to-control error, making it challenging for theoretical analysis. Our contributions In this paper, we generalize POS to a stochastic setting, and develop a novel analytical framework based on granular media equations [Malrieu, 2003, Cattiaux et al., 2008] to analyze its nonasymptotic convergence properties. Our contributions are summarized as follows: i) We \ufb01rst identify a pitfall of standard SVGD, where particles tend to collapse under certain conditions and measurement, indicating that developing non-asymptotic theory for SVGD is changing(if possible at all). ii) Based on the uni\ufb01ed framework in [Chen et al., 2018], we propose stochastic particle-optimization sampling (SPOS) by injecting Gaussian noise in particle updates to overcome the pitfall. iii) For the \ufb01rst time, we develop nonasymptotic convergence theory for the family of SPOS algorithms, considering both convexand nonconvex-energy targets. Di\ufb00erent from existing theory for SG-MCMC-based algorithms [Teh et al., 2016, Vollmer et al., 2016, Chen et al., 2015, Raginsky et al., 2017, Zhang et al., 2017, Xu et al., 2018], our development relies on the theory of nonlinear PDEs, which is more involved and less explored in the literature. In particular, we adopt tools from granular media equations [Malrieu, 2003, Cattiaux et al., 2008] to develop non-asymptotic error bounds in terms of 1-Wasserstein distance. More detailed distinctions between our work and existing work are discussed in Section N of the Supplementary Material (SM). Somewhat surprisingly, our theory indicates that utilizing more particles does not necessarily lead to better approximations, due to the numerical errors in the algorithms. This phenomenon is also observed for SVGD. iv) Our theory and advantages of the algorithm are veri\ufb01ed via various experiments, including experiments on synthetic data, Bayesian deep learning and Bayesian exploration for reinforcement learning. 2 Preliminaries Notation We use bold letters to denote variables in continuous-time di\ufb00usions and model de\ufb01nitions (no numerical methods included yet), e.g., \u03b8\u03c4 in (1) below (indexed by \u201ctime\u201d \u03c4). By contrast, unbold letters are used to denote parameters in algorithms (numerical solutions of continuous-time di\ufb00usions), e.g., \u03b8(i) k in (3) below (indexed by \u201citeration\u201d k). For conciseness, all proofs, extra experimental results and a discussion on algorithmic complexity are presented in the SM. 2.1 Stochastic gradient MCMC In Bayesian sampling, one aims to generate random samples from a posterior distribution p(\u03b8|X) \u221d p(X|\u03b8)p(\u03b8), where \u03b8 \u2208Rd represents the model parameter with a prior distribution p(\u03b8), and X \u225c {xq}N q=1 represents the observed data with likelihood p(X|\u03b8) = Q q p(xq |\u03b8). De\ufb01ne the potential energy as: U(\u03b8) \u225c\u2212log p(X|\u03b8) \u2212log p(\u03b8) = \u2212PN q=1 \u0000log p(xq |\u03b8) + 1 N log p(\u03b8) \u0001 \u225c PN q=1 Uq(\u03b8). SG-MCMC algorithms belong to di\ufb00usion-based sampling methods, where a continuous-time di\ufb00usion process is designed such that its stationary distribution matches the target posterior distribution. The di\ufb00usion process is driven by a speci\ufb01c SDE. For example, in stochastic gradient Langevin dynamic (SGLD) [Welling and Teh, 2011], the SDE endows the following form: d\u03b8\u03c4 = \u2212\u03b2\u22121F(\u03b8\u03c4)d\u03c4 + p 2\u03b2\u22121dW\u03c4 , (1) where F(\u03b8) \u225c \u2207\u03b8U(\u03b8) = PN q=1 \u2207\u03b8Uq(\u03b8) \u225c PN q=1 Fq(\u03b8); \u03c4 is the time index, \u03b2 > 0 is the temperature parameter, and W\u03c4 \u2208Rd is a d-dimensional Brownian motion. More instances of SDEs corresponding to other SG-MCMC algorithms can be de\ufb01ned by specifying di\ufb00erent forms of F and potentially other di\ufb00usion coe\ufb03cients. We focus on SGLD and (1) in this paper, and refer interested readers to [Ma et al., 2015] for a more detailed description of general SG-MCMC algorithms. Denote the probability density function of \u03b8\u03c4 in (1) as \u03bd\u03c4, and let a \u00b7 b \u225ca\u22a4b for two vectors a and b. It is known that \u03bdt is characterized by the following Fokker-Planck (FP) equation [Risken, 1989]: \u2202\u03c4\u03bd\u03c4 = \u2207\u03b8 \u00b7 (\u03b2\u22121\u03bd\u03c4F(\u03b8) + \u03b2\u22121\u2207\u03b8\u03bd\u03c4) . (2) According to [Chiang and Hwang, 1987], the stationary distribution \u03bd\u221eequals to our target distribution p(\u03b8|X). As a result, SGLD is designed to generates samples from p(\u03b8|X) by numerically solving the SDE (1). For scalability, it replaces F(\u03b8k) in each iteration with an unbiased evaluation by randomly sampling a subset of X, i.e., F(\u03b8k) is approximated by: Gk \u225c N Bk P q\u2208Ik Fq(\u03b8k), where Ik is a random subset of [1, 2, \u00b7 \u00b7 \u00b7 , N] with size Bk in each iteration. As a result, SGLD uses the Euler method with stepsize hk to numerically solve (1), resulting in the update equation: \u03b8k+1 = \u03b8k \u2212\u03b2\u22121Gkhk + p 2\u03b2\u22121hk\u03bek, with \u03bek \u223cN(0, I). 2.2 Stein variational gradient descent Di\ufb00erent from SG-MCMC, SVGD is a deterministic particle-optimization algorithm that is able to generate samples from a target distribution. In the algorithm, a set of particles interact with each other, driving them to high density regions in the parameter space while keeping them far away from each other with an induced repulsive force. The update equations of the particles follow the fastest descent direction of the KLdivergence between current particle distribution and \fJianyi Zhang1, Ruiyi Zhang1, Lawrence Carin1, Changyou Chen2k the target distribution, on a RKHS (reproducing kernel Hilbert space) induced by a kernel function \u03ba(\u00b7, \u00b7) [Liu and Wang, 2016]. Formally, [Liu and Wang, 2016] derived the following update rules for the particles {\u03b8(i) k }M i=1 at the k-th iteration with stepsize hk and G(i) k \u225c N Bk P q\u2208Ik Fq(\u03b8(i) k ): for \u2200i, \u03b8(i) k+1 = \u03b8(i) k + hk M M X j=1 h \u03ba(\u03b8(j) k , \u03b8(i) k )G(i) k + \u2207\u03b8(j) k \u03ba(\u03b8(j) k , \u03b8(i) k ) i (3) where the \ufb01rst term in the bracket encourages moving particles to the density modes, and the second term serves as repulsive force that pushes away di\ufb00erent particles. In particular, the particle evolution (3) are numerical solutions of the ODEs: d\u03b8(i) \u03c4 = 1 M PM j=1 h \u03ba(\u03b8(j) \u03c4 , \u03b8(i) \u03c4 )F(\u03b8(i) \u03c4 ) + \u2207\u03b8(j) \u03c4 \u03ba(\u03b8(j) \u03c4 , \u03b8(i) \u03c4 ) i d\u03c4. Di\ufb00erent from SG-MCMC, only particles at the current iteration, {\u03b8(i) k }M i=1, are used to approximate the target distribution. 2.3 Particle-optimization based sampling SG-MCMC and SVGD, though they may look closely related, behave very di\ufb00erently as algorithms, e.g., stochastic and noninteractive versus deterministic and interactive particle updates. Recently, [Chen et al., 2018] proposed a deterministic particle-optimization framework that uni\ufb01ed SG-MCMC and SVGD. Specifically, the authors viewed both SG-MCMC and SVGD as solutions of Wasserstein gradient \ufb02ows (WGFs) on the space of probabilistic measures, and derived several deterministic particle-optimization techniques for particle evolution, like SVGD. For SG-MCMC, the FP equation (2) for SGLD is a special type of WGF. Together with an interpretation of SVGD as a special case of the Vlasov equation in the nonlinear PDE literature, [Chen et al., 2018] proposed a general form of PDE to characterize the evolution of the density for the model parameter \u03b8, denoted as \u03bd\u03c4 at time \u03c4 with \u03bd\u221ematching our target (posterior) distribution, i.e., \u2202\u03c4\u03bd\u03c4 = \u2207\u03b8 \u00b7 \u0000\u03bd\u03c4\u03b2\u22121F(\u03b8) + \u03bd\u03c4 (K \u2217\u03bd\u03c4(\u03b8)) + \u03b2\u22121\u2207\u03b8\u03bd\u03c4 \u0001 , (4) where K is a function controlling the interaction of particles in the PDE system. For example, in SVGD, [Chen et al., 2018] showed that K and K \u2217\u03bd\u03c4(\u03b8) endow the following forms: K \u2217\u03bd\u03c4(\u03b8) \u225c Z K(\u03b8, \u03b8\u2032)\u03bd\u03c4(\u03b8\u2032)d\u03b8\u2032 , (5) where K(\u03b8, \u03b8\u2032) \u225cF(\u03b8\u2032)\u03ba(\u03b8\u2032, \u03b8)\u2212\u2207\u03b8\u2032\u03ba(\u03b8\u2032, \u03b8) and \u03ba(\u00b7, \u00b7) is a kernel function such as the RBF kernel. In the following, we introduce a new unary function K(\u03b8) = exp(\u2212\u2225\u03b8\u22252 \u03b72 ), thus \u03ba(\u03b8, \u03b8\u2032) can be rewritten as \u03ba(\u03b8, \u03b8\u2032) = K(\u03b8 \u2212\u03b8\u2032). Hence, (4) with K de\ufb01ned in (5) is equivalently written as: \u2202\u03c4\u03bd\u03c4 =\u2207\u03b8 \u00b7 (\u03bd\u03c4\u03b2\u22121F(\u03b8) + \u03bd\u03c4 (EY \u223c\u03bd\u03c4 K(\u03b8 \u2212Y )F(Y ) \u2212\u2207K \u2217\u03bd\u03c4(\u03b8)) + \u03b2\u22121\u2207\u03b8\u03bd\u03c4) , (6) where Y is a random sample from \u03bd\u03c4 independent of \u03b8. Note our formula here is signi\ufb01cantly di\ufb00erent from standard granular media equations in the literature. Section N of the SM provides more details. Proposition 1 ([Chen et al., 2018]) The stationary distribution of (6) equals to our target distribution, which means \u03bd\u221e(\u03b8) = p(\u03b8|X). [Chen et al., 2018] proposed to solve (4) numerically with deterministic particle-optimization algorithms, such as what is called the blob method. Speci\ufb01cally, the continuous density \u03bd\u03c4 is approximated by a set of M particles {\u03b8(i) \u03c4 }M i=1 that evolve over time \u03c4, i.e. \u03bd\u03c4 \u2248 1 M PM i=1 \u03b4\u03b8(i) \u03c4 (\u03b8), where \u03b4\u03b8(i) \u03c4 (\u03b8) = 1 if \u03b8 = \u03b8(i) \u03c4 and 0 otherwise. Note \u2207\u03b8\u03bd\u03c4 in (4) is no longer a valid de\ufb01nition when adopting particle approximation for \u03bd\u03c4. Consequently, \u2207\u03b8\u03bd\u03c4 needs nontrivial approximations, e.g., by discrete gradient \ufb02ows or blob methods proposed in [Chen et al., 2018]. We omit the details here for simplicity. 3 Stochastic Particle-Optimization Sampling (SPOS) We \ufb01rst introduce a pitfall of SVGD, which is overcame by SPOS. In the analysis for both SVGD and SPOS, we impose the following basic assumptions. Assumption 1 Assume F and K satisfy the following assumptions: 1.1 F is LF -Lipschitz continuous i.e., \u2225F(\u03b8) \u2212 F(\u03b8\u2032)\u2225\u2264LF \u2225\u03b8 \u2212\u03b8\u2032\u2225. 1.2 K is LK-Lipschitz continuous; \u2207K is L\u2207KLipschitz continuous. 1.3 F(0) = 0 and K is an even function, i.e., K(\u2212\u03b8) = K(\u03b8). A few remarks: i) Assumptions 1.1 is widely adopted in the other theoretical works such as [Dalalyan and Karagulyan, 2017, Chatterji et al., 2018] ii) F(0) = 0 in Assumption 1.3 is reasonable, as F in our setting corresponds to an unnormalized log-posterior, which can be shifted such that F(0) = 0 for a speci\ufb01c problem. The assumptions of K are satis\ufb01ed due to the properties of Gaussian Kernel. 3.1 A pitfall of SVGD We \ufb01rst motivate SPOS by discovering a pitfall of standard SVGD: particles in SVGD tend to collapse to a local mode under some particular conditions. Inspired by the work on analyzing granular media equations by [Malrieu, 2003, Cattiaux et al., 2008], we measure \fStochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory the collapse by calculating the expected distance between exact particles (without numerical errors), called expected particle distance (EPD) de\ufb01ned below. Assumption 2 Assume F and K satisfy the following assumptions: 2.1 There exists positive mK such that \u27e8\u2207K(\u03b8) \u2212 \u2207K(\u03b8\u2032), \u03b8 \u2212\u03b8\u2032\u27e9\u2264\u2212mK\u2225\u03b8 \u2212\u03b8\u2032\u22252. 2.2 F is bounded by HF i.e., \u2225F(\u03b8)\u2225\u2264HF For an RBF kernel, this assumption could be satis\ufb01ed by setting the bandwidth large enough and only considering the concave region for simplicity. This may seem restrictive. However, this assumption is imposed only for the analysis of the pitfall property. It is not needed in the non-asymptotic convergence analysis. Besides, we point out what will happen without this assumption in Remark 1. Theorem 2 Under Assumptions 1 and 2, for the particles \u03b8(i) \u03c4 de\ufb01ned in Section 2.2, the EPD for SVGD is bounded as: EPD \u225c qPM i,j E\u2225\u03b8(i) \u03c4 \u2212\u03b8(j) \u03c4 \u22252 \u2264C0e\u22122\u03bb\u03c4, where C0 = qPM i,j \u2225\u03b8(i) 0 \u2212\u03b8(j) 0 \u22252, \u03bb = mK \u2212HF LK. Remark 1 1) In the case of \u03bb \u22650, Theorem 2 indicates that particles in SVGD would collapse to a point when \u03c4 \u2192\u221e. In practice, we usually \ufb01nd that particles are trapped in a local mode instead of collapsing in practice. This might be due to two reasons: i) Particles in SVGD are numerical solutions instead of exact solutions as used in EPD, which induces extra numerical errors; ii) Some particles might be out of the concave region of K stated in Assumption 2 in SVGD, which is required for the theory to hold. These make the empirical EPD behave not exactly the same as the true particle distance. 2) Theorem 2 and its proof in the SM also apply to the case of non-convex energy functions. 3) Even if the kernel is not concave, the result would still indicate that particles in the concave regions would collapse. 4) The pitfall indicates a challenge for developing non-asymptotic SVGD theory (if possible at all), motivating the development of SPOS. 5) This is a complement to the result of [Liu et al., 2019], which proves SVGD is ill-pose under some conditions. 3.2 Stochastic particle-optimization sampling to mitigate the pitfall We argue the WGF framework proposed in [Chen et al., 2018], if solved appropriately, is able to overcome the pitfall of SVGD. Speci\ufb01cally, the original solution in [Chen et al., 2018] is based on a deterministic particle-approximation method for (4), which introduces hard-to-control approximation errors. Instead, we propose to solve (4) stochastically to replace the \u2207\u03b8\u03bd\u03c4 term in (4) with a Brownian motion. Speci\ufb01cally, \ufb01rst note that the term \u03b2\u22121\u2207\u03b8 \u00b7 \u2207\u03b8\u03bd\u03c4 is contributed from Brownian motion, i.e., solving the SDE, d\u03b8\u03c4 = p 2\u03b2\u22121dW\u03c4, is equivalent to solving the corresponding FP equation: \u2202\u03bd\u03c4 = \u03b2\u22121\u2207\u03b8 \u00b7 \u2207\u03b8\u03bd\u03c4. Consequently, we decompose the RHS of (4) into two parts: F1 \u225c\u2207\u03b8 \u00b7 \u0000\u03bd\u03c4\u03b2\u22121F(\u03b8\u03c4) + (K \u2217\u03bd\u03c4)\u03bd\u03c4 \u0001 and F2 \u225c\u03b2\u22121\u2207\u03b8 \u00b7\u2207\u03b8\u03bd\u03c4. Our idea is to solve F1 deterministically under a PDE setting, and solve F2 stochastically based on its corresponding SDE. When adopting particle approximation for the density \u03bd\u03c4, both solutions of F1 and F2 are represented in terms of particles {\u03b8(i) \u03c4 }. Thus we can combine the solutions from the two parts directly to approximate the original exact solution of (4). Similar to the results of SVGD in Section 3.3 in [Liu, 2017], we \ufb01rst formally show in Theorem 3 that when approximating \u03bd\u03c4 with particles, i.e., \u03bd\u03c4 \u2248 1 M PM i=1 \u03b4\u03b8(i) \u03c4 (\u03b8), the PDE can be transformed into a system of deterministic di\ufb00erential equations with interacting particles. Theorem 3 When approximating \u03bd\u03c4 in (4) with particles {\u03b8(i) \u03c4 }, the PDE \u2202\u03c4\u03bd\u03c4 = F1 reduces to the following system of di\ufb00erential equations describing evolutions of the particles over time: \u2200i d\u03b8(i) \u03c4 = \u2212\u03b2\u22121F(\u03b8(i) \u03c4 )d\u03c4 \u22121 M M X j=1 K(\u03b8(i) \u03c4 \u2212\u03b8(j) \u03c4 )F(\u03b8(j) \u03c4 )d\u03c4 + 1 M M X j=1 \u2207K(\u03b8(i) \u03c4 \u2212\u03b8(j) \u03c4 )d\u03c4 (7) Consequently, by solving \u2202\u03c4\u03bd\u03c4 = F2 stochastically from an SDE perspective, we arrive at the following differential equation system, describing evolution of the particles {\u03b8(i) \u03c4 } over time \u03c4: \u2200i d\u03b8(i) \u03c4 = \u2212\u03b2\u22121F(\u03b8(i) \u03c4 \u22121 M M X j=1 K(\u03b8(i) \u03c4 \u2212\u03b8(j) \u03c4 )F(\u03b8(j) \u03c4 ) + 1 M M X j=1 \u2207K(\u03b8(i) \u03c4 \u2212\u03b8(j) \u03c4 ))d\u03c4 + p 2\u03b2\u22121dW(i) \u03c4 (8) Algorithm 1 Stochastic Particle-Optimization Sampling Input: Initial particles {\u03b8(i) 0 }M i=1 with \u03b8(i) 0 \u2208Rd, step size hk, batch size Bk 1: for iteration k= 0,1,...,T do 2: Update \u03b8(i) k+1 with (9) for \u2200i. 3: end for Output:{\u03b8(i) T }M i=1 Our intuition is that if the particle evolution (8) can be solved exactly, the solution of (6) \u03bd\u03c4 will be wellapproximated by the particles {\u03b8(i) \u03c4 }M i=1. In our theory, we show this intuition is true. In practice, however, solving (8) is typically infeasible, and thus numerical methods are adopted. Furthermore, in the case of large \fJianyi Zhang1, Ruiyi Zhang1, Lawrence Carin1, Changyou Chen2k Figure 1: Comparison of SPOS (left) and SVGD (right) on a multi-mode distribution. The circles with di\ufb00erent colors are the \ufb01nal 100 particles, which are able to spread over all modes for SPOS. data sets, following SG-MCMC, F(\u03b8(i) k ) is typically replaced by a stochastic version G(i) k \u225c N Bk P q\u2208Ik Fq(\u03b8(i) k ) evaluated with a minibatch of data of size Bk for computational feasibility. Based on the Euler method [Chen et al., 2015] with a stepsize hk, (8) leads to the following updates for the particles at the k-th iteration: let \u03be(i) k \u223cN(0, I) for \u2200i, \u03b8(i) k+1 =\u03b8(i) k \u2212hk\u03b2\u22121G(i) k \u2212hk M M X j=1 K(\u03b8(i) k \u2212\u03b8(j) k )G(j) k + hk M M X j=1 \u2207K(\u03b8(i) k \u2212\u03b8(j) k ) + p 2\u03b2\u22121hk\u03be(i) k (9) We call the algorithm with particle update equations (9) stochastic particle-optimization sampling (Algorithm 1), in the sense that particles are optimized stochastically with extra random Gaussian noise. Intuitively, the added noise enhances the ability of the algorithm to jump out of local modes, leading to better exploration properties compared to standard SVGD. This serves as one of our motivations to generalize SVGD to SPOS. To illustrate the advantage of introducing the noise term, we compare SPOS and SVGD on sampling a di\ufb03cult multi-mode distribution, with the density function given in Section A of the SM. The particles are initialized on a local mode close to zero. Note that in this example there is always positive probability to jump between modes. Figure 1 plots the \ufb01nal locations of the particles along with the true density, which shows that particles in SPOS are able to reach di\ufb00erent modes, while they are all trapped at one mode in SVGD. Theorem 4 below bounds the EPD of SPOS, in contrast with that for SVGD in Theorem 2, which is intuitively obtained by taking the \u03b2 \u2192\u221elimit. Theorem 4 Under Assumption 1, further assume that every {\u03b8(i) \u03c4 } of (8) for approximating \u03bd\u03c4 in (4) has the same initial probability law \u03bd0 and \u0393 \u225c E\u03b8\u223c\u03bd0,\u03b8\u2032\u223c\u03bd0[\u2225\u03b8 \u2212\u03b8\u2032\u22252] < \u221e. Choose a \u03b2 such that \u03bb = mF \u03b2 + mK \u2212HF LK > 0. Then the EPD of SPOS is bounded as: EPD \u225c qPM i,j E\u2225\u03b8(i) \u03c4 \u2212\u03b8(j) \u03c4 \u22252 \u2264 C1e\u22122\u03bb\u03c4 + 4 q d \u03b2 M \u03bb , where C1 = M(M \u22121)\u0393 \u2212 4 p d\u03b2\u22121 M \u03bb . Remark 2 There are two interesting cases: i) When C1 > 0, the EPD would decrease to the bound 4 p d\u03b2\u22121M/\u03bb with time t. This represents the phenomenon of an attraction force between particles; ii) When C1 < 0, the EPD would increase to the same bound, which represents the phenomenon of a repulsive force between particles, e.g., when particles are initialized with the same value (\u0393 = 0), they would be pushed away from each other until the EPD increases to the aforementioned bound. 4 Non-Asymptotic Convergence Analysis In this section, we prove non-asymptotic convergence rates for the proposed SPOS algorithm under the 1Wasserstein metric W1, a special case of p-Wasserstein metric de\ufb01ned as Wp(\u00b5, \u03bd) = \u0012 inf \u03b6\u2208\u0393(\u00b5,\u03bd) Z Rd\u00d7Rd \u2225X\u00b5 \u2212X\u03bd\u2225pd\u03b6(X\u00b5, X\u03bd) \u00131/p where \u0393(\u00b5, \u03bd) is the set of joint distributions on Rd\u00d7Rd with marginal distribution \u00b5 and \u03bd. Note that SPOS reduces to SVGD when \u03b2 \u2192\u221e, thus our theory also sheds light on the convergence behavior of SVGD, where non-asymptotic theory is currently mostly missing, despite the asymptotic theory developed recently [Liu, 2017, Lu et al., 2018]. For analysis, we further impose the following assumptions. Assumption 3 Assume F and \u03bd0 satisfy the following assumptions: 3.1 There exists positive mF such that \u27e8F(\u03b8) \u2212 F(\u03b8\u2032), \u03b8 \u2212\u03b8\u2032\u27e9\u2265mF \u2225\u03b8 \u2212\u03b8\u2032\u22252. 3.2 The initial probability law of each particle has a bounded and strictly positive density \u03bd0 with respect to the Lebesgue measure on Rd, and \u03b30 \u225c log R Rd e\u2225\u03b8\u22252\u03bd0(\u03b8)d\u03b8 < \u221e A few remarks: i) Assumption 3.1 indicates U to be a convex function. Theory of non-convex U is presented in Section J of the SM with some extra assumptions. ii) Assumptions 3.1 is widely adopted in other theoretical works such as [Dalalyan and Karagulyan, 2017, Chatterji et al., 2018] iii) Assumptions 3.2 has also been adopted in [Raginsky et al., 2017] 4.1 Basic setup and extra notation Due to the exchangeability of the particle system {\u03b8(i) \u03c4 }M i=1 in (8), if we initialize all the particles \u03b8(i) \u03c4 with the same distribution \u03c10, they would endow the same distribution for each time \u03c4. We denote the distribution of each \u03b8(i) \u03c4 as \u03c1\u03c4. Similar arguments hold for the particle system {\u03b8(i) k }M i=1 in (9), and thus we denote \fStochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory the distribution of each \u03b8(i) k as \u00b5k (k = 1, 2, \u00b7 \u00b7 \u00b7 , T). To this end, our analysis aims at bounding W1(\u00b5T , \u03bd\u221e) since \u03bd\u221eis our target distribution p(\u03b8|X) according to Proposition 1. In the following, for conciseness, we use a summation of stepsizes to represent the \u201ctime index\u201d of some density, e.g., \u03c1PT \u22121 k=0 hk. The high-level idea of bounding W1(\u00b5T , \u03bd\u221e) in this section is to decompose it as follows: W1(\u00b5T , \u03bd\u221e) \u2264W1 \uf8eb \uf8ed\u00b5T , \u03c1T \u22121 P k=0 hk \uf8f6 \uf8f8 (10) + W1 \uf8eb \uf8ed\u03c1T \u22121 P k=0 hk , \u03bdT \u22121 P k=0 hk \uf8f6 \uf8f8+ W1 \uf8eb \uf8ed\u03bdT \u22121 P k=0 hk , \u03bd\u221e \uf8f6 \uf8f8. 4.2 Bounds with stochastic particle approximation In this section, we bound W1(\u03c1PT \u22121 k=0 hk, \u03bdPT \u22121 k=0 hk) and W1(\u03bdPT \u22121 k=0 hk, \u03bd\u221e) in (10). The \ufb01rst term corresponds to a variant of granular media equation, but is much more challenging to bound. Theorem 5 Under Assumption 1&3 and letting \u03c10 = \u03bd0, there exist positive constants c1 and c2 independent of (M, \u03c4) and satisfying c2 < \u03b2\u22121 such that W1(\u03c1\u03c4, \u03bd\u03c4) \u2264c1(\u03b2\u22121 \u2212c2)\u22121M \u22121/2, \u2200\u03c4. (11) Remark 3 According to Theorem 5, we can bound the W1(\u03c1PT \u22121 k=0 hk, \u03bdPT \u22121 k=0 hk) term as W1(\u03c1PT \u22121 k=0 hk, \u03bdPT \u22121 k=0 hk) \u2264 c1 \u221a M(\u03b2\u22121\u2212c2). Furthermore, by letting \u03c4 \u2192\u221e, we have W1(\u03c1\u221e, \u03bd\u221e) \u2264 c1 \u221a M(\u03b2\u22121\u2212c2), an important result to prove the following theorem. Theorem 6 Under Assumption 1&3, the following holds: W1(\u03bd\u03c4, \u03bd\u221e) \u2264c3e\u22122\u03bb1\u03c4, where \u03bb1 = \u03b2\u22121mF \u2212 LF \u22122LK and c3 is some positive constant independent of (M, \u03c4). Furthermore, the W1(\u03bdPT k=0 hk, \u03bd\u221e) term in (10) can be bounded as: W1(\u03bdPT \u22121 k=0 hk, \u03bd\u221e) \u2264c3 exp   \u22122\u03bb1( T \u22121 X k=0 hk) ! . (12) To ensure W1(\u03bdPT \u22121 k=0 hk, \u03bd\u221e) decreases over time, one needs to choose \u03b2 small enough such that \u03bb1 > 0. This also sheds light on a failure case of SVGD (where \u03b2 \u2192\u221e) discussed in Section 3.1. 4.3 Bounds with a numerical solution To bound the W1(\u00b5T , \u03c1PT \u22121 k=0 hk) term in (10), we adopt techniques from [Raginsky et al., 2017, Xu et al., 2018] on analyzing the behavior of SGLD, and derive the following results for our SPOS algorithm: Theorem 7 Under Assumptions 1&3, for a \ufb01xed step size hk = h (\u2200k) that is small enough, the corresponding W1(\u00b5T , \u03c1T h) is bounded as: W1(\u00b5T , \u03c1T h) \u2264c4Md 3 2 \u03b2\u22123(c5\u03b22B\u22121 + c6h) 1 2 T 1 2 h 1 2 (13) where B is the minibatch size and (c4, c5, c6) are some positive constants independent of (M, T, h). Combining bounds from Theorems 5 and (7), given T, the optimal bound over h can be seen to decrease at a rate of O(M \u22121/2). Furthermore, the dependence of T in the bound of Theorem 7 makes the bound relatively loose. Fortunately, the bound can be made independent of T by considering a decreasing-stepsize SPOS algorithm, as stated in Theorem 8. Theorem 8 Under Assumptions 1&3, for a decreasing step size hk = h0/(k + 1), and letting the minibatch size in each iteration k be Bk = B0 + [log(k + 1)]100/99 with B0 the initial batch size, the corresponding W1(\u00b5T , \u03c1PT \u22121 k=0 hk) term is bounded, for some \u03b2 small enough, as: W1 \uf8eb \uf8ed\u00b5T , \u03c1T \u22121 P k=0 hk \uf8f6 \uf8f8\u2264c4\u03b2\u22123Md 3 2 \u0000c7h3 0 + c8\u03b23h0/B0 +c9h2 0\u03b22\u00011/2 , (14) where (c4, c7, c8, c9) are positive constants independent of (M, T, h0). Note Bk increases at a very low speed, e.g., only by 15 after 105 iterations, thus it does not a\ufb00ect algorithm e\ufb03ciency. Consequently, W1(\u00b5T , \u03c1PT \u22121 k=0 hk) would approach zero when h1/2 0 M \u21920. The Overall Non-Asymptotic Bounds By directly combining results from Theorems 5\u20138, one can easily bound the target W1(\u00b5T , \u03bd\u221e), stated in Theorem 9 and Theorem 10. Theorem 9 (Fixed Stepsize) Under Assumption 1&3 and setting hk = h0, Bk = B0, W1(\u00b5T , \u03bd\u221e) is bounded as: W1(\u00b5T , \u03bd\u221e) \u2264 c1 \u221a M(\u03b2\u22121 \u2212c2) + c6Md 3 2 \u03b2\u22123(c4\u03b22B\u22121 + c5h) 1 2 T 1 2 h 1 2 + c3 exp \b \u22122 \u0000\u03b2\u22121mF \u2212LF \u22122LK \u0001 Th \t , (15) where (c1, c2, c3, c4, c5, c6, \u03b2) are positive constants such that 1 \u03b2 > c2 and mF \u03b2 > LF + 2LK. Theorem 10 (Decreasing Stepsize) Denote \u02dc hT \u225c PT \u22121 k=0 hk. Under Assumption 1&3, if we set hk = \fJianyi Zhang1, Ruiyi Zhang1, Lawrence Carin1, Changyou Chen2k 100 101 102 103 iteration 1 1.5 2 2.5 3 3.5 4 4.5 5 error 100 101 102 103 iteration 1 1.5 2 2.5 3 3.5 4 4.5 5 error Figure 2: Estimation errors versus number of iterations for SPOS (left) and SVGD (right). h0/(k+1) and Bk = B0+[log(k+1)]100/99, W1(\u00b5T , \u03bd\u221e) is bounded as: W1(\u00b5T , \u03bd\u221e) \u2264 c1 \u221a M(\u03b2\u22121 \u2212c2) (16) + c3 exp{\u22122 \u0000\u03b2\u22121mF \u2212LF \u22122LK \u0001 \u02dc hT } + c10\u03b2\u22123Md 3 2 (c9h3 0 + c7\u03b23h0/B0 + c8h2 0\u03b22) 1 2 . where (c1, c2, c3, c7, c7, c8, c9, c10, \u03b2) are positive constants such that 1 \u03b2 > c2 and mF \u03b2 > LF + 2LK. Remark 4 Four implications are highlighted from the theorems: i) M and T play a similar role when bounding the numerical errors (the third term in the RHS of (15)). The bound increases with increasing M and T, which seems unavoidable and is consistent with the latest result for SGLD, whose bound is proved to increase w.r.t. T [Raginsky et al., 2017]. ii) The increasing bound w.r.t. T can be compromised by using decreasing stepsizes shown in Theorem 10. Unfortunately, this does not seem to eliminate the e\ufb00ect of M. To accommodate this, one should either use a smaller h or a larger \u03b2. We believe future work is needed to improve the bound w.r.t. M. However, this is nontrivial as recent theory shows coordinate-wise SGLD scales linearly w.r.t. parameter dimension [Shen et al., 2019] (corresponding to scaling linearly w.r.t. M in our case, consistent with our theory). iii) When T \u00d7 M (proportional to computation cost) is not too large, the error is bounded above by O(M \u22121/2 + M), indicating the existence of an optimal M, i.e., one should not choose arbitrary many particles as it would induce larger numerical-error bounds. This is somewhat surprising and counter-intuitive compared with the asymptotic theory [Liu, 2017, Lu et al., 2018]. However, we will demonstrate this is true with experiments on synthesized data, where the phenomenon is also observed in SVGD. iv) When T \u00d7 M is large enough, the O(M) term dominates, indicating an increasing error w.r.t. M. This is veri\ufb01ed by the experiments in Section 5.1 (Figure 3), although the bound might not be strictly tight. 5 Experiments We use simple experiments with synthetic data to demonstrate the non-asymptotic convergence behaviors of SPOS indicated by our theory. In addition to the results presented below, more experiments and real applications are provided in Section O of the SM, considering comparisons of SPOS with SVGD and SGLD on Bayesian learning of deep neural network and Bayesian exploration in deep reinforcement learning (RL). 5.1 Sampling a Gaussian distribution We apply the algorithms to sample from a simple 1-D Gaussian distribution with mean 2 and variance 1. Since the 1-Wasserstein distance is infeasible to calculate, we follow [Vollmer et al., 2016, Chen et al., 2015] and measure the convergence using err \u225c |E\u03b8\u223c\u00b5T [f(\u03b8)] \u2212E\u03b8\u223cN (2,1)[f(\u03b8)]| with a test function f(\u03b8) \u225c\u03b82. We \ufb01x T = 1000 and h = 0.03. Particles are initialized as being drawn from N(0, 1). Figure 2 plots the estimation errors versus the number of iterations for di\ufb00erent particles M. For both SPOS and SVGD, it is observed that when T is not too large (\u2248100), the errors increase w.r.t. T, and the optimal M is around 300, consistent with our theory. When T is large enough, the errors decrease w.r.t. T, and larger M induces larger errors. This is also consistent with our theory, because the last term in Theorem 9 dominates when T is large, leading to increasing errors with larger M. The only concern seems to be the tightness of the bound, which might be due to technical di\ufb03culty as current techniques for SGLD also indicate an increasing bound w.r.t. T [Raginsky et al., 2017]. The large optimal M also suggests using a relatively large M should not be a problem in real applications. Impact of particle number M In addition to the above result to demonstrate the existence of an optimal M, we further verify that when T \u00d7 M is large enough, for a \ufb01xed T, we observe the errors increase with increasing M. We use the same setting as above. Figure 3 plots the curves of errors versus number of particles. We see that errors indeed increase w.r.t. particle numbers, consistent with our theory. Although \fStochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory 101 102 103 Number of particles 10-2 10-1 100 Error SPOS, iter = 1000 SVGD, iter = 1000 SPOS, iter = 800 SVGD, iter = 800 SPOS, iter = 600 SVGD, iter = 600 SPOS, iter = 400 SVGD, iter = 400 Figure 3: Errors versus Number of particles. Errors increase with increasing particle numbers. the rate of the bound from our theory might not match exactly with the experimental results, we believe this is still signi\ufb01cant as the problem has never been discovered before, which is somewhat counter-intuitive. On the other hand, the results are also reasonable, as more particles would need much more updates to \ufb01t a distribution well. The results indicate that to get a smaller error, one should increase number of iterations faster than increasing the number of particles. Table 1: Averaged RMSE with standard deviations. Test RMSE Dataset SGLD SVGD SPOS Boston 3.114 \u00b1 0.144 2.961 \u00b1 0.109 2.829 \u00b1 0.126 Concrete 5.508 \u00b1 0.275 5.157 \u00b1 0.082 5.071 \u00b1 0.150 Energy 0.842 \u00b1 0.060 1.291 \u00b1 0.029 0.752 \u00b1 0.029 Kin8nm 0.080 \u00b1 0.001 0.090 \u00b1 0.001 0.079 \u00b1 0.001 Naval 0.004 \u00b1 0.000 0.004 \u00b1 0.000 0.004 \u00b1 0.000 CCPP 4.059 \u00b1 0.080 4.127 \u00b1 0.027 3.939 \u00b1 0.049 Wine 0.632 \u00b1 0.022 0.604 \u00b1 0.007 0.598 \u00b1 0.014 Yacht 1.183 \u00b1 0.263 1.597 \u00b1 0.099 0.840 \u00b1 0.087 Protein 4.281 \u00b1 0.011 4.392 \u00b1 0.015 4.254 \u00b1 0.005 YearPredict 8.707 \u00b1 NA 8.684 \u00b1 NA 8.681 \u00b1 NA 5.2 BNNs for regression We next conduct experiments for Bayesian learning of deep neural networks (DNNs) to empirically compare SGLD, SVGD and SPOS for posterior sampling of BNN weights with standard Gaussian priors. We use a RBF kernel with the bandwidth set to the medium of particles. Following [Li et al., 2015], 10 UCI public datasets are considered, and we consider 100 hidden units for 2 large datasets (Protein and YearPredict), and 50 hidden units for the other 8 small datasets. We use the same setting as [Zhang et al., 2018b]. The datasets are randomly split into 90% training and 10% testing. For a fair comparison, we use the same split of data (train, val and test) for all methods. We report the root mean squared error (RMSE) in Table 1. The proposed SPOS outperforms both SVGD and SGLD. More detailed settings and results are given in Section O of the SM. 5.3 Bayesian exploration in deep RL It is well accepted that RL performance directly measures how well the uncertainty is learned, due to the need for exploration. We apply SPOS for RL, and compare it with SVPG, a SVGD version of the policy gradient method [Liu et al., 2017]. Following [Liu et al., 2017, Zhang et al., 2018a], we de\ufb01ne policies with Bayesian DNNs. This naturally introduces uncertainty into action selection, rendering Bayesian explorations to make policy learning more e\ufb00ective. We follow the same setting as in [Liu et al., 2017], except that we use simpler policy-network architectures, as in [Houthooft et al., 2016]. We conduct experiments on three classical continuous control tasks are considered: Cartpole Swing-Up, Double Pendulum, and Cartpole. Detailed experimental settings are given in the SM. Figure 4 plots the cumulative rewards over time on the Cartpole environment, which clearly shows the advantage of our method over SVPG. More results are provided in the SM. 0 200 400 600 800 1000 Episodes 100 50 0 50 100 150 200 Average\u00a0Reward Cartpole\u00a0Swing\u00a0Up\u00a0(A2C) SVPG SPOS\u00adPG 0 200 400 600 800 1000 Episodes 100 50 0 50 100 150 200 250 300 Average\u00a0Reward Cartpole\u00a0Swing\u00a0Up SVPG SPOS\u00adPG Figure 4: Policy learning with Bayesian exploration in policy-gradient methods with SVPG and SPOS-PG. 6"
+                }
+            ],
+            "Yiran Chen": [
+                {
+                    "url": "http://arxiv.org/abs/2108.07140v2",
+                    "title": "MTG: A Benchmark Suite for Multilingual Text Generation",
+                    "abstract": "We introduce MTG, a new benchmark suite for training and evaluating\nmultilingual text generation. It is the first-proposed multilingual multiway\ntext generation dataset with the largest human-annotated data (400k). It\nincludes four generation tasks (story generation, question generation, title\ngeneration and text summarization) across five languages (English, German,\nFrench, Spanish and Chinese). The multiway setup enables testing knowledge\ntransfer capabilities for a model across languages and tasks. Using MTG, we\ntrain and analyze several popular multilingual generation models from different\naspects. Our benchmark suite fosters model performance enhancement with more\nhuman-annotated parallel data. It provides comprehensive evaluations with\ndiverse generation scenarios. Code and data are available at\n\\url{https://github.com/zide05/MTG}.",
+                    "authors": "Yiran Chen, Zhenqiao Song, Xianze Wu, Danqing Wang, Jingjing Xu, Jiaze Chen, Hao Zhou, Lei Li",
+                    "published": "2021-08-13",
+                    "updated": "2022-06-10",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL"
+                    ],
+                    "main_content": "Introduction Natural language generation (NLG) aims to automatically generate meaningful texts with the input in different formats, such as images (Anderson et al., 2018), tables (Ye et al., 2020) or texts (Guan et al., 2019). The generated texts generally target at realizing an underlying communicative goal while remaining coherent with the input information and keeping grammatically correct. Multilingual text generation extends the natural language generation task to produce texts in multiple languages, which is important to overcome language barriers and enable universal information access for the world\u2019s citizens (Artetxe et al., 2020; Arivazhagan et al., 2019; Pan et al., 2021). To achieve this goal, various multilingual text generation datasets have been proposed. Some of them do not incorporate cross-lingual pairs (Liang et al., 2020; Ladhak et al., 2020). This limits the \u2217Corresponding author. \u2020Work is done while at ByteDance. knowledge transfer from one language to another. Others involve cross-lingual pairs while English is included on either source or target side in most cases (Zhu et al., 2019; Ladhak et al., 2020), leading to dif\ufb01cult transfer between low-resource or distant language pairs. Constructing a multilingual text generation dataset that can directly transfer knowledge between any two languages is still under-explored. To this end, we propose MTG, a humanannotated multilingual multiway dataset. Multiway means that the same sample is expressed in multiple languages. It covers four generation tasks (story generation, question generation, title generation and text summarization) across \ufb01ve languages (English, German, French, Spanish and Chinese). We do not include multilingual machine translation because MT itself is a standard task. The multiway parallel feature enables cross-lingual data construction between arbitrary language pairs. Such direct parallel signal promotes knowledge transfer and cross-lingual generation between any language pairs (even distant pairs such as Spanish-Chinese) without involving an intermediate language such as English (Leng et al., 2019). The multilingual multiway feature also enables various training and test scenarios. In this paper, we design four scenarios to verify the advantages of our MTG from different aspects. Several representative pretrained multilingual models are employed to test these scenarios, including multilingual BERT (M-BERT) (Devlin et al., 2019), XLM (Lample and Conneau, 2019), mBART (Liu et al., 2020) and mT5 (Xue et al., 2020). We leverage various metrics to assess the coherence and diversity of the outputs generated by these models. Besides, we also propose an ensemble metric, which mainly focuses on relevance, measuring to what degree is the generated text close to humanlevel. Human evaluation is also conducted to validate models\u2019 performances. arXiv:2108.07140v2  [cs.CL]  10 Jun 2022 \fIn summary, the contributions of this paper are listed as follows: (i) We propose a new human-annotated multilingual multiway text generation benchmark suite MTG. (ii) We design a new evaluation metric measuring how a text resembles human writing and prove that it has higher correlation scores with human scores compared with other automatic relevance metrics. (iii) We evaluate several representative pretrained multilingual models on our proposed MTG and make a rigorous analysis to verify its advantages. 2 Related Work A signi\ufb01cant body of works have been committed to the construction of multilingual datasets covering diverse tasks (Hu et al., 2020; Jiang et al., 2020; Longpre et al., 2020). XTREME (Hu et al., 2020) is a multilingual understanding benchmark across 40 languages and 9 tasks, but it does not cover any generation task. Jiang et al. (2020) propose X-FACTR, which is a cross-lingual factual retrieval benchmark. Longpre et al. (2020) propose MKQA, an opendomain question answering evaluation dataset covering 26 diverse languages. Ladhak et al. (2020) present WikiLingua, which is a large-scale, multilingual dataset for cross-lingual abstractive summarization systems. MLSUM (Wang et al., 2021) is a dataset for text summarization in 12 languages. Wiki-40B (Guo et al., 2020) is a multilingual language model dataset across 40+ languages. Although these datasets cover multiple languages, they either belong to natural language understanding tasks or a single, speci\ufb01c generation task, which limits researchers to obtain general \ufb01ndings incorporating a set of generation tasks. XGLUE (Liang et al., 2020) is a cross-lingual benchmark dataset for nine understanding tasks and two generation tasks. GEM (Gehrmann et al., 2021) is a newly-presented vision-language dataset covering 11 image-language and video-language tasks and 32 languages. These two datasets encompass multiple tasks and languages. However, a remarkable difference of our MTG from XGLUE and GEM is that MTG focuses on text-to-text generation tasks and is parallel across all languages, which facilitates easier knowledge transfer. 3 Dataset Collection and Methodology This section will introduce how to create the benchmark suite for multilingual text generation (MTG). In order to construct multiway parallel dataset, the initial dataset is translated into other languages by an off-the-shelf translation model. Part of the translated data is randomly selected for further human annotation to increase data quality. The selection of tasks, initial datasets and languages are based on several principles as described below. 3.1 Task and Dataset Selection It is important to select suitable tasks for our MTG benchmark to make it diverse and challenging. Thus, we de\ufb01ne several criteria during the task selection procedure: Task De\ufb01nition Tasks should be well-de\ufb01ned, which means that humans can easily determine whether the generated results meet the task requirements. Task Dif\ufb01culty Tasks should be solvable by most college-educated speakers. In the meantime, they should be challenging to current models, the performance of which in various test scenarios falls short of human performance. Task Diversity Tasks should cover a wide range of generation challenges that allow for \ufb01ndings to be as general as possible. Input Format The input format of the tasks needs to be as simple as possible to reduce the dif\ufb01culty of data processing. Besides, it should not contain anything but text (e.g., without any images or videos). In order to meet the above criteria, 8 domain experts are asked to vote from 10 typical generation tasks1. Finally, four generation tasks are selected for MTG, which are story generation, question generation, title generation and text summarization. Story generation (SG) aims to generate the end of a given story context, which requires the model to understand the story context and generate a reasonable and \ufb02uent ending (Guan et al., 2019). Question generation (QG) targets at generating a correct question for a given passage and its answer (Duan et al., 2017). For the same passage with different answers, the system should be able to generate different questions. Title generation (TG) converts a given article into a condensed sentence while preserving its main idea (Jin and Hauptmann, 2002). The title should be faithful to the original document and encourage users to read the news 1These generation tasks are story generation, commonsense generation, style transfer, question generation, question answering, dialogue generation, title generation, text summarization, image caption, and data-to-text generation. \fTask Corpus Domain Format Goal Story Generation ROCStories Daily life <story> Generate the end of the story Question Generation SQUAD 1.0 Wikipedia <passage,answer, question> Generate the question of the answer Title Generation ByteCup News <article, title> Generate the title of the document Text Summarization CNN/DailyMail News <article, summary> Generate the summary of the document Table 1: The description of tasks and English datasets included in MTG. For story generation, we use the last sentence as story end to be generated and the rest as input. at the same time. Text summarization (Summ) aims to condense the source document into a coherent, concise, and \ufb02uent summary (Mani, 2001). It is similar to title generation but the output of text summarization is relatively longer. These four tasks focus on different generative abilities and realize different goals. After con\ufb01rming the tasks, the next step is to choose the dataset for each task. The two selection principles are listed as follows:(1) License: Task data must be available under licenses that allow using and redistributing for research purposes. The dataset should be free and available for download. (2) Quality: The dataset size should be as large as possible and the quality should be checked. English datasets are chosen as the initial datasets because they are more accessible in all four tasks and have relatively larger size compared with datasets in other languages. We choose ROCStories (Mostafazadeh et al., 2016) for story generation, SQUAD 1.0 (Rajpurkar et al., 2016) for question generation, ByteCup 2 for title generation and CNN/DailyMail (Nallapati et al., 2016) for text summarization. These datasets are popular in the corresponding \ufb01elds and have been veri\ufb01ed to be high-quality by many works. Moreover, they are all under a permissive license. An overview of all task datasets is shown in Table 1. 3.2 Language Selection The original datasets are in English (en) only and we want to extend them into a multiway parallel form. This means that all English texts should be translated into other languages, which will lead to high annotation costs. Thus, a state-of-the-art translator is leveraged to do the translation and then annotators are asked to correct the translated text. Considering this construction method, MTG should contain languages that (1) have good English-to-X translators and (2) are diverse in language family. Finally, German (de), French (fr), Spanish (es) and Chinese (zh) are chosen. German is from the 2https://www.biendata.xyz/competition/bytecup2018/ same language branch as English while French and Spanish are from different ones. Chinese is more distant from the rest of languages in the language family tree. Task SG, QG, TG, Summ For each language Rough training size 76k/61k/270k/164k Annotated training size 15k/15k/15k/15k Annotated development size 2k/2k/2k/2k Annotated test size 3k/3k/3k/3k For \ufb01ve languages (en, de, fr, es, zh) Total Annotated size 400k Total dataset size 6.9m Table 2: The number of samples in MTG. MTG consists of four subsets: rough training, annotated training, development and test set. The rough training set is \ufb01ltered by back translating across \ufb01ve languages. The annotated training, development and test sets are corrected by human experts. 3.3 Data Collection After determining the tasks and languages, we introduce the data collection process to get the MTG. The Google Translate3 is used to translate the English datasets to the selected languages. To control the quality of translated texts, we back translate the text to English and \ufb01lter the samples whose n-gram overlap ratios with the original English texts are lower than a certain threshold. Different threshold values (from 0.3 to 0.6 with 0.1 as step length) are tested and if it is set to 0.6, the training data size of QG will drop more than 60%. Thus we decide to use 0.5 as the threshold number to improve the quality of the \ufb01ltered data while still maintaining more than 70% of the original training data.4 Samples in four languages are aligned to ensure that the dataset is multiway parallel. 20, 000 samples of each task and language are randomly selected for annotation under the premise 3https://translate.google.com/ 4The detailed sizes of the \ufb01ltered datasets with respect to different thresholds are included in appendix A. \fCorrelation AdaBoost DecisionTree ExtraTree GradientBoosting Kneighbors Linear RandomForest SVR Bagging Pearson 0.100 0.133 0.190 0.215 0.192 0.173 0.208 0.113 0.240 Correlation BLEU ROUGE-1 ROUGE-2 ROUGE-L METEOR BERTScore-P BERTScore-R BERTScore-F1 Bagging Pearson 0.180 0.142 0.163 0.144 0.122 0.142 0.176 0.162 0.344 Table 3: The correlation scores between automatic metric scores and human-annotated scores (the average scores of grammar, \ufb02uency and relevance). Upper part of the table shows the correlation scores of different regression algorithms in test set of all languages. The lower part demonstrates correlation scores of our ensemble score (the bagging regressor) and other classic automatic scores in test set without Chinese results because Meteor does not support Chinese. of ensuring inter-language alignment. The annotators are required to further check the translated results based on the following rules: (1) Semantic aligned Whether the target text is meaningful and is fully semantic aligned with the source text. (2) Fluency Whether the translated text is grammatically correct. (3) Style Whether the translation follows the norms of local culture, language conventions, and gender-related words. If the translated text contradicts any of the above rules, annotators will correct it accordingly. The annotated data is then split to 15k/2k/3k as training/development/test subsets. A team of 10 full-time experts5 are hired to do the annotation, who are paid daily. Some parttime workers6 are also employed to increase the annotation throughput, who are paid by the number of annotations. Each annotator is an expert in at least two languages (English and another target language). They are trained to correct translation errors according to the above rules, \ufb01rst a small number of samples for trial, these annotation results are re-checked by us and feedback is given to the annotators to help them understand the tasks better. After this annotation training process, the annotators start to annotate the dataset. For quality control, we sample 2% from the annotations and arrange for 9 experts to double-check them. Each example is assigned to two other experts and the data is quali\ufb01ed only if both of them agree on the annotation7. If more than 5% of the annotations fail, then all the data of that annotator for that day will be re-checked. Then the multiway parallel generation benchmark MTG is \ufb01nally completed. It contains four 5There are 3 language experts for German, 3 for French, 4 for Spanish and 4 for Chinese 6There are 16 part-time workers who are participated in the German annotation, 39 for French, 4 for Spanish and 15 for Chinese. 7The grammar, expressions, and punctuation of the annotated text are completely correct and the expressions are in accordance with the foreign language. different generation tasks in \ufb01ve languages and its quality is improved by the incorporation of human annotation. However, the number of humanannotated data is still small due to cost concerns. Introducing more human-annotated data or carrying out extra \ufb01ltering for machine-translated data can be future directions to further improve the quality of MTG. The statistics of MTG is shown in Table 2. 4 Experiments In this section, we conduct extensive experiments to benchmark the dif\ufb01culty of our proposed MTG via several state-of-the-art multilingual models under different scenarios. 4.1 Baseline Models The performance of the following four popular multilingual pretrained models is explored8: M-BERT Multilingual BERT (M-BERT) (Devlin et al., 2019) is a language model pretrained from monolingual corpora in 104 languages using Masked Language Modeling (MLM) task. XLM The Cross-Lingual Language Model (XLM) (Lample and Conneau, 2019) is pretrained with Masked Language Modeling (MLM) task using monolingual data and Translation Language Modeling (TLM) task using parallel data. mBART Multilingual BART (mBART) (Liu et al., 2020) is a pretrained encoder-decoder model using denoising auto-encoding objective on monolingual data over 25 languages. mT5 Multilingual T5 (mT5) (Xue et al., 2020) is a multilingual variant of T5 (Raffel et al., 2020) formatting all tasks as text-to-text generation problems. mT5 is pretrained on a span-corruption version of Masked Language Modeling objective over 101 languages. 8Detailed descriptions for models are included in Appendix B. \fen de fr es zh en de fr es zh 31.25 34.53 30.51 30.37 26.42 30.79 34.81 31.62 30.22 26.70 31.13 34.44 31.21 30.19 26.56 31.19 34.32 31.62 30.82 26.41 31.24 34.06 30.47 29.99 26.35 (a) SG M-BERT en de fr es zh en de fr es zh 31.70 29.68 27.74 28.35 26.67 31.97 29.76 27.69 28.06 26.49 32.03 29.39 27.14 28.12 26.28 31.26 30.00 27.97 27.17 26.15 33.84 30.24 27.00 28.58 26.06 (b) SG XLM en de fr es zh en de fr es zh 31.65 34.09 29.80 30.15 21.41 32.06 34.43 32.79 32.25 22.66 32.32 33.68 32.81 29.55 26.47 33.32 30.70 28.10 31.69 26.65 31.45 33.21 26.27 23.61 26.57 (c) SG mBART en de fr es zh en de fr es zh 31.41 34.57 31.99 31.76 26.87 32.24 35.68 32.46 31.90 26.65 32.83 34.33 32.89 32.56 26.99 31.21 34.70 32.18 31.80 27.04 31.67 34.15 32.16 32.05 26.35 10 15 20 25 30 35 (d) SG mT5 en de fr es zh en de fr es zh 30.82 30.59 29.27 31.24 26.36 30.34 31.74 29.69 30.60 27.83 29.54 31.18 29.64 31.43 26.19 30.08 31.31 29.17 32.23 26.00 29.01 30.00 28.73 30.45 28.35 (e) QG M-BERT en de fr es zh en de fr es zh 32.51 29.27 29.79 29.08 28.63 31.99 30.02 29.54 28.69 28.41 31.17 29.54 29.97 28.70 28.30 32.04 29.52 28.99 29.67 28.57 31.20 29.54 28.67 28.01 28.19 (f) QG XLM en de fr es zh en de fr es zh 33.57 28.61 28.66 31.53 27.97 30.63 33.42 30.04 32.58 26.31 29.22 31.39 32.52 33.47 24.29 30.12 32.50 29.20 34.08 28.11 30.20 32.13 28.44 32.11 29.01 (g) QG mBART en de fr es zh en de fr es zh 32.66 33.22 32.44 33.74 28.70 31.72 33.49 31.79 33.20 27.71 32.63 33.53 33.14 34.06 27.88 31.94 32.63 32.37 34.52 28.78 31.02 32.60 31.33 32.80 27.48 10 15 20 25 30 35 (h) QG mT5 en de fr es zh en de fr es zh 30.44 32.70 28.73 28.97 26.03 29.63 31.37 28.34 27.68 26.42 30.77 31.29 28.49 29.10 26.03 29.84 30.79 28.13 28.13 27.01 28.66 30.08 27.59 28.02 26.43 (i) TG M-BERT en de fr es zh en de fr es zh 31.02 30.61 28.47 29.46 27.14 30.66 30.08 27.62 28.23 28.16 29.66 30.47 27.81 28.42 27.01 30.55 29.97 28.19 27.95 27.39 30.08 30.35 28.28 27.99 27.16 (j) TG XLM en de fr es zh en de fr es zh 33.82 21.81 15.72 13.51 27.56 22.60 31.76 22.34 19.86 28.53 23.24 26.81 29.41 22.52 17.83 32.99 23.18 24.79 29.61 27.85 31.99 23.31 23.77 20.50 28.19 (k) TG mBART en de fr es zh en de fr es zh 32.20 33.14 29.33 30.70 27.66 31.71 31.45 30.04 29.47 27.80 31.07 31.08 28.48 30.16 26.81 31.11 32.60 29.29 29.50 27.73 30.06 31.78 28.54 29.11 26.14 10 15 20 25 30 35 (l) TG mT5 en de fr es zh en de fr es zh 21.35 25.97 22.52 20.58 19.38 20.94 26.28 22.39 21.01 19.93 21.22 26.21 21.83 20.92 19.51 20.61 25.96 22.31 20.92 19.63 20.78 26.30 22.45 21.14 20.23 (m) Summ M-BERT en de fr es zh en de fr es zh 22.98 21.47 20.30 20.68 19.61 22.01 21.51 21.14 19.70 19.38 22.26 21.36 20.56 20.13 18.91 21.57 21.16 20.62 19.63 20.14 21.27 22.15 20.06 19.33 11.73 (n) Summ XLM en de fr es zh en de fr es zh 21.86 6.79 16.02 6.27 19.80 21.01 24.97 22.36 20.57 20.07 21.34 24.53 22.26 20.16 19.53 21.00 20.63 22.65 21.21 19.89 20.51 24.84 21.73 21.74 19.56 (o) Summ mBART en de fr es zh en de fr es zh 23.72 23.37 21.16 19.61 20.68 23.39 21.37 20.32 20.42 20.90 22.80 22.57 18.49 19.37 21.63 23.13 23.16 20.85 17.84 18.78 23.05 21.73 20.28 19.74 21.08 10 15 20 25 30 35 (p) Summ mT5 Figure 1: The cross-lingual ensemble metric results for four models in four tasks. Every cell of row lang1 and column lang2 means the result when the languages of input and output are lang1 and lang2 respectively. Deeper red represents better cross-lingual performance while deeper gray indicates worse performance. 4.2 Evaluation Metrics In order to fully understand the model performance, the quality of generated texts is evaluated from different aspects, including metrics measuring the relevance between outputs and references (e.g., BLEU, ROUGE, and BERTScore) and metrics measuring the diversity of the generated texts (e.g., Distinct). Moreover, we propose a new ensemble metric leveraging relevance metrics to measure how close the generated text is to human writing. It not only has higher correlation scores with human judgments but also is capable of measuring model performances fairly between languages. N-gram based Metrics N-gram based metrics evaluate the text-overlapping scores between the outputs and references. The following three metrics are used: (1) BLEU (Papineni et al., 2002) is a popular metric that calculates the word-overlap scores between the generated texts and gold-standard ones. We use the BLEU-4, which is the average score for unigram, bigram, trigram, and 4-gram. (2) ROUGE (Lin, 2004) is a recall-oriented metric that counts the number of overlapping units such as n-gram and word sequences between the produced texts and gold-standard ones. (3) METEOR (Banerjee and Lavie, 2005) relies on semantic features to predict the similarity scores between system hypotheses and human references. Embedding based Metrics The embeddingbased metrics can, to a large extent, capture the semantic-level similarity between the generated text and the ground truth. BERTScore (Zhang et al., 2019) computes the similarity of candidate and reference as a sum of cosine similarities of tokens using BERT contextual embeddings. Diversity Metrics We also employ the distinct metric (Li et al., 2016), which calculates the proportion of the distinct n-grams in all the system hypotheses and can be used to evaluate the diversity of the generated texts. Human Evaluation Human evaluation is also leveraged to better estimate the quality of model outputs. Speci\ufb01cally, 30 cases are randomly sampled from the test set for each task and language while ensuring all 30 cases are aligned among \ufb01ve languages, and then they are presented to human annotators with the model outputs. The generated texts are evaluated under task-agnostic and taskspeci\ufb01c aspects. Task-agnostic aspects include Grammar, Fluency, Relevance and Language \fTask Model BLEU ROUGE-L METEOR BERTScore Distinct-1 Ensemble mono multi mono multi mono multi mono multi mono multi mono multi SG M-BERT 2.486 2.836 16.680 17.240 0.139 0.140 0.741 0.743 0.952 0.959 30.891 30.987 XLM 4.026 2.992 24.520 22.820 0.145 0.144 0.754 0.744 0.967 0.967 28.364 28.449 mBART 4.514 4.880 19.320 19.920 0.149 0.156 0.759 0.762 0.985 0.983 31.430 31.907 mT5 2.668 3.832 16.280 18.620 0.126 0.145 0.751 0.759 0.976 0.974 31.623 31.482 QG M-BERT 8.266 9.980 27.340 29.520 0.240 0.262 0.778 0.785 0.938 0.944 30.553 30.526 XLM 16.472 15.264 41.100 40.600 0.305 0.298 0.810 0.809 0.966 0.967 30.072 29.979 mBART 16.256 17.624 36.640 38.140 0.298 0.315 0.811 0.817 0.981 0.983 32.522 32.961 mT5 15.792 17.700 34.100 37.680 0.294 0.313 0.806 0.818 0.977 0.979 32.257 32.944 TG M-BERT 9.524 10.550 25.440 26.360 0.214 0.228 0.749 0.754 0.930 0.957 28.971 29.422 XLM 11.144 11.926 26.960 28.660 0.236 0.248 0.752 0.759 0.946 0.941 28.808 29.063 mBART 14.726 14.786 31.680 32.120 0.257 0.260 0.773 0.775 0.966 0.968 30.556 30.322 mT5 11.336 13.546 26.460 29.400 0.223 0.257 0.753 0.767 0.959 0.956 29.556 30.010 Summ M-BERT 9.766 10.956 31.280 32.220 0.221 0.232 0.748 0.751 0.787 0.815 22.122 22.018 XLM 9.486 11.830 30.160 34.740 0.235 0.235 0.729 0.755 0.814 0.772 19.281 20.770 mBART 12.858 12.792 32.940 32.920 0.256 0.257 0.750 0.750 0.796 0.803 21.972 22.292 mT5 5.022 6.090 25.060 27.980 0.145 0.162 0.724 0.741 0.826 0.870 20.499 21.826 Table 4: Automatic scores averaged across \ufb01ve languages for four models on four tasks. Mono and multi mean models are trained in monolingual and multilingual setting respectively. Higher scores between monolingual and multilingual results are bolded. Fusion. The former three aspects are scored from 1 to 5 while the language fusion score is set to 1 if all tokens of a model-generated text are in the target language and 0 otherwise. Besides task-agnostic aspects, the generated text is also evaluated under task-speci\ufb01c aspects. For title generation and summarization, coverage measures the degree to which the generated text covers the main content of the document. Correspondence for question generation measures the extent to which the generated question is matched with both document and answer. For story generation, we further evaluate whether the generated story is logically feasible. All task-speci\ufb01c aspects are scored from 1 to 5. Ensemble Metric Some N-gram based metrics such as BLEU and ROUGE largely depend on the tokenizer for speci\ufb01c languages. For example, BLEU scores for Chinese outputs are relatively high because it simply uses a character-level tokenizer. This causes unfair comparison between different languages. To this end, we propose an ensemble metric that evaluates the degree to which a piece of text resembles manual writing. It not only enables fair comparison between languages but is also proved to have a better correlation with humanannotated scores at the end of this subsection. We \ufb01rst average the grammar, \ufb02uency and relevance scores as targets, then normalize the automatic metrics and human scores among every language to eliminate the score discrepancy between languages. Three relevance metrics (BLEU, ROUGE-L, and BERTScore-F1) are gathered as features. The samples are split into training, development and test sets. After comparing different regression models\u2019 performance as shown in the upper part of Table 3 , we \ufb01nally choose bagging regression model (Breiman, 1996) as the ensemble metric. Moreover, the bagging ensemble metric shows a higher correlation with human-annotated scores compared with other relevance automatic metrics as shown in the lower part of Table 3. 4.3 Evaluation Scenarios To validate the effect of different experimental settings on model performance, several state-of-theart multilingual models are studied under four evaluation scenarios. Monolingual \ufb01ne-tuning The pretrained model is tuned for a downstream task using the training data for a speci\ufb01c language and evaluated on the test set for the same language. Multilingual \ufb01ne-tuning The pretrained model is jointly \ufb01ne-tuned with data in all languages for a speci\ufb01c task. Different from the monolingual \ufb01netuning setting, there is only one model for each downstream task, which can serve all languages. Cross-lingual generation Since MTG is multiway parallel, it can be reorganized to create inputoutput pairs that belong to different languages. In this paper, we make use of the multiway parallel data to do the supervised cross-lingual training, e.g., for English centric cross-lingual training, we take the English source as the input and the parallel German, French, Spanish, Chinese target as the out\fput. Then we evaluate the model on same setting (en->de, en->es, en->fr, en->zh). The cross-lingual generation performances on all 5 \u22174 directions are evaluated. Zero-shot transfer We also try to explore the zero-shot ability of multilingual pretrained models on the four tasks. The model is \ufb01ne-tuned on a speci\ufb01c task with English input and output. Then it is used to generate output in other languages with a given language tag. 5 Results 5.1 Monolingual and Cross-lingual This section displays the monolingual and crosslingual model comparison to explore their performances in different tasks and languages. Figure 1 contains the \ufb01ve language-centric cross-lingual and monolingual results. Several conclusions can be drawn from the results: The performance of Cross-lingual is better than monolingual in some cases. As shown in Figure 1, model performances on ensemble scores in cross-lingual setting exceed those in monolingual setting frequently (e.g., the monolingual result of French underperforms the English to French cross-lingual result in Figure 1(b) ). This is because the cross-lingual models are trained with more data (e.g., the English centric cross-lingual model is trained with en->de, en->fr, en->es, en>zh data), and the data from different cross-lingual directions can sometimes bene\ufb01t from each other thus improving the model performance. Chinese text generation is challenging in cross-lingual setting. As illustrated in Figure 1, nearly all models obtain inferior scores when generating Chinese text. Also, model results on Chinese inputs are usually worse than results on inputs in other languages. The wide discrepancies in grammar and vocabulary between Chinese and other languages lead to the poor performance of crosslingual generation when either the target language or source language is Chinese. Multilingual pretrained models obtain lower scores on the Summarization task. Compared with other tasks, summarization task requires longer output, which increases the dif\ufb01culty of text generation, thus causing poor performance both in cross-lingual and monolingual settings. Mono Multi Cross Zero 16 17 18 19 20 21 Ensemble Score Stage1 Stage2 Figure 2: The different stage performances averaged across \ufb01ve languages of XLM in summarization under various settings. Here stage1 represents models trained only on rough training data while stage2 represents models further trained on human-annotated training data based on models in stage1. 5.2 Monolingual and Multilingual In addition to cross-lingual analysis, we also explore the performance difference between models trained in monolingual and multilingual settings. Table 4 displays the monolingual and multilingual training results for four models in four tasks. In most cases, multilingual training can improve model performance on relevance. As shown in Table 4, 75 out of 96 multilingual results outperform the monolingual counterparts on various relevance metrics in different tasks. The reason is that the multilingual data in MTG is fully parallel across all \ufb01ve languages and every sample has semantically aligned counterparts in other languages. It makes better semantic fusion among different languages, thus boosting the multilingual training performance. The advantages of multilingual training are not obvious on diversity measured by distinct1. Especially in the story generation task, 3 out of 4 models obtain better distinct-1 scores in monolingual setting than in multilingual one. Diversity can not be improved by semantic sharing across languages especially when the samples of them are multiway parallel. This is because the multiway parallel dataset with the semantic aligned samples repeating in different languages encourages models to generate similar texts to some extent. 5.3 Zero-shot results To test the cross-lingual generation ability of multilingual pretrained models when no direct crosslingual training data are provided, we evaluate the zero-shot cross-lingual generation performance. Table 5 presents the zero-shot results for XLM in four tasks. It demonstrates that the multilin\fTask Language BLEU ROUGE-1 ROUGE-2 ROUGE-L METEOR BERTScore Distinct-1 Distinct-2 Ensemble SG en->de 0.02/3.20 7.20/27.20 0.20/4.00 7.20/25.80 0.05/0.14 0.63/0.73 0.47/0.96 0.50/1.00 18.90/29.70 en->fr 0.02/4.23 5.90/28.10 0.20/6.30 5.90/26.40 0.04/0.20 0.63/0.74 0.38/0.95 0.41/0.99 14.30/27.70 en->es 0.09/3.38 8.70/26.30 0.40/4.60 8.50/24.80 0.04/0.14 0.65/0.74 0.52/0.96 0.55/0.99 16.90/28.40 en->zh 0.00/5.79 0.00/28.80 0.00/8.80 0.00/26.80 0.45/0.67 0.61/0.99 0.57/0.34 16.60/26.70 QG en->de 1.96/10.41 18.10/38.70 2.40/14.70 17.60/37.20 0.10/0.25 0.73/0.78 0.94/0.97 0.98/1.00 29.80/29.30 en->fr 2.16/14.70 16.80/42.80 2.90/19.00 16.20/39.60 0.08/0.35 0.74/0.80 0.94/0.95 0.99/0.99 28.60/29.80 en->es 7.46/16.93 25.50/49.50 8.70/22.40 23.90/46.80 0.18/0.37 0.76/0.83 0.94/0.95 0.99/1.00 28.50/29.10 en->zh 0.00/16.07 0.00/43.10 0.00/22.90 0.00/37.90 0.44/0.73 0.10/1.00 0.08/1.00 16.40/28.60 TG en->de 2.58/9.15 13.40/26.90 4.40/11.10 12.50/24.30 0.12/0.22 0.67/0.73 0.83/0.95 0.88/0.99 26.30/30.60 en->fr 3.26/11.54 13.90/33.80 4.50/14.70 12.70/29.00 0.12/0.30 0.69/0.75 0.89/0.91 0.93/0.99 25.20/28.50 en->es 4.90/12.45 21.20/36.30 7.40/15.70 18.50/31.10 0.17/0.31 0.71/0.76 0.88/0.91 0.94/0.99 24.50/29.50 en->zh 0.01/15.44 0.00/34.50 0.00/19.40 0.00/29.90 0.45/0.69 0.37/0.98 0.22/0.58 16.70/27.10 Summ en->de 1.85/8.36 15.40/34.70 2.90/11.70 14.50/31.10 0.08/0.20 0.65/0.72 0.61/0.81 0.78/0.97 18.50/21.50 en->fr 1.29/11.79 13.70/39.90 2.60/15.80 13.00/35.50 0.07/0.29 0.68/0.75 0.64/0.75 0.82/0.94 18.60/20.30 en->es 4.18/11.93 22.50/41.00 5.80/15.60 20.30/36.60 0.14/0.29 0.69/0.75 0.64/0.74 0.82/0.95 17.30/20.70 en->zh 0.00/14.58 0.00/42.20 0.00/20.40 0.00/38.70 0.42/0.71 0.68/0.84 0.27/0.94 12.80/19.60 Table 5: English centric zero-shot and cross-lingual results for XLM on four tasks. Scores on the left and right side of each cell represent the zero-shot and cross-lingual results respectively. gual pretrained model XLM still lacks the ability to generate high-quality cross-lingual output in zero-shot scenario. Moreover, English to Chinese and French zero-shot generation shows inferior performance.9 The performance decline is rather salient when generating Chinese text. This is because Chinese and French (especially Chinese) are distant from English in the language family tree. On the other hand, zero-shot results underperform cross-lingual results which further emphasizes the importance of direct cross-lingual training data for cross-lingual text generation. 5.4 Pseudo and Annotated Data To answer the question \u201cDoes the 400k annotated training data help the model generate better? \u201d, we use the rough training data \ufb01ltered by back translation for the \ufb01rst stage \ufb01ne-tuning and the annotated training data for the second stage. The ablation study results on the two-step \ufb01netuning in summarization under all evaluation scenarios with XLM are illustrated in Figure 2. The extra human-annotated data boost model performance by at least 3.8% on the ensemble metric. We also make a T-test and prove that the improvement of annotated training data is signi\ufb01cant in all settings.10 It demonstrates that although the number of annotated data is small, it can signi\ufb01cantly improve the performance. It also highlights the necessity of human-annotated multilingual data compared with pseudo-parallel data via machine translation. 9Zero-shot results show the same trend as shown in Table 17 in Appendix. 10The t-test details are shown in Appendix C. Setting Model Gram. Flu. Rel. lang fuse task spec. SG mono 4.69 4.81 3.75 1.00 3.79 multi 4.71 4.80 3.67 1.00 4.02 cross 4.18 4.23 3.49 0.95 2.53 zero 4.15 4.18 3.27 0.18 3.00 QG mono 4.66 4.69 3.03 0.99 3.95 multi 4.69 4.67 3.06 0.97 4.11 cross 4.30 4.30 2.70 0.95 2.64 zero 3.35 4.26 3.18 0.19 3.09 TG mono 4.53 4.51 3.09 0.96 3.71 multi 4.66 4.65 3.18 0.93 3.17 cross 3.73 3.64 2.63 0.90 1.85 zero 3.52 4.15 3.51 0.18 1.43 Summ mono 4.19 3.99 3.71 0.68 3.71 multi 4.19 4.02 3.78 0.64 3.60 cross 2.14 2.22 2.23 0.68 2.05 zero 1.57 1.54 1.58 0.03 1.59 Table 6: Human evaluation scores averaged on \ufb01ve languages for mBART on four tasks. \u2018Gram.\u2019, \u2018Flu.\u2019, \u2018Rel.\u2019, \u2018Lang Fu.\u2019, \u2018Task Spec.\u2019 indicates Grammar, Fluency, Relevance, Language Fusion and Task Speci\ufb01c scores respectively. 5.5 Human evaluation Table 6 presents the human evaluation scores for mBART in four tasks. Multilingual training results can surpass the monolingual results in QG, TG and Summ on relevance. In terms of task-speci\ufb01c score, multilingual results are also superior in SG and QG. This is consistent with the conclusion in Sec. 5.2. On the other hand, language fusion scores in zero-shot setting are extremely low, indicating the pretrained models still lack the ability to generate texts in correct language in zero-shot setting. 6 Discussions Considering the annotation cost, it is not realistic to construct a multiway text generation dataset with all data annotated by human. As a consequence, most of the non-English data in MTG are automatically translated from their English counterparts. Although the n-gram consistency check \fwhen round-trip translating the data can guarantee the quality of them to some extent, some translation errors are inevitable. MTG with more annotated data and with data \ufb01ltered by more reliable methods will be explored in the future. On the other hand, human often gives an overall evaluation of a generated text rather than measuring it in \ufb01ne-grained aspects of grammar, \ufb02uency and relevance. Thus we try to propose a metric measuring how a text resembles human writing and consider grammar, \ufb02uency and relevance as a whole. This metric may not be perfect, but it is a promising direction as there does not exist a really reliable text generation metric nowadays. 7"
+                }
+            ],
+            "Changyou Chen": [
+                {
+                    "url": "http://arxiv.org/abs/1805.11659v2",
+                    "title": "A Unified Particle-Optimization Framework for Scalable Bayesian Sampling",
+                    "abstract": "There has been recent interest in developing scalable Bayesian sampling\nmethods such as stochastic gradient MCMC (SG-MCMC) and Stein variational\ngradient descent (SVGD) for big-data analysis. A standard SG-MCMC algorithm\nsimulates samples from a discrete-time Markov chain to approximate a target\ndistribution, thus samples could be highly correlated, an undesired property\nfor SG-MCMC. In contrary, SVGD directly optimizes a set of particles to\napproximate a target distribution, and thus is able to obtain good\napproximations with relatively much fewer samples. In this paper, we propose a\nprinciple particle-optimization framework based on Wasserstein gradient flows\nto unify SG-MCMC and SVGD, and to allow new algorithms to be developed. Our\nframework interprets SG-MCMC as particle optimization on the space of\nprobability measures, revealing a strong connection between SG-MCMC and SVGD.\nThe key component of our framework is several particle-approximate techniques\nto efficiently solve the original partial differential equations on the space\nof probability measures. Extensive experiments on both synthetic data and deep\nneural networks demonstrate the effectiveness and efficiency of our framework\nfor scalable Bayesian sampling.",
+                    "authors": "Changyou Chen, Ruiyi Zhang, Wenlin Wang, Bai Li, Liqun Chen",
+                    "published": "2018-05-29",
+                    "updated": "2018-07-10",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Bayesian methods have been playing an important role in modern machine learning, especially in unsupervised learning (Kingma and Welling, 2014; Li et al., 2017), and recently in deep reinforcement learning (Houthooft et al., 2016; Liu et al., 2017). When dealing with big data, two lines of research directions have been developed to scale up Bayesian methods, e.g., variational-Bayes-based and sampling-based methods. Stochastic gradient Markov chain Monte Carlo (SG-MCMC) is a family of scalable Bayesian learning algorithms designed to ef\ufb01ciently sample from a target distribution such as a posterior distribution (Welling and Teh, 2011; Chen et al., 2014; Ding et al., 2014; Chen et al., 2015). In principle, SG-MCMC generates samples from a Markov chain, which are used to approximate a target distribution. Under a standard setting, samples from SG-MCMC are able to match a target distribution exactly with an in\ufb01nite number of samples (Teh et al., 2016; Chen et al., 2015). However, this is practically infeasible, as only a \ufb01nite number of samples are obtained. Although nonasymptotic approximation bounds w.r.t.the number of samples have been investigated (Teh et al., 2016; Vollmer et al., 2016; Chen et al., 2015), there are no theory/algorithms to guide learning an optimal set of \ufb01xed-size samples/particles. This is an undesirable property of SG-MCMC, because in practice one often seeks to learn the optimal samples of a \ufb01nite size that best approximate a target distribution. A remedy for this issue is to adopt the idea of particlebased sampling methods, where a set of particles (or samples) are initialized from some simple distribution, followed by iterative updates to better approximate a target distribution. The updating procedure is usually done by optimizing some metrics such as a distance measure between the target distribution and the current approximation. There is not much work in this direction for large-scale Bayesian sampling, with an outstanding representative being the Stein variational gradient descent (SVGD) (Liu and Wang, 2016a). In SVGD, the update of particles are done by optimizing the KL-divergence between the empirical particle distribution and a target distribution, thus the samples are designed to be updated optimally to reduce the KL-divergence in each iteration. Because of this property, SVGD is found to perform better than SG-MCMC when the number of samples used to approximate a target distribution is limited, and has been applied to other problems such as deep generative models arXiv:1805.11659v2  [stat.ML]  10 Jul 2018 \f(Feng et al., 2017) and deep reinforcement learning (Liu et al., 2017; Haarnoja et al., 2017; Zhang et al., 2018b). Though often achieving comparable performance in practice, little work has been done on investigating connections between SG-MCMC and SVGD, and on developing particle-optimization schemes for SG-MCMC. In this paper, adopting ideas from Waserstein-gradient-\ufb02ow literature, we propose a uni\ufb01ed particle-optimization framework for scalable Bayesian sampling. The idea of our framework is to work directly on the evolution of a density functions on the space of probability measures, e.g., the Fokker-Planck equation in SG-MCMC. To make the evolution solution computationally feasible, particle approximations are adopted for densities, where particles can be optimized during the evolution process. Both SG-MCMC and SVGD are special cases of our framework, and are shown to be highly related. Notably, sampling with SG-MCMC becomes a deterministic particleoptimization problem as SVGD on the space of probability measures, overcoming the aforementioned correlatedsample issue. Furthermore, we are able to develop new uni\ufb01ed particle-optimization algorithms by combing SGMCMC and SVGD, which is less prone to high-dimension space and thus obtains better performance for large-scale Bayesian sampling. We conduct extensive experiments on both synthetic data and Bayesian learning of deep neural networks, verifying the effectiveness and ef\ufb01ciency of our proposed framework. 2 PRELIMINARIES In this section, we review related concepts and algorithms for SG-MCMC, SVGD, and Wasserstein gradient \ufb02ows (WGF) on the space of probability measures. 2.1 Stochastic gradient MCMC Diffusion-based sampling methods Generating random samples from a distribution (e.g., a posterior distribution) is one of the fundamental problems in Bayesian statistics, which has many important applications in machine learning. Traditional Markov Chain Monte Carlo methods (MCMC), such as the Metropolis\u2013Hastings algorithm (Metropolis et al., 1953) produces unbiased samples from a desired distribution when the density function is known up to a normalizing constant. However, most of these methods are based on random walk proposals which suffer from high dimensionality and often lead to highly correlated samples. On the other hand, dynamicsbased sampling methods such as the Metropolis adjusted Langevin algorithm (MALA) (Xifara et al., 2014) avoid this high degree of correlation by combining dynamical systems with the Metropolis step. In fact, these dynamical systems are derived from a more general mathematical technique called diffusion process, or more speci\ufb01cally, It\u00b4 o diffusion (\u00d8ksendal, 1985). Speci\ufb01cally, our objective is to generate random samples from a posterior distribution p(\u03b8| X) \u221dp(X |\u03b8)p(\u03b8), where \u03b8 \u2208Rr represents the model parameter, and X \u225c{xi}N i=1 represents the data. The canonical form is p(\u03b8| X) = (1/Z) exp(U(\u03b8)), where U(\u03b8) = log p(X |\u03b8) + log p(\u03b8) \u225c N X i=1 log p(xi |\u03b8) + log p(\u03b8) is referred to as the potential energy based on an i.i.d. assumption of the model, and Z is the normalizing constant. In Bayesian sampling, the posterior distribution corresponds to the (marginal) stationary distribution of a (continuous-time) It\u00b4 o diffusion, de\ufb01ned as a stochastic differential equation of the form: d\u0398t = F(\u0398t)dt + g(\u0398t)dWt , (1) where t is the time index; \u0398t \u2208Rp represents the full variables in a dynamical system, and \u0398t \u2287\u03b8t (thus p \u2265r) is potentially an augmentation of model parameter \u03b8; Wt \u2208Rp is p-dimensional Brownian motion. Functions F : Rp \u2192Rp and g : Rp \u2192 Rp \u00d7 p are assumed to satisfy the Lipschitz continuity condition (Ghosh, 2011). By Fokker-Planck equation (or the forward Kolmogorov equation) (Kolmogoroff, 1931; Risken, 1989), when appropriately designing the diffusion-coef\ufb01cient functions F(\u00b7) and g(\u00b7), the stationary distribution of the corresponding It\u00b4 o diffusion equals the posterior distribution of interest, p(\u03b8| X). For example, the 1st-order Langevin dynamic de\ufb01nes \u0398 = \u03b8, and F(\u0398t) = 1 2\u2207\u03b8U(\u03b8), g(\u0398t) = Ir; the 2nd-order Langevin diffusion de\ufb01nes \u0398 = (\u03b8, q), and F(\u0398t) = \u0010 q \u2212B q \u2212\u2207\u03b8U(\u03b8) \u0011 , g(\u0398t) = \u221a 2B \u0010 0 0 0 Ir \u0011 for a scalar B > 0; q is an auxiliary variable known as the momentum (Chen et al., 2014; Ding et al., 2014). Let the density of \u0398t be \u00b5t, it is known \u00b5t is characterized by the Fokker-Planck (FP) equation (Risken, 1989): \u2202\u00b5t \u2202t = \u2212\u2207\u0398 \u00b7 (\u00b5tF(\u0398t)) + \u2207\u0398\u2207\u0398 :(\u00b5t\u03a3(\u0398t)) (2) where \u03a3(\u0398t) \u225cg(\u0398t)g\u22a4(\u0398t), a \u00b7 b \u225ca\u22a4b for vectors a and b, A : B \u225ctrace(A\u22a4B) for matrices A and B. The FP equation is the key to develop our particleoptimization framework for SG-MCMC. In the following, we focus on the simplest case of 1st-order Langevin dynamics if not stated explicitly, though the derivations apply to other variants. Stochastic gradient MCMC SG-MCMC algorithms are discretized numerical approximations of the It\u00b4 o dif\ffusion (1). They mitigate the slow mixing and nonscalability issues encountered in traditional MCMC algorithms by i) adopting gradient information of the posterior distribution, ii) using minibatches of the data in each iteration of the algorithm to generate samples, and iii) ignoring the rejection step as in standard MCMC. To make the algorithms scalable in a big-data setting, three developments will be implemented based on the It\u00b4 o diffusion: i) de\ufb01ne appropriate functions F and g in the It\u00b4 o-diffusion formula so that the (marginal) stationary distributions coincide with the target posterior distribution p(\u03b8| X); ii) replace F or g with unbiased stochastic approximations to reduce the computational complexity, e.g., approximating F with a random subset of the data instead of using the full data. For example, in the 1st-order Langevin dynamics, \u2207\u03b8U(\u03b8) could be approximated by an unbiased estimator with a subset of data: \u2207\u03b8 \u02dc U(\u03b8) \u225c\u2207log p(\u03b8) + N n n X i=1 \u2207\u03b8 log p(x\u03c0i |\u03b8) (3) where \u03c0 is a size-n random subset of {1, 2, \u00b7 \u00b7 \u00b7 , N}, leading to the \ufb01rst SG-MCMC algorithm in machine learning \u2013 stochastic gradient Langevin dynamics (SGLD) (Welling and Teh, 2011); and iii) solve the generally intractable continuous-time It\u02c6 o diffusions with a numerical method, e.g., the Euler method (Chen et al., 2015). For example, this leads to the following update in SGLD: \u03b8\u2113= \u03b8\u2113\u22121 + \u2207\u03b8 \u02dc U(\u03b8\u2113\u22121)h + \u221a 2h \u03b4\u2113, where h means the stepsize, \u2113indexes the samples, \u03b4\u2113\u223c N(0, I) is a random sample from an isotropic normal distribution. After running the algorithm for L steps, the collection of samples {\u03b8\u2113}L \u2113=1 are used to approximate the unknown posterior distribution 1 Z eU(\u03b8). 2.2 Stein variational gradient descent Different from SG-MCMC, SVGD initializes a set of particles which are iteratively updated so that the empirical particle distribution approximates the posterior distribution. Speci\ufb01cally, we consider a set of particles {\u03b8(i)}M i=1 drawn from some distribution q. SVGD tries to update these particles by doing gradient descent on the interactive particle system via \u03b8(i) \u2190\u03b8(i) + h\u03c6(\u03b8(i)), \u03c6 = arg max \u03c6\u2208F { \u2202 \u2202hKL(q[h\u03c6]||p)} where \u03c6 is a function perturbation direction chosen to minimize the KL divergence between the updated density q[h\u03c6] estimated by the particles and the posterior p(\u03b8| X) (p for short). Since KL(q\u2225p) is convex in q, global optimum of q = p can be guaranteed. SVGD considers F as the unit ball of a vector-valued reproducing kernel Hilbert space (RKHS) H associated with a kernel \u03ba(\u03b8, \u03b8\u2032). In such as setting, Liu and Wang (2016b) shown: \u2212\u2202 \u2202hKL(q[h\u03c6]\u2225p)|h=0 = E\u03b8\u223cq[trace(\u0393p\u03c6(\u03b8))], (4) with \u0393p\u03c6(\u03b8) \u225c\u2207\u03b8U(\u03b8)\u22a4\u03c6(\u03b8) + \u2207\u03b8 \u00b7 \u03c6(\u03b8), where \u0393p is called the Stein operator. Assuming that the update function \u03c6(\u03b8) is in a RKHS with kernel \u03ba(\u00b7, \u00b7), it was shown in (Liu and Wang, 2016b) that (4) is maximized with: \u03c6(\u03b8) = E\u03b8\u223cq[\u03ba(\u03b8, \u03b8\u2032)\u2207\u03b8U(\u03b8) + \u2207\u03b8\u03ba(\u03b8, \u03b8\u2032)]. (5) When approximating the expectation E\u03b8\u223cq[\u00b7] with empirical particle distribution and adopting stochastic gradients, we arrive at the following updates for the particles (\u2113 denotes the iteration number): \u03b8(i) \u2113+1 = \u03b8(i) \u2113+ h M M X j=1 h \u03ba(\u03b8(j) \u2113, \u03b8(i) \u2113)\u2207\u03b8(j) \u2113 \u02dc U(\u03b8(j) \u2113) + \u2207\u03b8(j) \u2113\u03ba(\u03b8(j) \u2113, \u03b8(i) \u2113) i (6) SVGD applies updates (6) repeatedly, moving the samples to a target distribution p. 2.3 Wasserstein Gradient Flows For a better motivation of WGF, we start from gradient \ufb02ows de\ufb01ned on the Euclidean space. Gradient \ufb02ows on the Euclidean space For a smooth function E : Rr \u2192R, and a starting point \u03b80 \u2208Rr, the gradient \ufb02ow of E(\u03b8) is de\ufb01ned as the solution of the differential equation: d\u03b8 dt = \u2212\u2207E(\u03b8(t)), s.t. \u03b8(0) = \u03b80. This is a standard Cauchy problem (Rulla, 1996), endowed with a unique solution if \u2207E is Lipschitz continuous. When E is non-differentiable, the gradient is replaced with its subgradient, de\ufb01ned as \u2202E(\u03b8) \u225c{p \u2208 Rr : F(\u03b8\u2032) \u2265F(\u03b8) + p \u00b7(\u03b8\u2032 \u2212\u03b8), \u2200\u03b8\u2032 \u2208Rr}. Note \u2202E(\u03b8) = {\u2207E(\u03b8)} if E is differentiable at \u03b8. In this case, the gradient \ufb02ow formula above is replaced with: d\u03b8 dt \u2208\u2212\u2202E(\u03b8(t)). Wasserstein gradient \ufb02ows Let P(\u2126) denote the space of probability measures on \u2126\u2282Rr. WGF is an extension of gradient \ufb02ows in Euclidean space by lifting the de\ufb01nition onto the space of probability measures. Formally, let P(\u2126) be endowed with a Riemannian geometry induced by the 2nd-order Wasserstein distance, i.e., the curve length between two elements (two distributions) is de\ufb01ned as: W 2 2 (\u00b5, \u03bd) \u225cinf \u03b3 \u001aZ \u2126\u00d7\u2126 \u2225\u03b8 \u2212\u03b8\u2032\u22252 2d\u03b3(\u03b8, \u03b8\u2032) : \u03b3 \u2208\u0393(\u00b5, \u03bd) \u001b where \u0393(\u00b5, \u03bd) is the set of joint distributions over (\u03b8, \u03b8\u2032) such that the two marginals equal \u00b5 and \u03bd, respectively. The Wasserstein distance can be explained as an optimaltransport problem, where one wants to transform elements \fin the domain of \u00b5 to \u03bd with a minimum cost (Villani, 2008). The term \u2225\u03b8\u2212\u03b8\u2032\u22252 2 represents the cost to transport \u03b8 in \u00b5 to \u03b8\u2032 in \u03bd, and can be replaced by a general metric c(\u03b8, \u03b8\u2032) in a metric space. If \u00b5 is absolutely continuous w.r.t. the Lebesgue measure, there is a unique optimal transport plan from \u00b5 to \u03bd, i.e., a mapping T : Rr \u2192Rr pushing elements in the domain of \u00b5 onto \u03bd satisfying T#\u00b5 = \u03bd. Here T#\u00b5 denotes the pushforward measure of \u00b5. The Wasserstein distance is equivalently reformulated as: W 2 2 (\u00b5, \u03bd) \u225cinfT \bR \u2126c(\u03b8, T (\u03b8))d\u00b5(\u03b8) \t . Let {\u00b5t}t\u2208[0,1] be an absolutely continuous curve in P(\u2126) with \ufb01nite second-order moments. We consider to de\ufb01ne the change of \u00b5t\u2019s by investigating W 2 2 (\u00b5t, \u00b5t+h). Motivated by the Euclidean-space case, this is re\ufb02ected by a vector \ufb01eld, vt(\u03b8) \u225climh\u21920 T (\u03b8t)\u2212\u03b8t h called the velocity of the particle. A gradient \ufb02ow can be de\ufb01ned on P(\u2126) correspondingly (Ambrosio et al., 2005). Lemma 1 Let {\u00b5t}t\u2208[0,1] be an absolutely-continuous curve in P(\u2126) with \ufb01nite second-order moments. Then for a.e. t \u2208[0, 1], the above vector \ufb01eld vt de\ufb01nes a gradient \ufb02ow on P(\u2126) as: \u2202t\u00b5t + \u2207\u00b7 (vt \u00b5t) = 0. The gradient \ufb02ow describes the evolution of a functional E, which is a lifted version of the function in the case of Euclidean space in Section 2.3 to the space of probability measures. E maps a probability measure \u00b5 to a real value, i.e., E : P(\u2126) \u2192R. We will focus on the case where E is convex in this paper, which is enough considering gradient \ufb02ows for SG-MCMC and SVGD, though the theory applies to a more general \u03bb-convex energy functional setting (Ambrosio et al., 2005). It can be shown that vt in Lemma 1 has the form vt = \u2212\u2207\u03b4E \u03b4\u00b5t (\u00b5t) (Ambrosio et al., 2005), where \u03b4E \u03b4\u00b5t is called the \ufb01rst variation of E at \u00b5t. Based on this, gradient \ufb02ows on P(\u2126) can be written \u2202t\u00b5t = \u2212\u2207\u00b7 (vt \u00b5t) = \u2207\u00b7 \u0012 \u00b5t\u2207( \u03b4E \u03b4\u00b5t (\u00b5t)) \u0013 . (7) Remark 1 Intuitively, an energy functional E characterizes the landscape structure (appearance) of the corresponding manifold in P(\u2126), and the gradient \ufb02ow (7) de\ufb01nes a geodesic path on this manifold. Usually, by choosing appropriate E, the landscape is convex, e.g., for the cases of both SG-MCMC and SVGD described below. This provides a theoretical guarantee on the optimal convergence of a gradient \ufb02ow. 3 PARTICLE-OPTIMIZATIONBASED SAMPLING In this section, we interpret the continuous versions of both SG-MCMC and SVGD as WGFs, followed by several techniques for particle optimization in the next section. In the following, \u00b5t denotes the distribution of \u03b8t. 3.1 SVGD as WGF The continuous-time and in\ufb01nite-particle limit of SVGD with full gradients, denoted as SVGD\u221e, is known to be a special instance of the Vlasov equation in nonlinear partial-differential-equation literature (Liu, 2017): \u2202t\u00b5t = \u2207\u00b7 ((W \u2217\u00b5t)\u00b5t) , (8) where (W \u2217\u00b5t)(\u03b8) \u225c R W(\u03b8 \u2212\u03b8\u2032)\u00b5t(\u03b8\u2032)d\u03b8\u2032 is the convolutional operator applied for some function W : Rr \u2192 R. To specify SVGD\u221e, we generalize the convolutional operator, and consider W as a function with two input arguments, i.e., (W \u2217\u00b5t)(\u03b8) \u225c Z W(\u03b8, \u03b8\u2032)\u00b5t(\u03b8\u2032)d\u03b8\u2032 . Under this setting, we can specify the function W(\u00b7, \u00b7) for SVGD\u221eas W(\u03b8, \u03b8\u2032) \u225c\u2207\u03b8\u2032 log p(\u03b8\u2032| X)\u03ba(\u03b8\u2032, \u03b8) + \u2207\u03b8\u2032\u03ba(\u03b8\u2032, \u03b8) = \u2207\u03b8\u2032 [p(\u03b8\u2032| X)\u03ba(\u03b8, \u03b8\u2032)] /p(\u03b8\u2032) . (9) As will be shown in Section 4, W in (9) naturally leads to the SVGD algorithm, without the need to derive from an RKHS perspective. Proposition 2 The stationary distribution of (8) is limt\u2192\u221e\u00b5t \u225c\u00b5 = p(\u03b8| X). To interpret SVGD\u221eas a WGF, we need to specify two quantities, the energy functional and an underlying metric to measure distances between density functions. Energy functional and distance metric of SVGD\u221e There are two ways to derive energy functionals for SVGD\u221e, depending on the underlying metrics for probability distributions. When adopting the WGF framework where W2 is used as the underlying metric, according to (7), the energy functional Es must satisfy \u2207\u03b8 \u0012\u03b4Es \u03b4\u00b5t (\u00b5t) \u0013 = W(\u03b8, \u03b8\u2032) \u2217\u00b5t (10) = E\u03b8\u2032\u223c\u00b5t \u0002 \u2207\u03b8\u2032 \u0002 p(\u03b8\u2032| X)K(\u03b8, \u03b8\u2032) \u0003 /p(\u03b8\u2032| X) \u0003 . In general, there is no close-form solution for the above equation. Alternatively, Liu (2017) proved another form of the energy functional by de\ufb01ning a different distance metric on the space of probability measures, called HWasserstein distance: WH(q1, q2) \u225cinf \u03c6t,\u00b5t \u001aZ 1 0 \u2225\u03c6t\u2225Hdt, s.t. \u00b5t = \u2212\u2207\u03b8 \u00b7 (\u03c6t\u00b5t), \u00b50 = q1, \u00b51 = q2\u2225} , (11) where \u03c6t \u225cW \u2217\u00b5t, and \u2225\u00b7 \u2225H is the norm in the Hilbert space induced by \u03ba(\u00b7, \u00b7). Under this metric, the underlying energy functional is proved to be the standard KLdivergence between \u00b5t and p, e.g., Es = KL(\u00b5t, p(\u00b7| X)). As can be seen in Section 4, this interpretation allows one to derive SVGD, a particle-optimization-based algorithm to approximate the continuous-time equation (8). \f3.2 SG-MCMC as WGF The continuous-time limit of SG-MCMC, when considering gradients to be exact, corresponds to standard It\u00b4 o diffusions. We consider the It\u00b4 o diffusion of SGLD for simplicity, e.g., d\u03b8t = 1 2\u2207U(\u03b8t)dt + dW . (12) Energy functional The energy functional for SGMCMC is easily seen by noting that the corresponding FP equation (2) is in the gradient-\ufb02ow form of (7). Speci\ufb01cally, the energy functional E is de\ufb01ned as: E(\u00b5) \u225c\u2212 Z U(\u03b8)\u00b5(\u03b8)d\u03b8 | {z } E1 + Z \u00b5(\u03b8) log \u00b5(\u03b8)d\u03b8 | {z } E2 (13) Note E2 is the energy functional of a pure Brownian motion (e.g., U(\u03b8) = 0 in (12)). We can verify (13) by showing that it satis\ufb01es that FP equation. According to (7), the \ufb01rst variation of E1 and E2 is calculated as \u03b4E1 \u03b4\u00b5 = \u2212U, \u03b4E2 \u03b4\u00b5 = log \u00b5 + 1 . (14) Substituting (14) into (7) recovers the FP equation (2) for the It\u00b4 o diffusion (12). 4 PARTICLE OPTIMIZATION An ef\ufb01cient way to solve the generally infeasible WGF formula (7) is to adopt numerical methods with particle approximation. With a little abuse of notation but for conciseness, we do not distinguish subscripts t and \u2113for the particle \u03b8, i.e., \u03b8t denotes the continuous-time version of the particle, while \u03b8\u2113denotes the discrete-time version. We develop several techniques to approximate different types of WGF for SG-MCMC and SVGD. In particle approximation, the continuous density \u00b5t is approximated by a set of M particles (\u03b8(i) t )M i=1 that evolve over time t with weights (mi)M i=1 such that PM i=1 mi = 1, i.e., \u00b5t \u2248PM i=1 mi\u03b4(\u03b8(i) t ), where \u03b4(\u03b8(i) t ) = 1 when \u03b8 = \u03b8(i) t and 0 otherwise. Typically m\u2032 is are chosen at the beginning and \ufb01xed over time, thus we assume mi = 1 M and rewrite \u00b5t \u2248 1 M PM i=1 \u03b4(\u03b8(i) t ) in the following for simplicity. We investigate two types of particleapproximation methods in the following, discrete gradient \ufb02ows and by blob methods. Particle approximation by discrete gradient \ufb02ows Denote Ps(Rr) be the space of probability measures with \ufb01nite 2nd-order moments. De\ufb01ne the following optimization problem with stepsize h: Jh(\u00b5) \u225carg min \u03bd\u2208Ps(Rd) \u001a 1 2hW 2 2 (\u00b5, \u03bd) + E(\u03bd) \u001b . (15) A discrete gradient \ufb02ow of the continuous one in (7) up to time T is the composition of a sequence of the solutions (\u02dc \u00b5\u2113)T/h \u2113=1 of (15), i.e., \u02dc \u00b5\u2113\u225cJh(\u02dc \u00b5\u2113\u22121) = Jh(Jh(\u00b7 \u00b7 \u00b7 \u00b50)) \u225cJ\u2113 h\u00b50 . (16) One can show that when h \u21920, the discrete gradient \ufb02ow (16) converges to the true \ufb02ow (7) for all \u2113. Speci\ufb01cally, let \u2202E(\u00b5) be the set of Wasserstein subdifferential of E at \u00b5, i.e., \u03be \u2208\u2202E(\u00b5) if \u2202t\u00b5 = \u03be is satis\ufb01ed. De\ufb01ne |\u2202E|(\u00b5) = min{\u2225\u03be\u2225L2(\u00b5) : \u03be \u2208\u2202E(\u00b5)} to be the minimum norm of the elements in \u2202E(\u00b5). We have Lemma 3 (Craig (2014)) Assume E is proper, coercive and lower semicontinuous (specify in Section B of the Supplementary Material (SM)). For an \u00b50 and t \u22650, as T h \u2192\u221e, the discrete gradient sequence \u02dc \u00b5T/h \u225cJT/h h \u00b50 converge uniformly in t to a compact subset of [0, +\u221e), and W 2 2 (\u02dc \u00b5T/h, \u00b5T ) \u2264 \u221a 3|\u2202E|(\u00b5) \u221a Th. Lemma 3 suggests the discrete gradient \ufb02ow can approximate the original WGF arbitrarily well if a small enough stepsize h is adopted. Consequently, one solves (16) through a sequence of optimization procedures to update the particles. We will derive a particle-approximation method for the W2 term in (15), which allows us to solve SG-MCMC ef\ufb01ciently. However, this technique is not applicable to SVGD, as we neither have an explicit form of the energy functional in (10) when adopting the W2 metric, nor have an explicit form for the metric WH in (11) when adopting the KL-divergence as the energy functional. Fortunately, this can be solved by the second approximation method called blob methods. Particle approximation by blob methods The name of blob methods comes from the classical \ufb02uids literature, where instead of evolving the density in (7), one evolves all particles on a grid with time-spacing h (Carrillo et al., 2017). Speci\ufb01cally, note the function vt in (7) represents velocity of particles via transportation map T , thus solving a WGF is equivalent to evolving the particles along their velocity in each iteration. Formally, one can prove Proposition 4 (Craig and Bertozzi (2016)) Let \u00b50 \u2248 1 M PM i=1 \u03b4(\u03b8(i) 0 ). Assume vt in (7) is well-de\ufb01ned and continuous w.r.t.each \u03b8(i) t at time t. Then solving the PDE (7) reduces to solving a system of ordinary differential equations for the locations of the Dirac masses: d\u03b8(i) t /dt = \u2212vt(\u03b8(i) t ) . (17) Proposition 4 suggests evolving each particle along the directions de\ufb01ned by vt, eliminating the requirement to know an explicit form of the energy functional. In the following, we apply the above particle-optimization techniques to derive algorithms for SVGD and SG-MCMC. \f4.1 A particle-optimization algorithm for SVGD As mentioned above, discrete-gradient-\ufb02ow approximation does not apply to SVGD. We thus rely on the blob method. From Section 3.1, vt in SVGD is de\ufb01ned as vt(\u03b8) = (W \u2217\u00b5t)(\u03b8). When \u00b5t(\u03b8) is approximated by particles, vt(\u03b8(i) t ) is simpli\ufb01ed as: vt(\u03b8(i) t ) = 1 M M X j=1 W(\u03b8(i) t , \u03b8(j) t ) . As a result, with the de\ufb01nition of W in (9), updating {\u03b8(i) t } by time discretizing (17) recovers the update equations for standard SVGD in (6). 4.2 Particle-optimization algorithms for SG-MCMC Both the discrete-gradient-\ufb02ow and the blob methods can be applied for SG-MCMC, which are detailed below. Particle optimization with discrete gradient \ufb02ows We \ufb01rst specify Lemma 3 in the case of SG-MCMC in Lemma 5, which is known as the Jordan-KinderlehrerOtto scheme (Jordan et al., 1998). Lemma 5 (Jordan et al. (1998)) Assume that p(\u03b8t| X) \u2264 C1 is in\ufb01nitely differentiable, and \u2225\u2207\u03b8 log p(\u03b8| X)\u2225 \u2264 C2 (1 + C1 \u2212log p(\u03b8| X)) (\u2200\u03b8) for some constants {C1, C2}. Let T = hK with K the number of iterations, \u02dc \u00b50 be an arbitrary distribution with same support as p(\u03b8| X), and {\u02dc \u00b5k}K k=1 be the solution of the functional optimization problem: \u02dc \u00b5k = arg min \u00b5\u2208Ps(Rr) KL (\u00b5\u2225p) + 1 2hW 2 2 (\u02dc \u00b5k\u22121, \u00b5) . (18) Then \u02dc \u00b5K converges to \u00b5T in the limit of h \u21920, i.e., limh\u21920 \u02dc \u00b5K = \u00b5T , where \u00b5T is the solution of the FP equation (2) at time T. According to Lemma 5, it is apparent that SG-MCMC can be implemented by iteratively solving the optimization problem in (18). However, particle approximations for both terms in (18) are challenging. In the following, we develop ef\ufb01cient techniques to solve the problem. First, rewrite the optimization problem in (18) as min \u00b5\u2208Ps(Rr) \u2212E\u00b5[log p(\u03b8| X)] | {z } F1 + E\u00b5[log \u00b5] + 1 2hW 2 2 (\u02dc \u00b5k\u22121, \u00b5) | {z } F2 We aim at deriving gradient formulas for both the F1 and F2 terms under a particle approximation in order to perform gradient descent for the particles. Let \u00b5 \u2248 1 M PM i=1 \u03b4(\u03b8(i)). The gradient of F1 is easily approximated as \u2202F1 \u2202\u03b8(i) \u2248\u2212\u2207\u03b8(i) log p(\u03b8(i)| X) . (19) To approximate the gradient for F2, let pij denote the joint distribution of the particle-pair (\u03b8(i), \u03b8(j) k\u22121). Note E\u00b5[log \u00b5] is minimized when the particles {\u03b8(i)} are uniformly distributed. In other words, the marginal distribution vector (P j pij)i is a uniform distribution. Combining E\u00b5[log \u00b5] with the de\ufb01nition of W2, calculating F2 is equivalent to solving the following optimization problem: P \u225c{pij} = arg min pi,j X i,j pijdij (20) s.t. X j pij = 1 M , X i pij = 1 M , where dij \u225c\u2225\u03b8(i) \u2212\u03b8(j) k\u22121\u22252. We can further enforce the joint distribution {pij} to have maximum entropy by introducing a regularization term Epij[log pij], which is stronger than the regularizer enforced for the marginal distribution above. After introducing Lagrangian multipliers {\u03b1i, \u03b2i} to deal with the constraints in (20), we arrive at the dual problem: maxLD({pij}, {\u03b1i}, {\u03b2i}) = \u03bb X i,j pij log pij + pijdij + X i \u03b1i( X j pij \u22121 M ) + X j \u03b2j( X i pij \u22121 M ) , where \u03bb is the weight for the regularizer. The optimal pij\u2019s can be obtained by applying KKT conditions to set the derivative w.r.t. pij to be zero, ending up with the following form: p\u2217 ij = uie\u2212dij/\u03bbvj , where ui \u225ce\u22121 2 \u2212\u03b1i \u03bb , vj = e\u22121 2 \u2212 \u03b2j \u03bb . As a result, the particle gradients on F2 can be approximated as \u2202F2 \u2202\u03b8(i) \u2248\u2212 P j uivjdije\u2212dij/\u03bb \u2202\u03b8(i) (21) = X j 2uivj(dij \u03bb \u22121)e\u2212dij/\u03bb(\u03b8(i) \u2212\u03b8(j) k\u22121) . Theoretically, we need to adaptively update {ui, vj} as well to ensure the constraints in (20). In practice, however, we use a \ufb01xed scaling factor \u03b3 to approximate uivj for the sake of simplicity. Particle gradients are obtained by combining (19) and (21), which are then used to update the particles {\u03b8(i)} by standard gradient descent. Intuitively, (19) encourages particles move to local modes while (21) regularizes particle interactions. Different from SVGD, our scheme imposes both attractive and repulsive forces for the particles. Speci\ufb01cally, by inspecting (21), we can conclude that: i) When \u03b8(i) is far from a previous particle \u03b8(j) k , i.e., dij \u03bb > 1, \u03b8(i) is pulled close to {\u03b8(j) k } with force proportional to ( dij \u03bb \u22121)e\u2212dij/\u03bb; ii) when \u03b8(i) is close enough to a previous particle \u03b8(j) k , i.e., dij \u03bb < 1, \u03b8(i) is pushed away, preventing it from collapsing to \u03b8(j) k . \fParticle optimization with blob methods The idea of blob methods can also be applied to particle approximation for SG-MCMC, which require the velocity vector \ufb01eld vt. According to (13), this is calculated as: vt(\u03b8) = \u2212\u2207\u03b8 \u03b4(E1+E2) \u03b4\u00b5 = \u2212\u2207\u03b8U \u2212\u2207\u03b8\u00b5/\u00b5. Unfortunately, direct application of particle approximation is infeasible because the term \u2207\u03b8\u00b5 is unde\ufb01ned with discrete \u00b5. To tackle this problem, we adopt the idea in Carrillo et al. (2017) to approximate the energy functional E2 in (13) as: E2 \u2248 R \u00b5(\u03b8) log(\u00b5 \u2217K)(\u03b8)d\u03b8, where K(\u00b7, \u00b7) is another kernel function to smooth out \u00b5. Consequently, based on Carrillo et al. (2017), the velocity vt can be calculated as (details in Section C of the SM): vt(\u03b8) = \u2212\u2207\u03b8U \u2212 n X j=1 \u2207\u03b8(j) t K(\u03b8, \u03b8(j) t )/ X k K(\u03b8(j) t , \u03b8(k) t ) \u2212 n X j=1 \u2207\u03b8(j) t K(\u03b8, \u03b8(j) t )/ n X k=1 K(\u03b8, \u03b8(k) t ) (22) Given vt, particle updates can be obtained by solving (17) numerically as in SVGD. By inspecting the formula of vt in (22), the last two terms both act as repulsive forces. Interestingly, the mechanism is similar to SVGD, but with adaptive force between different particle pairs. 5 THE GENERAL RECIPE Based on the above development, a more general particleoptimization framework is proposed by combining the PDEs of both SG-MCMC and SVGD. As a result, we propose the following PDE to drive evolution of densities \u2202\u00b5t \u2202t = \u2212\u2207\u03b8 \u00b7 (\u00b5tF(\u03b8t)) + \u03bb1\u2207\u03b8 \u00b7 ((W \u2217\u00b5t)\u00b5t) + \u03bb2\u2207\u03b8\u2207\u03b8 : \u0000\u00b5tg(\u03b8t)g\u22a4(\u03b8t) \u0001 , (23) where \u03bb1 and \u03bb2 are two constants. It is easily seen that to ensure the stationary distribution of (23) to be equal to p(\u03b8| X), the following condition must be satis\ufb01ed: \u2207\u03b8\u00b7 (p(\u03b8| X)F(\u03b8)) = \u03bb1\u2207\u03b8 \u00b7 ((W \u2217p(\u03b8| X))p(\u03b8| X)) + \u03bb2\u2207\u03b8\u2207\u03b8 : \u0000p(\u03b8| X)g(\u03b8)g\u22a4(\u03b8) \u0001 (24) There are many feasible choices for the functions and parameters {F(\u03b8), W, g(\u03b8), \u03bb1, \u03bb2} to satisfy (24). However, the veri\ufb01cation procedure might be complicated given the present of a convolutional term in (24). We recommend the following choices for simplicity: \u2022 F(\u03b8) = 1 2U(\u03b8), W = 0, g(\u03b8) = I and \u03bb2 = 1: this reduces to the Wasserstein-based SGLD with particle optimization. Speci\ufb01cally, when the discretegradient-\ufb02ow approximation is adopted, the algorithm is denoted as w-SGLD; whereas when the blob method is adopted, it is denoted as w-SGLD-B. \u2022 F(\u03b8) = 0, g(\u03b8) = 0, W is de\ufb01ned as (9): this reduces to standard SVGD. \u2022 F(\u03b8) = 1 2U(\u03b8), g(\u03b8) = I, W is de\ufb01ned as (9), and \u03bb2 = 1: this is the combination of SGLD and SVGD, and is called particle interactive SGLD, denoted as PI-SGLD or \u03c0-SGLD. It is easy to verify that condition (24) is satis\ufb01ed for all the above three particle-optimization algorithms. Furthermore, particle updates are readily developed by applying either the discrete-gradient-\ufb02ow or blob-based methods. 6 RELATED PARTICLE-BASED MCMC METHODS There have been related particle-based MCMC algorithms. Representative methods are sequential Monte Carlo (SMC) (Moral et al., 2006), particle MCMC (PMCMC) (Andrieu et al., 2010) and many variants. In SMC, particles are sample from a proposal distribution, and the corresponding weights are updated by a resampling step. PMCMC extends SMC by sampling from an extended distribution interacted with a MH-rejection step. Compared to our framework, their proposal distributions are typically hard to choose; furthermore, optimality of the particles from both methods can not be guaranteed. Furthermore, the methods are typically much more computationally expensive. Recently, Dai et al. (2016) proposed a particle-based MCMC algorithm by approximating a target distribution with weighted kernel density estimator, which updates particle weights based on likelihoods of the corresponding particles. This approach is theoretically sound but lacks an underlying geometry interpretation. Finally, we note that w-SGLD has been successfully applied to reinforcement learning recently for improved policy optimization (Zhang et al., 2018a). 7 EXPERIMENTS We verify our framework on a set of experiments, including a number of toy experiments and applications to Bayesian sampling of deep neural networks (DNNs). 7.1 Demonstrations Toy Distributions We compare various sampling methods on multi-mode toy examples, i.e., SGLD, SVGD, w-SGLD, w-SGLD-B and \u03c0-SGLD. We aim to sample from four unnormalized 2D densities p(z) \u221dexp{U(z)}, with detailed functional form provided in the SM. We optimize/sample 2000 particles to approximate target distributions. The results are shown in Figure 1. It can be seen from Figure 1 that though SGLD maintains good asymptotic properties, it is inaccurate to approximate distributions with only a few samples; in some case, the \fsamples cannot even cover all the modes. Interestingly, all other particle-optimization-based algorithms successfully \ufb01nd all the modes and \ufb01t the distributions well. wSGLD is good at \ufb01nding modes, but worse at modeling the correct variance due to dif\ufb01culty of controlling the balance between attractive and repulsive forces between particles. w-SGLD-B is better than w-SGLD at modeling the distribution variance, performing similarly to SVGD and \u03c0-SGLD. Even though, we note that w-SGLD is very useful when the number of particles is small, which \ufb01ts a distribution better, as shown in Section E of the SM. Figure 1: Illustration of different algorithms on toy distributions. Each column is a distribution case. 1st row: Ground truth; 2nd row: standard SGLD; 3rd row: w-SGLD; 4th row: w-SGLD-B; 5th row: SVGD; 6th row: \u03c0-SGLD. Bayesian Logistic regression We next compare the three variants of our framework (i.e.SVGD, w-SGLD and w-SGLD-B) on a simple logistic-regression task with quantitative evaluations. We use the same model, data and experimental settings as Liu and Wang (2016a). The Covertype dataset contains 581,012 data points and 54 features. We perform 5 runs for each setting and report the mean of testing accuracies/log-likelihoods. Figure 2 plots both test accuracies and test log-likelihoods w.r.t.the Iteration Figure 2: Test accuracies (left) and log-likelihoods (right) v.s.iterations for SVGD, w-SGLD and w-SGLD-B. number of training iterations. It is clearly that while all methods converge to the same accuracy/likelihood level, both w-SGLD and w-SGLD-B converge slightly faster than SVGD. In addition, w-SGLD and w-SGLD-B have similar convergence behaviors, thus we only use w-SGLD in the DNN experiments below. 0 0.1 0.2 0.5 1 W2 2 0.735 0.740 0.745 0.750 0.755 0.760 Accuracy M=3 M=5 M=10 M=20 M=50 Figure 3: Impact of W 2 2 factor \u03b3 and particle number M. Parameter Sensitivity Now we study the role of hyperparameters in \u03c0-SGLD: the number of particles M and the scaling factor \u03b3 to replace the uivj-term in (21). We use the same dataset and model as the above experiment. Figure 3 plots test accuracies along with different parameter settings. As expected, the best performance is achieved with appropriate scale of W 2 2 . The performance keep improving with increasing particles. Interestingly, the Wasserstein regularization is more important when the number of particles is small, demonstrating the superiority when approximate distributions with very few particles. 7.2 Applications on deep neural networks We conduct experiments for Bayesian learning of DNNs. Different from traditional optimization for DNNs, we are interested in modeling weight uncertainty of neural networks, an important topic that has been well explored (Hern\u00b4 andez-Lobato and Adams, 2015; Blundell et al., 2015a; Li et al., 2016; Louizos and Welling, 2016). We assign priors to the weights, which are simple isotropic Gaussian priors in our case, and perform posterior sampling with the proposed particle-optimization-based algorithms, as well as other standard algorithms such as SGLD and SGD. We use the RMSprop optimizer for feed-forward networks (FNN), and Adam for for convolutional neural networks (CNNs) and recurrent neural \fnetworks (RNNs). For all methods, we use a RBF kernel K(\u03b8, \u03b8\u2032) = exp(\u2212\u2225\u03b8 \u2212\u03b8\u2032\u22252 2/h), with the bandwidth set to h = med2/ log M. Here med is the median of the pairwise distance between particles. All experiments are conducted on a single TITAN X GPU. Feed-forward Neural Networks We perform the classi\ufb01cation tasks on the standard MNIST dataset. A twolayer model 784-X-X-10 with ReLU activation function is used, with X being the number of hidden units for each layer. The training epoch is set to 100. The test errors are reported in Table 1. Not surprisingly, Bayesian methods generally perform better than their optimization counterparts. The new \u03c0-SGLD which combines w-SGLD and SVGD improves both methods with little computational overhead. In additional, w-SGLD seems to perform better than SVGD in this case, partially due to a better asymptotic property mentioned in (Liu, 2017). Furthermore, standard SGLD which is based on MCMC obtains higher test errors compared to particle-optimization-based algorithms, partially due to the correlated-sample issue discussed in the introduction. See (Blundell et al., 2015b) for details on the other methods in Table 1. Table 1: Classi\ufb01cation error of FNN on MNIST. Test Error Method 400-400 800-800 \u03c0-SGLD 1.36% 1.30% w-SGLD 1.44% 1.37% SVGD 1.53% 1.40% SGLD 1.64% 1.41% RMSprop 1.59% 1.43% RMSspectral 1.65% 1.56% SGD 1.72% 1.47% BPB, Gaussian 1.82% 1.99% SGD, dropout 1.51% 1.33% Convolution Neural Networks We use the CIFAR-10 dataset to test our framework on CNNs. We adopt a CNN of three convolution layers, using 3\u00d73 \ufb01lter size with C64-C128-C256 channels, and 2\u00d72 max-pooling after each convolution layer. Our implementation adopts batch normalization, drop out and data augmentation to improve the performance. Training losses and test accuracies are presented in Table 2. Consistently, \u03c0-SGLD outperforms all other algorithms in terms of test accuracy. ADAM obtains a better training loss but worse test accuracy, indicating worse generalization ability of the optimization-based methods compared to Bayesian methods. Recurrent Neural Networks For RNNs, we run standard language models. Experiments are presented on three publicly available corpora: APNEWS, IMDB and BNC. APNEWS is a collection of Associated Press news articles from 2009 to 2016. IMDB is a set of movie reTable 2: Classi\ufb01cation error of CNN on CIFAR-10. Method Training Loss Test Accuracy ADAM 23.80 86.76% SVGD 30.57 88.72% SGLD 28.52 88.64% w-SGLD 31.26 88.80% \u03c0-SGLD 25.06 89.52% views collected by Maas et al. (2011), and BNC BNC Consortium (2007) is the written portion of the British National Corpus, which contains excerpts from journals, books, letters, essays, memoranda, news and other types of text. These datasets can be downloaded from Github\u2217. Table 3: Perplexity of language model on three corpora. Method APNEWS IMDB BNC SGD 64.13 72.14 102.89 SGLD 63.01 68.12 95.13 SVGD 61.64 69.25 94.99 w-SGLD 61.22 67.41 93.68 \u03c0-SGLD 59.83 67.04 92.33 We follow the standard set up as Wang et al. (2017). Speci\ufb01cally, we lower case all the word tokens and \ufb01lter out word tokens that occur less than 10 times. All the datasets are divided into training, development and testing sets. For the language model set up, we consider a 1-layer LSTM model with 600 hidden units. The sequence length is \ufb01xed to be 30. In order to alleviate over\ufb01tting, dropout with a rate of 0.4 is used in each LSTM layer. Results in terms of test perplexities are presented in Table 3. Again, we see that \u03c0-SGLD performs best among all algorithms, and w-SGLD is slightly better than SVGD, both of which are better than other algorithms. 8"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1711.10927v1",
+                    "title": "Particle Optimization in Stochastic Gradient MCMC",
+                    "abstract": "Stochastic gradient Markov chain Monte Carlo (SG-MCMC) has been increasingly\npopular in Bayesian learning due to its ability to deal with large data. A\nstandard SG-MCMC algorithm simulates samples from a discretized-time Markov\nchain to approximate a target distribution. However, the samples are typically\nhighly correlated due to the sequential generation process, an undesired\nproperty in SG-MCMC. In contrary, Stein variational gradient descent (SVGD)\ndirectly optimizes a set of particles, and it is able to approximate a target\ndistribution with much fewer samples. In this paper, we propose a novel method\nto directly optimize particles (or samples) in SG-MCMC from scratch.\nSpecifically, we propose efficient methods to solve the corresponding\nFokker-Planck equation on the space of probability distributions, whose\nsolution (i.e., a distribution) is approximated by particles. Through our\nframework, we are able to show connections of SG-MCMC to SVGD, as well as the\nseemly unrelated generative-adversarial-net framework. Under certain\nrelaxations, particle optimization in SG-MCMC can be interpreted as an\nextension of standard SVGD with momentum.",
+                    "authors": "Changyou Chen, Ruiyi Zhang",
+                    "published": "2017-11-29",
+                    "updated": "2017-11-29",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Bayesian methods have been playing an important role in modern machine learning, especially in an unsupervised-learning setting. When facing with big data, two lines of research directions have been developed to scale up Bayesian methods, e.g., variational-Bayes-based and sampling-based methods. Stochastic gradient Markov chain Monte Carlo (SG-MCMC) is a family of scalable Bayesian learning algorithms designed to ef\ufb01ciently sample from a target distribution such as a posterior distribution [1\u2013 4]. In principle, SG-MCMC generates samples from a Markov chain, which are used to approximate a target distribution. Under a standard setting, samples from SG-MCMC are able to match a target distribution exactly in an in\ufb01nite-sample regime [5, 4]. However, this case never occurs in practice, as only a \ufb01nite amount of samples are available. Although nonasymptotic bounds w.r.t.the number of samples have been investigated [5, 6, 4], there are no theory/algorithms to guide learning an optimal set of \ufb01xed-size samples/particles. This is an undesirable property of SG-MCMC, because given a \ufb01xed number of samples, one often wants ot learn the optimal samples that best approximate a target distribution. A remedy for this issue is to adopt the idea of particle-based sampling methods, where a set of particles (or samples) are initialized from some simple distribution, and they are updated iteratively such that they approximate a target distribution better and better. The updating procedure is usually done by optimizing some objective function. There is not much work in this direction for Bayesian sampling, with an outstanding representative being the Stein variational gradient descent (SVGD) [7]. In SVGD, the update of particles are done by optimizing the KL-divergence between the empiricalparticle distribution and a target distribution, thus the samples can be guaranteed to be optimal in each update. Because of this property, SVGD is found to perform better than SG-MCMC when the number of samples used to approximate a target distribution is limited [7]. arXiv:1711.10927v1  [stat.ML]  29 Nov 2017 \fLittle research has been done on investigating the particle-optimization idea in SG-MCMC. Inspired by SVGD, we develop a similar particle-optimization procedure for SG-MCMC for more ef\ufb01cient sampling. To achieve this goal, we propose a novel technique to directly optimize particles based on a variational reformulation of the corresponding Fokker-Planck equation of an SG-MCMC algorithm, adapted from [8]. In this way, instead of sampling from a Markov chain sequentially, we evolve particles through an optimization procedure, obtaining both optimal particles and faster convergence speed compared to standard SG-MCMC. Furthermore, under some relaxations, we are able to show particle optimization in SG-MCMC can be regarded as an extension of SVGD with momentum. To the best of our knowledge, this is the \ufb01rst time particles can be optimized in SG-MCMC algorithms. 2 Preliminaries 2.1 Stochastic gradient MCMC Diffusion-based sampling methods Generating random samples from a posterior distribution is a pervasive problem in Bayesian statistics which has many important applications in machine learning. The Markov Chain Monte Carlo method (MCMC), proposed by Metropolis et al. [9], produces unbiased samples from a desired distribution when the density function is known up to a normalizing constant. However, traditional MCMC methods are based on random walk proposals which often lead to highly correlated samples. On the other hand, dynamics-based sampling methods, e.g., Hybrid Monte Carlo (HMC) [10, 11], avoid this high degree of correlation by combining dynamical systems with the Metropolis step. The dynamical system uses information from the gradient of the log density to reduce the random walk effect, and the Metropolis step serves as a correction of the discretization error introduced by the numerical integration of the dynamical systems. In fact, these dynamical systems are derived from a more general mathematical technique called diffusion process (or more speci\ufb01cally, It\u00f3 diffusion) [12]. Speci\ufb01cally, our objective is to generate random samples from a posterior distribution p(\u03b8| X) \u221d p(X |\u03b8)p(\u03b8), where \u03b8 \u2208Rr represents the model parameter, and X \u225c{xi}N i=1 represents the data. The canonical form is p(\u03b8| X) = (1/Z) exp(\u2212U(\u03b8)), where U(\u03b8) = \u2212log p(X |\u03b8) \u2212log p(\u03b8) \u225c\u2212 N X i=1 log p(xi |\u03b8) \u2212log p(\u03b8) (1) is referred to as the potential energy based on an i.i.d. assumption of the model, and Z is the normalizing constant. In general, the posterior distribution can be corresponding to the (marginal) stationary distribution of a (continuous-time) It\u00f3 diffusion, de\ufb01ned as a stochastic differential equation of the form: d\u0398t = F(\u0398t)dt + g(\u0398t)dWt , (2) where t is the time index; \u0398t \u2208Rp represents the full variables in a dynamical system, and \u0398t \u2287\u03b8t (thus p \u2265r) is potentially an augmentation of model parameter \u03b8; Wt \u2208Rp is pdimensional Brownian motion. Functions F : Rp \u2192Rp and g : Rp \u2192Rp \u00d7 p are assumed to satisfy the Lipschitz continuity condition [13]. By Fokker-Planck equation (or the forward Kolmogorov equation) [14, 15], when appropriately designing the diffusion-coef\ufb01cient functions F(\u00b7) and g(\u00b7), the stationary distribution of the corresponding It\u00f3 diffusion equals the posterior distribution of interest, p(\u03b8| X). For example, the 1st-order Langevin dynamic de\ufb01nes \u0398 = \u03b8, and F(\u0398t) = \u2212\u2207\u03b8U(\u03b8), g(\u0398t) = \u221a 2 Ir; the 2nd-order Langevin diffusion de\ufb01nes \u0398 = (\u03b8, q), and F(\u0398t) = \u0010 q \u2212B q \u2212\u2207\u03b8U(\u03b8) \u0011 , g(\u0398t) = \u221a 2B \u0010 0 0 0 In \u0011 for a scalar B > 0; q is an auxiliary variable known as the momentum [2, 3]. Denoting the distribution of \u0398t as \u03c1t, it is well known [15] that \u03c1t is characterized by the FokkerPlanck (FP) equation: \u2202\u03c1t \u2202t = \u2212\u2207\u0398 \u00b7 (\u03c1tF(\u0398t)) + \u2207\u0398 \u2207\u0398 : \u0000\u03c1tg(\u0398t)g\u22a4(\u0398t) \u0001 , (3) where a \u00b7 b \u225ca\u22a4b for vectors a and b, A : B \u225ctrace(A\u22a4B) for matrices A and B. The FP equation is the key to develop our particle-optimization framework in SG-MCMC. 2 \fStochastic gradient MCMC SG-MCMC algorithms are discretized numerical approximations of the It\u00f3 diffusions. They mitigate the slow mixing and non-scalability issues encountered by traditional MCMC algorithms by i) adopting gradient information of the posterior distribution, and ii) using minibatches of the data in each iteration of the algorithm to generate samples. To make the algorithms scalable in a big-data setting, three developments will be implemented based on the It\u00f3 diffusion: i) de\ufb01ne appropriate functions F and g in the It\u00f3-diffusion formula so that the (marginal) stationary distributions coincide with the target posterior distribution p(\u03b8| X); ii) replace F or g with unbiased stochastic approximations to reduce the computational complexity, e.g., approximating F with a random subset of the data instead of using the full data. For example, in the 1st-order Langevin dynamics, \u2207\u03b8U(\u03b8) could be approximated by an unbiased estimator with a subset of data: \u2207\u03b8 \u02dc U(\u03b8) \u225c\u2207log p(\u03b8) + N n n X i=1 \u2207\u03b8 log p(x\u03c0i |\u03b8) (4) where \u03c0 is a size-n random subset of {1, 2, \u00b7 \u00b7 \u00b7 , N}, leading to the \ufb01rst SG-MCMC algorithm in machine learning \u2013 stochastic gradient Langevin dynamics (SGLD) [1]; and iii) solve the generally intractable continuous-time It\u00f4 diffusions with a numerical method, e.g., the Euler method [4]. For example, this leads to the following update in SGLD: \u03b8(\u2113) = \u03b8(\u2113\u22121) \u2212\u2207\u03b8 \u02dc U(\u03b8(\u2113\u22121))h + \u221a 2h \u03b4\u2113, where h means the stepsize, \u2113indexes the samples, \u03b4\u2113\u223cN(0, I) is a random sample from an isotropic normal distribution. After running the algorithm for L steps, the collection of samples {\u03b8(\u2113)}L \u2113=1, which are collected from a Markov chain, are used to approximate the unknown posterior distribution 1 Z e\u2212U(\u03b8). 2.2 Stein variational gradient descent Different from SG-MCMC, SVGD initializes a set of particles and iteratively updates them so that the empirical particle distribution approximates the posterior distribution. Speci\ufb01cally, considers a set of particles {\u03b8i}M i=1 drawn from distribution q. SVGD tries to update these particles by doing gradient descent on the space of probability distributions via \u03b8i \u2190\u03b8i + \u03f5\u03c6(\u03b8i), \u03c6 = arg max \u03c6\u2208F \u001a \u2202 \u2202\u03f5KL(q[\u03f5\u03c6]||p) \u001b , (5) where \u03c6 is a function perturbation direction chosen to minimize the KL divergence between the updated empirical distribution q[\u03f5\u03c6] and the posterior p(\u03b8| X), p for short. Since KL(q\u2225p) is convex in q, global optimum of q = p can be guaranteed. SVGD considers F as the unit ball of a vector-valued reproducing kernel Hilbert space (RKHS) H associated with a kernel \u03ba(\u03b8, \u03b8\u2032). In such as setting, it was shown in [16] that: \u2212\u2202 \u2202\u03f5KL(q[\u03f5\u03c6]\u2225p)|\u03f5=0 = E\u03b8\u223cq[trace(\u0393p\u03c6(\u03b8))], (6) with \u0393p\u03c6(\u03b8) \u225c\u2207\u03b8 log p(\u03b8| X)\u22a4\u03c6(\u03b8) + \u2207\u03b8 \u00b7 \u03c6(\u03b8), where \u0393p is called the Stein operator. Assuming that the update function \u03c6(\u03b8) is in a RKHS with kernel \u03ba(\u00b7, \u00b7), it was shown in [16] that (6) is maximized with: \u03c6(\u03b8) = E\u03b8\u223cq[\u03ba(\u03b8, \u03b8\u2032)\u2207\u03b8 log p(\u03b8| X) + \u2207\u03b8\u03ba(\u03b8, \u03b8\u2032)]. (7) When approximating the expectation E\u03b8\u223cq[\u00b7] with empirical particle distribution, we arrive at the following updates for the particles at the \u2113-th iteration: \u03b8(\u2113) i = \u03b8(\u2113\u22121) i + \u03f5 M M X j=1 h \u03ba(\u03b8(\u2113\u22121) j , \u03b8(\u2113\u22121) i )\u2207\u03b8j log p(\u03b8(\u2113\u22121) j | X) + \u2207\u03b8j\u03ba(\u03b8(\u2113\u22121) j , \u03b8(\u2113\u22121) i ) i . (8) SVGD applies the updates in (8) repeatedly, and the samples move closer to the target distribution p in each iteration. 3 \f2.3 Comparing SG-MCMC with SVGD SG-MCMC is a Markov-chain-based sampling methods, in the sense that samples are generated from a Markov chain, with potentially highly correlated samples. Furthermore, it often requires a large number of samples in order to approximate a target distribution reasonably well [4]. In contrast, SVGD directly updates the particles to their optimum guided by an objective function, thus requires much less samples to approximate a target distribution. On the other hand, SVGD has been explained as gradient \ufb02ows whose gradient operator is de\ufb01ned on the RKHS [17]; whereas SG-MCMC are \ufb02ows with \ufb02ow operator de\ufb01ned on the L2 space the functional space that is square integrable. Since RKHS is smaller than L2, SG-MCMC can potentially obtain better asymptotic properties than SVGD in theory [17]. The above arguments motivate us to combine goods from both sides, i.e., we aim to developed a particle-based SG-MCMC algorithm similar to what SVGD does. 3 Particle Optimization in SG-MCMC To develop our particle-optimization framework, we \ufb01rst introduce the following lemma adapted from [18], viewing SG-MCMC from an optimization perspective. Lemma 1 Assume that p(\u03b8t| X) \u2264C1 is in\ufb01nitely differentiable, and \u2225\u2207\u03b8 log p(\u03b8| X)\u2225\u2264 C2 (1 + C1 \u2212log p(\u03b8| X)) (\u2200\u03b8) for some constants {C1, C2}. Let T = \u03f5K with K being an integer, \u02dc \u03c10 is an arbitrary distribution with same support as p(\u03b8| X), and {\u02dc \u03c1k}K k=1 be the solution of the functional optimization problem: \u02dc \u03c1k = arg min \u03c1\u2208K KL (\u03c1\u2225p(\u03b8| X)) + 1 2hW 2 2 (\u02dc \u03c1k\u22121, \u03c1) , (9) where W 2 2 (\u00b51, \u00b52) \u225cinfp\u2208P(\u00b51,\u00b52) R \u2225x \u2212y\u22252 2 p(d x, d y), W2 (\u00b51, \u00b52) is the 2nd-order Wasserstein distance, with P(\u00b51, \u00b52) being the space of joint distributions of {\u00b51, \u00b52}; K is the space of probability distributions with \ufb01nite 2nd-order moment. Then \u02dc \u03c1K converges to \u03c1T in the limit of h \u21920, i.e., limh\u21920 \u02dc \u03c1K = \u03c1T , where \u03c1T is the solution of the FP equation (3) at time T. Lemma 1 reveals an interesting way to compute \u03c1T via a sequence of functional optimization problems. By comparing it with the objective of SVGD, which minimizes the KL-divergence between \u03c1k and p(\u03b8| X), at each sub-optimization-problem in Lemma 1, it minimizes the KL-divergence, plus a regularization term as the Wasserstein distance between \u02dc \u03c1k\u22121 and \u02dc \u03c1k. The extra Wassersteindistance term arises naturally due to the fact that the corresponding diffusion is a gradient \ufb02ow equipped with a geometric associated with the Wasserstein distance [19]. From another point of view, it is known that the Wasserstein distance is a better metric for probability distributions than the KL-divergence, especially in the case of non-overlapping domains [20, 21]. According to Lemma 1, it is now apparent that SG-MCMC can be achieved by alternatively solving the optimization problem in (9) for each iteration. 3.1 Optimizing on the space of probability distributions Our idea of particle-optimization is inspired by Lemma 1, in that we can obtain the optimal distribution \u02dc \u03c1k for each iteration (which will be approximated by particles) by optimizing (9). Consequently, instead of doing simulation based on the original It\u00f3 diffusion, we propose to directly solve (9) on the space of probability distributions K. As will be shown, this allows us to derive algorithms which directly optimize particles in SG-MCMC. However, this also bring challenges for the optimization, as the probability-distribution space K is too \ufb02exible to derive exact solutions. In the following, we propose techniques to approximate the corresponding terms in (9). We denote \u03b8k as a sample from \u02dc \u03c1k, i.e., \u03b8k \u223c\u02dc \u03c1k. 3.1.1 Approximating the KL-divergence Given \u02dc \u03c1k\u22121, the solution \u02dc \u03c1k of (9) can be considered as an unknown transformation G from \u02dc \u03c1k\u22121 to \u02dc \u03c1k, i.e., \u02dc \u03c1k = G(\u02dc \u03c1k\u22121) , (10) 4 \funder the constraint that \u02dc \u03c1k still lies in K after the transformation. Directly solving the unknown transformation is challenging, we propose two methods to solve it approximately, detailed below. Optimizing G with adversarial learning Optimizing the KL divergence with an unknown transformation G is generally infeasible. Instead, we approximate it with the Jensen-Shanon divergence, which appears to have the same optimality solution, thus they are equivalent. Formally, we \ufb01rst introduce the following lemma. Lemma 2 Let p1 and p2 be probability distributions on (X, \u03a3) with the same support. Then KL divergence KL(p1\u2225p2) is equivalent to the Jensen-Shanon divergence (JSD) JSD(p1\u2225p2) in the sense that they are both convex in (p1, p2), and achieve the same minimum value of zero at p1 = p2. Based on Lemma 2, we can replace the KL term in (9) with the JSD, resulting in \u02dc \u03c1k = arg min \u03c1\u2208K JSD (\u03c1\u2225p(\u03b8| X)) + \u03bb 2hW 2 2 (\u02dc \u03c1k\u22121, \u03c1) , (11) where \u03bb > 0 is introduced to balance the difference between the scales of KL and JSD. It is well known that JSD is the metric for measuring the distance between two probability distributions in generative adversarial networks (GANs) [22]. According to the properties of GAN, it is easy to see that the unknown transformation G is equivalent to the generator network in GANs. The only difference is that in our case, the latent space of GAN will be the same as the data space, which does not impact the learning algorithm. Consequently, we can update the transformation G in each iteration by running a GAN update on its generator, which are then used to generate samples from \u02dc \u03c1k. Optimizing G via kernelized Stein operator Optimizing G with adversarial learning described above brings us a little computation overhead to update the transformation. Here we propose a more ef\ufb01cient method based on the kernelized Stein operator [7]. We \ufb01rst introduce a theoretical result from [7] in Lemma 3. Lemma 3 ([7]) Let H denotes the reproducing kernel Hilbert space (RKHS), and Hr the space of vector functions f = [f1, \u00b7 \u00b7 \u00b7 , fr] with fi \u2208H. Let G(\u03b8) = \u03b8 + f(\u03b8), where f \u2208Hr. Denote \u03c1 to be the density of \u02dc \u03b8 = G(\u03b8), then we have \u2207fKL(\u03c1\u2225p(\u03b8| X)) f\u2208Hr |f=0 = \u2212\u03c6\u2217 \u03c1,p(\u03b8) , where \u03c6\u2217 \u03c1,p(\u03b8) = E\u03b8\u223c\u03c1 [k(\u03b8, \u00b7)\u2207\u03b8 log p(\u03b8| X) + \u2207\u03b8k(\u03b8, \u00b7)]. Lemma 3 essentially says that the functional gradient of the KL divergence is \u2212\u03c6\u2217 q,p(\u03b8), when the transformation is in the form of G(\u03b8) = \u03b8 + f(\u03b8) with f restricted to Hr. The result seems to be applicable in our problem (9) except that we require fi to be in a large space L2 instead of H. To compromise this, we propose to inject noise into the functional gradient, leading to the following approximation: \u2207fKL(\u03c1\u2225p(\u03b8| X)) f\u2208Lr |f=0 \u2248\u2212\u03c6\u2217 \u03c1,p(\u03b8) + \u03c3 \u03b4 , (12) where Lr 2 \u225cL2 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 L2, \u03b4 \u223cN(0, I), and \u03c3 controls the variance of the injected noise, which is typically decreasing in the algorithm to ensure convergence. Note this resembles stochastic gradient descent, where full gradients are replaced with noisy gradients, thus convergence can still be guaranteed. 3.1.2 Approximating the Wasserstein distance In order to calculate the Wasserstein term on the RHS of (9), we adapt results from optimal transport theory [23, 24] to rewrite W 2 2 (\u02dc \u03c1k\u22121, \u03c1) as W 2 2 (\u02dc \u03c1k\u22121, \u03c1) (13) =E\u02dc \u03c1k\u22121 \u2225\u03b8t\u22121\u22252 + E\u03c1 \u2225\u03b8\u22252 + 2 sup \u03c8(\u00b7) convex \u2212E\u02dc \u03c1k\u22121 [\u03c8(\u03b8t\u22121)] \u2212E\u03c1 [\u03c8\u2217(\u03b8)] , 5 \fwhere \u03c8\u2217(\u03b8) \u225csupv \u0000vT \u03b8 \u2212\u03c8(v) \u0001 is the convex-conjugate of the function \u03c8. Optimizing W 2 2 (\u02dc \u03c1k\u22121, \u03c1) is in general infeasible due to the need to search on the space of convex functions. However, we can approximate it by restricting \u03c8 on some nice convex functions. We describe some preliminary results below. Restricting \u03c8 to be quadratic We can de\ufb01ned \u03c8 to be in the form of \u03c8(\u03b8) \u225c1 2\u27e8\u03b8, A \u03b8\u27e9+ \u27e8b, \u03b8\u27e9+ c with parameters {A, b, c}. In this case, the convex-conjugate is in a nice form of \u03c8\u2217(\u03b8) \u225c1 2\u27e8\u03b8 \u2212b, A\u22121(\u03b8 \u2212b)\u27e9\u2212c . Substituting these forms into the \u201csup\u201d part and maximizing w.r.t.{A, b, c} by setting the derivatives to zero, we have b = \u03b8 \u2212A \u03b8t\u22121 Substituting the above formula into (13) and simplifying, we have W 2 2 (\u02dc \u03c1k\u22121, \u03c1) = E\u02dc \u03c1k\u22121 \u2225\u03b8t\u22121\u22252 + E\u03c1 \u2225\u03b8\u22252 \u22122E\u02dc \u03c1k\u22121,\u03c1 \u0002 \u03b8T t\u22121\u03b8 \u0003 = E\u02dc \u03c1k\u22121,\u03c1 (\u03b8 \u2212\u03b8t\u22121)2 . (14) Restricting \u03c8 to other forms It is also interesting to parameterize \u03c8 with other convex functions. We are currently investigating on this direction. 3.2 Particle optimization The above sections describes how to optimize the distribution \u02dc \u03c1k directly from (9). In practice, one must adopt some representation for \u02dc \u03c1k in order to perform optimization. A simple representation is to use particles for approximation, i.e., \u02dc \u03c1k \u2248 1 M PM i=1 \u03b4\u03b8(k) i where \u03b4\u03b8 is a point mass at \u03b8, {\u03b8(k) i } is the set of particles at the k-th iteration. Denote our objective to be F({\u03b8(\u2113) i }) where \u2113indexes the iteration. We can update the particles by calculating their gradients and perform gradient descent. For example, when adopting the approximations (12) and (14), we have \u2202 \u2202\u03b8(\u2113) i F({\u03b8(\u2113) i }) \u2248\u2212\u02c6 \u03c6\u2217(\u03b8(\u2113) i ) + \u03c3\u2113\u03b4 + 1 h(\u03b8(\u2113) i \u2212\u03b8(\u2113\u22121) i ) , (15) where \u02c6 \u03c6\u2217(\u03b8) = 1 M PM i=1 h k(\u03b8(\u2113) i , \u03b8)\u2207\u03b8(\u2113) i log p(\u03b8(\u2113) i | X) + \u2207\u03b8(\u2113) i k(\u03b8(\u2113) i , \u03b8) i . SG-MCMC vs SVGD with Polyak\u2019s momentum gradient descent Let f(\u03b8) be the objective function to be optimized. Polyak\u2019s momentum gradient descent update [25] is given by \u03b8(\u2113) = \u03b8(\u2113\u22121) \u2212h\u2207\u03b8f(\u03b8) + \u00b5 \u0010 \u03b8(\u2113\u22121) \u2212\u03b8(\u2113\u22122)\u0011 . (16) To see the relation of particle optimization in SG-MCMC of (15) with SVGD, \ufb01rst note that because (13) is an upper bound for the 2nd-order Wasserstein distance, we should scale it by some constant \u00b5 \u2208(0, 1) in the implementation to approximate the true W 2 2 (\u02dc \u03c1k\u22121, \u03c1). Based on the gradient formula in (15), the update equation for \u03b8(\u2113) i then becomes \u03b8(\u2113) i = \u03b8(\u2113\u22121) i \u2212h \u0010 \u02c6 \u03c6\u2217(\u03b8(\u2113\u22121) i ) + \u03c3\u2113\u03b4 \u0011 + \u00b5 \u0010 \u03b8(\u2113\u22121) i \u2212\u03b8(\u2113\u22122) i \u0011 , which is in the same form of (16) where the gradient is represented as noisy gradient. Thus particle optimization in SG-MCMC with (15) can be regarded as SVGD with Polyak\u2019s momentum method. 6 \f500 1500 2500 3500 4500 5500 #iterations 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 accuracy SG-MCMC SVGD 500 1500 2500 3500 4500 5500 #iteration -0.66 -0.64 -0.62 -0.6 -0.58 -0.56 -0.54 -0.52 -0.5 log-likelihood SG-MCMC SVGD Figure 1: Test accuracies (left) and log-likelihoods (right) v.s.training iterations for SG-MCMC and SVGD. 4 Empirical Veri\ufb01cation We test our algorithm against SVGD on a simple logistic-regression task. We use the same model and data as [7]. For SVGD, we adopt the same setting as in [7]. Note we use the authors\u2019 implementation for SVGD [7], which uses Adagrad for optimization, thus it is not strictly the standard SVGD (should perform better). For our particle optimization in SG-MCMC, we simply set the noise \u03b4 to be standard isotropic normal. We scale W 2 2 (\u02dc \u03c1k\u22121, \u03c1) with 0.1, and use (15) to update particles in SG-MCMC. Figure 1 plots both test accuracies and test log-likelihoods w.r.t. the number of training iterations. It is clearly that without tuning, SG-MCMC already obtains slightly faster convergence speed than SVGD in terms of both test accuracy and test log-likelihood. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1709.01180v1",
+                    "title": "A Convergence Analysis for A Class of Practical Variance-Reduction Stochastic Gradient MCMC",
+                    "abstract": "Stochastic gradient Markov Chain Monte Carlo (SG-MCMC) has been developed as\na flexible family of scalable Bayesian sampling algorithms. However, there has\nbeen little theoretical analysis of the impact of minibatch size to the\nalgorithm's convergence rate. In this paper, we prove that under a limited\ncomputational budget/time, a larger minibatch size leads to a faster decrease\nof the mean squared error bound (thus the fastest one corresponds to using full\ngradients), which motivates the necessity of variance reduction in SG-MCMC.\nConsequently, by borrowing ideas from stochastic optimization, we propose a\npractical variance-reduction technique for SG-MCMC, that is efficient in both\ncomputation and storage. We develop theory to prove that our algorithm induces\na faster convergence rate than standard SG-MCMC. A number of large-scale\nexperiments, ranging from Bayesian learning of logistic regression to deep\nneural networks, validate the theory and demonstrate the superiority of the\nproposed variance-reduction SG-MCMC framework.",
+                    "authors": "Changyou Chen, Wenlin Wang, Yizhe Zhang, Qinliang Su, Lawrence Carin",
+                    "published": "2017-09-04",
+                    "updated": "2017-09-04",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction With the increasing size of datasets of interest to machine learning, stochastic gradient Markov Chain Monte Carlo (SGMCMC) has been established as an effective tool for largescale Bayesian learning, with applications in topic modeling (Gan et al. 2015; Liu, Zhu, and Song 2016), matrix factorization (Chen, Fox, and Guestrin 2014; Ding et al. 2014; S \u00b8ims \u00b8ekli et al. 2016), differential privacy (Wang, Fienberg, and Smola 2015), Bayesian optimization (Springenberg et al. 2016) and deep neural networks (Li et al. 2016). Typically, in each iteration of an SG-MCMC algorithm, a minibatch of data is used to generate the next sample, yielding computational ef\ufb01ciency comparable to stochastic optimization. While a large number of SG-MCMC algorithms have been proposed, their optimal convergence rates generally appear to share the same form, and are typically slower than stochastic gradient descent (SGD) (Chen, Ding, and Carin 2015). The impact of stochastic gradient noise comes from a higherorder term (see Lemma 1 below), which was omitted in the analysis of (Chen, Ding, and Carin 2015). In other words, current theoretical analysis for SG-MCMC does not consider the impact of minibatch size (corresponding to stochastic gradient noise), making the underlying convergence theory w.r.t.minibatch size unclear. Recent work by (Dubey et al. 2016) on applying variance reduction in stochastic gradient Langevin dynamics (SGLD) claims to improve the convergence rate of standard SGLD (Welling and Teh 2011; Teh, Thiery, and Vollmer 2016; Vollmer, Zygalakis, and Teh 2016). The theoretical analysis in (Dubey et al. 2016) omits certain aspects of variance reduction in SGLD, that we seek to address here: i) how does the minibatch size (or equivalently the stochastic gradient noise) affect the convergence rate of an SG-MCMC algorithm? and ii) how can one effectively reduce the stochastic gradient noise in SG-MCMC to improve its convergence rate, from both an algorithmic and a theoretical perspective? For (i), we provide theoretical results on the convergence rates of SG-MCMC w.r.t.minibatch size. For (ii), we propose a practical variance-reduction technique for SG-MCMC, as well as theory to analyze improvements of the corresponding convergence rates. The resulting SG-MCMC algorithm is referred to as variance-reduction SG-MCMC (vrSG-MCMC). For a clearer description, we \ufb01rst de\ufb01ne notation. In a Bayesian model, our goal is typically to evaluate the posterior average of a test function \u03c6(x), de\ufb01ned as \u00af \u03c6 \u225c R X \u03c6(x)\u03c1(x)d x, where \u03c1(x) is the target posterior distribution with x the possibly augmented model parameters (see Section ). Let {xl}L l=1 be the samples generated from an SG-MCMC algorithm. We use the sample average, \u02c6 \u03c6L \u225c 1 L PL l=1 \u03c6(xl), to approximate \u00af \u03c6. The corresponding bias and mean square error (MSE) are de\ufb01ned as |E\u02c6 \u03c6L \u2212\u00af \u03c6| and E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 , respectively. In vrSG-MCMC, unbiased estimations of full gradients are used, leading to the same bias bound as standard SG-MCMC (Chen, Ding, and Carin 2015). As a result, we focus here on analyzing the MSE bound for vrSG-MCMC. Speci\ufb01cally, we \ufb01rst analyze how minibatch size affects the MSE convergence rate of standard SG-MCMC, summarized in two cases: i) for a limited computation budget, the optimal MSE bound is achieved when using full gradients in the algorithm; ii) for a large enough computational budget, i.e., in a long-run setting, stochastic gradients with minibatches of size one are preferable. This indicates that stochastic gradient noise hurts SG-MCMC at the beginning arXiv:1709.01180v1  [stat.ML]  4 Sep 2017 \fof the algorithm. While it is computationally infeasible to use full gradients in practice, a remedy to overcome this issue is to use relatively small minibatches with variance reduction techniques to reduce stochastic gradient noise. Consequently, we propose a practical variance-reduction scheme, making SG-MCMC computationally ef\ufb01cient in a big-data setting. Finally, we develop theory to analyze the bene\ufb01t of the proposed variance-reduction technique and empirically show improvements of vrSG-MCMC over standard SG-MCMC algorithms. Preliminaries SG-MCMC is a family of scalable Bayesian sampling algorithms, developed recently to generate approximate samples from a posterior distribution p(\u03b8| D). Here \u03b8 \u2208Rr represents a model parameter vector and D = {d1, \u00b7 \u00b7 \u00b7 , dN} represents the data available to learn the model. In general, SG-MCMC algorithms are discretized numerical approximations of continuous-time It\u02c6 o diffusions (Chen, Ding, and Carin 2015; Ma, Chen, and Fox 2015), which are equipped with stationary distributions coincident with the target posterior distributions. An It\u02c6 o diffusion is written as d xt = F(xt)dt + g(xt)dwt , (1) where x \u2208Rd is the state variable, t is the time index, and wt \u2208Rd is d-dimensional Brownian motion. Typically, x \u2287 \u03b8 is an augmentation of the model parameters, so r \u2264d. Functions F : Rd \u2192Rd and g : Rd \u2192Rd\u00d7d are assumed to satisfy the Lipschitz continuity condition (Ghosh 2011). According to (Ma, Chen, and Fox 2015), all SG-MCMC algorithms can be formulated by de\ufb01ning appropriate functions F and g in (1). For example, the stochastic gradient Langevin dynamic (SGLD) model corresponds to x = \u03b8, and F(xt) = \u2212\u2207\u03b8U(\u03b8), g(xt) = \u221a 2 Ir, where U(\u03b8) \u225c\u2212log p(\u03b8) \u2212PN i=1 log p(di |\u03b8) denotes the unnormalized negative log-posterior. Similar formula can be de\ufb01ned for other SG-MCMC algorithms, such as stochastic gradient Hamiltonian Monte Carlo (SGHMC) (Chen, Fox, and Guestrin 2014) and stochastic gradient thermostats (SGNHT) (Ding et al. 2014). An SG-MCMC algorithm is usually developed by numerically solving the corresponding It\u02c6 o diffusion and replacing the full gradient \u2207\u03b8U(\u03b8) with an unbiased estimate from a minibatch of data \u2207\u03b8 \u02dc U(\u03b8) in each iteration. For example, in SGLD, this yields an update equation of \u03b8l = \u03b8l\u22121 \u2212\u2207\u03b8 \u02dc U(\u03b8l\u22121)hl + \u221a2hl\u03b6l for the l-th iteration, where hl is the stepsize, \u03b6l \u223cN(0, Ir). This brings two sources of error into the chain: numerical error (from discretization of the differential equation) and stochastic noise error from use of minibatches. In particular, (Chen, Ding, and Carin 2015) proved the following bias and MSE bounds for general SG-MCMC algorithms: Lemma 1 ((Chen, Ding, and Carin 2015)) Under Assumption 2 in Appendix , the bias and MSE of SG-MCMC with a Kth-order integrator\u2217at time t = hL are bounded \u2217The order characterizes the accuracy of a numerical integrator, e.g., the Euler method is a 1st-order integrator. as: \f \f \fE\u02c6 \u03c6L \u2212\u00af \u03c6 \f \f \f = O \u0012P l \u2225E\u2206Vl\u2225 L + 1 Lh + hK \u0013 E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 = O   1 L P l E \u2225\u2206Vl\u22252 L + 1 Lh + h2K ! Here \u2206Vl \u225c (L \u2212\u02dc Ll)\u03c6, where L is the in\ufb01nitesimal generator of the It\u02c6 o diffusion (1) de\ufb01ned as Lf(xt) = \u0000F(xt) \u00b7 \u2207x + 1 2 \u0000g(xt)g(xt)T \u0001 :\u2207x\u2207T x \u0001 f(xt), for any compactly supported twice differentiable function f : Rd \u2192R. a \u00b7 b \u225caT b for two vectors a and b, A : B \u225ctr{AT B} for two matrices A and B. \u2225\u00b7 \u2225is de\ufb01ned as the standard operator norm acting on the space of bounded functions, e.g., \u2225f\u2225\u225csupx f(x) for a function f. \u02dc Ll is the same as L except for the substitution of the stochastic gradient \u2207\u02dc Ul(\u03b8) for the full gradient due to the usage of a stochastic gradient in the l-th iteration. By substituting the de\ufb01nition of \u2206Vl and L, typically we have \u2206Vl = (\u2207\u03b8Ul(\u03b8) \u2212\u2207\u03b8 \u02dc Ul(\u03b8))\u00b7\u2207\u03c6. By using an unbiased estimate of the true gradient, the term E\u2206Vl in the bias bound in Lemma 1 vanishes, indicating that stochastic gradients (or equivalently minibatch size) only affect the MSE bound. Consequently, we focus on improving the MSE bound with the proposed variance-reduction SGMCMC framework. Practical Variance-Reduction SG-MCMC We \ufb01rst motivate the necessity of variance reduction in SGMCMC, by analyzing how minibatch size affects the MSE bound. A practical variance reduction scheme is then proposed, which is ef\ufb01cient from both computational and storage perspectives. Comparison with existing variance-reduction SG-MCMC approaches is also highlighted. Previous research has revealed that the convergence of diffusion-based MCMC scales at an order of O(d1/3) w.r.t. dimension d (Durmus et al. 2016). For the interest of SG-MCMC, we following standard analysis (Vollmer, Zygalakis, and Teh 2016) and do not consider the impact of d in our analysis. The necessity of variance reduction: a theoretical perspective It is clear from Lemma 1 that the variance of noisy stochastic gradients plays an important role in the MSE bound of an SG-MCMC algorithm. What is unclear is how exactly minibatch size affects the convergence rate. Intuitively, minibatch size appears to play the following roles in SG-MCMC: i) smaller minibatch sizes introduce larger variance into stochastic gradients; ii) smaller minibatch sizes allow an algorithm to run faster (thus more samples can be obtained in a given amount of computation time). To balance the two effects, in addition to using the standard assumptions for SG-MCMC (which basically requires the coef\ufb01cients of It\u02c6 o diffusions to be smooth and bounded, and is deferred to Assumption 2 in the Appendix), we assume that the algorithms with different minibatch sizes all run for a \ufb01xed computational time/budget T in the analysis, as stated in Assumption 1. \fAssumption 1 For a fair comparison, all SG-MCMC algorithms with different minibatch sizes are assumed to run for a \ufb01xed amount of computation time/budget T. Further, we assume that T linearly depends on the minibatch size n and the sample size L, i.e., T \u221dnL. For simplicity, we rewrite the gradient of the log-likelihood for data di in the l-th iteration as: \u03b1li = \u2207\u03b8 log p(di |\u03b8l). We \ufb01rst derive the following lemma about the property of {\u03b1li}, which is useful in the subsequent developments, e.g., to guarantee a positive bound in Theorem 3 and an improved bound for the proposed vrSG-MCMC (Theorem 6). Lemma 2 Under Assumption 2, given \u03b8l in the l-th iteration, \u0393l \u225c 1 N 2 PN i=1 PN j=1 E \u0002 \u03b1T li\u03b1lj \u0003 \u2212 P i\u0338=j E\u03b1T li\u03b1lj N(N\u22121) \u22650, where the expectation is taken over the randomness of an SG-MCMC algorithm\u2020. We next generalize Lemma 1 by incorporating the minibatch size n into the MSE bound. The basic idea in our derivation is to associate with each data di a binary random variable, zi, to indicate whether data di is included in the current minibatch or not. These {zi} depend on each other such that PN i=1 zi = n in order to guarantee minibatches of size n. Consequently, the stochastic gradient in the l-th iteration can be rewritten as: \u2207\u03b8 \u02dc Ul(\u03b8) = \u2212\u2207\u03b8 log p(\u03b8l) \u2212N n PN i=1 \u2207\u03b8 log p(di |\u03b8l)zi. Substituting the above gradient formula into the proof of standard SGMCMC (Chen, Ding, and Carin 2015) and further summing out {zi} results in an alternative MSE bound for SG-MCMC, stated in Theorem 3. In the analysis, we assume to use a 1st-order numerical integrator for simplicity, e.g. the Euler method, though the results generalize to Kth-order integrators easily. Theorem 3 Under Assumption 2, let the minibatch size of an SG-MCMC be n, \u0393M \u225cmaxl \u0393l. The \ufb01nite-time MSE is bounded, for a constant C independent of {h, L, n}, as: E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 \u2264C \u00122(N \u2212n)N 2\u0393M nL + 1 Lh + h2 \u0013 . Theorem 3 represents the bound in terms of minibatch size n and sample size L. Note in our \ufb01nite-time setting, L and N are considered to be constants. Consequently, \u0393M is also a bounded constant in our analysis. To bring in the computational budget T, based on Assumption 1, e.g., T \u221dnL, the optimal MSE bound w.r.t.stepsize h in Theorem 3 can be written as: E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 = O \u0010 (N\u2212n)N 2\u0393M T + n2/3 T 2/3 \u0011 . After further optimizing the bound w.r.t.n by setting the derivative of the above MSE bound to zero, the optimal minibatch size can be written as n = O \u0010 8T 27N 6\u03933 M \u0011 . To guarantee this bound for n to be \ufb01nite and integers, it is required that the computational budget T to scale at the order of O(N 6) when varying N. When considering both T and N as impact factors, the \u2020The same meaning goes for other expectations in the paper if not explicitly speci\ufb01ed. optimal n becomes more interesting, and is concluded in Corollary 4\u2021. Corollary 4 Under Assumption 1 and 2, we have three cases of optimal minibatch sizes, each corresponding to different levels of computational budget. 1) When the computational budget is small, e.g., T < O \u0000 27 8 \u03933 MN 6\u0001 , the optimal MSE bound is decreasing w.r.t. n in range [1, N]. The minimum MSE bound is achieved at n = N. 2) When the computational budget is large, e.g., T > O \u0000 27 8 \u03933 MN 7\u0001 , the optimal MSE bound is increasing w.r.t. n in range [1, N]. The minimum MSE bound is achieved at n = 1. 3) When the computational budget is in between the above two cases, the optimal MSE bound \ufb01rst increases then decreases w.r.t.n in range [1, N]. The optimal MSE bound is obtained either at n = 1 or at n = N, depending on (N, T, \u0393M). In many machine learning applications, the computational budget is limited, leading the algorithm to the \ufb01rst case of Corollary 4, i.e., T < O \u0000 27 8 \u03933 MN 6\u0001. According to Corollary 4, processing full data (i.e., no minibatch) is required to achieve the optimal MSE bound, which is computationally infeasible when N is large (which motivated use of minibatches in the \ufb01rst place). A practical way to overcome this is to use small minibatches and adopt variance-reduction techniques to reduce the stochastic gradient noise. A practical variance reduction algorithm For practical use, we require that a variance-reduction method should achieve both computational and storage ef\ufb01ciency. While variance reduction has been studied extensively in stochastic optimization, it is applied much less often in SGMCMC. In this section we propose a vrSG-MCMC algorithm, a simple extension of the algorithm in (Dubey et al. 2016), but is more computationally practical in large-scale applications. A convergence theory is also developed in Section . The proposed vrSG-MCMC is illustrated in Algorithm 1. Similar to stochastic optimization (Schmidt, Le Roux, and Bach 2016), the idea of variance reduction is to balance the gradient noise with a less-noisy old gradient, i.e., a stochastic gradient is calculated based on a previous sample, as well as using a larger minibatch than that of the current stochastic gradient, resulting in a less noisy estimation. In each iteration of our algorithm, an unbiased stochastic gradient is obtained by combining the above two versions of gradients in an appropriate way (see gl+1 in Algorithm 1). Such a construction of stochastic gradients essentially inherits a low variance with theoretical guarantees (detailed in Section ). In Algorithm 1, the whole parameter x is decomposed into the model parameter \u03b8 and the remaining algorithm-speci\ufb01c parameter \u03c4, e.g., the momentum parameter. The expression \u201c\u03b8 \u2190x\u201d means assigning the corresponding model parameter from x to \u03b8. The old gradient is denoted as \u02dc g, calculated \u2021Note we only have that T = C1nL for some unknown constant C1, i.e., the speci\ufb01c value of T is unknown. \fwith a minibatch of size n1. The current stochastic gradient is calculated on a minibatch of size n2 < n1. We use xl+1 = NextS (xl, gl+1, hl) to denote a function which generates the next sample xl+1 with an SG-MCMC algorithm, based on the current sample xl, input stochastic gradient gl+1, and step size hl. Algorithm 1 Practical Variance-Reduction SG-MCMC. Input: \u00af x = x0 = (\u03b80, \u03c40) \u2208Rd, minibatch sizes (n1, n2) such that n1 > n2, update interval m, total iterations L, stepsize {hl}L l=1 Output: approximate samples {xl}L l=1 for l = 0 to L \u22121 do if (l mod m) = 0 then Sample w/t replacement {\u03c0i}n1 i=1 \u2286{1, \u00b7 \u00b7 \u00b7 , N}; \u00af x = xl; \u02dc \u03b8l \u2190\u00af x; \u02dc g = N n1 P i\u2208\u03c0 \u2207\u03b8 log p(di |\u02dc \u03b8l); end if \u03b8l \u2190xl; \u02dc \u03b8l \u2190\u00af x; Sample w/t replacement {\u02dc \u03c0i}n2 i=1 \u2286{1, \u00b7 \u00b7 \u00b7 , N}; gl+1 = \u02dc g + \u2207\u03b8 log p(\u03b8l) + N n2 P i\u2208\u02dc \u03c0 \u0010 \u2207\u03b8 log p(di |\u03b8l) \u2212\u2207\u03b8 log p(di |\u02dc \u03b8l) \u0011 ; xl+1 = NextS (xl, gl+1, hl+1); end for One should note that existing variance-reduction algorithms, e.g.(Johnson and Zhang 2013), use a similar concept to construct low-variance gradients. However, most algorithms use the whole training data to compute \u02dc g in Algorithm 1, which is computationally infeasible in large-scale settings. Moreover, we note that like in stochastic optimization (Reddi et al. 2016; Allen-Zhu and Hazan 2016), instead of using a single parameter sample to compute \u02dc g, similar methods can be adopted to compute \u02dc g based on an average of old parameter samples. The theoretical analysis can be readily adopted for such cases, which is omitted here for simplicity. More references are discussed in Section . Comparison with existing variance-reduction SG-MCMC algorithms The most related variance-reduction SG-MCMC algorithm we are aware of is a recent work on variance-reduction SGLD (SVRG-LD) (Dubey et al. 2016). SVRG-LD shares a similar \ufb02avor to our scheme from the algorithmic perspective, except that when calculating the old gradient \u02dc g, the whole training data set is used in SVRG-LD. As mentioned above, this brings a computational challenge for largescale learning. Although the problem is mitigated by using a moving average estimation of the stochastic gradient, this scheme does not match their theory. A more distinctive advantage of vrSG-MCMC over SVRG-LD (Dubey et al. 2016) is in terms of theoretical analysis. Concerning SVRG-LD, i) the authors did not show theoretically in which case variance reduction is useful in SGLD, and ii) it is not clear in their theory whether SVRG-LD is able to speed up the convergence rate compared to standard SGLD. Specifically, the MSE of SVRG-LD was shown to be bounded by O \u0010 N 2 min{2\u03c32,m2(D2h2\u03c32+hd)} nL + 1 Lh + h2\u0011 , compared to O \u0010 N 2\u03c32 nL + 1 Lh + h2\u0011 for SGLD, where (d, D, \u03c3) are constants. By inspecting the above bounds, it is not clear whether SVRG-LD improves SGLD because the two bounds are not directly comparable\u00a7. More detailed explanations are provided in Appendix . Convergence rate We derive convergence bounds for Algorithm 1 and analyze the improvement of vrSG-MCMC over the corresponding standard SG-MCMC. Using a similar approach as in Section , we \ufb01rst introduce additional binary random variables, {bi}N i=1, to indicate which data points are included in calculating the old gradient \u02dc g in Algorithm 1. This results in the expression for the stochastic gradient used in the l-th iteration: \u2207\u03b8 \u02dc U(\u03b8l) = N n2 PN i=1 \u0010 \u2207\u03b8 log p(di |\u03b8l) \u2212\u2207\u03b8 log p(di |\u02dc \u03b8l) \u0011 zi + N n1 PN i=1 PN i=1 \u2207\u03b8 log p(di |\u02dc \u03b8l)bi. It is easy to verify that the above stochastic gradient is an unbiased estimation of the true gradient in the l-th iteration (see Appendix ). In order to see how Algorithm 1 reduces the variance of stochastic gradients, from Lemma 1, it suf\ufb01ces to study \u2206Vl, as the minibatch size only impacts this term. For notational simplicity, similar to the \u03b1li de\ufb01ned in Section , we denote \u03b2 \u03b2 \u03b2li \u225c\u2207\u03b8 log p(di |\u02dc \u03b8l), which is similar to \u03b1li but evaluated on the old parameter \u02dc \u03b8l. Intuitively, since the old gradient \u02dc g is calculated from \u03b2 \u03b2 \u03b2 to balance the stochastic gradient noise (calculated from \u03b1), \u03b1 and \u03b2 \u03b2 \u03b2 are expected to be close to each other. Lemma 5 formulates the intuition, a key result in proving our main theorem, where we only consider the update interval m and stepsize h as factors. In the lemma below, following (Chen et al. 2016) (Assumption 1), we further assume the gradient function \u2207\u03b8U(\u03b8) to be Lipschitz. Lemma 5 Under Assumption 2 and assume \u2207\u03b8U(\u03b8) to be Lipschitz (Assumption 1 in (Chen et al. 2016)), \u03b1li and \u03b2 \u03b2 \u03b2li are close to each other in expectation, i.e., E\u03b1li = E\u03b2 \u03b2 \u03b2li +O(mh). In the Appendix, we further simplify E\u2225\u2206Vl\u22252 in the MSE bound by decomposing it into several terms. Finally, we arrive at our main theorem for the proposed vrSG-MCMC framework. Theorem 6 Under the setting of Lemma 5, let AM \u225cmaxl Al, and Al = \u0010 N n2 \u22121 \u0011 P ij E\u03b1T li\u03b1lj \u2212 2 N(N\u2212n2) n2(N\u22121) P i<j E\u03b1T li\u03b1lj. The MSE of vrSG-MCMC with a Kth-order integrator is bounded as: E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 = O \u0012AM L + 1 Lh + h2K + mh L \u2212\u03bbM L \u0013 , \u00a7The \ufb01rst term in the min of the SVRG-LD bound is strictly larger than the \ufb01rst term of the SGLD bound (if the term 2\u03c32 is used in the \u201cmin\u201d), making the bounds not easily compared. \fwhere \u03bbM = minl \u03bbl, and \u03bbl \u225c \u0010 N n1 \u2212N n2 \u0011 P ij E\u03b2 \u03b2 \u03b2T li \u03b2 \u03b2 \u03b2li \u22122 \u0010 N(N\u2212n2) n2(N\u22121) \u2212N(N\u2212n1) n1(N\u22121) \u0011 P i<j E\u03b2 \u03b2 \u03b2T li \u03b2 \u03b2 \u03b2lj. Furthermore, we have \u03bbl > 0 for \u2200l, so that \u03bbM > 0. Note that for a \ufb01xed m, mh L in the above bound is a highorder term relative to 1 Lh. As a result, the MSE is bounded by O \u0000 AM L + 1 Lh + h2K \u2212\u03bbM L \u0001 . Because the MSE of standard SG-MCMC is bounded by O \u0000 AM L + 1 Lh + h2K\u0001 (see Appendix ) and \u03bbM > 0 from Theorem 6, we conclude that vrSG-MCMC induces a lower MSE bound compared to the corresponding SG-MCMC algorithm, with an improvement of O \u0000 \u03bbM L \u0001 . It is worth noting that in Algorithm 1, the minibatch for calculating the old gradient \u02dc g is required to be larger than that for calculating the current stochastic gradient, i.e., n1 > n2. Otherwise, \u03bbl in Theorem 6 would become negative, leading to an increased MSE bound compared to standard SG-MCMC. This matches the intuition that old gradients need to be more accurate (thus with larger minibatches) than current stochastic gradients in order to reduce the stochastic gradient noise. Remark 7 In the special case of (Dubey et al. 2016) where n1 = N for SGLD, Theorem 6 gives a MSE bound of O \u0000 AM L + 1 Lh + h2 + mh L \u2212maxl \u03bbl L \u0001 , with \u03bbl = \u0010 N n2 \u22121 \u0011 P ij E\u03b2 \u03b2 \u03b2T li \u03b2 \u03b2 \u03b2li \u22122N(N\u2212n2) n2(N\u22121) P i<j E\u03b2 \u03b2 \u03b2T li \u03b2 \u03b2 \u03b2lj. According to Lemma 2, \u03bbl is also positive, thus leading to a reduced MSE bound. However, the bound is not necessarily better than that of vrSG-MCMC, where a minibatch is used instead of the whole data set to calculate \u02dc g, leading to a signi\ufb01cant decrease of computational time. Remark 8 Following Corollary 4, Theorem 6 can also be formulated in terms of the computational budget T. Specifically, according to Algorithm 1, the computational budget T would be proportional to n1 m + n2. Substituting this into the MSE bound of Theorem 6 gives a reformulated bound of O \u0012 (AM+mh\u2212\u03bbM)( n1 m +n2) T + 1 Lh + h2 \u0013 . The optimal MSE w.r.t.n1 and n2 would be complicated since both AM and \u03bbM depend on n1 and n2. We omit the details here for simplicity. Nevertheless, our experiments indicate that our algorithm always improves standard SG-MCMC algorithms for the same computational time. Related Work Variance reduction was \ufb01rst introduced in stochastic optimization, which quickly became a popular research topic and has been actively developed in recent years. (Schmidt, Le Roux, and Bach 2013; Schmidt, Le Roux, and Bach 2016) introduced perhaps the \ufb01rst variance reduction algorithm, called stochastic average gradient (SAG), where historical gradients are stored and continuously updated in each iteration. Later, stochastic variance reduction gradient (SVRG) was developed to reduce the storage bottleneck of SAG, at the cost of an increased computational time (Johnson and Zhang 2013; Zhang, Mahdavi, and Jin 2013). (Defazio, Bach, and LacosteJulien 2014) combined ideas of SAG and SVRG and proposed the SAGA algorithm, which improves SAG by using a better and unbiased stochastic-gradient estimation. Variance reduction algorithms were \ufb01rst designed for convex optimization problems, followed by a number of recent works extending the techniques for non-convex optimization (Reddi et al. 2016; Reddi, Sra, and B. P\u00b4 oczos 2016; Allen-Zhu and Hazan 2016; Allen-Zhu et al. 2016), as well as for distributed learning (Reddi et al. 2015). All these algorithms are mostly based on SVRG and are similar in algorithmic form, but differ in the techniques for proving the rigorous theoretical results. For scalable Bayesian sampling with SG-MCMC, however, this topic has been studied little until a recent work on variance reduction for SGLD (Dubey et al. 2016). In this work, the authors adapted the SAG and SVRG ideas to SGLD. Although they provided corresponding convergence results, some fundamental problems, such as how minibatch size affects the convergence rate, were not fully studied. Furthermore, their algorithms suffer from an either high computational or storage cost in a big-data setting, because the whole data set needs to be accessed frequently. To reduce the computational cost of SVRG-based algorithms, the idea of using a minibatch of data to calculate the old gradient (corresponding to the \u02dc g in Algorithm 1) has also been studied in stochastic optimization. Representative works include, but are not limited to (Harikandeh et al. 2015; Frostig et al. 2015; Shah et al. 2016; Lei and Jordan 2016; Lian, Wang, and Liu 2017). The proposed approach adopts similar ideas, with the following main differences: i) Our algorithm represents the \ufb01rst work for large-scalable Bayesian sampling with a practical (computationally cheap) variance reduction technique; ii) the techniques used here for analysis are different and appear to be simpler than those used for stochastic optimization; iii) our theory addresses fundamental questions for variance reduction in SG-MCMC, such as those raised in the Introduction. Experiments A synthetic experiment We \ufb01rst test the conclusion of the long-run setting in Corollary 4, which indicates that vrSG-MCMC with minibatches of size 1 achieve the optimal MSE bound. To make the algorithm go into the long-run setting regime as suf\ufb01cient as possible, we test vrSG-MCMC on a simple Gaussian model, which runs very fast so that a little actual walk-clock time is regarded as a large computational budget. The model is de\ufb01ned as: xi \u223cN(\u03b8, 1), \u03b8 \u223cN(0, 1). We generate N = 1000 data samples {xi}, and calculate the the MSE for minibatch sizes of n = 1, 10, 100. The test function is \u03c6(\u03b8) = \u03b82. The results are ploted in Figure 1. We can see from the \ufb01gure that n = 1 achieves the lowese MSE, consistent with the theory (Corollary 4). Applications on deep neural networks We apply the proposed vrSG-MCMC framework to Bayesian learning of deep neural networks, including the multilayer \f10 -4 Time 10 -3 10 -2 10 -1 10 0 10 1 10 2 MSE batch-1 batch-10 batch-100 Figure 1: MSE vs. wall-clock time for different minibatch sizes. perceptron (MLP), convolutional neural network (CNN), and recurrent neural network (RNN). The latter two have not been empirically evaluated in previous work. Experiments with Bayesian logistic regression are given in Appendix . In the experiments, we are interested in modeling weight uncertainty of neural networks, which is an important topic and has been well studied (Hern\u00b4 andez-Lobato and Adams 2015; Blundell et al. 2015; Li et al. 2016; Louizos and Welling 2016). We achieve this goal by applying priors to the weights (in our case, we use simple isotropic Gaussian priors) and performing posterior sampling with vrSG-MCMC or SGMCMC. We implement vrSG-MCMC based on SGLD, and compare it to the standard SGLD and SVRG-LD (Dubey et al. 2016) in our experiments\u00b6. For this reason, comparisons to other optimization-based methods such as the maximum likelihood are not considered. For simplicity, we set the update interval for the old gradient \u02dc g in Algorithm 1 to m = 10. For all the experiments, the minibatch sizes for vrSG-MCMC are set to n1 = 100 and n2 = 10. To be fair, this corresponds to a minibatch size of n = 10 in SGLD and SVRG-LD. Sensitivity of model performance w.r.t.minibatch size n1 is tested in Section . For a fair comparison, following convention (Allen-Zhu and Hazan 2016; Dubey et al. 2016), we plot the number of data passes versus error in the \ufb01gures\u2225. Results on the number of data passes versus loss are given in the Appendix. In addition, we use \ufb01xed stepsizes in our algorithm for all except for the ResNet model speci\ufb01ed below. Following relevant literature (Johnson and Zhang 2013; Dubey et al. 2016), we tune the stepsizes and plot the best results for all the algorithms to ensure fairness. Note in our Bayesian setup, it is enough to run an algorithm for once since the uncertainty is encoded in the samples. Multilayer perceptron We follow conventional settings (Reddi et al. 2016; AllenZhu and Hazan 2016) and use a single-layer MLP with 100 hidden units, using the sigmoid activation function as the nonlinear transformation. We test the MLP on the MNIST and CIFAR-10 datasets. The stepsizes for both vrSG-MCMC and SGLD are set to 0.25 and 0.01 in the two datasets, respectively. Figure 2 plots the number of passes through the data \u00b6The SAGA-LD algorithm in (Dubey et al. 2016) is not compared here because it is too storage-expensive thus is not fair. \u2225Since true posterior averages are infeasible, we plot sample averages in terms of accuracy/loss. versus test error/loss. Results on the training datasets, including training results for the CNN and RNN-based deep learning models described below, are provided in Appendix . It is clear that vrSG-MCMC leads to a much faster convergence speed than SGLD, resulting in much lower test errors and loss at the end, especially on the CIFAR-10 dataset. SVRGLD, though it leads to potential lower errors/loss, converges slower than vrSG-MCMC, due to the high computational cost in calculating the old gradient \u02dc g. As a result, we do not compare vrSG-MCMC with SVRG-LD in the remaining experiments. 50 100 150 200 #Data Pass 1.6 1.8 2 2.2 2.4 2.6 2.8 Error Rate (%) SGLD SVRG-LD vrSG-MCMC 50 100 150 200 #Data Pass 0.06 0.07 0.08 0.09 0.1 Loss SGLD SVRG-LD vrSG-MCMC 100 200 300 400 500 600 #Data Pass 46 48 50 52 54 Error Rate (%) SGLD SVRG-LD vrSG-MCMC 100 200 300 400 500 600 #Data Pass 1.3 1.4 1.5 1.6 1.7 Loss SGLD SVRG-LD vrSG-MCMC Figure 2: Number of passes through data vs. testing error (left) / loss (right) on MNIST (top) and CIFAR-10 (bottom) datasets. Convolutional neural networks We use the CIFAR-10 dataset, and test two CNN architectures for image classi\ufb01cation. The \ufb01rst architecture is a deep convolutional neural networks with 4 convolutional layers, denoted as C32-C32-C64-C32, where max-pooling is applied on the output of the \ufb01rst three convolutional layers, and a Dropout layer is applied on the output of the last convolutional layer. The second architecture is a 20-layers deep residual network (ResNet) with the same setup as in (He et al. 2016). Speci\ufb01cally, we use a step-size-decrease scheme as hl = 1 10+1.8e-3\u00d7l for both vrSG-MCMC and SGLD, where l is the number of iterations so far. Figure 3 plots the number of passes through the data versus test error/loss on both models. Similar to the results on MLP, vrSG-MCMC converges much faster than SGLD, leading to lower test errors and loss. Interestingly, the gap seems larger in the more complicated ResNet architecture; furthermore, the learning curves look much less noisy (smoother) for vrSG-MCMC because of the reduced variance in stochastic gradients. Recurrent neural networks The recurrent neural network with LSTM units (Hochreiter and Schmidhuber 1997) is a powerful architecture used for modeling sequence-to-sequence data. We consider the task \f0 20 40 60 80 100 #Data Pass 30 35 40 45 Error Rate (%) SGLD vrSG-MCMC 20 40 60 80 100 #Data Pass 0.8 1 1.2 1.4 1.6 Loss SGLD vrSG-MCMC 0 50 100 150 200 #Data Pass 10 20 30 40 50 Test Error (%) SGLD vrSG-MCMC 0 50 100 150 200 #Data Pass 0 0.5 1 1.5 2 Loss SGLD vrSG-MCMC Figure 3: Number of passes through data vs. testing error (left) / loss (right) with CNN-4 (top) and ResNet (bottom) on CIFAR-10. of language modeling on two datasets, i.e., the Penn Treebank (PTB) dataset and WikiText-2 dataset (Merity et al. 2016). PTB is the smaller dataset among the two, containing a vocabulary of size 10,000. We use the default setup of 887,521 tokens for training, 70,390 for validation and 78,669 for testing. WikiTest-2 is a large dataset with 2,088,628 tokens from 600 Wiki articles for training, 217,649 tokens from 60 Wiki articles for validation, and 245,569 tokens from an additional 60 Wiki articles for testing. The total vocabulary size is 33,278. We adopt the hierarchical LSTM achitecture (Zaremba, Sutskever, and Vinyals 2014). The hierarchy depth is set to 2, with each LSTM containing 200 hidden unites. The step size is set to 0.5 for both datasets. For more stable training, standard gradient clipping is adopted, where gradients are clipped if the norm of the parameter vector exceeds 5. Figure 4 plots the number of passes through the data versus test perplexity on both datasets. The results are consistent with the previous experiments on MLPs and CNNs, where vrSGMCMC achieves faster convergence than SGLD; its learning curves in terms of testing error/loss are also much smoother. 0 1 2 3 4 5 #Data Pass 100 200 300 400 500 Testing Perplexisty SGLD vrSG-MCMC 0 1 2 3 #Data Pass 100 200 300 400 500 600 Testing Perplexisty SGLD vrSG-MCMC Figure 4: Number of passes through data vs. testing perplexity on the PTB dataset (top) and WikiTest-2 dataset (bottom). Parameter sensitivity Note that one of the main differences between vrSG-MCMC and the recently proposed SVRG-LD (Dubey et al. 2016) is that the former uses minibatches of size n1 to calculate the old gradient \u02dc g in Algorithm 1, leading to a much more computationally ef\ufb01cient algorithm, with theoretical guarantees. This section tests the sensitivity of model performance to the parameter n1. 100 200 300 400 500 600 #Data Pass 45 50 55 60 Error Rate (%) SGLD vrSG-MCMC-G10 vrSG-MCMC-G20 vrSG-MCMC-G30 vrSG-MCMC-G40 vrSG-MCMC-G50 vrSG-MCMC-G60 vrSG-MCMC-G70 vrSG-MCMC-G100 vrSG-MCMC-G200 100 200 300 400 500 600 #Data Pass 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Loss SGLD vrSG-MCMC-G10 vrSG-MCMC-G20 vrSG-MCMC-G30 vrSG-MCMC-G40 vrSG-MCMC-G50 vrSG-MCMC-G60 vrSG-MCMC-G70 vrSG-MCMC-G100 vrSG-MCMC-G200 Figure 5: Number of passes through data vs. testing errors (top) / loss (bottom) on the CIFAR-10 dataset, with varying n1 values. For simplicity, we run on the same MLP model described in Section on the CIFAR-10 dataset, where the same parameter settings are used, but varying n1 in {100, 200, 300, 400, 500, 600, 700, 1000, 2000}. Figure 5 plots the number of passes through data versus test errors/loss, where we use \u201cvrSG-MCMC-Gn\u201d to denote vrSG-MCMC with n1 = n. Interestingly, vrSG-MCMC outperforms the baseline SGLD on all n1 values. Notably, when n1 is large enough (n1 = 200 in our case), their corresponding test errors and loss are very close. This agrees with the intuition that computing the old gradient using the whole training data is not necessarily a good choice in order to balance the stochastic gradient noise and computational time."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1709.01179v4",
+                    "title": "Continuous-Time Flows for Efficient Inference and Density Estimation",
+                    "abstract": "Two fundamental problems in unsupervised learning are efficient inference for\nlatent-variable models and robust density estimation based on large amounts of\nunlabeled data. Algorithms for the two tasks, such as normalizing flows and\ngenerative adversarial networks (GANs), are often developed independently. In\nthis paper, we propose the concept of {\\em continuous-time flows} (CTFs), a\nfamily of diffusion-based methods that are able to asymptotically approach a\ntarget distribution. Distinct from normalizing flows and GANs, CTFs can be\nadopted to achieve the above two goals in one framework, with theoretical\nguarantees. Our framework includes distilling knowledge from a CTF for\nefficient inference, and learning an explicit energy-based distribution with\nCTFs for density estimation. Both tasks rely on a new technique for\ndistribution matching within amortized learning. Experiments on various tasks\ndemonstrate promising performance of the proposed CTF framework, compared to\nrelated techniques.",
+                    "authors": "Changyou Chen, Chunyuan Li, Liqun Chen, Wenlin Wang, Yunchen Pu, Lawrence Carin",
+                    "published": "2017-09-04",
+                    "updated": "2018-08-01",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Ef\ufb01cient inference and robust density estimation are two important goals in unsupervised learning. In fact, they can be uni\ufb01ed from the perspective of learning desired target distributions. In inference problems, one targets to learn a tractable distribution for a latent variable that is close to a given unnormalized distribution (e.g., a posterior distribution in a Bayesian model). In density estimation, one tries to learn an unknown data distribution only based on the samples from it. It is also helpful to make a distinction between two types of representations for learning distributions: explicit and implicit methods (Mohamed & Lakshminarayanan, 2017). Explicit methods provide a prescribed parametric form for the distribution, while implicit methods 1SUNY at Buffalo 2Duke University. Correspondence to: Changyou Chen <cchangyou@gmail.com>. Proceedings of the 35 th International Conference on Machine Learning, Stockholm, Sweden, PMLR 80, 2018. Copyright 2018 by the author(s). learn a stochastic procedure to directly generate samples from the unknown distribution. Existing deep generative models can easily be identi\ufb01ed from this taxonomy. For example, the standard variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014) is an important example of an explicit inference method. Within the inference arm (encoder) of a VAE, recent research has focused on improving the accuracy of the approximation to the posterior distribution on latent variables (codes) using normalizing \ufb02ow (NF) (Rezende & Mohamed, 2015). NF is particularly interesting due to its ability to approximate the posterior distribution arbitrarily well, while maintaining explicit parametric forms. On the other hand, Stein VAE (Feng et al., 2017; Pu et al., 2017) is an implicit inference method, as it only learns to draw samples to approximate posteriors, without assuming an explicit form for the distribution.. For density estimation on observed data, the generative adversarial network (GAN) can be regarded as an implicit density estimation method (Ranganath et al., 2016; Husz\u00e1r, 2017; Mohamed & Lakshminarayanan, 2017), in the sense that one may sample from the distribution (regarded as a representation of the unknown distribution), but an explicit form for the distribution is not estimated. GAN has recently been augmented by Flow-GAN (Grover et al., 2017) to incorporate a likelihood term for explicit density estimation. Further, the real-valued non-volume preserving (real NVP) transformations algorithm (Dinh et al., 2017) was proposed to perform inference within the implicit density estimation framework. Some aforementioned methods rely on the concept of \ufb02ows. A \ufb02ow de\ufb01nes a series of transformations for a random variable (RV), such that the distribution of the RV evolves from a simple distribution to a more complex distribution. When the sequence of transformations are indexed on a discrete-time domain (e.g., indexed with integers) with a \ufb01nite number of transformations, this method is referred to as a normalizing \ufb02ow (Rezende & Mohamed, 2015). Various ef\ufb01cient implementations of NFs have been proposed, such as the planar, radial (Rezende & Mohamed, 2015), Householder (Tomczak & Welling, 2016), and inverse autoregressive \ufb02ows (Kingma et al., 2016). One theoretical limitation of existing normalizing \ufb02ows is that there is no guarantee on the approximation accuracy due to the \ufb01nite number of transformations. arXiv:1709.01179v4  [stat.ML]  1 Aug 2018 \fContinuous-Time Flows By contrast, little work has explored the applicability of continuous-time \ufb02ows (CTFs) in deep generative models, where a sequence of transformations are indexed on a continuous-time domain (e.g., indexed with real numbers). There are at least two reasons encouraging research in this direction: i) CTFs are more general than traditional normalizing \ufb02ows in terms of modeling \ufb02exibility, due to the intrinsic in\ufb01nite number of transformations; ii) CTFs are more theoretically grounded, in the sense that they are guaranteed to approach a target distribution asymptotically (details provided in Section 2.2). In this paper, we propose ef\ufb01cient ways to apply CTFs for the two motivating tasks. Based on the CTF, our framework learns to drawn samples directly from desired distributions (e.g., the unknown posterior and data distributions) for both inference and density estimation tasks via amortization. In addition, we are able to learn an explicit form of the unknown data distribution for density estimation\u2217. The core idea of our framework is the amortized learning, where knowledge in a CTF is distilled sequentially into another neural network (called inference network in the inference task, and generator in density estimation). The distillation relies on the distribution matching technique proposed recently via adversarial lerning (Li et al., 2017a). We conduct various experiments on both synthetic ad real datasets, demonstrating excellent performance of the proposed framework, relative to representative approaches. 2. Preliminaries 2.1. Ef\ufb01cient inference and density estimation Ef\ufb01cient inference with normalizing \ufb02ows Consider a probabilistic generative model with observation x \u2208RD and latent variable z \u2208RL such that x | z \u223cp\u03b8(x | z) with z \u223cp(z). For ef\ufb01cient inference of z, the VAE (Kingma & Welling, 2014) introduces the concept of an inference network (recognition model or encoder), q\u03c6 \u03c6 \u03c6(z | x), as a variational distribution in the VB framework. An inference network is typically a stochastic (nonlinear) mapping from the input x to the latent z, with associated parameters \u03c6 \u03c6 \u03c6. For example, one of the simplest inference networks is de\ufb01ned as q\u03c6 \u03c6 \u03c6(z | x) = N(z; \u00b5\u03c6 \u03c6 \u03c6(x), diag(\u03c32 \u03c6 \u03c6 \u03c6(x))), where the mean function \u00b5\u03c6 \u03c6 \u03c6(x) and the standard-derivation function \u03c3\u03c6 \u03c6 \u03c6(x) are speci\ufb01ed via deep neural networks parameterized by \u03c6 \u03c6 \u03c6. Parameters are learned by minimizing the negative evidence lower bound (ELBO), i.e., the KL divergence between p\u03b8(x, z) and q\u03c6 \u03c6 \u03c6(z | x): KL (q\u03c6 \u03c6 \u03c6(z | x)\u2225p\u03b8(x, z)) \u225c Eq\u03c6 \u03c6 \u03c6(z | x) [log q\u03c6 \u03c6 \u03c6(z | x) \u2212log p\u03b8(x, z)], via stochastic gradient descent (Bottou, 2012). One limitation of the VAE framework is that q\u03c6 \u03c6 \u03c6(z | x) is \u2217Although the density is represented as an energy-based distribution with an intractable normalizer. often restricted to simple distributions for feasibility, e.g., the normal distribution discussed above, and thus the gap between q\u03c6 \u03c6 \u03c6(z | x) and p\u03b8(z | x) is typically large for complicated posterior distributions. NF is a recently proposed VB-based technique designed to mitigate this problem (Rezende & Mohamed, 2015). The idea is to augment z via a sequence of deterministic invertible transformations {Tk : RL \u2192RL}K k=1, such that: z0 \u223cq\u03c6 \u03c6 \u03c6(\u00b7| x), z1 = T1(z0), \u00b7 \u00b7 \u00b7 , zK = TK(zK\u22121). Note the transformations {Tk} are typically endowed with different parameters, and we absorb them into \u03c6 \u03c6 \u03c6. Because the transformations are deterministic, the distribution of zK can be written as q(zK) = q\u03c6 \u03c6 \u03c6(z0 | x) QK k=1 \f \f \fdet \u2202Tk \u2202zk \f \f \f \u22121 via the change of variable formula. As a result, the negative ELBO for normalizing \ufb02ows becomes: KL (q\u03c6 \u03c6 \u03c6(zK | x)\u2225p\u03b8(x, z)) = Eq\u03c6 \u03c6 \u03c6(z0 | x) [log q\u03c6 \u03c6 \u03c6(z0 | x)] (1) \u2212Eq\u03c6 \u03c6 \u03c6(z0 | x) [log p\u03b8(x, zK)] \u2212Eq\u03c6 \u03c6 \u03c6(z0 | x)[ K X k=1 log |det \u2202Tk \u2202zk |]. Typically, transformations Tk of a simple parametric form are employed to make the computations tractable (Rezende & Mohamed, 2015). Our method generalizes these discretetime transformations to continuous-time ones, ensuring convergence of the transformations to a target distribution. Related density-estimation methods There exist implicit and explicit density-estimation methods. Implicit density models such as GAN provide a \ufb02exible way to draw samples directly from unknown data distributions (via a deep neural network (DNN) called a generator with stochastic inputs) without explicitly modeling their density forms; whereas explicit models such as the pixel RNN/CNN (van den Oord et al., 2016) de\ufb01ne and learn explicit forms of the unknown data distributions. This gives the advantage that the likelihood for a test data point can be explicitly evaluated. However, the generation of samples is typically time-consuming due to the sequential generation nature. Similar to Wang & Liu (2017), our CTF-based approach in Section 4 provides an alternative way for this problem, by simultaneously learning an explicit energy-based data distribution (estimated density) and a generator whose generated samples match the learned data distribution. This not only gives us the advantage of explicit density modeling but also provides an ef\ufb01cient way to generate samples. Note that our technique differs from that of Wang & Liu (2017) in that distribution matching is adopted to learn an accurate generator, which is a key component in our framework. 2.2. Continuous-time \ufb02ows We notice two potential limitations with traditional normalizing \ufb02ows: i) given speci\ufb01ed transformations {Tk}, there is no guarantee that the distribution of zK could exactly match p\u03b8(x, z); ii) the randomness is only introduced in z0 (from the inference network), limiting the representation \fContinuous-Time Flows power. We specify CTFs where transformations are indexed by real numbers, thus they could be considered as consisting of in\ufb01nite transformations. Further, we consider stochastic \ufb02ows where randomness is injected in a continuous-time manner. In fact, the concept of CTFs (such as the Hamiltonian \ufb02ow) has been introduced by Rezende & Mohamed (2015), without further development on ef\ufb01cient inference. We consider a \ufb02ow on RL, de\ufb01ned as the mapping\u2020 T : RL \u00d7 R \u2192RL such that\u2021 we have T (Z, 0) = z and T (T (Z, t), s) = T (Z, s + t), for all Z \u2208RL and s, t \u2208R. A speci\ufb01c form consider here is de\ufb01ned as T (Z, t) = Zt, where Zt is driven by a diffusion of the form: d Zt = F(Zt)dt + V (Zt)dW . (2) Here F : RL \u2192RL, V : RL\u00d7L \u2192RL are called the drift term and diffusion term, respectively; W is the standard L-dimensional Brownian motion. In the context of inference, we seek to make the stationary distribution of Zt approach p\u03b8(z | x). One solution for this is to set F(Zt) = 1 2\u2207z log p\u03b8(x, z = Zt) and V (Zt) = IL with IL the L \u00d7 L identity matrix. The resulting diffusion is called Langevin dynamics (Welling & Teh, 2011). Denoting the distribution of Zt as \u03c1t, it is well known (Risken, 1989) that \u03c1t is characterized by the Fokker-Planck (FP) equation: \u2202t\u03c1t = \u2212\u2207z \u00b7 (\u03c1tF(Zt) + \u2207z \u00b7 (\u03c1tV (Zt)V \u22a4(Zt))) , (3) where a \u00b7 b \u225ca\u22a4b for vectors a and b. For simplicity, we consider the \ufb02ow de\ufb01ned by the Langevin dynamics speci\ufb01ed above, though our results generalize to other stochastic \ufb02ows (Dorogovtsev & Nishchenko, 2014). In fact, CTF has been applied for scalable Bayesian sampling (Ding et al., 2014; Li et al., 2016a; Chen et al., 2016; Li et al., 2016b; Zhang et al., 2017). In this paper, we generalize it by specifying an ELBO under a CTF, which can then be readily solved by a discretized numerical scheme, based on the results from Jordan et al. (1998). An approximation error bound for the scheme is also derived. We defer proofs of our theoretical results to the Supplementary Material (SM) for conciseness. 3. Continuous-Time Flows for Inference For this task, we adopt the VAE/normalizing-\ufb02ow framework with an encoder-decoder structure. An important difference is that instead of feeding data to an encoder and sampling a latent representation in the output as in VAE, we concatenate the data with independent noise as input and directly generate output samples\u00a7, constituting an implicit \u2020We reuse the notation T as transformations from the discrete case above for simplicity, and use Z instead of z (reserved for the discrete-time setting) to denote the random variable in the continuous-time setting. \u2021Note we de\ufb01ne continuous-time \ufb02ows in terms of latent variable Z in order to incorporate it into the setting of inference. However, the same description applies when we de\ufb01ne the \ufb02ow in data space, which is the setting of density estimation in Section 4. \u00a7Such structure can represent much more complex distributions than a parametric form, useful for the follow up procedures. In model. These outputs are then driven by the CTF to approach the true posterior distribution. In the following, we \ufb01rst show that directly optimizing the ELBO is infeasible. We then propose an amortized-learning process that sequentially distills the implicit transformations from the CTF an inference network by distribution matching in Section 3.2. 3.1. The ELBO and discretized approximation We \ufb01rst incorporate CTF into the NF framework by writing out the corresponding ELBO. Note that there are two steps in the inference process. First, an initial z0 is drawn from the inference network q\u03c6 \u03c6 \u03c6(\u00b7| x); second, z0 is evolved via a diffusion such as (2) for time T (via the transformation ZT = T (z0, T)). Consequently, the negative ELBO for CTF can be written as F(x) = Eq\u03c6 \u03c6 \u03c6(z0 | x)E\u03c1T [log \u03c1T \u2212log p\u03b8(x, ZT ) + log \f \f \f \fdet\u2202ZT \u2202z0 \f \f \f \f \u0015 \u225cEq\u03c6 \u03c6 \u03c6(z0 | x) [F1(x, z0)] . (4) Note the term F1(x, z0) is intractable to calculate, in that i) \u03c1T does not have an explicit form; ii) the Jacobian \u2202ZT \u2202z0 is generally infeasible. In the following, we propose an approximate solution for problem i). Learning by avoiding problem ii) is presented in Section 3.2 via amortization. For problem i), a reformulation of the results from Jordan et al. (1998) leads to a nice way to approximate \u03c1t in Lemma 1. Note in practice we adopt an implicit method which uses samples to approximate the solution in Lemma 1 for feasibility, detailed in (6). Lemma 1 Assume that log p\u03b8(x, z) \u2264 C1 is in\ufb01nitely differentiable, and \u2225\u2207z log p\u03b8(x, z)\u2225 \u2264 C2 (1 + C1 \u2212log p\u03b8(x, z)) (\u2200x, z) for some constants {C1, C2}. Let T = hK (h is the stepsize and K is the number of transformations), \u03c10 \u225cq\u03c6 \u03c6 \u03c6(z0 | x), and {\u02dc \u03c1k}K k=1 be the solution of the functional optimization problem: \u02dc \u03c1k = arg min \u03c1\u2208K KL (\u03c1\u2225p\u03b8(x, z)) + 1 2hW 2 2 (\u02dc \u03c1k\u22121, \u03c1) , (5) with W 2 2 (\u00b51, \u00b52) \u225cinfp\u2208P(\u00b51,\u00b52) R \u2225x \u2212y\u22252 2 p(d x, d y) the square of 2nd-order Wasserstein distance, and P(\u00b51, \u00b52) the set of joint distributions on {\u00b51, \u00b52}. K is the space of distributions with \ufb01nite 2nd-order moment. Then \u02dc \u03c1K converges to \u03c1T in the limit of h \u21920, i.e., limh\u21920 \u02dc \u03c1K = \u03c1T , where \u03c1T is the solution of the FP equation (3) at time T. Lemma 1 reveals an interesting way to compute \u03c1T via a sequence of functional optimization problems. By comparing it with the objective of the traditional NF, which minimizes the KL-divergence between \u03c1K and p\u03b8(x, z), at each sub-optimization-problem in Lemma 1, it minimizes the KL-divergence between \u02dc \u03c1k and p\u03b8(x, z), plus a regularization term as the Wasserstein distance between \u02dc \u03c1k\u22121 contrast, we will de\ufb01ne an explicit energy-based distribution for the density in density-estimation tasks. \fContinuous-Time Flows and \u02dc \u03c1k. The extra Wasserstein-distance term arises naturally due to the fact that the Langevin diffusion can be explained as a gradient \ufb02ow whose geometry is equipped with the Wasserstein distance (Otto, 1998). The optimization problem in Lemma 1 is, however, dif\ufb01cult to deal with directly. In practice, we instead approximate the discretization in an equivalent way by simulation from the CTF. Starting from z0, each zk (k = 0, \u00b7 \u00b7 \u00b7 , K \u22121) is fed into a transformation Tk (speci\ufb01ed below), resulting in zk+1 whose distribution coincides with \u02dc \u03c1k+1 in Lemma 1. The discretization procedure is illustrated in Figure 1. We must specify the transformations Tk. For each k, let t = hk; we can conclude from Lemma 1 that limh\u21920 \u02dc \u03c1k = \u03c1t. From FP theory, \u03c1t is obtained by solving the diffusion (2) with initial condition Z0 = z0. It is thus reasonable to specify the transformation Tk as the k-th step of a numerical integrator for (2). Speci\ufb01cally, we specify Tk stochastically: zk = Tk(zk\u22121) \u225czk\u22121 +F(zk\u22121)h + V (zk\u22121)\u03b6k , (6) where \u03b6k \u223cN(0, hIL) is drawn from an isotropic normal. Note the transformation de\ufb01ned here is stochastic, thus we only get samples from \u02dc \u03c1K at the end. A natural way to approximate \u02dc \u03c1K is to use the empirical sample distribution, i.e., \u02dc \u03c1K \u2248 1 K PK k=1 \u03b4zk \u225c\u00af \u03c1T with \u03b4z a point mass at z. Afterwards, \u02dc \u03c1K (thus \u00af \u03c1T ) will be used to approximate the true \u03c1T from (3). 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 continuous-timeflows \u22ef \"#(%&|() \u22ef *+((|%) Inference network Generative model \u22ef \"#(%&|() \u22ef *+((|%) Inference network Generative model \u22ef \u22ef \u22ef \u22ef\u22ef Z0 ZT z0 z1 z2 zK\u22121 zK T1 T2 TK\u22121 Figure 1. Discretized approximation (right) of a continuous-time \ufb02ow (left). Densities {\u02dc \u03c1k} of {zk} evolve via transformations {Tk}, with \u02dc \u03c1k \u2192\u03c1hk when h \u21920 for each k due to Lemma 1. Better ways to approximate \u03c1T might be possible, e.g., by assigning more weights to the more recent samples. However, the problem becomes more challenging in theoretical analysis, an interesting point left for future work. In the following, we study how well \u00af \u03c1T approximates \u03c1T . Following literature on numerical approximation for It\u00f4 diffusions (Vollmer et al., 2016; Chen et al., 2015), we consider a 1-Lipschitz test function \u03c8 : RL \u2192R, and use the mean square error (MSE) bound to measure the closeness of \u00af \u03c1T and \u03c1T , de\ufb01ned as: MSE(\u00af \u03c1T , \u03c1T ; \u03c8) \u225cE \u0000R \u03c8(z)(\u02dc \u03c1T \u2212\u03c1T )(z)d z \u00012, where the expectation is taken over all the randomness in the construction of \u02dc \u03c1T . Note that our goal is related but different from the standard setup as in (Vollmer et al., 2016; Chen et al., 2015), which studies the closeness of \u00af \u03c1T to p\u03b8(x, z). We need to adopt the assumptions from Vollmer et al. (2016); Chen et al. (2015), which are described in the SM. The assumptions are somewhat involved but essentially require coef\ufb01cients of the diffusion (2) to be well-behaved. We derive the following bound for the MSE of the sampled approximation, \u00af \u03c1T , and the true distribution. Theorem 2 Under Assumption 1 in the SM, assume that R \u03c1T (z)p\u22121 \u03b8 (x, z)d z < \u221eand there exists a constant C such that dW 2 2 (\u03c1T ,p\u03b8(x,z)) dt \u2265CW 2 2 (\u03c1T , p\u03b8(x, z)), the MSE is bounded as MSE(\u00af \u03c1T , \u03c1T ; \u03c8) = O \u0012 1 hK + h2 + e\u22122ChK \u0013 . The last assumption in Theorem 2 requires \u03c1T to evolve fast through the FP equation, which is a standard assumption used to establish convergence to equilibrium for FP equations (Bolley et al., 2012). The MSE bound consists of three terms, the \ufb01rst two terms come from numerical approximation of the continuous-time diffusion, whereas the third term comes from the convergence bound of the FP equation in terms of the Wasserstein distance (Bolley et al., 2012). When the time T = hK is large enough, the third term may be ignored due to its exponential-decay rate. Moreover, in the in\ufb01nite-time limit, the bound endows a bias proportional to h; this, however, can be removed by adopting a decreasing-step-size scheme in the numerical method, as in standard stochastic gradient MCMC methods (Teh et al., 2016; Chen et al., 2015). Remark 3 To examine the optimal bound in Theorem 2, we drop out the term e\u22122ChK in the long-time case (when hK is large enough) for simplicity because it is in a much lower order term than the other terms. The optimal MSE bound (over h) decreases at a rate of O \u0000K\u22122/3\u0001 , meaning that O \u0000\u03f5\u22123/2\u0001 steps of transformations in Figure 1 (right) are needed to reach an \u03f5-accurate approximation, i.e., MSE \u2264\u03f5. This is computationally expensive. An ef\ufb01cient way for inference is thus imperative, developed in the next section. 3.2. Ef\ufb01cient inference via amortization Even though we approximate \u03c1T with \u00af \u03c1T , it is still infeasible to directly apply it to the ELBO in (4) as \u00af \u03c1T is discrete. To deal with this problem, we adopt the idea of \u201camortized learning\u201d (Gershman & Goodman, 2014) for inference, by alternatively optimizing the two sets of parameters \u03c6 \u03c6 \u03c6 and \u03b8. Updating \u03c6 \u03c6 \u03c6 To explain the idea, \ufb01rst note that the negative ELBO can be equivalently written as F(x) = E\u03c10\u225cq\u03c6 \u03c6 \u03c6(z0 | x)E\u03c1T [log \u03c10 \u2212log p\u03b8(x, ZT )] . (7) When \u03c10 = \u03c1T , it is easy to see that: F(x) = E\u03c10 [log \u03c10 \u2212log p\u03b8(ZT | x)]+log p(x) = log p(x), which essentially makes the gap between q\u03c6 \u03c6 \u03c6(z0 | x) and p\u03b8(ZT | x) vanished. As a result, our goal is to learn \u03c6 \u03c6 \u03c6 such that q\u03c6 \u03c6 \u03c6(z0 | x) approaches p\u03b8(ZT | x). This is a distribution matching problem (Li et al., 2017a). As mentioned previously, we will learn an implicit distribution of q\u03c6 \u03c6 \u03c6(z0 | x) \fContinuous-Time Flows !\"($%|') \u22ef *+('|$) $% \u22ef step 1 step K !\"($%|') \u22ef *+('|$) $, $% \u22ef $$. \u22ef \u22ef \u22ee !\"($%|') \u22ef *+('|$) $, $% \u22ef $. \u22ef \u22ee Figure 2. Amortized learning of CTFs. From left to right: the initial architecture with K-step transformations; For each step k, q\u03c6 \u03c6 \u03c6(\u00b7) is trained to match the distributin of zk in CTFs; In the end, the CTF is distilled into q\u03c6 \u03c6 \u03c6(\u00b7) with distribution matching as in (8). (i.e., learn how to draw samples from q\u03c6 \u03c6 \u03c6(z0 | x) instead of its explicit form), as it allows us to chose a candidate distribution from a much larger distribution space, compared to explicitly de\ufb01ning q\u03c6 \u03c6 \u03c6\u00b6. Consequently, q\u03c6 \u03c6 \u03c6(z0 | x) is implemented by a stochastic generator (a DNN parameterized by \u03c6 \u03c6 \u03c6) Q\u03c6 \u03c6 \u03c6(x, \u03c9) with input as the concatenation of x and \u03c9, where \u03c9 is a sample from an isotropic Gaussian distribution q0(\u03c9). Our goal is now translated to update the parameter \u03c6 \u03c6 \u03c6 of Q\u03c6 \u03c6 \u03c6(x, \u03c9) to \u03c6 \u03c6 \u03c6\u2032 such that the distribution of {z\u2032 0 = Q\u03c6 \u03c6 \u03c6\u2032(x, \u03c9)} with \u03c9 \u223cq0(\u03c9) matches that of z1 in the original generating process with \u03c6 \u03c6 \u03c6 in Figure 1. In this way, the generating process of z1 via T1 is distilled into the parameterized generator Q\u03c6 \u03c6 \u03c6(\u00b7), eliminating the need to do a speci\ufb01c transformation via T1 in testing, and thus is very ef\ufb01cient. Speci\ufb01cally, we update \u03c6 \u03c6 \u03c6\u2032 such that \u03c6 \u03c6 \u03c6\u2032 = arg min \u03c6 \u03c6 \u03c6 D({z\u2032(i) 0 }, {z(i) 1 }) , (8) where {z\u2032(i) 0 }S i=1 are a set of samples generated from q\u03c6 \u03c6 \u03c6\u2032(z\u2032 0 | x) via Q\u03c6(\u00b7), and {z(i) 1 }S i=1 are samples drawn by \u03c9i \u223cq0(\u03c9), \u02dc zi 0 = Q\u03c6 \u03c6 \u03c6(x, \u03c9i), z(i) 1 \u223cT1(\u02dc zi 0); D(\u00b7, \u00b7) is a metric between samples speci\ufb01ed below. We call this procedure distilling knowledge from T1 to Q\u03c6 \u03c6 \u03c6(\u00b7). In practice, one can choose to distill knowledge for several steps (e.g., Tk) instead of one step (e.g., T1) to Q\u03c6 \u03c6 \u03c6(\u00b7) each time. Note the distillation idea is related to Bayesian dark knowledge (Korattikara et al., 2015), but with different goal and approach. After distilling knowledge from T1, we apply the same procedure for other transformations Tk(k > 1) sequentially. The \ufb01nal inference network, represented by q\u03c6 \u03c6 \u03c6(\u00b7| x), can then well approximate the CTF, e.g., the distribution of z0 \u223cq\u03c6 \u03c6 \u03c6(\u00b7| x) is close to \u03c1T from the CTF. This concept is illustrated in Figure 2. We note choosing an appropriate D in (8) is important in order to make Theorem 2 applicable. Amortized SVGD (Wang & Liu, 2017) de\ufb01nes D as standard Euclidean distance between samples. We show in Proposition 4 that this would induce a large error in terms of approximation accuracy. Proposition 4 Fix \u03b8. If D in (8) is de\ufb01ned as the summation of pairwise Euclidean distance between samples, then samples generated from Q\u03c6 \u03c6 \u03c6 converge to local modes of log p\u03b8(z | x). \u00b6This is distinct from our density-estimation framework described in the next section, where an explicit form is assumed at the beginning for practical needs. Consequently, it is crucial to impose more general distance for D. As GAN has been interpreted as distribution matching (Li et al., 2017a), we de\ufb01ne D using the Wasserstein distance, implemented as a discriminator parameterized by a neural network. Speci\ufb01cally, we adopt the ALICE framework (Li et al., 2017a), and use {(x, z\u2032(i) 0 )} as fake data and {(z(i) 1 , xi \u223cp\u03b8(\u00b7| z(i) 1 ))} as real data to train a discriminator. More details are discussed in Section C of the SM. Updating \u03b8 Given \u03c6 \u03c6 \u03c6, \u03b8 can be updated by simply optimizing the ELBO in (7), where \u03c1T is approximated by \u00af \u03c1T from the discretized CTF. Speci\ufb01cally, the expectation w.r.t. \u03c1T in (7) is approximated by a sample average from: z0 \u223cq\u03c6 \u03c6 \u03c6(z0 | x), z1 \u223cT1(z0), z2 \u223cT2(z1), \u00b7 \u00b7 \u00b7 , zK \u223cTK(zK\u22121) To sum up, there are three steps to learn a CTF-based VAE: 1. Generate (z0, \u00b7 \u00b7 \u00b7 , zK) according to q\u03c6 \u03c6 \u03c6(z0 | x) and the discretized \ufb02ow with transformations {Tk}; 2. Update \u03c6 \u03c6 \u03c6 according to (8); 3. Optimize \u03b8 by minimizing the ELBO (7) with the generated sample path. In testing, we use only the \ufb01nally learned q\u03c6 \u03c6 \u03c6(z0 | x) for inference (into which the CTF has been distilled), and hence testing is like the standard VAE. Since the discretized-CTF model is essentially a Markov chain, we call our model Markov-chain-based VAE (MacVAE). 4. CTFs for Energy-based Density Estimation We assume that the density of the observation x is characterized by a parametric Gibbsian-style probability model p\u03b8(x) = 1 Z(\u03b8) \u02dc p\u03b8(x) \u225c 1 Z(\u03b8)eU(x;\u03b8), where \u02dc p\u03b8(x) is an unnormalized version of p\u03b8(x) with parameter \u03b8, U(x; \u03b8) \u225c log \u02dc p\u03b8(x) is called the energy function (Zhao et al., 2017), and Z(\u03b8) \u225c R \u02dc p\u03b8(x)d x is the normalizer. Note this form of distributions constitutes a very large class of distributions as long as the capacity of the energy function is large enough. This can be easily achieved by adopting a DNN to implement U(x; \u03b8), the setting we considered in this paper. Note our model can be placed in between existing implicit and explicit density estimation methods, because we model the data density with an explicit distribution form up to an intractable normalizer. Such distributions have been proved to be useful in real applications, e.g., (Haarnoja et al., 2017) used them to model policies in deep reinforcement learning. \fContinuous-Time Flows 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 continuous-timeflows \u22ef \"#(%&;() \u22ef *+(%) %, %& Generator Energy-based model Figure 3. Learning a generator with CTF. The goal is to match the samples x0 from q\u03c6 \u03c6 \u03c6 to those after a CTF (xT ). Our goal is to learn \u03b8 given {xi}N i=1, which can be achieved via standard maximum likelihood estimation (MLE): \u03b8 = arg max\u03b8 PN i=1 log p\u03b8(xi) \u225carg max\u03b8 M({xi}; \u03b8). The optimization can be achieved by standard stochastic gradient descent (SGD), with the following gradient formula: \u2202M \u2202\u03b8 = 1 N N X i=1 \u2202U(xi; \u03b8) \u2202\u03b8 \u2212Ep\u03b8(x) \u0014\u2202U(x; \u03b8) \u2202\u03b8 \u0015 (9) The above formula requires an integration over the model distribution p\u03b8(x), which can be approximated by Monte Carlo integration with samples. Here we adopt the idea of CTFs and propose to use a DNN guided by a CTF, which we call a generator, to generate approximate samples from the original model p\u03b8(x). Speci\ufb01cally, we require that samples from the generator should well approximate the target p\u03b8(x). This can be done by adopting the CTF idea above, i.e., distilling knowledge of a CTF (which approaches p\u03b8(x)) to the generator. In testing, instead of generating samples from p\u03b8(x) via MCMC (which is complicated and time consuming), we generate samples from the generator directly. Furthermore, when evaluating the likelihood for test data, the constant Z(\u03b8) can also be approximated by Monte Carlo integration with samples drawn from the generator. Note the \ufb01rst term on the RHS of (9) is a model \ufb01t to observed data, and the second term is a model \ufb01t to synthetic data drawn from p\u03b8(x); this is similar to the discriminator in GANs (Arjovsky et al., 2017), but derived directly from the MLE. More connections are discussed below. 4.1. Learning via Amortization Our goal is to learn a generator whose generated samples match those from the original model p\u03b8(x).Similar to inference setting, the generator is learned implicitly. However, we also learn an explicit density model for the data by SGD, with samples from the implicit generator to estimate gradients in (9). Note that in this case, the CTF is performed directly on the data space, instead of on latent-variable space as in previous sections. Speci\ufb01cally, the sampling procedure from the generator plus a CTF are written as: x0 \u223cq\u03c6 \u03c6 \u03c6(x0), xT \u223cT (x0, T) . Here T (\u00b7, \u00b7) is the continuous-time \ufb02ow; a sample x0 from q\u03c6 \u03c6 \u03c6(\u00b7) is implemented by a deep neural network (generator) G\u03c6 \u03c6 \u03c6(\u03c9) with input \u03c9 \u223cq0(\u03c9), where q0 is a simple distribution for a noise random variable, e.g., the isotropic normal distribution. The procedure is illustrated in Figure 3. Speci\ufb01cally, denote the parameters in the k-th step with subscript \u201c(k)\u201d. For ef\ufb01cient sample generation, in the k-th step, we again adopt the amortization idea from Section 3.2 to update \u03c6 \u03c6 \u03c6(k\u22121) of the generator network G\u03c6 \u03c6 \u03c6(\u00b7), such that samples from the updated generator match those from the current generator followed by a one-step transformation T1(\u00b7). After that, \u03b8 is updated by drawing samples from q\u03c6 \u03c6 \u03c6(\u00b7) to estimate the expectation in (9). The algorithm is presented in Algorithm 1 in Section E of the SM. 4.2. Connections to WGAN and MLE There is an interesting relation between our model and the WGAN framework (Arjovsky et al., 2017). To see this, let pr be the data distribution. Substituting p\u03b8(x) with q\u03c6 \u03c6 \u03c6(x) for the expectation in the gradient formula (9) and integrating out \u03b8, we have that our objective is max Ex\u223cpr [U(x; \u03b8)] \u2212Ex\u223cq\u03c6 \u03c6 \u03c6 [U(x; \u03b8)] (10) This is an instance of the integral probability metrics (Arjovsky & Bottou, 2017). When U is chosen to be 1-Lipschitz functions, it recovers WGAN. This connection motivates us to introduce weight clipping (Arjovsky et al., 2017) or alternative regularizers (Gulrajani et al., 2017) when updating \u03b8 for a better theoretical property. For this reason, we call our model Markov-chain-based GAN (MacGAN). Furthermore, it can be shown by Jensen\u2019s inequality that the MLE is bounded by (detailed derivations are provided in Section F of the SM) max 1 N N X i=1 log p\u03b8(xi) (11) \u2264max Ex\u223cpr [U(x; \u03b8)] \u2212Ex\u223cq\u03c6 \u03c6 \u03c6 [U(x; \u03b8)] \u2212Ex\u223cq\u03c6 \u03c6 \u03c6 [log q\u03c6 \u03c6 \u03c6] . By inspecting (10) and (11), it is clear that: i) when learning the energy-based model parameters \u03b8, the objective can be interpreted as maximizing an upper bound of the MLE shown in (11); ii) when optimizing the parameter \u03c6 \u03c6 \u03c6 of the inference network, we adopt the amortized learning procedure presented in Algorithm 1, whose objective is min\u03c6 \u03c6 \u03c6 KL (q\u03c6 \u03c6 \u03c6\u2225p\u03b8), coinciding with the last two terms in (11). In other words, both \u03b8 and \u03c6 \u03c6 \u03c6 are optimized by maximizing the same upper bound of the MLE, guaranteeing convergence of the algorithm, although previous work has pointed out maximizing an upper bound is not a well-posed problem in general (Salakhutdinov & Hinton, 2012). Proposition 5 The optimal solution of MacGAN is the maximum likelihood estimator. Note another difference between MacGAN and standard GAN framework is the way of learning the generator q\u03c6 \u03c6 \u03c6. \fContinuous-Time Flows We adopt the amortization idea, which directly guides q\u03c6 \u03c6 \u03c6 to approach p\u03b8; whereas in GAN, the generator is optimized via a min-max procedure to make it approach the empirical data distribution pr. By explicitly learning p\u03b8, MacGAN is able to evaluate likelihood for test data up to a constant. 5. Related Work Our framework extends the idea of normalizing \ufb02ows (Rezende & Mohamed, 2015) and gradient \ufb02ows (Altieri & Duvenaud, 2015) to continuous-time \ufb02ows, by developing theoretical properties on the convergence behavior. Inference based on CTFs has been studied in (Sohl-Dickstein et al., 2015) based on maximum likelihood and (Salimans et al., 2015) based on the auxiliary-variable technique. However, they directly uses discrete approximations for the \ufb02ow, and the approximation accuracy is unclear. Moreover, the inference network requires simulating a long Markov chain for the auxiliary model, thus is less ef\ufb01cient than ours. Finally, the inference network is implemented as a parametric distribution (e.g., the Gaussian distribution), limiting the representation power, a common setting in existing auxiliaryvariable based models (Tran et al., 2016). The idea of amortization (Gershman & Goodman, 2014) has recently been explored in various research topics for Bayesian inference such as in variational inference (Kingma & Welling, 2014; Rezende et al., 2014) and Markov chain Monte Carlo (Wang & Liu, 2017; Li et al., 2017b; Pu et al., 2017). Both (Wang & Liu, 2017) and (Pu et al., 2017) extend the idea of Stein variational gradient descent (Liu & Wang, 2016) with amortized inference for a GAN-based and a VAE-based model, respectively, which resemble our proposed MacVAE and MacGAN in concept. Li et al. (2017b) applies amortization to distill knowledge from MCMC to learn a student network. The ideas in (Li et al., 2017b) are similar to ours, but the motivation and underlying theory are different from that developed here. The authors proposed several divergence measures for distribution matching including the JensenShannon divergence, similar to our method. 6. Experiments We conduct experiments to test our CTF-based framework for ef\ufb01cient inference and density estimation problems, and compared them with related methods. Some experiments are based on the excellent code for SteinGAN\u2225(Wang & Liu, 2017), where their default parameter setting are adopted. The discretization stepsize h is robust as long as it is set in a reasonable range, e.g., we set it the same as the stepsize in SGD. More experimental results are given in the SM, including a sensitiveness experiment on model parameters in Section G.4. \u2225https://github.com/DartML/SteinGAN 6.1. CTFs for inference Synthetic experiment We examine our amortized learning framework with three toy experiments. Particularly, we want to verify the necessity of distribution matching de\ufb01ned in (8), i.e., we test D implemented as a discriminator for Wasserstein distance (adversarial-CTF) against that implemented with standard Euclidean distance (\u21132-CTF), which can be considered as an instance of the amortized MCMC (Li et al., 2017b) with a Langevin-dynamic transition function and a Euclidean-distance-based divergence measure for samples. Two 2D distributions similar to (Rezende & Mohamed, 2015) are considered, de\ufb01ned in Section D of the SM. The inference network q\u03c6 \u03c6 \u03c6 is de\ufb01ned to be a 2-layer MLP with isotropic normal random variables as input. Figure 4 plots the densities estimated with the samples from transformations {TK=100} (before optimizing \u03c6 \u03c6 \u03c6), as well as with samples generated directly from q\u03c6 \u03c6 \u03c6 (after optimizing \u03c6 \u03c6 \u03c6). It is clear that the amortized learning with Wasserstein distance is able to distill knowledge from the CTF to the inference network, while the algorithm fails when Euclidean distance is adopted. Next, we test MacVAE on a VAE setting on a simple synthetic dataset containing 4 data points, each is a 4D one-hot vector, with the non-zero elements at different positions. The prior of latent code is a 2D standard Normal. Figure 5 plots the distribution of the learned latent code for VAE, adversarial-CTF and \u21132-CTF. Each color means the codes for one particular observation. It is observed that VAE divides the space into a mixture of 4 Gaussians (consistent with VAE theory), the adversarial-CTF learns complex posteriors, while the \u21132-CTF converges to the mode of each posterior (consistent with Proposition 4). MacVAE on MNIST Following (Rezende & Mohamed, 2015; Tomczak & Welling, 2016), we de\ufb01ne the inference network as a deep neural network with two fully connected layers of size 300 with softplus activation functions. We compare MacVAE with the standard VAE and the VAE with normalizing \ufb02ow, where testing ELBOs are reported (Section G.1 of the SM describes how to calculate the ELBO). We do not compare with other state-of-the-art methods such as the inverse autoregressive \ufb02ow (Kingma et al., 2016), because they typically endowed more complicated inference networks (with more parameters), unfair for comparison. We use the same inference network architecture for all the models. Figure 7 (left) plots the testing ELBO versus training epochs. MacVAE outperforms VAE and normalizing \ufb02ows with a better ELBO (around -85.62). 6.2. CTFs for density estimation We test MacGAN on three datasets: MNIST, CIFAR-10 and CelabA. Following GAN-related methods, the model is evaluated by observing its ability to draw samples from the learned data distribution. Inspiring by (Wang & Liu, 2017), \fContinuous-Time Flows Figure 4. Illustration of CTF on toy distributions. Each row is a distribution case. 1st column: true distributions; 2nd column: MCMC results; 3rd column: approximations via adversarial-CTF; 4th column: approximations via \u21132-CTF. (a) Standard VAE (b) Adversarial CTF (c) L2 CTF Figure 5. Comparison of the learned latent space with standard VAE (left), adversarial-CTF (middle) and \u21132-CTF (right). we de\ufb01ne a parametric form of the energy-based model as p\u03b8(x) \u221d exp{\u2212\u2225x \u2212DEC\u03b8 (ENC\u03b8(x))\u22252}, where ENC\u03b8(\u00b7) and DEC\u03b8(\u00b7) are encoder and decoder de\ufb01ned by using deep convolutional neural networks and deconvolutional neural networks, respectively, parameterized by \u03b8. For simplicity, we adopt the popular DCGAN architecture (Radford et al., 2016) for the encoder and decoder. The generator G\u03c6 \u03c6 \u03c6 is de\ufb01ned as a 3-layer CNN with the ReLU activation function (except for the top layer which uses tanh as the activation function, see SM G for details). Following (Wang & Liu, 2017), the stepsizes are set to (me\u2212e)\u00d7lr me\u221250 , where e indexes the epoch, me is the total number of epochs, lr = 1e-4 when updating \u03b8, and lr = 1e-3 when updating \u03c6 \u03c6 \u03c6. The stepsize in L1 is set to 1e-3. We compare MacGAN with DCGAN (Radford et al., 2016), the improved WGAN (WGAN-I) (Gulrajani et al., 2017) and SteinGAN (Wang & Liu, 2017). We plot images generated with MacGAN and its most related method SteinGAN in Figure 6 for CelebA and CIFAR-10 datasets. More results are provided in SM Section G. We observe that visually MacGAN is able to generate clear-looking images. Following (Wang & Liu, 2017), we also plot the images generated by a random walk in the \u03c9 space in Figure 6. Qualitatively evaluating a GAN-like model is challenging. We follow literature and use the inception score (Salimans et al., 2016) to measure the quantity of the generated images. Figure 7 (right) plots inception scores vs epochs for different models. MacGAN obtains competitive inception scores with the popular DCGAN model. Quantitatively, we get a \ufb01nal inception score of 6.49 for MacGAN, compared to 6.35 for airplane automobile bird cat deer dog frog horse ship truck ! Figure 6. Generated images for CIFAR-10 (top) and CelebA (middle) datasets with MacGAN (left) and SteinGAN (right). The bottom are images generated by a random walk on the \u03c9 space for the generator of MacGAN, i.e., \u03c9t = \u03c9t\u22121 +0.03\u00d7rand([\u22121, 1]). 0 20 40 60 80 100 Epoch -96 -94 -92 -90 -88 -86 ELBO MacVAE VAE VAE+NF(T=1) VAE+NF(T=10) 0 20 40 60 80 100 Epoch 4 4.5 5 5.5 6 6.5 Inception Score MacGAN WGAN-I SteinGAN DCGAN Figure 7. ELBO on MNIST vs epochs (left), and Inception score versus epochs (right) for different models. VAE with 80-layer NF is not included because it has much more parameters. SteinGAN, 6.25 for WGAN-I and 6.58 for DCGAN. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1610.06665v1",
+                    "title": "On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators",
+                    "abstract": "Recent advances in Bayesian learning with large-scale data have witnessed\nemergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic\ngradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC\n(SGHMC), and the stochastic gradient thermostat. While finite-time convergence\nproperties of the SGLD with a 1st-order Euler integrator have recently been\nstudied, corresponding theory for general SG-MCMCs has not been explored. In\nthis paper we consider general SG-MCMCs with high-order integrators, and\ndevelop theory to analyze finite-time convergence properties and their\nasymptotic invariant measures. Our theoretical results show faster convergence\nrates and more accurate invariant measures for SG-MCMCs with higher-order\nintegrators. For example, with the proposed efficient 2nd-order symmetric\nsplitting integrator, the {\\em mean square error} (MSE) of the posterior\naverage for the SGHMC achieves an optimal convergence rate of $L^{-4/5}$ at $L$\niterations, compared to $L^{-2/3}$ for the SGHMC and SGLD with 1st-order Euler\nintegrators. Furthermore, convergence results of decreasing-step-size SG-MCMCs\nare also developed, with the same convergence rates as their fixed-step-size\ncounterparts for a specific decreasing sequence. Experiments on both synthetic\nand real datasets verify our theory, and show advantages of the proposed method\nin two large-scale real applications.",
+                    "authors": "Changyou Chen, Nan Ding, Lawrence Carin",
+                    "published": "2016-10-21",
+                    "updated": "2016-10-21",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction In large-scale Bayesian learning, diffusion based sampling methods have become increasingly popular. Most of these methods are based on It\u02c6 o diffusions, de\ufb01ned as: d Xt = F(Xt)dt + \u03c3(Xt)dWt . (1) Here Xt \u2208Rn represents model states, t the time index, Wt is Brownian motion, functions F : Rn \u2192Rn and \u03c3 : Rn \u2192Rn\u00d7m (m not necessarily equal to n) are assumed to satisfy the usual Lipschitz continuity condition. In a Bayesian setting, the goal is to design appropriate functions F and \u03c3, so that the stationary distribution, \u03c1(X), of the It\u02c6 o diffusion has a marginal distribution that is equal to the posterior distribution of interest. For example, 1st-order Langevin dynamics (LD) correspond to X = \u03b8, F = \u2212\u2207\u03b8U and \u03c3 = \u221a 2 In, with In being the n\u00d7n identity matrix; 2nd-order Langevin dynamics correspond to X = (\u03b8, p), F = \u0010 p \u2212D p \u2212\u2207\u03b8U \u0011 , and \u03c3 = \u221a 2D \u0010 0 0 0 In \u0011 for some D > 0. Here U is the unnormalized negative log-posterior, and p is known as the momentum [1, 2]. Based on the Fokker-Planck equation [3], the stationary distributions of these dynamics exist and their marginals over \u03b8 are equal to \u03c1(\u03b8) \u221dexp(\u2212U(\u03b8)), the posterior distribution we are interested in. Since It\u02c6 o diffusions are continuous-time Markov processes, exact sampling is in general infeasible. As a result, the following two approximations have been introduced in the machine learning liter1 arXiv:1610.06665v1  [stat.ML]  21 Oct 2016 \fature [1, 2, 4], to make the sampling numerically feasible and practically scalable: 1) Instead of analytically integrating in\ufb01nitesimal increments dt, numerical integration over small step h is used to approximate the integration of the true dynamics. Although many numerical schemes have been studied in the SDE literature, in machine learning only the 1st-order Euler scheme is widely applied. 2) During every integration, instead of working with the gradient of the full negative log-posterior U(\u03b8), a stochastic-gradient version of it, \u02dc Ul(\u03b8), is calculated from the l-th minibatch of data, important when considering problems with massive data. In this paper, we call algorithms based on 1) and 2) SG-MCMC algorithms. To be complete, some recently proposed SG-MCMC algorithms are brie\ufb02y reviewed in Appendix A. SG-MCMC algorithms often work well in practice, however some theoretical concerns about the convergence properties have been raised [5\u20137]. Recently, [5, 6, 8] showed that the SGLD [4] converges weakly to the true posterior. In [7], the author studied the sample-path inconsistency of the Hamiltonian PDE with stochastic gradients (but not the SGHMC), and pointed out its incompatibility with data subsampling. However, real applications only require convergence in the weak sense, i.e., instead of requiring sample-wise convergence as in [7], only laws of sample paths are of concern\u2217. Very recently, the invariance measure of an SGMCMC with a speci\ufb01c stochastic gradient noise was studied in [9]. However, the technique is not readily applicable to our general setting. In this paper we focus on general SG-MCMCs, and study the role of their numerical integrators. Our main contributions include: i) From a theoretical viewpoint, we prove weak convergence results for general SG-MCMCs, which are of practical interest. Speci\ufb01cally, for a Kth-order numerical integrator, the bias of the expected sample average of an SG-MCMC at iteration L is upper bounded by L\u2212K/(K+1) with optimal step size h \u221dL\u22121/(K+1), and the MSE by L\u22122K/(2K+1) with optimal h \u221dL\u22121/(2K+1). This generalizes the results of the SGLD with an Euler integrator (K = 1) in [5, 6, 8], and is better when K \u22652; ii) From a practical perspective, we introduce a numerically ef\ufb01cient 2nd-order integrator, based on symmetric splitting schemes [9]. When applied to the SGHMC, it outperforms existing algorithms, including the SGLD and SGHMC with Euler integrators, considering both synthetic and large real datasets. 2 Preliminaries & Two Approximation Errors in SG-MCMCs In weak convergence analysis, instead of working directly with sample-paths in (1), we study how the expected value of any suitably smooth statistic of Xt evolves in time. This motivates the introduction of an (in\ufb01nitesimal) generator. Formally, the generator L of the diffusion (1) is de\ufb01ned for any compactly supported twice differentiable function f : Rn \u2192R, such that, Lf(Xt) \u225clim h\u21920+ E [f(Xt+h)] \u2212f(Xt) h = \u0012 F(Xt) \u00b7 \u2207+ 1 2 \u0000\u03c3(Xt)\u03c3(Xt)T \u0001 :\u2207\u2207T \u0013 f(Xt) , where a \u00b7 b \u225caT b, A : B \u225ctr(AT B), h \u21920+ means h approaches zero along the positive real axis. L is associated with an integrated form via Kolmogorov\u2019s backward equation\u2020 : E [f(Xe T )] = eT Lf(X0), where Xe T denotes the exact solution of the diffusion (1) at time T. The operator eT L is called the Kolmogorov operator for the diffusion (1). Since diffusion (1) is continuous, it is generally infeasible to solve analytically (so is eT L). In practice, a local numerical integrator is used for every small step h, with the corresponding Kolmogorov operator Ph approximating ehL. Let Xn lh denote the approximate sample path from such a numerical integrator; similarly, we have E[f(Xn lh)] = Phf(Xn (l\u22121)h). Let A \u25e6B denote the composition of two operators A and B, i.e., A is evaluated on the output of B. For time T = Lh, we have the following approximation E [f(Xe T )] A1 = ehL \u25e6. . . \u25e6ehLf(X0) A2 \u2243Ph \u25e6. . . \u25e6Phf(X0) = E[f(Xn T )], with L compositions, where A1 is obtained by decomposing TL into L sub-operators, each for a minibatch of data, while approximation A2 is manifested by approximating the infeasible ehL with Ph from a feasible integrator, e.g., the symmetric splitting integrator proposed later, such that \u2217For completeness, we provide mean sample-path properties of the SGHMC (similar to [7]) in Appendix J. \u2020More details of the equation are provided in Appendix B. Speci\ufb01cally, under mild conditions on F, we can expand the operator ehL up to the mth-order (m \u22651) such that the remainder terms are bounded by O(hm+1). Refer to [10] for more details. We will assume these conditions to hold for the F\u2019s in this paper. 2 \fE [f(Xn T )] is close to the exact expectation E [f(Xe T )]. The latter is the \ufb01rst approximation error introduced in SG-MCMCs. Formally, to characterize the degree of approximation accuracies for different numerical methods, we use the following de\ufb01nition. De\ufb01nition 1. An integrator is said to be a Kth-order local integrator if for any smooth and bounded function f, the corresponding Kolmogorov operator Ph satis\ufb01es the following relation: Phf(x) = ehLf(x) + O(hK+1) . (2) The second approximation error is manifested when handling large data. Speci\ufb01cally, the SGLD and SGHMC use stochastic gradients in the 1st and 2nd-order LDs, respectively, by replacing in F and L the full negative log-posterior U with a scaled log-posterior, \u02dc Ul, from the l-th minibatch. We denote the corresponding generators with stochastic gradients as \u02dc Ll, e.g., the generator in the l-th minibatch for the SGHMC becomes \u02dc Ll = L +\u2206Vl, where \u2206Vl = (\u2207\u03b8 \u02dc Ul\u2212\u2207\u03b8U)\u00b7\u2207p. As a result, in SG-MCMC algorithms, we use the noisy operator \u02dc P l h to approximate eh\u02dc Ll such that E[f(Xn,s lh )] = \u02dc P l hf(X(l\u22121)h), where Xn,s lh denotes the numerical sample-path with stochastic gradient noise, i.e., E [f(Xe T )] B1 \u2243eh\u02dc LL \u25e6. . . \u25e6eh\u02dc L1f(X0) B2 \u2243\u02dc P L h \u25e6. . . \u25e6\u02dc P 1 hf(X0) = E[f(Xn,s T )]. (3) Approximations B1 and B2 in (3) are from the stochastic gradient and numerical integrator approximations, respectively. Similarly, we say \u02dc P l h corresponds to a Kth-order local integrator of \u02dc Ll if \u02dc P l hf(x) = eh\u02dc Llf(x) + O(hK+1). In the following sections, we focus on SG-MCMCs which use numerical integrators with stochastic gradients, and for the \ufb01rst time analyze how the two introduced errors affect their convergence behaviors. For notational simplicity, we henceforth use Xt to represent the approximate sample-path Xn,s t . 3 Convergence Analysis This section develops theory to analyze \ufb01nite-time convergence properties of general SG-MCMCs with both \ufb01xed and decreasing step sizes, as well as their asymptotic invariant measures. 3.1 Finite-time error analysis Given an ergodic\u2021 It\u02c6 o diffusion (1) with an invariant measure \u03c1(x), the posterior average is de\ufb01ned as: \u00af \u03c6 \u225c R X \u03c6(x)\u03c1(x)d x for some test function \u03c6(x) of interest. For a given numerical method with generated samples (Xlh)L l=1, we use the sample average \u02c6 \u03c6 de\ufb01ned as \u02c6 \u03c6 = 1 L PL l=1 \u03c6(Xlh) to approximate \u00af \u03c6. In the analysis, we de\ufb01ne a functional \u03c8 that solves the following Poisson Equation: L\u03c8(Xlh) = \u03c6(Xlh) \u2212\u00af \u03c6, or equivalently, 1 L L X l=1 L\u03c8(Xlh) = \u02c6 \u03c6 \u2212\u00af \u03c6. (4) The solution functional \u03c8(Xlh) characterizes the difference between \u03c6(Xlh) and the posterior average \u00af \u03c6 for every Xlh, thus would typically possess a unique solution, which is at least as smooth as \u03c6 under the elliptic or hypoelliptic settings [12]. In the unbounded domain of Xlh \u2208Rn, to make the presentation simple, we follow [6] and make certain assumptions on the solution functional, \u03c8, of the Poisson equation (4), which are used in the detailed proofs. Extensive empirical results have indicated the assumptions to hold in many real applications, though extra work is needed for theoretical veri\ufb01cations for different models, which is beyond the scope of this paper. Assumption 1. \u03c8 and its up to 3rd-order derivatives, Dk\u03c8, are bounded by a function\u00a7 V, i.e., \u2225Dk\u03c8\u2225\u2264CkVpk for k = (0, 1, 2, 3), Ck, pk > 0. Furthermore, the expectation of V on {Xlh} is bounded: supl EVp(Xlh) < \u221e, and V is smooth such that sups\u2208(0,1) Vp (s X + (1 \u2212s) Y) \u2264 C (Vp (X) + Vp (Y)), \u2200X, Y, p \u2264max{2pk} for some C > 0. \u2021See [6, 11] for conditions to ensure (1) is ergodic. \u00a7The existence of such function can be translated into \ufb01nding a Lyapunov function for the corresponding SDEs, an important topic in PDE literatures [13]. See Assumption 4.1 in [6] and Appendix C for more details. 3 \fWe emphasize that our proof techniques are related to those of the SGLD [6, 12], but with signi\ufb01cant distinctions in that, instead of expanding the function \u03c8(Xlh) [6], whose parameter Xlh does not endow an explicit form in general SG-MCMCs, we start from expanding the Kolmogorov\u2019s backward equation for each minibatch. Moreover, our techniques apply for general SG-MCMCs, instead of for one speci\ufb01c algorithm. More speci\ufb01cally, given a Kth-order local integrator with the corresponding Kolmogorov operator \u02dc P l h, according to De\ufb01nition 1 and (3), the Kolmogorov\u2019s backward equation for the l-th minibatch can be expanded as: E[\u03c8(Xlh)] = \u02dc P l h\u03c8(X(l\u22121)h) = eh\u02dc Ll\u03c8(X(l\u22121)h) + O(hK+1) = \u0010 I + h\u02dc Ll \u0011 \u03c8(X(l\u22121)h) + K X k=2 hk k! \u02dc L k l \u03c8(X(l\u22121)h) + O(hK+1) , (5) where I is the identity map. Recall that \u02dc Ll = L +\u2206Vl, e.g., \u2206Vl = (\u2207\u03b8 \u02dc Ul \u2212\u2207\u03b8U)\u00b7\u2207p in SGHMC. By further using the Poisson equation (4) to simplify related terms associated with L, after some algebra shown in Appendix D, the bias can be derived from (5) as: |E\u02c6 \u03c6 \u2212\u00af \u03c6| = \f \f \f \f \f E[\u03c8(Xlh)] \u2212\u03c8(X0) Lh \u22121 L X l E[\u2206Vl\u03c8(X(l\u22121)h)] \u2212 K X k=2 hk\u22121 k!L L X l=1 E[\u02dc L k l \u03c8(X(l\u22121)h)] \f \f \f \f \f + O(hK) . All terms in the above equation can be bounded, with details provided in Appendix D. This gives us a bound for the bias of an SG-MCMC algorithm in Theorem 2. Theorem 2. Under Assumption 1, let \u2225\u00b7\u2225be the operator norm. The bias of an SG-MCMC with a Kth-order integrator at time T = hL can be bounded as: \f \f \fE\u02c6 \u03c6 \u2212\u00af \u03c6 \f \f \f = O \u0012 1 Lh + P l \u2225E\u2206Vl\u2225 L + hK \u0013 . Note the bound above includes the term P l \u2225E\u2206Vl\u2225/L, measuring the difference between the expectation of stochastic gradients and the true gradient. It vanishes when the stochastic gradient is an unbiased estimation of the exact gradient, an assumption made in the SGLD. This on the other hand indicates that if the stochastic gradient is biased, |E\u02c6 \u03c6 \u2212\u00af \u03c6| might diverge when the growth of P l \u2225E\u2206Vl\u2225is faster than O(L). We point this out to show our result to be more informative than that of the SGLD [6], though this case might not happen in real applications. By expanding the proof for the bias, we are also able to bound the MSE of SG-MCMC algorithms, given in Theorem 3. Theorem 3. Under Assumption 1, and assume \u02dc Ul is an unbiased estimate of Ul. For a smooth test function \u03c6, the MSE of an SG-MCMC with a Kth-order integrator at time T = hL is bounded, for some C > 0 independent of (L, h), as E \u0010 \u02c6 \u03c6 \u2212\u00af \u03c6 \u00112 \u2264C   1 L P l E \u2225\u2206Vl\u22252 L + 1 Lh + h2K ! . Compared to the SGLD [6], the extra term 1 L2 P l E \u2225\u2206Vl\u22252 relates to the variance of noisy gradients. As long as the variance is bounded, the MSE still converges with the same rate. Speci\ufb01cally, when optimizing bounds for the bias and MSE, the optimal bias decreases at a rate of L\u2212K/(K+1) with step size h \u221dL\u22121/(K+1); while this is L\u22122K/(2K+1) with step size h \u221dL\u22121/(2K+1) for the MSE\u00b6. These rates decrease faster than those of the SGLD [6] when K \u22652. The case of K = 2 for the SGHMC with our proposed symmetric splitting integrator is discussed in Section 4. 3.2 Stationary invariant measures The asymptotic invariant measures of SG-MCMCs correspond to L approaching in\ufb01nity in the above analysis. According to the bias and MSE above, asymptotically (L \u2192\u221e) the sample average \u02c6 \u03c6 is a random variable with mean E\u02c6 \u03c6 = \u00af \u03c6+O(hK), and variance E(\u02c6 \u03c6\u2212E\u02c6 \u03c6)2 \u2264E(\u02c6 \u03c6\u2212\u00af \u03c6)2+E(\u00af \u03c6\u2212E\u02c6 \u03c6)2 = O(h2K), close to the true \u00af \u03c6. This section de\ufb01nes distance between measures, and studies more formally how the approximation errors affect the invariant measures of SG-MCMC algorithms. \u00b6To compare with the standard MCMC convergence rate of 1/2, the rate needs to be taken a square root. 4 \fFirst we note that under mild conditions, the existence of a stationary invariant measure for an SGMCMC can be guaranteed by application of the Krylov\u2013Bogolyubov Theorem [14]. Examining the conditions is beyond the scope of this paper. For simplicity, we follow [12] and assume stationary invariant measures do exist for SG-MCMCs. We denote the corresponding invariant measure as \u02dc \u03c1h, and the true posterior of a model as \u03c1. Similar to [12], we assume our numerical solver is geometric ergodic, meaning that for a test function \u03c6, we have R X \u03c6(x)\u02dc \u03c1h(d x) = R X Ex\u03c6(Xlh)\u02dc \u03c1h(d x) for any l \u22650 from the ergodic theorem, where Ex denotes the expectation conditional on X0 = x. The geometric ergodicity implies that the integration is independent of the starting point of an algorithm. Given this, we have the following theorem on invariant measures of SG-MCMCs. Theorem 4. Assume that a Kth-order integrator is geometric ergodic and its invariance measures \u02dc \u03c1h exist. De\ufb01ne the distance between the invariant measures \u02dc \u03c1h and \u03c1 as: d(\u02dc \u03c1h, \u03c1) \u225c sup\u03c6 \f \fR X \u03c6(x)\u02dc \u03c1h(d x) \u2212 R X \u03c6(x)\u03c1(d x) \f \f. Then any invariant measure \u02dc \u03c1h of an SG-MCMC is close to \u03c1 with an error up to an order of O(hK), i.e., there exists some C \u22650 such that: d(\u02dc \u03c1h, \u03c1) \u2264ChK. For a Kth-order integrator with full gradients, the corresponding invariant measure has been shown to be bounded by an order of O(hK) [9, 12]. As a result, Theorem 4 suggests only orders of numerical approximations but not the stochastic gradient approximation affect the asymptotic invariant measure of an SG-MCMC algorithm. This is also re\ufb02ected by experiments presented in Section 5. 3.3 SG-MCMCs with decreasing step sizes The original SGLD was \ufb01rst proposed with a decreasing-step-size sequence [4], instead of \ufb01xing step sizes, as analyzed in [6]. In [5], the authors provide theoretical foundations on its asymptotic convergence properties. We demonstrate in this section that for general SG-MCMC algorithms, decreasing step sizes for each minibatch are also feasible. Note our techniques here are different from those used for the decreasing-step-size SGLD [5], which interestingly result in similar convergence patterns. Speci\ufb01cally, by adapting the same techniques used in the previous sections, we establish conditions on the step size sequence to ensure asymptotic convergence, and develop theory on their \ufb01nite-time ergodic error as well. To guarantee asymptotic consistency, the following conditions on decreasing step size sequences are required. Assumption 2. The step sizes {hl} are decreasing\u2225, i.e., 0 < hl+1 < hl, and satisfy that 1) P\u221e l=1 hl = \u221e; and 2) limL\u2192\u221e PL l=1 hK+1 l PL l=1 hl = 0. Denote the \ufb01nite sum of step sizes as SL \u225cPL l=1 hl. Under Assumption 2, we need to modify the sample average \u00af \u03c6 de\ufb01ned in Section 3.1 as a weighted summation of {\u03c6(Xlh)}: \u02dc \u03c6 = PL l=1 hl SL \u03c6(Xlh). For simplicity, we assume \u02dc Ul to be an unbiased estimate of U such that E\u2206Vl = 0. Extending techniques in previous sections, we develop the following bounds for the bias and MSE. Theorem 5. Under Assumptions 1 and 2, for a smooth test function \u03c6, the bias and MSE of a decreasing-step-size SG-MCMC with a Kth-order integrator at time SL are bounded as: BIAS: \f \f \fE\u02dc \u03c6 \u2212\u00af \u03c6 \f \f \f = O   1 SL + PL l=1 hK+1 l SL ! (6) MSE: E \u0010 \u02dc \u03c6 \u2212\u00af \u03c6 \u00112 \u2264C  X l h2 l S2 L E \u2225\u2206Vl\u22252 + 1 SL + (PL l=1 hK+1 l )2 S2 L ! . (7) As a result, the asymptotic bias approaches 0 according to the assumptions. If further assuming\u2217\u2217 P\u221e l=1 h2 l S2 L = 0, the MSE also goes to 0. In words, the decreasing-step-size SG-MCMCs are consistent. Among the kinds of decreasing step size sequences, a commonly recognized one is hl \u221dl\u2212\u03b1 for 0 < \u03b1 < 1. We show in the following corollary that such a sequence leads to a valid sequence. Corollary 6. Using the step size sequences hl \u221dl\u2212\u03b1 for 0 < \u03b1 < 1, all the step size assumptions in Theorem 5 are satis\ufb01ed. As a result, the bias and MSE approach zero asymptotically, i.e., the sample average \u02dc \u03c6 is asymptotically consistent with the posterior average \u00af \u03c6. \u2225Actually the sequence need not be decreasing; we assume it is decreasing for simplicity. \u2217\u2217The assumption of P\u221e l=1 h2 l < \u221esatis\ufb01es this requirement, but is weaker than the original assumption. 5 \fRemark 7. Theorem 5 indicates the sample average \u02dc \u03c6 asymptotically converges to the true posterior average \u00af \u03c6. It is possible to \ufb01nd out the optimal decreasing rates for the speci\ufb01c decreasing sequence hl \u221dl\u2212\u03b1. Speci\ufb01cally, using the bounds for PL l=1 l\u2212\u03b1 (see the proof of Corollary 6), for the two terms in the bias (6) in Theorem 5, 1 SL decreases at a rate of O(L\u03b1\u22121), whereas (PL l=1 hK+1 l )/SL decreases as O(L\u2212K\u03b1). The balance between these two terms is achieved when \u03b1 = 1/(K + 1), which agrees with Theorem 2 on the optimal rate of \ufb01xed-step-size SG-MCMCs. Similarly, for the MSE (7), the \ufb01rst term decreases as L\u22121, independent of \u03b1, while the second and third terms decrease as O(L\u03b1\u22121) and O(L\u22122K\u03b1), respectively, thus the balance is achieved when \u03b1 = 1/(2K+ 1), which also agrees with the optimal rate for the \ufb01xed-step-size MSE in Theorem 3. According to Theorem 5, one theoretical advantage of decreasing-step-size SG-MCMCs over \ufb01xedstep-size variants is the asymptotically unbiased estimation of posterior averages, though the bene\ufb01t might not be signi\ufb01cant in large-scale real applications where the asymptotic regime is not reached. 4 Practical Numerical Integrators Given the theory for SG-MCMCs with high-order integrators, we here propose a 2nd-order symmetric splitting integrator for practical use. The Euler integrator is known as a 1st-order integrator; the proof and its detailed applications on the SGLD and SGHMC are given in Appendix I. The main idea of the symmetric splitting scheme is to split the local generator \u02dc Ll into several sub-generators that can be solved analytically\u2020\u2020. Unfortunately, one cannot easily apply a splitting scheme with the SGLD. However, for the SGHMC, it can be readily split into: \u02dc Ll = LA+LB+LOl, where LA = p \u00b7\u2207\u03b8, LB = \u2212D p \u00b7\u2207p, LOl = \u2212\u2207\u03b8 \u02dc U(\u03b8) \u00b7 \u2207p + 2D In : \u2207p\u2207T p . (8) These sub-generators correspond to the following SDEs, which are all analytically solvable: A : \u001a d\u03b8 = p dt d p = 0 , B : \u001a d\u03b8 = 0 d p = \u2212D p dt , O : \u001a d\u03b8 = 0 d p = \u2212\u2207\u03b8 \u02dc Ul(\u03b8)dt + \u221a 2DdW (9) Based on these sub-SDEs, the local Kolmogorov operator \u02dc P l h is de\ufb01ned as: E[f(Xlh)] = \u02dc P l hf(X(l\u22121)h), where, \u02dc P l h \u225ce h 2 LA \u25e6e h 2 LB \u25e6ehLOl \u25e6e h 2 LB \u25e6e h 2 LA, so that the corresponding updates for Xlh = (\u03b8lh, plh) consist of the following 5 steps: \u03b8(1) lh = \u03b8(l\u22121)h + p(l\u22121)h h/2 \u21d2p(1) lh = e\u2212Dh/2 p(l\u22121)h \u21d2p(2) lh = p(1) lh \u2212\u2207\u03b8 \u02dc Ul(\u03b8(1) lh )h + \u221a 2Dh\u03b6l \u21d2 plh = e\u2212Dh/2 p(2) lh \u21d2 \u03b8lh = \u03b8(1) lh + plh h/2 , where (\u03b8(1) lh , p(1) lh , p(2) lh ) are intermediate variables. We denote such a splitting method as the ABOBA scheme. From the Markovian property of a Kolmogorov operator, it is readily seen that all such symmetric splitting schemes (with different orders of \u2018A\u2019, \u2018B\u2019 and \u2018O\u2019) are equivalent [15]. Lemma 8 below shows the symmetric splitting scheme is a 2nd-order local integrator. Lemma 8. The symmetric splitting scheme is a 2nd-order local integrator, i.e., the corresponding Kolmogorov operator \u02dc P l h satis\ufb01es: \u02dc P l h = eh \u02dc Ll + O(h3). When this integrator is applied to the SGHMC, the following properties can be obtained. Remark 9. Applying Theorem 2 to the SGHMC with the symmetric splitting scheme (K = 2), the bias is bounded as: |E\u02c6 \u03c6 \u2212\u00af \u03c6| = O( 1 Lh + P l\u2225E\u2206Vl\u2225 L + h2). The optimal bias decreasing rate is L\u22122/3, compared to L\u22121/2 for the SGLD [6]. Similarly, the MSE is bounded by: E(\u02c6 \u03c6 \u2212\u00af \u03c6)2 \u2264 C( 1 L P l E\u2225\u2206Vl\u22252 L + 1 Lh + h4), decreasing optimally as L\u22124/5 with step size h \u221dL\u22121/5, compared to the MSE of L\u22122/3 for the SGLD [6]. This indicates that the SGHMC with the splitting integrator converges faster than the SGLD and SGHMC with 1st-order Euler integrators. Remark 10. For a decreasing-step-size SGHMC, based on Remark 7, the optimal step size decreasing rate for the bias is \u03b1 = 1/3, and \u03b1 = 1/5 for the MSE. These agree with their \ufb01xed-step-size counterparts in Remark 9, thus are faster than the SGLD/SGHMC with 1st-order Euler integrators. \u2020\u2020This is different from the traditional splitting in SDE literatures[9, 15], where L instead of \u02dc Ll is split. 6 \f#iterations 10 1 10 2 10 3 10 4 BIAS 10 -2 10 -1 10 0 10 1 , = 0:1 , = 0:2 , = 0:33 , = 0:7 #iterations 10 1 10 2 10 3 10 4 MSE 10 -4 10 -2 10 0 10 2 , = 0:1 , = 0:2 , = 0:3 , = 0:4 Figure 2: Bias of SGHMC-D (left) and MSE of SGHMC-F (right) with different step size rates \u03b1. Thick red curves correspond to theoretically optimal rates. 5 Experiments We here verify our theory and compare with related algorithms on both synthetic data and large-scale machine learning applications. step size 0.001 0.005 0.01 0.02 0.05 0.1 BIAS 10 -4 10 -3 10 -2 10 -1 Splitting Euler Figure 1: Comparisons of symmetric splitting and Euler integrators. Synthetic data We consider a standard Gaussian model where xi \u223cN(\u03b8, 1), \u03b8 \u223cN(0, 1). 1000 data samples {xi} are generated, and every minibatch in the stochastic gradient is of size 10. The test function is de\ufb01ned as \u03c6(\u03b8) \u225c\u03b82, with explicit expression for the posterior average. To evaluate the expectations in the bias and MSE, we average over 200 runs with random initializations. First we compare the invariant measures (with L = 106) of the proposed splitting integrator and Euler integrator for the SGHMC. Results of the SGLD are omitted since they are not as competitive. Figure 1 plots the biases with different step sizes. It is clear that the Euler integrator has larger biases in the invariant measure, and quickly explodes when the step size becomes large, which does not happen for the splitting integrator. In real applications we also \ufb01nd this happen frequently (shown in the next section), making the Euler scheme an unstable integrator. Next we examine the asymptotically optimal step size rates for the SGHMC. From the theory we know these are \u03b1 = 1/3 for the bias and \u03b1 = 1/5 for the MSE, in both \ufb01xed-step-size SGHMC (SGHMC-F) and decreasing-step-size SGHMC (SGHMC-D). For the step sizes, we did a grid search to select the best prefactors, which resulted in h=0.033\u00d7L\u2212\u03b1 for the SGHMC-F and hl=0.045 \u00d7l\u2212\u03b1 for the SGHMC-D, with different \u03b1 values. We plot the traces of the bias for the SGHMC-D and the MSE for the SGHMC-F in Figure 2. Similar results for the bias of the SGHMC-F and the MSE of the SGHMC-D are plotted in Appendix K. We \ufb01nd that when rates are smaller than the theoretically optimal rates, i.e., \u03b1 = 1/3 (bias) and \u03b1 = 1/5 (MSE), the bias and MSE tend to decrease faster than the optimal rates at the beginning (especially for the SGHMC-F), but eventually they slow down and are surpassed by the optimal rates, consistent with the asymptotic theory. This also suggests that if only a small number of iterations were feasible, setting a larger step size than the theoretically optimal one might be bene\ufb01cial in practice. Finally, we study the relative convergence speed of the SGHMC and SGLD. We test both \ufb01xedstep-size and decreasing-step-size versions. For \ufb01xed-step-size experiments, the step sizes are set to h = CL\u2212\u03b1, with \u03b1 chosen according to the theory for SGLD and SGHMC. To provide a fair comparison, the constants C are selected via a grid search from 10\u22123 to 0.5 with an interval of 0.002 for L = 500, it is then \ufb01xed in the other runs with different L values. The parameter D in the SGHMC is selected within (10, 20, 30) as well. For decreasing-step-size experiments, an initial step size is chosen within [0.003, 0.05] with an interval of 0.002 for different algorithms\u2021\u2021, and then it decreases according to their theoretical optimal rates. Figure 3 shows a comparison of the biases for the SGHMC and SGLD. As indicated by both theory and experiments, the SGHMC with the splitting integrator yields a faster convergence speed than the SGLD with an Euler integrator. Large-scale machine learning applications For real applications, we test the SGLD with an Euler integrator, the SGHMC with the splitting integrator (SGHMC-S), and the SGHMC with an \u2021\u2021Using the same initial step size is not fair because the SGLD requires much smaller step sizes. 7 \f#iterations 20 100 250 400 550 700 BIAS 0 0.05 0.1 0.15 0.2 SGLD SGHMC #iterations 20 100 250 400 550 700 BIAS 0 0.02 0.04 0.06 0.08 SGLD SGHMC Figure 3: Biases for the \ufb01xed-step-size (left) and decreasing-step-size (right) SGHMC and SGLD. Euler integrator (SGHMC-E). First we test them on the latent Dirichlet allocation model (LDA) [16]. The data used consists of 10M randomly downloaded documents from Wikipedia, using scripts provided in [17]. We randomly select 1K documents for testing and validation, respectively. As in [17, 18], the vocabulary size is 7,702. We use the Expanded-Natural reparametrization trick to sample from the probabilistic simplex [19]. The step sizes are chosen from {2, 5, 8, 20, 50, 80} \u00d7 10\u22125, and parameter D from {20, 40, 80}. The minibatch size is set to 100, with one pass of the whole data in the experiments (and therefore L = 100K). We collect 300 posterior samples to calculate test perplexities, with a standard holdout technique as described in [18]. Next a recently studied sigmoid belief network model (SBN) [20] is tested, which is a directed counterpart of the popular RBM model. We use a one layer model where the bottom layer corresponds to binary observed data, which is generated from the hidden layer (also binary) via a sigmoid function. As shown in [18], the SBN is readily learned by SG-MCMCs. We test the model on the MNIST dataset, which consists of 60K hand written digits of size 28 \u00d7 28 for training, and 10K for testing. Again the step sizes are chosen from {3, 4, 5, 6}\u00d710\u22124, D from {0.9, 1, 5}/ \u221a h. The minibatch is set to 200, with 5000 iterations for training. Like applied for the RBM [21], an advance technique called anneal importance sampler (AIS) is adopted for calculating test likelihoods. step size 2e-05 5e-05 0.0002 0.0008 Perplexity 1200 1400 1600 1800 2000 SGHMC-Euler SGHMC-Splitting Figure 4: SGHMC with 200 topics. The Euler explodes with large step sizes. We brie\ufb02y describe the results here, more details are provided in Appendix K. For LDA with 200 topics, the best test perplexities for the SGHMC-S, SGHMC-E and SGLD are 1168, 1180 and 2496, respectively; while these are 1157, 1187 and 2511, respectively, for 500 topics. Similar to the synthetic experiments, we also observed SGHMC-E crashed when using large step sizes. This is illustrated more clearly in Figure 4. For the SBN with 100 hidden units, we obtain negative test log-likelihoods of 103, 105 and 126 for the SGHMC-S, SGHMC-E and SGLD, respectively; and these are 98, 100, and 110 for 200 hidden units. Note the SGHMC-S on SBN yields state-of-the-art results on test likelihoods compared to [22], which was 113 for 200 hidden units. A decrease of 2 units in the neg-log-likelihood with AIS is considered to be a reasonable gain [20], which is approximately equal to the gain from a shallow to a deep model [22]. SGHMC-S is more accuracy and robust than SGHMC-E due to its 2nd-order splitting integrator. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1610.06664v1",
+                    "title": "Stochastic Gradient MCMC with Stale Gradients",
+                    "abstract": "Stochastic gradient MCMC (SG-MCMC) has played an important role in\nlarge-scale Bayesian learning, with well-developed theoretical convergence\nproperties. In such applications of SG-MCMC, it is becoming increasingly\npopular to employ distributed systems, where stochastic gradients are computed\nbased on some outdated parameters, yielding what are termed stale gradients.\nWhile stale gradients could be directly used in SG-MCMC, their impact on\nconvergence properties has not been well studied. In this paper we develop\ntheory to show that while the bias and MSE of an SG-MCMC algorithm depend on\nthe staleness of stochastic gradients, its estimation variance (relative to the\nexpected estimate, based on a prescribed number of samples) is independent of\nit. In a simple Bayesian distributed system with SG-MCMC, where stale gradients\nare computed asynchronously by a set of workers, our theory indicates a linear\nspeedup on the decrease of estimation variance w.r.t. the number of workers.\nExperiments on synthetic data and deep neural networks validate our theory,\ndemonstrating the effectiveness and scalability of SG-MCMC with stale\ngradients.",
+                    "authors": "Changyou Chen, Nan Ding, Chunyuan Li, Yizhe Zhang, Lawrence Carin",
+                    "published": "2016-10-21",
+                    "updated": "2016-10-21",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction The pervasiveness of big data has made scalable machine learning increasingly important, especially for deep models. A basic technique is to adopt stochastic optimization algorithms [1], e.g., stochastic gradient descent and its extensions [2]. In each iteration of stochastic optimization, a minibatch of data is used to evaluate the gradients of the objective function and update model parameters (errors are introduced in the gradients, because they are computed based on minibatches rather than the entire dataset; since the minibatches are typically selected at random, this yields the term \u201cstochastic\u201d gradient). This is highly scalable because processing a minibatch of data in each iteration is relatively cheap compared to analyzing the entire (large) dataset at once. Under certain conditions, stochastic optimization is guaranteed to converge to a (local) optima [1]. Because of its scalability, the minibatch strategy has recently been extended to Markov Chain Monte Carlo (MCMC) Bayesian sampling methods, yielding SG-MCMC [3, 4, 5]. In order to handle large-scale data, distributed stochastic optimization algorithms have been developed, for example [6], to further improve scalability. In a distributed setting, a cluster of machines with multiple cores cooperate with each other, typically through an asynchronous scheme, for scalability [7, 8, 9]. A downside of an asynchronous implementation is that stale gradients must be used in parameter updates (\u201cstale gradients\u201d are stochastic gradients computed based on outdated parameters, instead of the latest parameters; they are easier to compute in a distributed system, but introduce additional errors relative to traditional stochastic gradients). While some theory has been developed to guarantee the convergence of stochastic optimization with stale gradients [10, 11, 12], little analysis has been done in a Bayesian setting, where SG-MCMC is applied. Distributed SG-MCMC algorithms share characteristics with distributed stochastic optimization, and thus are highly scalable and suitable for large-scale Bayesian learning. Existing Bayesian distributed systems with traditional MCMC 29th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain. arXiv:1610.06664v1  [stat.ML]  21 Oct 2016 \fmethods, such as [13], usually employ stale statistics instead of stale gradients, where stale statistics are summarized based on outdated parameters, e.g., outdated topic distributions in distributed Gibbs sampling [13]. Little theory exists to guarantee the convergence of such methods. For existing distributed SG-MCMC methods, typically only standard stochastic gradients are used, for limited problems such as matrix factorization, without rigorous convergence theory [14, 15, 16]. In this paper, by extending techniques from standard SG-MCMC [17], we develop theory to study the convergence behavior of SG-MCMC with Stale gradients (S2G-MCMC). Our goal is to evaluate the posterior average of a test function \u03c6(x), de\ufb01ned as \u00af \u03c6 \u225c R X \u03c6(x)\u03c1(x)d x, where \u03c1(x) is the desired posterior distribution with x the possibly augmented model parameters (see Section 2). In practice, S2G-MCMC generates L samples {xl}L l=1 and uses the sample average \u02c6 \u03c6L \u225c 1 L PL l=1 \u03c6(xl) to approximate \u00af \u03c6. We measure how \u02c6 \u03c6L approximates \u00af \u03c6 in terms of bias, MSE and estimation variance, de\ufb01ned as |E\u02c6 \u03c6L \u2212\u00af \u03c6|, E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 and E \u0010 \u02c6 \u03c6L \u2212E\u02c6 \u03c6L \u00112 , respectively. From the de\ufb01nitions, the bias and MSE characterize how accurately \u02c6 \u03c6L approximates \u00af \u03c6, and the variance characterizes how fast \u02c6 \u03c6L converges to its own expectation (for a prescribed number of samples L). Our theoretical results show that while the bias and MSE depend on the staleness of stochastic gradients, the variance is independent of it. In a simple asynchronous Bayesian distributed system with S2G-MCMC, our theory indicates a linear speedup on the decrease of the variance w.r.t. the number of workers used to calculate the stale gradients, while maintaining the same optimal bias level as standard SG-MCMC. We validate our theory on several synthetic experiments and deep neural network models, demonstrating the effectiveness and scalability of the proposed S2G-MCMC framework. Related Work Using stale gradients is a standard setup in distributed stochastic optimization systems. Representative algorithms include, but are not limited to, the ASYSG-CON [6] and HOGWILD! algorithms [18], and some more recent developments [19, 20]. Furthermore, recent research on stochastic optimization has been extended to non-convex problems with provable convergence rates [12]. In Bayesian learning with MCMC, existing work has focused on running parallel chains on subsets of data [21, 22, 23, 24], and little if any effort has been made to use stale stochastic gradients, the setting considered in this paper. 2 Stochastic Gradient MCMC Throughout this paper, we denote vectors as bold lower-case letters, and matrices as bold uppercase letters. For example, N(m, \u03a3) means a multivariate Gaussian distribution with mean m and covariance \u03a3. In the analysis we consider algorithms with \ufb01xed-stepsizes for simplicity; decreasingstepsize variants can be addressed similarly as in [17]. The goal of SG-MCMC is to generate random samples from a posterior distribution p(\u03b8| D) \u221d p(\u03b8) QN i=1 p(di |\u03b8), which are used to evaluate a test function. Here \u03b8 \u2208Rn represents the parameter vector and D = {d1, \u00b7 \u00b7 \u00b7 , dN} represents the data, p(\u03b8) is the prior distribution, and p(di |\u03b8) the likelihood for di. SG-MCMC algorithms are based on a class of stochastic differential equations, called It\u00f4 diffusion, de\ufb01ned as d xt = F(xt)dt + g(xt)dwt , (1) where x \u2208Rm represents the model states, typically x augments \u03b8 such that \u03b8 \u2286x and n \u2264m; t is the time index, wt \u2208Rm is m-dimensional Brownian motion, functions F : Rm \u2192Rm and g : Rm \u2192Rm\u00d7m are assumed to satisfy the usual Lipschitz continuity condition [25]. For appropriate functions F and g, the stationary distribution, \u03c1(x), of the It\u00f4 diffusion (1) has a marginal distribution equal to the posterior distribution p(\u03b8| D) [26]. For example, denoting the unnormalized negative log-posterior as U(\u03b8) \u225c\u2212log p(\u03b8) \u2212PN i=1 log p(di |\u03b8), the stochastic gradient Langevin dynamic (SGLD) method [3] is based on 1st-order Langevin dynamics, with x = \u03b8, and F(xt) = \u2212\u2207\u03b8U(\u03b8), g(xt) = \u221a 2 In, where In is the n \u00d7 n identity matrix. The stochastic gradient Hamiltonian Monte Carlo (SGHMC) method [4] is based on 2nd-order Langevin dynamics, with x = (\u03b8, q), and F(xt) = \u0010 q \u2212B q \u2212\u2207\u03b8U(\u03b8) \u0011 , g(xt) = \u221a 2B \u0010 0 0 0 In \u0011 for a scalar B > 0; q is an auxiliary variable known as the momentum [4, 5]. Diffusion forms for other 2 \fSG-MCMC algorithms, such as the stochastic gradient thermostat [5] and variants with Riemannian information geometry [27, 26, 28], are de\ufb01ned similarly. In order to ef\ufb01ciently draw samples from the continuous-time diffusion (1), SG-MCMC algorithms typically apply two approximations: i) Instead of analytically integrating in\ufb01nitesimal increments dt, numerical integration over small step size h is used to approximate the integration of the true dynamics. ii) Instead of working with the full gradient \u2207\u03b8U(\u03b8lh), a stochastic gradient \u2207\u03b8 \u02dc Ul(\u03b8lh), de\ufb01ned as \u2207\u03b8 \u02dc Ul(\u03b8) \u225c\u2212\u2207\u03b8 log p(\u03b8) \u2212N J J X i=1 \u2207\u03b8 log p(d\u03c0i |\u03b8), (2) is calculated from a minibatch of size J, where {\u03c01, \u00b7 \u00b7 \u00b7 , \u03c0J} is a random subset of {1, \u00b7 \u00b7 \u00b7 , N}. Note that to match the time index t in (1), parameters have been and will be indexed by \u201clh\u201d in the l-th iteration. 3 Stochastic Gradient MCMC with Stale Gradients In this section, we extend SG-MCMC to the stale-gradient setting, commonly met in asynchronous distributed systems [7, 8, 9], and develop theory to analyze convergence properties. 3.1 Stale stochastic gradient MCMC (S2G-MCMC) The setting for S2G-MCMC is the same as the standard SG-MCMC described above, except that the stochastic gradient (2) is replaced with a stochastic gradient evaluated with outdated parameter \u03b8(l\u2212\u03c4l)h instead of the latest version \u03b8lh (see Appendix A for an example): \u2207\u03b8 \u02c6 U\u03c4l(\u03b8) \u225c\u2212\u2207\u03b8 log p(\u03b8(l\u2212\u03c4l)h) \u2212N J J X i=1 \u2207\u03b8 log p(d\u03c0i |\u03b8(l\u2212\u03c4l)h), (3) where \u03c4l \u2208Z+ denotes the staleness of the parameter used to calculate the stochastic gradient in the l-th iteration. A distinctive difference between S2G-MCMC and SG-MCMC is that stale stochastic gradients are no longer unbiased estimations of the true gradients. This leads to additional challenges in developing convergence bounds, one of the main contributions of this paper. Algorithm 1 State update of SGHMC with the stale stochastic gradient \u2207\u03b8 \u02c6 U\u03c4l(\u03b8) Input: xlh = (\u03b8lh, qlh), \u2207\u03b8 \u02c6 U\u03c4l(\u03b8), \u03c4l, \u03c4, h, B Output: x(l+1)h = (\u03b8(l+1)h, q(l+1)h) if \u03c4l \u2264\u03c4 then Draw \u03b6l \u223cN(0, I); q(l+1)h = (1\u2212Bh) qlh \u2212\u2207\u03b8 \u02c6 U\u03c4l(\u03b8)h+ \u221a 2Bh\u03b6l; \u03b8(l+1)h = \u03b8lh + q(l+1)h h; end if We assume a bounded staleness for all \u03c4l\u2019s, i.e., max l \u03c4l \u2264\u03c4 for some constant \u03c4. As an example, Algorithm 1 describes the update rule of the stale-SGHMC in each iteration with the Euler integrator, where the stale gradient \u2207\u03b8 \u02c6 U\u03c4l(\u03b8) with staleness \u03c4l is used. 3.2 Convergence analysis This section analyzes the convergence properties of the basic S2G-MCMC; an extension with multiple chains is discussed in Section 3.3. It is shown that the bias and MSE depend on the staleness parameter \u03c4, while the variance is independent of it, yielding signi\ufb01cant speedup in Bayesian distributed systems. Bias and MSE In [17], the bias and MSE of the standard SG-MCMC algorithms with a Kth order integrator were analyzed, where the order of an integrator re\ufb02ects how accurately an SG-MCMC algorithm approximates the corresponding continuous diffusion. Speci\ufb01cally, if evolving xt with a numerical integrator using discrete time increment h induces an error bounded by O(hK), the integrator is called a Kth order integrator, e.g., the popular Euler method used in SGLD [3] is a 1st-order integrator. In particular, [17] proved the bounds stated in Lemma 1. 3 \fLemma 1 ([17]). Under standard assumptions (see Appendix B), the bias and MSE of SG-MCMC with a Kth-order integrator at time T = hL are bounded as: Bias: \f \f \fE\u02c6 \u03c6L \u2212\u00af \u03c6 \f \f \f = O \u0012P l \u2225E\u2206Vl\u2225 L + 1 Lh + hK \u0013 MSE: E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 = O   1 L P l E \u2225\u2206Vl\u22252 L + 1 Lh + h2K ! Here \u2206Vl \u225cL \u2212\u02dc Ll, where L is the generator of the It\u00f4 diffusion (1) de\ufb01ned as Lf(xt) \u225clim h\u21920+ E [f(xt+h)] \u2212f(xt) h = \u0012 F(xt) \u00b7 \u2207x + 1 2 \u0000g(xt)g(xt)T \u0001 :\u2207x\u2207T x \u0013 f(xt) , (4) for any compactly supported twice differentiable function f : Rn \u2192R, h \u21920+ means h approaches zero along the positive real axis. \u02dc Ll is the same as L except using the stochastic gradient \u2207\u02dc Ul instead of the full gradient. We show that the bounds of the bias and MSE of S2G-MCMC share similar forms as SG-MCMC, but with additional dependence on the staleness parameter. In addition to the assumptions in SG-MCMC [17] (see details in Appendix B), the following additional assumption is imposed. Assumption 1. The noise in the stochastic gradients is well-behaved, such that: 1) the stochastic gradient is unbiased, i.e., \u2207\u03b8U(\u03b8) = E\u03be\u2207\u03b8 \u02dc U(\u03b8) where \u03be denotes the random permutation over {1, \u00b7 \u00b7 \u00b7 , N}; 2) the variance of stochastic gradient is bounded, i.e., E\u03be \r \r \rU(\u03b8) \u2212\u02dc U(\u03b8) \r \r \r 2 \u2264\u03c32; 3) the gradient function \u2207\u03b8U is Lipschitz (so is \u2207\u03b8 \u02dc U), i.e., \u2225\u2207\u03b8U(x) \u2212\u2207\u03b8U(y)\u2225\u2264C \u2225x \u2212y\u2225, \u2200x, y. In the following theorems, we omit the assumption statement for conciseness. Due to the staleness of the stochastic gradients, the term \u2206Vl in S2G-MCMC is equal to L \u2212\u02dc Ll\u2212\u03c4l, where \u02dc Ll\u2212\u03c4l arises from \u2207\u03b8 \u02c6 U\u03c4l. The challenge arises to bound these terms involving \u2206Vl. To this end, de\ufb01ne flh \u225c \r \rxlh \u2212x(l\u22121)h \r \r, and \u03c8 to be a functional satisfying the Poisson Equation\u2217: 1 L L X l=1 L\u03c8(xlh) = \u02c6 \u03c6L \u2212\u00af \u03c6 . (5) Theorem 2. After L iterations, the bias of S2G-MCMC with a Kth-order integrator is bounded, for some constant D1 independent of {L, h, \u03c4}, as: \f \f \fE\u02c6 \u03c6L \u2212\u00af \u03c6 \f \f \f \u2264D1 \u0012 1 Lh + M1\u03c4h + M2hK \u0013 , where M1 \u225cmaxl |Lflh| maxl \u2225E\u2207\u03c8(xlh)\u2225C, M2 \u225cPK k=1 P l E\u02dc L k+1 l \u03c8(x(l\u22121)h) (k+1)!L are constants. Theorem 3. After L iterations, the MSE of S2G-MCMC with a Kth-order integrator is bounded, for some constant D2 independent of {L, h, \u03c4}, as: E \u0010 \u02c6 \u03c6L \u2212\u00af \u03c6 \u00112 \u2264D2 \u0012 1 Lh + \u02dc M1\u03c4 2h2 + \u02dc M2h2K \u0013 , where constants \u02dc M1 \u225cmaxl \u2225E\u2207\u03c8(xlh)\u22252 maxl (Lflh)2 C2, \u02dc M2 \u225cE( P l \u02dc L K+1 l \u03c8(x(l\u22121)h) L(K+1)! )2. The theorems indicate that both the bias and MSE depend on the staleness parameter \u03c4. For a \ufb01xed computational time, this could possibly lead to unimproved bounds, compared to standard SG-MCMC, when \u03c4 is too large, i.e., the terms with \u03c4 would dominate, as is the case in the distributed system discussed in Section 4. Nevertheless, better bounds than standard SG-MCMC could be obtained if the decrease of 1 Lh is faster than the increase of the staleness in a distributed system. \u2217The existence of a nice \u03c8 is guaranteed in the elliptic/hypoelliptic SDE settings when x is on a torus [25]. 4 \fVariance Next we investigate the convergence behavior of the variance, Var(\u02c6 \u03c6L) \u225c E \u0010 \u02c6 \u03c6L \u2212E\u02c6 \u03c6L \u00112 . Theorem 4 indicates the variance is independent of \u03c4, hence a linear speedup in the decrease of variance is always achievable when stale gradients are computed in parallel. An example is discussed in the Bayesian distributed system in Section 4. Theorem 4. After L iterations, the variance of S2G-MCMC with a Kth-order integrator is bounded, for some constant D, as: Var \u0010 \u02c6 \u03c6L \u0011 \u2264D \u0012 1 Lh + h2K \u0013 . The variance bound is the same as for standard SG-MCMC, whereas L could increase linearly w.r.t.the number of workers in a distributed setting, yielding signi\ufb01cant variance reduction. When optimizing the the variance bound w.r.t.h, we get an optimal variance bound stated in Corollary 5. Corollary 5. In term of estimation variance, the optimal convergence rate of S2G-MCMC with a Kth-order integrator is bounded as: Var \u0010 \u02c6 \u03c6L \u0011 \u2264O \u0000L\u22122K/(2K+1)\u0001 . In real distributed systems, the decrease of 1/Lh and increase of \u03c4, in the bias and MSE bounds, would typically cancel, leading to the same bias and MSE level compared to standard SG-MCMC, whereas a linear speedup on the decrease of variance w.r.t. the number of workers is always achievable. More details are discussed in Section 4. 3.3 Extension to multiple parallel chains This section extends the theory to the setting with S parallel chains, each independently running an S2G-MCMC algorithm. After generating samples from the S chains, an aggregation step is needed to combine the sample average from each chain, i.e., {\u02c6 \u03c6Ls}M s=1, where Ls is the number of iterations on chain s. For generality, we allow each chain to have different step sizes, e.g., (hs)S s=1. We aggregate the sample averages as \u02c6 \u03c6S L \u225cPS s=1 Ts T \u02c6 \u03c6Ls, where Ts \u225cLshs, T \u225cPS s=1 Ts. Interestingly, with increasing S, using multiple chains does not seem to directly improve the convergence rate for the bias, but improves the MSE bound, as stated in Theorem 6. Theorem 6. Let Tm \u225cmaxl Tl, hm \u225cmaxl hl, \u00af T = T/S, the bias and MSE of S parallel S2GMCMC chains with a Kth-order integrator are bounded, for some constants D1 and D2 independent of {L, h, \u03c4}, as: Bias: \f \f \fE\u02c6 \u03c6S L \u2212\u00af \u03c6 \f \f \f \u2264D1 \u0012 1 \u00af T + Tm \u00af T \u0000M1\u03c4hs + M2hK s \u0001\u0013 MSE: E \u0010 \u02c6 \u03c6S L \u2212\u00af \u03c6 \u00112 \u2264D2 \u00121 \u22121/ \u00af T T + 1 \u00af T 2 + T 2 m \u00af T 2 \u0000M 2 1 \u03c4 2h2 s + M 2 2 h2K s \u0001\u0013 . Assume that \u00af T = T/S is independent of the number of chains. As a result, using multiple chains does not directly improve the bound for the bias\u2020. However, for the MSE bound, although the last two terms are independent of S, the \ufb01rst term decreases linearly with respect to S because T = \u00af TS. This indicates a decreased estimation variance with more chains. This matches the intuition because more samples can be obtained with more chains in a given amount of time. The decrease of MSE for multiple-chain is due to the decrease of the variance as stated in Theorem 7. Theorem 7. The variance of S parallel S2G-MCMC chains with a Kth-order integrator is bounded, for some constant D independent of {L, h, \u03c4}, as: E \u0010 \u02c6 \u03c6S L \u2212E\u02c6 \u03c6S L \u00112 \u2264D   1 T + S X s=1 T 2 s T 2 h2K s ! . When using the same step size for all chains, Theorem 7 gives an optimal variance bound of O \u0000(P s Ls)\u22122K/(2K+1)\u0001 , i.e. a linear speedup with respect to S is achieved. In addition, Theorem 6 with \u03c4 = 0 and K = 1 provides convergence rates for the distributed SGLD algorithm in [14], i.e., improved MSE and variance bounds compared to the single-server SGLD. \u2020It means the bound does not directly relate to low-order terms of S, though constants might be improved. 5 \f4 Applications to Distributed SG-MCMC Systems Our theory for S2G-MCMC is general, serving as a basic analytic tool for distributed SG-MCMC systems. We propose two simple Bayesian distributed systems with S2G-MCMC in the following. Single-chain distributed SG-MCMC Perhaps the simplest architecture is an asynchronous distributed SG-MCMC system, where a server runs an S2G-MCMC algorithm, with stale gradients computed asynchronously from W workers. The detailed operations of the server and workers are described in Appendix A. With our theory, now we explain the convergence property of this simple distributed system with SG-MCMC, i.e., a linear speedup w.r.t. the number of workers on the decrease of variance, while maintaining the same bias level. To this end, rewrite L = W \u00af L from Theorems 2 and 3, where \u00af L is the average number of iterations on each worker. We can observe from the theorems that when M1\u03c4h > M2hK in the bias and \u02dc M1\u03c4 2h2 > \u02dc M2h2K in the MSE, the terms with \u03c4 dominate. Optimizing the bounds with respect to h yields a bound of O((\u03c4/W \u00af L)1/2) for the bias, and O((\u03c4/W \u00af L)2/3) for the MSE. In practice, we usually observe \u03c4 \u2248W, making W in the optimal bounds cancels, i.e., the same optimal bias and MSE bounds as standard SG-MCMC are obtained, no theoretical speedup is achieved when increasing W. However, from Corollary 5, the variance is independent of \u03c4, thus a linear speedup on the variance bound can be always obtained when increasing the number of workers, i.e., the distributed SG-MCMC system convergences a factor of W faster than standard SG-MCMC with a single machine. We are not aware of similar conclusions from optimization, because most of the research focuses on the convex setting, thus only variance (equivalent to MSE) is studied. Multiple-chain distributed SG-MCMC We can also adopt multiple servers based on the multiplechain setup in Section 3.3, where each chain corresponds to one server. The detailed architecture is described in Appendix A. This architecture trades off communication cost with convergence rates. As indicated by Theorems 6 and 7, the MSE and variance bounds can be improved with more servers. Note that when only one worker is associated with one server, we recover the setting of S independent servers. Compared to the single-server architecture described above with S workers, from Theorems 2\u20137, while the variance bound is the same, the single-server arthitecture improves the bias and MSE bounds by a factor of S. More advanced architectures More complex architectures could also be designed to reduce communication cost, for example, by extending the downpour [7] and elastic SGD [29] architectures to the SG-MCMC setting. Their convergence properties can also be analyzed with our theory since they are essentially using stale gradients. We leave the detailed analysis for future work. 5 Experiments Our primal goal is to validate the theory, comparing with different distributed architectures and algorithms, such as [30, 31], is beyond the scope of this paper. We \ufb01rst use two synthetic experiments to validate the theory, then apply the distributed architecture described in Section 4 for Bayesian deep learning. To quantitatively describe the speedup property, we adopt the the iteration speedup [12], de\ufb01ned as: iteration speedup \u225c#iterations with a single worker average #iterations on a worker , where # is the iteration count when the same level of precision is achieved. This speedup best matches with the theory. We also consider the time speedup, de\ufb01ned as: running time for a single worker running time for W worker , where the running time is recorded at the same accuracy. It is affected signi\ufb01cantly by hardware, thus is not accurately consistent with the theory. 5.1 Synthetic experiments Impact of stale gradients A simple Gaussian model is used to verify the impact of stale gradients on the convergence accuracy, with di \u223cN(\u03b8, 1), \u03b8 \u223cN(0, 1). 1000 data samples {di} are generated, with minibatches of size 10 to calculate stochastic gradients. The test function is \u03c6(\u03b8) \u225c\u03b82. The distributed SGLD algorithm is adopted in this experiment. We aim to verify that the optimal MSE bound \u221d\u03c4 2/3L\u22122/3, derived from Theorem 3 and discussed in Section 4 (with W = 1). The optimal stepsize is h = C\u03c4 \u22122/3L\u22121/3 for some constant C. Based on the optimal bound, setting 6 \f10 0 10 1 10 2 10 3 #iterations 10 -8 10 -6 10 -4 10 -2 10 0 Var 1 worker 2 workers 3 workers 4 workers 5 workers 6 workers 7 workers 8 workers 9 workers 10 -1 10 0 Time (s) 10 -6 10 -4 10 -2 Var 1 worker 2 workers 3 workers 4 workers 5 workers 6 workers 7 workers 8 workers 9 workers 2 4 6 8 #workers 1 2 3 4 5 6 7 8 9 Speedup linear speedup iteration-speedup time-speedup (a) Variance vs. Iteration \u00af L (b) Variance vs. Time (c) Speedup Figure 2: Variance with increasing number of workers. L = L0 \u00d7 \u03c4 for some \ufb01xed L0 and varying \u03c4\u2019s would result in the same MSE, which is \u221dL\u22122/3 0 . In the experiments we set C = 1/30, L0 = 500, \u03c4 = {1, 2, 5, 10, 15, 20}, and average over 200 runs to approximate the expectations in the MSE formula. As indicated in Figure 1, approximately the same MSE\u2019s are obtained after L0\u03c4 iterations for different \u03c4 values, consistent with the theory. Note since the stepsizes are set to make end points of the curves reach the optimal MSE\u2019s, the curves would not match the optimal MSE curves of \u03c4 2/3L\u22122/3 in general, except for the end points, i.e., they are lower bounded by \u03c4 2/3L\u22122/3. 10 1 10 2 10 3 10 4 #iterations: L 10 0 10 1 MSE = = 1 = = 2 = = 5 = = 10 = = 15 = = 20 Achieving the  same MSE level Figure 1: MSE vs. # iterations (L = 500 \u00d7 \u03c4) with increasing staleness \u03c4. Resulting in roughly the same MSE. Convergence speedup of the variance A Bayesian logistic regression model (BLR) is adopted to verify the variance convergence properties. We use the Adult dataset\u2021, a9a, with 32,561 training samples and 16,281 test samples. The test function is de\ufb01ned as the standard logistic loss. We average over 10 runs to estimate the expectation E\u02c6 \u03c6L in the variance. We use the single-server distributed architecture in Section 4, with multiple workers computing stale gradients in parallel. We plot the variance versus the average number of iterations on the workers (\u00af L) and the running time in Figure 2 (a) and (b), respectively. We can see that the variance drops faster with increasing number of workers. To quantitatively relate these results to the theory, Corollary 5 indicates that L1 L2 = W1 W2 , where (Wi, Li)2 i=1 means the number of workers and iterations at the same variance, i.e., a linear speedup is achieved. The iteration speedup and time speedup are plotted in Figure 2 (c), showing that the iteration speedup approximately scales linearly worker numbers, consistent with Corollary 5; whereas the time speedup deteriorates when the worker number is large due to high system latency. 5.2 Applications to deep learning We further test S2G-MCMC on Bayesian learning of deep neural networks. The distributed system is developed based on an MPI (message passing interface) extension of the popular Caffe package for deep learning [32]. We implement the SGHMC algorithm, with the point-to-point communications between servers and workers handled by the MPICH library.The algorithm is run on a cluster of \ufb01ve machines. Each machine is equipped with eight 3.60GHz Intel(R) Core(TM) i7-4790 CPU cores. We evaluate S2G-MCMC on the above BLR model and two deep convolutional neural networks (CNN). In all these models, zero mean and unit variance Gaussian priors are employed for the weights to capture weight uncertainties, an effective way to deal with over\ufb01tting [33]. We vary the number of servers S among {1, 3, 5, 7}, and the number of workers for each server from 1 to 9. LeNet for MNIST We modify the standard LeNet to a Bayesian setting for the MNIST dataset.LeNet consists of 2 convolutional layers, 2 max pool layers and 2 ReLU nonlinear layers, followed by 2 fully connected layers [34]. The detailed speci\ufb01cation can be found in Caffe. For simplicity, we use the default parameter setting speci\ufb01ed in Caffe, with the additional parameter B in SGHMC (Algorithm 1) set to (1 \u2212m), where m is the moment variable de\ufb01ned in the SGD algorithm in Caffe. \u2021http://www.csie.ntu.edu.tw/ cjlin/libsvmtools/datasets/binary.html. 7 \f#iterations 100 101 102 103 104 Loss 0.35 0.4 0.45 0.5 0.55 0.6 1 worker 2 workers 4 workers 6 workers 9 workers #iterations 2000 4000 6000 8000 10000 Loss 10-1 100 1 worker 2 workers 4 workers 6 workers 9 workers #iterations #104 0.5 1 1.5 2 Loss 0.8 1 1.2 1.4 1.6 1.8 2 2.2 1 worker 2 workers 4 workers 6 workers 9 workers Time (s) 10-1 100 101 Loss 0.35 0.4 0.45 0.5 0.55 0.6 1 worker 2 workers 4 workers 6 workers 9 workers Time (s) 0 100 200 300 400 500 600 Loss 10-1 100 1 worker 2 workers 4 workers 6 workers 9 workers Time (s) 0 1000 2000 3000 4000 Loss 0.8 1 1.2 1.4 1.6 1.8 2 2.2 1 worker 2 workers 4 workers 6 workers 9 workers Figure 3: Testing loss vs. #workers. From left to right, each column corresponds to the a9a, MNIST and CIFAR dataset, respectively. The loss is de\ufb01ned in the text. Cifar10-Quick net for CIFAR10 The Cifar10-Quick net consists of 3 convolutional layers, 3 max pool layers and 3 ReLU nonlinear layers, followed by 2 fully connected layers. The CIFAR-10 dataset consists of 60,000 color images of size 32\u00d732 in 10 classes, with 50,000 for training and 10,000 for testing.Similar to LeNet, default parameter setting speci\ufb01ed in Caffe is used. In these models, the test function is de\ufb01ned as the cross entropy of the softmax outputs {o1, \u00b7 \u00b7 \u00b7 , oN} for test data {(d1, y1), \u00b7 \u00b7 \u00b7 , (dN, yN)} with C classes, i.e., loss = \u2212PN i=1 oyi +N log PC c=1 eoc. Since the theory indicates a linear speedup on the decrease of variance w.r.t. the number of workers, this means for a single run of the models, the loss would converge faster to its expectation with increasing number of workers. The following experiments verify this intuition. 5.2.1 Single-server experiments We \ufb01rst test the single-server architecture in Section 4 on the three models. Because the expectations in the bias, MSE or variance are not analytically available in these complex models, we instead plot the loss versus average number of iterations (\u00af L de\ufb01ned in Section 4) on each worker and the running time in Figure 3. As mentioned above, faster decrease of the loss with more workers is expected. For the ease of visualization, we only plot the results with {1, 2, 4, 6, 9} workers; more detailed results are provided in Appendix I. We can see that generally the loss decreases faster with increasing number of workers. In the CIFAR-10 dataset, the \ufb01nal losses of 6 and 9 workers are worst than the one with 4 workers. It shows that the accuracy of the sample average suffers from the increased staleness due to the increased number of workers. Therefore a smaller step size h should be considered to maintain high accuracy when using a large number of workers. Note the 1-worker curves correspond to the standard SG-MCMC, whose loss decreases much slower due to high estimation variance, though in theory it has the same level of bias as the single-server architecture for a given number of iterations (they will converge to the same accuracy). 5.2.2 Multiple-server experiments Finally, we test the multiple-servers architecture on the same models. We use the same criterion as the single-server setting to measure the convergence behavior. The loss versus average number of iterations on each worker (\u00af L de\ufb01ned in Section 4) for the three datasets are plotted in Figure 4, where we vary the number of servers among {1, 3, 5, 7}, and use 2 workers for each server. The plots of loss versus time and using different number of workers for each server are provided in the Appendix. We can see that in the simple BLR model, multiple servers do not seem to show signi\ufb01cant speedup, probably due to the simplicity of the posterior, where the sample variance is too small for multiple 8 \fservers to take effect; while in the more complicated deep neural networks, using more servers results in a faster decrease of the loss, especially in the MNIST dataset. #iterations 100 101 102 103 104 Loss 0.35 0.4 0.45 0.5 0.55 0.6 1 server 3 servers 5 servers 7 servers #iterations 2000 4000 6000 8000 10000 Loss 10-1 100 1 server 3 servers 5 servers 7 servers #iterations #104 0.5 1 1.5 2 Loss 1 1.2 1.4 1.6 1.8 2 2.2 1 server 3 servers 5 servers 7 servers Figure 4: Testing loss vs. #servers. From left to right, each column corresponds to the a9a, MNIST and CIFAR dataset, respectively. The loss is de\ufb01ned in the text. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1512.07962v3",
+                    "title": "Bridging the Gap between Stochastic Gradient MCMC and Stochastic Optimization",
+                    "abstract": "Stochastic gradient Markov chain Monte Carlo (SG-MCMC) methods are Bayesian\nanalogs to popular stochastic optimization methods; however, this connection is\nnot well studied. We explore this relationship by applying simulated annealing\nto an SGMCMC algorithm. Furthermore, we extend recent SG-MCMC methods with two\nkey components: i) adaptive preconditioners (as in ADAgrad or RMSprop), and ii)\nadaptive element-wise momentum weights. The zero-temperature limit gives a\nnovel stochastic optimization method with adaptive element-wise momentum\nweights, while conventional optimization methods only have a shared, static\nmomentum weight. Under certain assumptions, our theoretical analysis suggests\nthe proposed simulated annealing approach converges close to the global optima.\nExperiments on several deep neural network models show state-of-the-art results\ncompared to related stochastic optimization algorithms.",
+                    "authors": "Changyou Chen, David Carlson, Zhe Gan, Chunyuan Li, Lawrence Carin",
+                    "published": "2015-12-25",
+                    "updated": "2016-08-05",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Machine learning has made signi\ufb01cant recent strides due to large-scale learning applied to \u201cbig data\u201d. Large-scale learning is typically performed with stochastic optimization, and the most common method is stochastic gradient descent (SGD) (Bottou, 2010). Stochastic optimization methods are devoted to obtaining a (local) optima of an objective function. Alternatively, Bayesian methods aim to compute the expectation of a test function over the posterior disAppearing in Proceedings of the 19th International Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 41. Copyright 2016 by the authors. tribution. At \ufb01rst glance, these methods appear to be distinct, independent approaches to learning. However, even the celebrated Gibbs sampler was \ufb01rst introduced to statistics as a simulated annealing method for maximum a posteriori estimation (i.e., \ufb01nding an optima) (Geman and Geman, 1984). Recent work on large-scale Bayesian learning has focused on incorporating the speed and low-memory costs from stochastic optimization. These approaches are referred to as stochastic gradient Markov chain Monte Carlo (SG-MCMC) methods. Well-known SGMCMC methods include stochastic gradient Langevin dynamics (SGLD) (Welling and Teh, 2011), stochastic gradient Hamiltonian Monte Carlo (SGHMC) (Chen et al., 2014), and stochastic gradient thermostats (SGNHT) (Ding et al., 2014). SG-MCMC has become increasingly popular in the literature due to practical successes, ease of implementation, and theoretical convergence properties (Teh et al., 2014; Vollmer et al., 2015; Chen et al., 2015). There are obvious structural similarities between SGMCMC algorithms and stochastic optimization methods. For example, SGLD resembles SGD with additive Gaussian noise. SGHMC resembles SGD with momentum (Rumelhart et al., 1986), adding additive Gaussian noise when updating the momentum terms (Chen et al., 2014). These similarities are detailed in Section 2. Despite these structural similarities, the theory is unclear on how additive Gaussian noise di\ufb00erentiates a Bayesian algorithm from its optimization analog. Just as classical sampling methods were originally used for optimization (Geman and Geman, 1984), we directly address using SG-MCMC algorithms for optimization. A major bene\ufb01t of adapting these schemes is that Bayesian learning is (in theory) able to fully explore the parameter space. Thus it may \ufb01nd a better local optima, if not the global optima, for a non-convex objective function. Speci\ufb01cally, in this work we \ufb01rst extend the recently proposed multivariate stochastic gradient thermostat arXiv:1512.07962v3  [stat.ML]  5 Aug 2016 \fBridging the Gap between SG-MCMC and Stochastic Optimization algorithm (Gan et al., 2015) with Riemannian information geometry, which results in an adaptive preconditioning and momentum scheme with analogs to Adam (Kingma and Ba, 2015) and RMSprop (Tieleman and Hinton, 2012). We propose an annealing scheme on the system temperature to move from a Bayesian method to a stochastic optimization method. We call the proposed algorithm Stochastic AnNealing Thermostats with Adaptive momentum (Santa). We show that in the temperature limit, Santa recovers the SGD with momentum algorithm except that: i) adaptive preconditioners are used when updating both model and momentum parameters; ii) each parameter has an individual, learned momentum parameter. Adaptive preconditioners and momentums are desirable in practice because of their ability to deal with uneven, dynamic curvature (Dauphin et al., 2015). For completeness, we \ufb01rst review related algorithms in Section 2, and present our novel algorithm in Section 3. We develop theory to analyze convergence properties of our algorithm, suggesting that Santa is able to \ufb01nd a solution for an (non-convex) objective function close to its global optima, shown in Section 4. The theory is based on the analysis from stochastic di\ufb00erential equations (Teh et al., 2014; Chen et al., 2015), and presents results on bias and variance of the annealed Markov chain. This is a fundamentally di\ufb00erent approach from the traditional convergence explored in stochastic optimization, or the regret bounds used in online optimization. We note we can adapt the regret bound of Adam (Kingma and Ba, 2015) for our zero-temperature algorithm (with a few trivial modi\ufb01cations) for a convex problem, as shown in Supplementary Section F. However, this neither addresses non-convexity nor the annealing scheme that our analysis does. In addition to theory, we demonstrate e\ufb00ective empirical performance on a variety of deep neural networks (DNNs), achieving the best performance compared to all competing algorithms for the same model size. This is shown in Section 5. The code is publicly available at https://github.com/cchangyou/Santa. 2 Preliminaries Throughout this paper, we denote vectors as bold, lower-case letters, and matrices as bold, upper-case letters. We use \u2299for element-wise multiplication, and \u2298as element-wise division; \u221a\u00b7 denotes the elementwise square root when applied to vectors or matricies. We reserve (\u00b7)1/2 for the standard matrix square root. Ip is the p \u00d7 p identity matrix, 1 is an all-ones vector. The goal of an optimization algorithm is to minimize an objective function U(\u03b8) that corresponds to a (non-convex) model of interest. In a Bayesian model, this corresponds to the potential energy de\ufb01ned as the negative log-posterior, U(\u03b8) \u225c\u2212log p(\u03b8)\u2212 PN n=1 log p(xn |\u03b8). Here \u03b8 \u2208Rp are the model parameters, and {xn}n=1,...,N are the d-dimensional observed data; p(\u03b8) corresponds to the prior and p(xn |\u03b8) is a likelihood term for the nth observation. In optimization, \u2212PN n=1 log p(xn |\u03b8) is typically referred to as the loss function, and \u2212log p(\u03b8) as a regularizer. In large-scale learning, N is prohibitively large. This motivates the use of stochastic approximations. We denote the stochastic approximation \u02dc Ut(\u03b8) \u225c \u2212log p(\u03b8) \u2212N m Pm j=1 log p(xij |\u03b8), where (i1, \u00b7 \u00b7 \u00b7 , im) is a random subset of the set {1, 2, \u00b7 \u00b7 \u00b7 , N}. The gradient on this minibatch is denoted as \u02dc f t(\u03b8) = \u2207\u02dc Ut(\u03b8), which is an unbiased estimate of the true gradient. A standard approach to learning is SGD, where parameter updates are given by \u03b8t = \u03b8t\u22121 \u2212\u03b7t\u02dc f t\u22121(\u03b8) with \u03b7t the learning rate. This is guaranteed to converge to a local minima under mild conditions (Bottou, 2010). The SG-MCMC analog to this is SGLD, with updates \u03b8t = \u03b8t\u22121 \u2212\u03b7t\u02dc f t\u22121(\u03b8) + \u221a2\u03b7t\u03b6t. The additional term is a standard normal random vector, \u03b6t \u223cN(0, Ip) (Welling and Teh, 2011). The SGLD method draws approximate posterior samples instead of obtaining a local minima. Using momentum in stochastic optimization is important in learning deep models (Sutskever et al., 2013). This motivates SG-MCMC algorithms with momentum. The standard SGD with momentum (SGD-M) approach introduces an auxiliary variable ut \u2208Rp to represent the momentum. Given a momentum weight \u03b1, the updates are \u03b8t = \u03b8t\u22121 + \u03b7tut and ut = (1 \u2212\u03b1)ut\u22121 \u2212\u02dc f t\u22121(\u03b8). A Bayesian analog is SGHMC (Chen et al., 2014) or multivariate SGNHT (mSGNHT) (Gan et al., 2015). In mSGNHT, each parameter has a unique momentum weight \u03b1t \u2208Rp that is learned during the sampling sequence. The momentum weights are updated to maintain the system temperature 1/\u03b2. An inverse temperature of \u03b2 = 1 corresponds to the posterior. This algorithm has updates \u03b8t = \u03b8t\u22121 + \u03b7tut, ut = (1 \u2212\u03b7t\u03b1t\u22121) \u2299 ut\u22121 \u2212\u03b7t\u02dc f t\u22121(\u03b8) + p 2\u03b7t/\u03b2\u03b6t. The main di\ufb00erence is the additive Gaussian noise and step-size dependent momentum update. The weights have updates \u03b1t = \u03b1t\u22121 + \u03b7t((ut \u2299ut) \u22121/\u03b2), which matches the kinetic energy to the system temperature. A recent idea in stochastic optimization is to use an adaptive preconditioner, also known as a variable metric, to improve convergence rates. Both ADAgrad (Duchi et al., 2011) and Adam (Kingma and Ba, 2015) adapt to the local geometry with a regret bound of O( \u221a N). Adam adds momentum as well through moment smoothing. RMSprop (Tieleman and Hinton, 2012), Adadelta (Zeiler, 2012), and RMSspectral \fC. Chen, D. Carlson, Z. Gan, C. Li, and L. Carin Algorithm 1: Santa with the Euler scheme Input: \u03b7t (learning rate), \u03c3, \u03bb, burnin, \u03b2 = {\u03b21, \u03b22, \u00b7 \u00b7 \u00b7 } \u2192\u221e, {\u03b6t \u2208Rp} \u223cN(0, Ip). Initialize \u03b80, u0 = \u221a\u03b7 \u00d7 N(0, I), \u03b10 = \u221a\u03b7C, v0 = 0 ; for t = 1, 2, . . . do Evaluate \u02dc f t \u225c\u2207\u03b8 \u02dc U(\u03b8t\u22121) on the tth mini-batch; vt = \u03c3 vt\u22121 + 1\u2212\u03c3 N 2 \u02dc f t \u2299\u02dc f t ; gt = 1 \u2298 p \u03bb + \u221avt ; if t < burnin then /* exploration */ \u03b1t = \u03b1t\u22121 + (ut\u22121 \u2299ut\u22121 \u2212\u03b7/\u03b2t); ut = \u03b7 \u03b2t \u00001 \u2212gt\u22121 \u2298gt \u0001 \u2298ut\u22121 + q 2\u03b7 \u03b2t gt\u22121 \u2299\u03b6t else /* refinement */ \u03b1t = \u03b1t\u22121; ut = 0; end ut = ut + (1 \u2212\u03b1t) \u2299ut\u22121 \u2212\u03b7 gt \u2299\u02dc f t; \u03b8t = \u03b8t\u22121 + gt \u2299ut; end (Carlson et al., 2015) are similar methods with preconditioners. Our method introduces adaptive momentum and preconditioners to the SG-MCMC. This di\ufb00ers from stochastic optimization in implementation and theory, and is novel in SG-MCMC. Simulated annealing (Kirkpatrick et al., 1983; \u02c7 Cern\u00b4 y, 1985) is well-established as a way of acquiring a local mode by moving from a high-temperature, \ufb02at surface to a low-temperature, peaky surface. It has been explored in the context of MCMC, including reversible jump MCMC (Andrieu et al., 2000), annealed important sampling (Neal, 2001) and parallel tempering (Li et al., 2009). Traditional algorithms are based on Metropolis\u2013Hastings sampling, which require computationally expensive accept-reject steps. Recent work has applied simulated annealing to large-scale learning through mini-batch based annealing (van de Meent et al., 2014; Obermeyer et al., 2014). Our approach incorporates annealing into SG-MCMC with its inherent speed and mini-batch nature. 3 The Santa Algorithm Santa extends the mSGNHT algorithm with preconditioners and a simulated annealing scheme. A simple pseudocode is shown in Algorithm 1, or a more complex, but higher accuracy version, is shown in Algorithm 2, and we detail the steps below. The \ufb01rst extension we consider is the use of adaptive preconditioners. Preconditioning has been proven critical for fast convergence in both stochastic optimization (Dauphin et al., 2015) and SG-MCMC algorithms (Patterson and Teh, 2013). In the MCMC literature, preconditioning is alternatively referred to as Riemannian information geometry (Patterson and Teh, 2013). We denote the preconditioner as {Gt \u2208Rp\u00d7p}. A popular choice in SG-MCMC is the Fisher information matrix (Girolami and Calderhead, 2011). Unfortunately, this approach is computationally prohibitive for many models of interest. To avoid this problem, we adopt the preconditioner from RMSprop and Adam, which uses a vector {gt \u2208Rp} to approximate the diagonal of the Fisher information matrixes (Li et al., 2016a). The construction sequentially updates the preconditioner based on current and historical gradients with a smoothing parameter \u03c3, and is shown as part of Algorithm 1. While this approach will not capture the Riemannian geometry as e\ufb00ectively as the Fisher information matrix, it is computationally e\ufb03cient. Santa also introduces an annealing scheme on system temperatures. As discussed in Section 2, mSGNHT naturally accounts for a varying temperature by matching the particle momentum to the system temperature. We introduce \u03b2 = {\u03b21, \u03b22, \u00b7 \u00b7 \u00b7 }, a sequence of inverse temperature variables with \u03b2i < \u03b2j for i < j and limi\u2192\u221e\u03b2i = \u221e. The in\ufb01nite case corresponds to the zero-temperature limit, where SGMCMCs become deterministic optimization methods. The annealing scheme leads to two stages: the exploration and the re\ufb01nement stages. The exploration stage updates all parameters based on an annealed sequence of stochastic dynamic systems (see Section 4 for more details). This stage is able to explore the parameter space e\ufb03ciently, escape poor local modes, and \ufb01nally converge close to the global mode. The re\ufb01nement stage corresponds to the temperature limit, i.e., \u03b2n \u2192\u221e. In the temperature limit, the momentum weight updates vanish and it becomes a stochastic optimization algorithm. We propose two update schemes to solve the corresponding stochastic di\ufb00erential equations: the Euler scheme and the symmetric splitting scheme (SSS). The Euler scheme has simpler updates, as detailed in Algorithm 1; while SSS endows increased accuracy (Chen et al., 2015) with a slight increase in overhead computation, as shown in Algorithm 2. Section 4.1 elaborates on the details of these two schemes. We recommend the use of SSS, but the Euler scheme is simpler to implement and compare to known algorithms. Practical considerations According to Section 4, the exploration stage helps the algorithm traverse the parameter space following the posterior curve as accurate as possible. For optimization, slightly biased samples do not a\ufb00ect the \ufb01nal solution. As a result, \fBridging the Gap between SG-MCMC and Stochastic Optimization Algorithm 2: Santa with SSS Input: \u03b7t (learning rate), \u03c3, \u03bb, burnin, \u03b2 = {\u03b21, \u03b22, \u00b7 \u00b7 \u00b7 } \u2192\u221e, {\u03b6t \u2208Rp} \u223cN(0, Ip). Initialize \u03b80, u0 = \u221a\u03b7 \u00d7 N(0, I), \u03b10 = \u221a\u03b7C, v0 = 0 ; for t = 1, 2, . . . do Evaluate \u02dc f t \u225c\u2207\u03b8 \u02dc U(\u03b8t\u22121) on the tth mini-batch; vt = \u03c3 vt\u22121 + 1\u2212\u03c3 N 2 \u02dc f t \u2299\u02dc f t ; gt = 1 \u2298 p \u03bb + \u221avt ; \u03b8t = \u03b8t\u22121 + gt \u2299ut\u22121 /2; if t < burnin then /* exploration */ \u03b1t = \u03b1t\u22121 + (ut\u22121 \u2299ut\u22121 \u2212\u03b7/\u03b2t) /2; ut = exp (\u2212\u03b1t/2) \u2299ut\u22121; ut = ut \u2212gt \u2299\u02dc f t\u03b7 + p 2 gt\u22121 \u03b7/\u03b2t \u2299\u03b6t + \u03b7/\u03b2t \u00001 \u2212gt\u22121 \u2298gt \u0001 \u2298ut\u22121; ut = exp (\u2212\u03b1t/2) \u2299ut; \u03b1t = \u03b1t + (ut \u2299ut \u2212\u03b7/\u03b2t) /2; else /* refinement */ \u03b1t = \u03b1t\u22121; ut = exp (\u2212\u03b1t/2) \u2299ut\u22121; ut = ut \u2212gt \u2299\u02dc f t\u03b7; ut = exp (\u2212\u03b1t/2) \u2299ut; end \u03b8t = \u03b8t + gt \u2299ut /2; end the term consisting of (1 \u2212gt\u22121 \u2298gt) in the algorithm (which is an approximation term, see Section 4.1) is ignored. We found no decreasing performance in our experiments. Furthermore, the term gt\u22121 associated with the Gaussian noise could be replaced with a \ufb01xed constant without a\ufb00ecting the algorithm. 4 Theoretical Foundation In this section we present the stochastic di\ufb00erential equations (SDEs) that correspond to the Santa algorithm. We \ufb01rst introduce the general SDE framework, then describe the exploration stage in Section 4.1 and the re\ufb01nement stage in Section 4.2. We give the convergence properties of the numerical scheme in Section 4.3. This theory uses tools from the SDE literature and extends the mSGNHT theory (Ding et al., 2014; Gan et al., 2015). The SDEs are presented with re-parameterized p = u /\u03b71/2, \u039e = diag(\u03b1)/\u03b71/2, as in Ding et al. (2014). The SDEs describe the motion of a particle in a system where \u03b8 is the location and p is the momentum. In mSGNHT, the particle is driven by a force \u2212\u2207\u03b8 \u02dc Ut(\u03b8) at time t. The stationary distribution of \u03b8 corresponds to the model posterior (Gan et al., 2015). Our critical extension is the use of Riemannian information geometry, important for fast convergence (Patterson and Teh, 2013). Given an inverse temperature \u03b2, the system is described by the following SDEs1: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 d\u03b8 = G1(\u03b8) p dt d p = \u0010 \u2212G1(\u03b8)\u2207\u03b8U(\u03b8) \u2212\u039e p + 1 \u03b2 \u2207\u03b8G1(\u03b8) +G1(\u03b8)(\u039e \u2212G2(\u03b8))\u2207\u03b8G2(\u03b8)) dt + ( 2 \u03b2 G2(\u03b8)) 1 2 dw d\u039e = \u0010 Q \u22121 \u03b2 I \u0011 dt , (1) where Q = diag(p \u2299p), w is standard Brownian motion, G1(\u03b8) encodes geometric information of the potential energy U(\u03b8), and G2(\u03b8) characterizes the manifold geometry of the Brownian motion. Note G2(\u03b8) may be the same as G1(\u03b8) for the same Riemannian manifold. We call G1(\u03b8) and G2(\u03b8) Riemannian metrics, which are commonly de\ufb01ned by the Fisher information matrix (Girolami and Calderhead, 2011). We use the RMSprop preconditioner (with updates from Algorithm 1) for computational feasibility. Using the Fokker-Plank equation (Risken, 1989), we show that the marginal stationary distribution of (1) corresponds to the posterior distribution. Lemma 1. Denote A : B \u225ctr n AT B o .The stationary distribution of (1) is: p\u03b2(\u03b8, p, \u039e) \u221d e\u2212\u03b2U(\u03b8)\u2212\u03b2 2 pT p \u2212\u03b2 2 (\u039e\u2212G2(\u03b8)):(\u039e\u2212G2(\u03b8)) . (2) An inverse temperature \u03b2 = 1 corresponds to the standard Bayesian posterior. We note that p in (1) has additional dependencies on G1 and G2 compared to Gan et al. (2015) that must be accounted for. \u039e p introduces friction into the system so that the particle does not move too far away by the random force; the terms \u2207\u03b8G1(\u03b8) and \u2207\u03b8G2(\u03b8) penalize the in\ufb02uences of the Riemannian metrics so that the stationary distribution remains invariant. 4.1 Exploration The \ufb01rst stage of Santa, exploration, explores the parameter space to obtain parameters near the global mode of an objective function2. This approach applies ideas from simulated annealing (Kirkpatrick et al., 1983). Speci\ufb01cally, the inverse temperature \u03b2 is slowly annealed to temperature zero to freeze the particles at the global mode. Minimizing U(\u03b8) is equivalent to sampling from the zero-temperature limit p\u03b2(\u03b8) \u225c 1 Z\u03b2 e\u2212\u03b2U(\u03b8) (propor1We abuse notation for conciseness. Here, \u2207\u03b8 G(\u03b8) is a vector with the i-th element being P j \u2207\u03b8j Gij(\u03b8). 2This requires an ergodic algorithm. While ergodicity is not straightforward to check, we follow most MCMC work and assume it holds in our algorithm. \fC. Chen, D. Carlson, Z. Gan, C. Li, and L. Carin tional to (2)), with Z\u03b2 being the normalization constant such that p\u03b2(\u03b8) is a valid distribution. We construct a Markov chain that sequentially transits from high temperatures to low temperatures. At the state equilibrium, the chain reaches the temperature limit with marginal stationary distribution \u03c10(\u03b8) \u225c lim\u03b2\u2192\u221ee\u2212\u03b2U(\u03b8), a point mass3 located at the global mode of U(\u03b8). Speci\ufb01cally, we \ufb01rst de\ufb01ne a sequence of inverse temperatures, (\u03b21, \u03b22, \u00b7 \u00b7 \u00b7 , \u03b2L), such that \u03b2L is large enough4. For each time t, we generate a sample according to the SDE system (1) with temperature 1 \u03b2t , conditioned on the sample from the previous temperature, 1 \u03b2t\u22121 . We call this procedure annealing thermostats to denote the analog to simulated annealing. Generating approximate samples Generating exact samples from (1) is infeasible for general models. One well-known numerical approach is the Euler scheme in Algorithm 1. The Euler scheme is a 1storder method with relatively high approximation error (Chen et al., 2015). We increase accuracy by implementing the symmetric splitting scheme (SSS) (Chen et al., 2015; Li et al., 2016b). The idea of SSS is to split an infeasible SDE into several sub-SDEs, where each sub-SDE is analytically solvable; approximate samples are generated by sequentially evolving parameters via these sub-SDEs. Speci\ufb01cally, in Santa, we split (1) into the following three sub-SDEs: A : \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 d\u03b8 = G1(\u03b8) p dt d p = 0 d\u039e = \u0010 Q \u22121 \u03b2 I \u0011 dt , B : \uf8f1 \uf8f2 \uf8f3 d\u03b8 = 0 d p = \u2212\u039e p dt d\u039e = 0 , O : \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 d\u03b8 = 0 d p = \u0010 \u2212G1(\u03b8)\u2207\u03b8U(\u03b8) + 1 \u03b2 \u2207\u03b8G1(\u03b8) +G1(\u03b8)(\u039e \u2212G2(\u03b8))\u2207\u03b8G2(\u03b8)) dt + ( 2 \u03b2 G2(\u03b8)) 1 2 dw d\u039e = 0 We then update the sub-SDEs in order A-B-O-B-A to generative approximate samples (Chen et al., 2015). This uses half-steps h/2 on the A and B updates5, and full steps h in the O update. This is analogous to the leapfrog steps in Hamiltonian Monte Carlo (Neal, 2011). Update equations are given in the Supplementary Section A. The resulting parameters then serve as an approximate sample from the posterior distribution with the inverse temperature of \u03b2. Replacing G1 and G2 with the RMSprop preconditioners gives Algorithm 2. These updates require approximations to \u2207\u03b8G1(\u03b8) and \u2207\u03b8G2(\u03b8), addressed below. 3The sampler samples a uniform distribution over global modes, or a point mass if the mode is unique. We assume uniqueness and say point mass for clarity henceforth. 4Due to numerical issues, it is impossible to set \u03b2L to in\ufb01nity; we thus assign a large enough value for it and handle the in\ufb01nity case in the re\ufb01nement stage. 5As in Ding et al. (2014), we de\ufb01ne h = \u221a\u03b7. Approximate calculation for \u2207\u03b8 G1(\u03b8) We propose a computationally e\ufb03cient approximation for calculating the derivative vector \u2207\u03b8 G1(\u03b8) based on the de\ufb01nition. Speci\ufb01cally, for the i-th element of \u2207\u03b8 G1(\u03b8) at the t-th iteration, denoted as (\u2207\u03b8 Gt 1(\u03b8))i, it is approximated as: (\u2207\u03b8 Gt 1(\u03b8))i A1 \u2248 X j (Gt 1(\u03b8))ij \u2212(Gt\u22121 1 (\u03b8))ij \u03b8tj \u2212\u03b8(t\u22121)j A2 = X j (\u2206Gt 1)ij (Gt 1(\u03b8) pt\u22121)jh = X j (\u2206Gt 1)ij (Gt 1(\u03b8) ut\u22121)j where \u2206Gt 1 \u225cGt 1 \u2212Gt\u22121 1 . Step A1 follows by the de\ufb01nition of a derivative, and A2 by using the update equation for \u03b8t, i.e., \u03b8t = \u03b8t\u22121+Gt 1(\u03b8) pt\u22121 h. According to Taylor\u2019s theory, the approximation error for the \u2207\u03b8 G1(\u03b8) is O(h), e.g., X i \f \f \f \f \f X j (\u2206Gt 1)ij (Gt 1(\u03b8) ut\u22121)j \u2212(\u2207\u03b8 Gt 1(\u03b8))i \f \f \f \f \f \u2264Bth , (3) for some positive constant Bt. The approximation error is negligible in term of convergence behaviors because it can be absorbed into the stochastic gradients error. Formal theoretical analysis on convergence behaviors with this approximation is given in later sections. Using similar methods, \u2207\u03b8Gt 2(\u03b8) is also approximately calculated. 4.2 Re\ufb01nement The re\ufb01nement stage corresponds to the zerotemperature limit of the exploration stage, where \u039e is learned. We show that in the limit Santa gives signi\ufb01cantly simpli\ufb01ed updates, leading to an stochastic optimization algorithm similar to Adam or SGD-M. We assume that the Markov chain has reached its equilibrium after the exploration stage. In the zerotemperature limit, some terms in the SDE (1) vanish. First, as \u03b2 \u2192\u221e, the term 1 \u03b2 \u2207\u03b8 G1(\u03b8) and the variance term for the Brownian motion approach 0. As well, the thermostat variable \u039e approaches G2(\u03b8), so the term G1(\u03b8)(\u039e \u2212G2(\u03b8))\u2207\u03b8 G2(\u03b8) vanishes. The stationary distribution in (2) implies E Qii \u225cE p2 i \u2192 0, which makes the SDE for \u039e in (1) vanish. As a result, in the re\ufb01nement stage, only \u03b8 and p need to be updated. The Euler scheme for this is shown in Algorithm 1, and the symmetric splitting scheme is shown in Algorithm 2. Relation to stochastic optimization algorithms In the re\ufb01nement stage Santa is a stochastic optimization algorithm. This relation is easier seen with the Euler scheme in Algorithm 1. Compared with SGD-M (Rumelhart et al., 1986), Santa has both adaptive gradient and adaptive momentum updates. Unlike Adagrad (Duchi et al., 2011) and RMSprop (Tieleman and \fBridging the Gap between SG-MCMC and Stochastic Optimization Hinton, 2012), re\ufb01nement Santa is a momentum based algorithm. The recently proposed Adam algorithm (Kingma and Ba, 2015) incorporates momentum and preconditioning in what is denoted as \u201cadaptive moments.\u201d We show in Supplementary Section F that a constant step size combined with a change of variables nearly recovers the Adam algorithm with element-wise momentum weights. For these reasons, Santa serves as a more general stochastic optimization algorithm that extends all current algorithms. As well, for a convex problem and a few trivial algorithmic changes, the regret bound of Adam holds for re\ufb01nement Santa, which is O( \u221a T), as detailed in Supplementary Section F. However, our analysis is focused on non-convex problems that do not \ufb01t in the regret bound formulation. 4.3 Convergence properties Our convergence properties are based on the framework of Chen et al. (2015). The proofs for all theorems are given in the Supplementary Material. We focus on the exploration stage of the algorithm. Using the Monotone Convergence argument (Schechter, 1997), the re\ufb01nement stage convergence is obtained by taking the temperature limit from the results of the exploration stage. We emphasize that our approach differs from conventional stochastic optimization or online optimization approaches. Our convergence rate is weaker than many stochastic optimization methods, including SGD; however, our analysis applies to nonconvex problems, whereas traditionally convergence rates only apply to convex problems. The goal of Santa is to obtain \u03b8\u2217such that \u03b8\u2217= argmin\u03b8 U(\u03b8). Let {\u03b81, \u00b7 \u00b7 \u00b7 , \u03b8L} be a sequence of parameters collected from the algorithm. De\ufb01ne \u02c6 U \u225c 1 L PL t=1 U(\u03b8t) as the sample average, \u00af U \u225cU(\u03b8\u2217) the global optima of U(\u03b8). As in Chen et al. (2015), we require certain assumptions on the potential energy U. To show these assumptions, we \ufb01rst de\ufb01ne a functional \u03c8t for each t that solves the following Poisson equation: Lt\u03c8t(\u03b8t) = U(\u03b8t) \u2212\u00af U , (4) Lt is the generator of the SDE system (1) in the t-th iteration, de\ufb01ned Ltf(xt) \u225climh\u21920+ E[f(xt+h)]\u2212f(xt) h where xt \u225c(\u03b8t, pt, \u039et), f : R3p \u2192R is a compactly supported twice di\ufb00erentiable function. The solution functional \u03c8t(\u03b8t) characterizes the di\ufb00erence between U(\u03b8t) and the global optima \u00af U for every \u03b8t. As shown in Mattingly et al. (2010), (4) typically possesses a unique solution, which is at least as smooth as U under the elliptic or hypoelliptic settings. We assume \u03c8t is bounded and smooth, as described below. Assumption 1. \u03c8t and its up to 3rd-order derivatives, Dk\u03c8t, are bounded by a function V(\u03b8, p, \u039e), i.e., \u2225Dk\u03c8\u2225\u2264CkVrk for k = (0, 1, 2, 3), Ck, rk > 0. Furthermore, the expectation of V is bounded: supt EVr(\u03b8, p, \u039e) < \u221e, and V is smooth such that sups\u2208(0,1) Vr (sx + (1 \u2212s) y) \u2264C (Vr (x) + Vr (y)), \u2200x \u2208R3p, y \u2208R3p, r \u2264max{2rk} for some C > 0. Let \u2206U(\u03b8) \u225cU(\u03b8) \u2212U(\u03b8\u2217). Further de\ufb01ne an operator \u2206Vt = \u0010 G1(\u03b8)(\u2207\u03b8 \u02dc Ut \u2212\u2207\u03b8U) + Bt \u03b2t \u0011 \u00b7 \u2207p for each t, where Bt is from (3). Theorem 2 depicts the closeness of \u02c6 U to the global optima \u00af U in term of bias and mean square error (MSE) de\ufb01ned below. Theorem 2. Let \u2225\u00b7\u2225be the operator norm. Under Assumption 1, the bias and MSE of the exploration stage in Santa with respect to the global optima for L steps with stepsize h is bounded, for some constants C > 0 and D > 0, with: Bias: \f \f \fE \u02c6 U \u2212\u00af U \f \f \f \u2264Ce\u2212U(\u03b8\u2217)   1 L L X t=1 Z e\u2212\u03b2t\u2206U(\u03b8)d\u03b8 ! + D \u0012 1 Lh + P t \u2225E\u2206Vt\u2225 L + h2 \u0013 . MSE: E \u0010 \u02c6 U \u2212\u00af U \u00112 \u2264C2e\u22122U(\u03b8\u2217)   1 L L X t=1 Z e\u2212\u03b2t\u2206U(\u03b8)d\u03b8 !2 + D2   1 L P t E \u2225\u2206Vt\u22252 L + 1 Lh + h4 ! . Both bounds for the bias and MSE have two parts. The \ufb01rst part contains integration terms, which characterizes the distance between the global optima, e\u2212U(\u03b8\u2217), and the unnormalized annealing distributions, e\u2212\u03b2tU(\u03b8), decreasing to zero exponentially fast with increasing \u03b2; the remaining part characterizes the distance between the sample average and the annealing posterior average. This shares a similar form as in general SG-MCMC algorithms (Chen et al., 2015), and can be controlled to converge. Furthermore, the term P t\u2225E\u2206Vt\u2225 L in the bias vanishes as long as the sum of the annealing sequence {\u03b2t} is \ufb01nite6, indicating that the gradient approximation for \u2207\u03b8G1(\u03b8) in Section 4.1 does not a\ufb00ect the bias of the algorithm. Similar arguments apply for the MSE bound. To get convergence results right before the re\ufb01nement stage, let a sequence of functions {gm} be de\ufb01ned as gm \u225c\u22121 L PL+m\u22121 l=m e\u2212\u03b2l \u02c6 U(\u03b8); it is easy to see that {gm} satis\ufb01es gm1 < gm2 for m1 < m2, and limm\u2192\u221egm = 0. According to the Monotone Convergence Theorem (Schechter, 1997), the bias and MSE in the limit exists, leading to Corollary 3. Corollary 3. Under Assumptions 1, the bias and MSE of the re\ufb01nement stage in Santa with respect 6In practice we might not need to care about this constraint because a small bias in the exploration stage does not a\ufb00ect convergence of the re\ufb01nement stage. \fC. Chen, D. Carlson, Z. Gan, C. Li, and L. Carin to the global optima for L steps with stepsize h are bounded, for some constants D1 > 0, D2 > 0, as Bias: \f \f \fE \u02c6 U \u2212\u00af U \f \f \f \u2264D1 \u0012 1 Lh + P t \u2225E\u2206Vt\u2225 L + h2 \u0013 MSE: E \u0010 \u02c6 U \u2212\u00af U \u00112 \u2264D2   1 L P t E \u2225\u2206Vt\u22252 L + 1 Lh + h4 ! Corollary 3 implies that in the re\ufb01nement stage, the discrepancy between annealing distributions and the global optima vanishes, leaving only errors from discretized simulations of the SDEs, similar to the result of general SG-MCMC (Chen et al., 2015). We note that after exploration, Santa becomes a pure stochastic optimization algorithm, thus convergence results in term of regret bounds can also be derived; refer to Supplementary Section F for more details. 5 Experiments 5.1 Illustration In order to demonstrate that Santa is able to achieve the global mode of an objective function, we consider the double-well potential (Ding et al., 2014), U(\u03b8) = (\u03b8 + 4)(\u03b8 + 1)(\u03b8 \u22121)(\u03b8 \u22123)/14 + 0.5 . As shown in Figure 1 (left), the double-well potential has two modes, located at \u03b8 = \u22123 and \u03b8 = 2, with the global optima at \u03b8 = \u22123. We use a decreasing learning rate ht = t\u22120.3/10, and the annealing sequence is set to \u03b2t = t2. To make the optimization more challenging, we initialize the parameter at \u03b80 = 4, close to the local mode. The evolution of \u03b8 with respect to iterations is shown in Figure 1(right). As can be seen, \u03b8 \ufb01rst moves to the local mode but quickly jumps out and moves to the global mode in the exploration stage (\ufb01rst half iterations); in the re\ufb01nement stage, \u03b8 quickly converges to the global mode and sticks to it afterwards. In contrast, RMSprop is trapped on the local optima, and convergences slower than Santa at the beginning. 6 4 2 0 2 4 6 \u03b8 0 5 10 15 20 exp(\u2212U(\u03b8)) 0 1000 2000 3000 4000 5000 iterations 6 4 2 0 2 4 \u03b8 Santa RMSprop Figure 1: (Left) Double-well potential. (Right) The evolution of \u03b8 using Santa and RMSprop algorithms. 0 20 40 60 80 100 Epochs 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Test Error (%) SGD SGD-M SGLD RMSprop Adam Santa 0 20 40 60 80 100 Epochs 0.4 0.6 0.8 1.0 1.2 1.4 Test Error (%) Figure 2: Learning curves of di\ufb00erent algorithms on MNIST. (Left) FNN with size of 400. (Right) CNN. 5.2 Feedforward neural networks We \ufb01rst test Santa on the Feedforward Neural Network (FNN) with recti\ufb01ed linear units (ReLU). We test two-layer models with network sizes 784-X-X-10, where X is the number of hidden units for each layer; 100 epochs are used. For variants of Santa, we denote Santa-E as Santa with a Euler scheme illustrated in Algorithm 1, Santa-r as Santa running only on the re\ufb01nement stage, but with updates on \u03b1 as in the exploration stage. We compare Santa with SGD, SGDM, RMSprop, Adam, SGD with dropout, SGLD and Bayes by Backprop (Blundell et al., 2015). We use a grid search to obtain good learning rates for each algorithm, resulting in 4 \u00d7 10\u22126 for Santa, 5 \u00d7 10\u22124 for RMSprop, 10\u22123 for Adam, and 5 \u00d7 10\u22121 for SGD, SGD-M and SGLD. We choose an annealing schedule of \u03b2t = At\u03b3 with A = 1 and \u03b3 selected from 0.1 to 1 with an interval of 0.1. For simplicity, the exploration is set to take half of total iterations. We test the algorithms on the standard MNIST dataset, which contains 28 \u00d7 28 handwritten digital images from 10 classes with 60, 000 training samples and 10, 000 test samples. The network size (X-X) is set to 400-400 and 800-800, and test classi\ufb01cation errors are shown in Table 1. Santa show improved stateof-the-art performance amongst all algorithms. The Euler scheme shows a slight decrease in performance, due to the integration error when solving the SDE. Santa without exploration (i.e., Santa-r) still performs relatively well. Learning curves are plotted in Figure 2, showing that Santa converges as fast as other algorithms but to a better local optima7. 5.3 Convolution neural networks We next test Santa on the Convolution Neural Network (CNN). Following Jarrett et al. (2009), a standard network con\ufb01guration with 2 convolutional layers followed by 2 fully-connected layers is adopted. Both convolutional layers use 5\u00d75 \ufb01lter size with 32 and 64 7Learning curves of FNN with size of 800 are provided in Supplementary Section G. \fBridging the Gap between SG-MCMC and Stochastic Optimization Algorithms FNN-400 FNN-800 CNN Santa 1.21% 1.16% 0.47% Santa-E 1.41% 1.27% 0.58% Santa-r 1.45% 1.40% 0.49% Adam 1.53% 1.47% 0.59% RMSprop 1.59% 1.43% 0.64% SGD-M 1.66% 1.72% 0.77% SGD 1.72% 1.47% 0.81% SGLD 1.64% 1.41% 0.71% BPB\u22c4 1.32% 1.34% \u2212 SGD, Dropout\u22c4 1.51% 1.33% \u2212 Stoc. Pooling\u25b7 \u2212 \u2212 0.47% NIN, Dropout\u25e6 \u2212 \u2212 0.47% Maxout, Dropout\u22c6 \u2212 \u2212 0.45% Table 1: Test error on MNIST classi\ufb01cation using FNN and CNN. (\u22c4) taken from Blundell et al. (2015). (\u25b7) taken from Zeiler and Fergus (2013). (\u25e6) taken from Lin et al. (2014). (\u22c6) taken from Goodfellow et al. (2013). channels, respectively; 2 \u00d7 2 max pooling is used after each convolutional layer. The fully-connected layers have 200-200 hidden nodes with ReLU activation. The same parameter setting and dataset as in the FNN are used. The test errors are shown in Table 1, and the corresponding learning curves are shown in Figure 2. Similar trends as in FNN are obtained. Santa signi\ufb01cantly outperforms other algorithms with an error of 0.45%. This result is comparable or even better than some recent state-of-the-art CNN-based systems, which have much more complex architectures. 5.4 Recurrent neural networks We test Santa on the Recurrent Neural Network (RNN) for sequence modeling, where a model is trained to minimize the negative log-likelihood of training sequences: min \u03b8 1 N N X n=1 Tn X t=1 \u2212log p(xn t | xn 1, . . . , xn t\u22121; \u03b8) (5) where \u03b8 is a set of model parameters, {xn t } is the observed data. The conditional distributions in (5) are modeled by the RNN. The hidden units are set to gated recurrent units (Cho et al., 2014). We consider the task of sequence modeling on four di\ufb00erent polyphonic music sequences of piano, i.e., Piano-midi.de (Piano), Nottingham (Nott), MuseData (Muse) and JSB chorales (JSB). Each of these datasets are represented as a collection of 88-dimensional binary sequences, that span the whole range of piano from A0 to C8. The number of hidden units is set to 200. Each model is trained for at most 100 epochs. According to the experiments and their results on the validation set, we 0 10 20 30 40 50 60 70 80 Epochs 6 7 8 9 10 11 12 13 14 Negative Log-likelihood SGD SGD-M RMSprop Adam Santa Santa-s 0 10 20 30 40 50 60 70 80 Epochs 6 7 8 9 10 11 12 13 14 Negative Log-likelihood Figure 3: Learning curves of di\ufb00erent algorithms on Piano using RNN. (Left) training set. (Right) validation set. Algorithms Piano. Nott. Muse. JSB. Santa 7.60 3.39 7.20 8.46 Adam 8.00 3.70 7.56 8.51 RMSprop 7.70 3.48 7.22 8.52 SGD-M 8.32 3.60 7.69 8.59 SGD 11.13 5.26 10.08 10.81 HF\u22c4 7.66 3.89 7.19 8.58 SGD-M\u22c4 8.37 4.46 8.13 8.71 Table 2: Test negative log-likelihood results on polyphonic music datasets using RNN. (\u22c4) taken from Boulanger-Lewandowski et al. (2012). use a learning rate of 0.001 for all the algorithms. For Santa, we consider an additional experiment using a learning rate of 0.0002, denoted Santa-s. The annealing coe\ufb03cient \u03b3 is set to 0.5. Gradients are clipped if the norm of the parameter vector exceeds 5. We do not perform any dataset-speci\ufb01c tuning other than early stopping on validation sets. Each update is done using a minibatch of one sequence. The best log-likelihood results on the test set are achieved by using Santa, shown in Table 2. Learning curves on the Piano dataset are plotted in Figure 3. We observe that Santa achieves fast convergence, but is over\ufb01tting. This is straightforwardly addressed through early stopping. The learning curves for all the other datasets are provided in Supplementary Section G. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1206.4671v1",
+                    "title": "Dependent Hierarchical Normalized Random Measures for Dynamic Topic Modeling",
+                    "abstract": "We develop dependent hierarchical normalized random measures and apply them\nto dynamic topic modeling. The dependency arises via superposition, subsampling\nand point transition on the underlying Poisson processes of these measures. The\nmeasures used include normalised generalised Gamma processes that demonstrate\npower law properties, unlike Dirichlet processes used previously in dynamic\ntopic modeling. Inference for the model includes adapting a recently developed\nslice sampler to directly manipulate the underlying Poisson process.\nExperiments performed on news, blogs, academic and Twitter collections\ndemonstrate the technique gives superior perplexity over a number of previous\nmodels.",
+                    "authors": "Changyou Chen, Nan Ding, Wray Buntine",
+                    "published": "2012-06-18",
+                    "updated": "2012-06-18",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Dirichlet processes and their variants are popular in recent years, with applications found in diverse discrete domains such as topic modeling (Teh et al., 2006), ngram modeling (Teh, 2006), clustering (Socher et al., 2011), and image modeling (Li et al., 2011). These models take as input a base distribution and produce as output another distribution which is somewhat similar. Moreover, they can be used hierarchically. Together this makes them ideal for modeling structured data such as text and images. Appearing in Proceedings of the 29 th International Conference on Machine Learning, Edinburgh, Scotland, UK, 2012. Copyright 2012 by the author(s)/owner(s). When modeling dynamic data or data from multiple sources, dependent nonparametric Bayesian models (MacEachern, 1999) are needed in order to harness related or previous information. Among these models, the hierarchical Dirichlet process (HDP) (Teh et al., 2006) is the most popular one. However, a basic assumption underlying the HDP is the full exchangeability of the sample path, which is often violated in practice, e.g., we could assume the content of ICML depends on previous years\u2019 so order is important. To overcome the full exchangeability limitation, several dependent Dirichlet process models have been proposed, for example, the dynamic HDP (Ren et al., 2008), the evolutionary HDP (Zhang et al., 2010), and the recurrent Chinese Restaurant process (Ahmed & Xing, 2010). Dirichlet processes are used because of simplicity and conjugacy (James et al., 2006). These models are constructed by incorporating the previous DP\u2019s into the base distribution of the current DP. Dependent DPs have also been constructed using the underlying Poisson processes (Lin et al., 2010). However, recent research has shown that many real datasets have the power-law property, e.g., in images (Sudderth & Jordan, 2008), in topic-word distributions (Teh, 2006) and in document topic (label) distributions (Rubin et al., 2011). This makes the Dirichlet process an improper tool for modeling these datasets. Although there also exists some dependent nonparametric models with power-law phenomena, their dependencies are limited. For example, Bartlett et al. (2010) proposed a dependent hierarchical PitmanYor process that only allows deletion of atoms, while Sudderth & Jordan (2008) construct the dependent Pitman-Yor process by only allowing dependencies between atoms. \fDHNRM Dynamic Topic Modeling In this paper, we use a larger class of stochastic processes called normalized random measures with independent increments (NRM) (James et al., 2009). While this includes the Dirichlet process as a special case, some other versions of NRMs have the powerlaw property. This class of discrete random measures can also be constructed from Poisson processes. Given this, following (Lin et al., 2010), we analogically de\ufb01ne superposition, subsampling and point transition on these normalized random measures, and construct a time dependent hierarchical model for dynamic topic modeling. By this, the dependencies are \ufb02exibly controlled between both jumps and atoms of the NRMs. All proofs and some extended theories are available in (Chen et al., 2012). 2. Normalized Random Measures 2.1. Background and De\ufb01nitions This background on random measures follows (James et al., 2009). Let (S, S) be a measure space where S is the \u03c3-algebra of S. Let \u03bd be a measure on it. A Poisson process on S is a random subset \u03a0 \u2208S such that if N(A) is the number of points of \u03a0 in the measurable set A \u2286S, then N(A) is a Poisson random variable with mean \u03bd(A), and N(A1), \u00b7 \u00b7 \u00b7 , N(An) are independent if A1, \u00b7 \u00b7 \u00b7 , An are disjoint. Based on the de\ufb01nition, we de\ufb01ne a complete random measure (CRM) on (X, B(X)) to be a linear functional of the Poisson random measure N(\u00b7), with mean measure \u03bd(dt, dx) de\ufb01ned on a product space S = R+ \u00d7X: \u02dc \u00b5(B) = Z R+\u00d7B tN(dt, dx), \u2200B \u2208B(X). (1) Here \u03bd(dt, dx) is called the L\u00b4 evy measure of \u02dc \u00b5. It is worth noting that the CRM is usually written in the form \u02dc \u00b5(B) = P\u221e k=1 Jk\u03b4xk(B), where J1, J2, \u00b7 \u00b7 \u00b7 > 0 are called the jumps of the process, and x1, x2, \u00b7 \u00b7 \u00b7 are a sequence of independent random variables drawn from a base measurable space (X, B(X))1. A normalized random measure (NRM) on (X, B(X)) is de\ufb01ned as \u00b5 = \u02dc \u00b5 \u02dc \u00b5(X). We always use \u00b5 to denote an NRM, and \u02dc \u00b5 its unnormalized counterpart. Taking di\ufb00erent L\u00b4 evy measures \u03bd(dt, dx), we can obtain di\ufb00erent NRMs, and the form we consider is described in Section 2.3. Here we consider the case \u03bd(dt, dx) = M\u03c1\u03b7(dt)H(dx), where H(dx) is the base probability measure, M is the total mass acting as a 1B(X) means the \u03c3-algebra of X, we sometimes omit this and use X to denote the measurable space. concentration parameter, and \u03b7 is the set of other hyperparameters, depending on the speci\ufb01c NRM\u2019s. We use NRM(M, \u03b7, H) to denote the corresponding normalized random measure. 2.2. Slice sampling NRMs We brie\ufb02y introduce the ideas of slice sampling normalized random measures discussed in (\u201cSlice 1\u201d version, Gri\ufb03n & Walker, 2011). It deals with the normalized random measure mixture of the type \u00b5(\u00b7) = \u221e X k=1 rk\u03b4\u03b8k(\u00b7), \u03b8si \u223c\u00b5(\u00b7), xi \u223cg0(\u00b7|\u03b8si) (2) where rk = Jk/ P\u221e l=1 Jl, \u03b8k\u2019s are the component of the mixture model drawn i.i.d. from a parameter space H(\u00b7), si denotes the component that xi belongs to, and g0(\u00b7|\u03b8k) is the density function to generate data from component k. Given the observations \u20d7 x, a slice latent variable ui is introduced for each xi so that it only considers those components whose jump sizes Jk\u2019s are larger than the corresponding ui\u2019s. Furthermore, an auxiliary variable v is introduced to decouple each individual jump Jk and their in\ufb01nite sum of the jumps P\u221e l=1 Jl appeared in the denominators of rk\u2019s. It is shown in (Gri\ufb03n & Walker, 2011) that the posterior of the in\ufb01nite mixture model (2) with the above auxiliary variables is proportional to P\u00b5(\u20d7 \u03b8, J1, \u00b7 \u00b7 \u00b7 , JK, K, \u20d7 u, L,\u20d7 s, v|\u20d7 x, H, \u03c1\u03b7) \u221d exp ( \u2212v K X k=1 Jk ) exp ( \u2212M Z L 0 (1 \u2212exp {\u2212vt}) \u03c1\u03b7(t)dt ) vN\u22121p(J1, \u00b7 \u00b7 \u00b7 , JK) K Y k=1 h(\u03b8k) N Y i=1 1(Jsi > ui)g0(xi|\u03b8si), (3) where 1(a) is an indicator function returning 1 if a is true and 0 otherwise, h(\u00b7) is the density of H(\u00b7), L = min{\u20d7 u}, and p(J1, \u00b7 \u00b7 \u00b7 , JK) = QK k=1 \u03c1\u03b7(Jk) R \u221e L \u03c1\u03b7(t)dt is the distribution for the jumps which are larger than L derived from the underlying Poisson process. Sampling for this mixture model iteratively cycles over {\u20d7 \u03b8, (J1, \u00b7 \u00b7 \u00b7 , JK), K, \u20d7 u,\u20d7 s, v} based on (3). Please refer to (Section 1.3 Chen et al., 2012) for more details. 2.3. Normalized generalized Gamma processes In this paper, we consider the normalized generalized Gamma processes. Generalized Gamma processes (Lijoi et al., 2007) (GGP) are random measures with the L\u00b4 evy measure \u03bd(dt, dx) = M e\u2212bt t1+a H(dx), b > 0, 0 < a < 1. (4) \fDHNRM Dynamic Topic Modeling By normalizing the GGP, we obtain the normalized generalized Gamma process (NGG)2. One of the most familiar special cases is the Dirichlet process, which is a normalized Gamma process where a \u21920 and b = 1 and the concentration parameter appears as M. Crucially, unlike the DP, the NGG can produce the power-law phenomenon. Proposition 1 ((Lijoi et al., 2007)) Let Kn be the number of components induced by the NGG with parameters a an b or the Dirichlet process with total mass M. Then for the NGG, Kn/na \u2192Sab almost surely, where Sab is a strictly positive random variable parameterized by a and b. For the DP, Kn/ log(n) \u2192M. Therefore, in order to better analyze certain kinds of real data, we propose to use the NGG in place of the Dirichlet process. In the next section, we propose a dynamic topic model which extends two major advances of the Dirichlet process: the HDP (Teh et al., 2006) and the dependent Dirichlet process (Lin et al., 2010), to normalized random measures. 3. Dynamic topic modeling with dependent hierarchical NRMs Our main interest is to construct a dynamic topic model that inherits partial exchangeability, meaning that the documents within each time frame are exchangeable, while between time frames they are not. To achieve this, it is crucial to model the dependency of the topics between di\ufb00erent time frames. In particular, a topic can either inherit from the topics of earlier time frames with certain transformation, or be a completely new one which is \u201dborn\u201d in the current time frame. The above idea can be modeled by a series of hierarchical NRMs, one per time frame. Between the time frames, these hierarchical NRMs depend on each other through three dependency operators: superposition, subsampling and point transition, which will be de\ufb01ned below. The corresponding graphical model is shown in Figure 1(left) and the generating process for the model is as follows: \u2022 Generating independent NRMs \u00b5m for time frame m = 1, \u00b7 \u00b7 \u00b7 , n: \u00b5m|H, \u03b70 \u223cNRM(M0, \u03b70, P0) (5) where H(\u00b7) = M0P0(\u00b7). M0 is the total mass for \u00b5m and P0 is the base distribution. In this paper, P0 is the Dirichlet distribution, \u03b70 is the set of 2In NGG, b can be absolved into M, thus we usually set b = 1, see (Chen et al., 2012) for detail. hyperparameters of the corresponding NRM, e.g., in NGG, \u03b70 = {a, b}. \u2022 Generating dependent NRMs \u00b5\u2032 m (from \u00b5m and \u00b5\u2032 m\u22121), for time frame m > 1: \u00b5\u2032 m = T(Sq(\u00b5\u2032 m\u22121)) \u2295\u00b5m . (6) where the three dependency operators superposition (\u2295), subsampling (Sq(\u00b7)) with acceptance rate q, and point transition (T(\u00b7)) are generalized from those of Dirichlet process (Lin et al., 2010). We will discuss them in more details in the following subsection. \u2022 Generating hierarchical NRM mixtures (\u00b5mj, \u03b8mji, xmji) for time frame m = 1, \u00b7 \u00b7 \u00b7 , n, document j = 1, \u00b7 \u00b7 \u00b7 , Nm, word i = 1, \u00b7 \u00b7 \u00b7 , Wmj: \u00b5mj = NRM(Mm, \u03b7m, \u00b5\u2032 m), (7) \u03b8mji|\u00b5mj \u223c\u00b5mj, xmji|\u03b8mji \u223cg0(\u00b7|\u03b8mji) where Mm is the total mass for \u00b5mj, g0(\u00b7|\u03b8mji) denotes the density function to generate data xmji from atom \u03b8mji. 3.1. The three dependency operators Adapting from the dependent Dirichlet process (Lin et al., 2010), the three dependency operators for the NRMs are de\ufb01ned as follows. Superposition of normalized random measures Given n independent NRMs \u00b51, \u00b7 \u00b7 \u00b7 , \u00b5n on X, the superposition (\u2295) is: \u00b51 \u2295\u00b52 \u2295\u00b7 \u00b7 \u00b7 \u2295\u00b5n := c1\u00b51 + c2\u00b52 + \u00b7 \u00b7 \u00b7 + cn\u00b5n . where the weights cm = \u02dc \u00b5m(X) P j \u02dc \u00b5j(X) and \u02dc \u00b5m is the unnormalized version of \u00b5m. Subsampling of normalized random measures Given a NRM \u00b5 = P\u221e k=1 rk\u03b4\u03b8k on X, and a Bernoulli parameter q \u2208[0, 1], the subsampling of \u00b5, is de\ufb01ned as Sq(\u00b5) := X k:zk=1 rk P j zjrj \u03b4\u03b8k, (8) where zk \u223cBernoulli(q) are Bernoulli random variables with acceptance rate q. Point transition of normalized random measures Given a NRM \u00b5 = P\u221e k=1 rk\u03b4\u03b8k on X, the point transition of \u00b5, is to draw atoms \u03b8\u2032 k from a transformed base measure to yield a new NRM as T(\u00b5) := P\u221e k=1 rk\u03b4\u03b8\u2032 k. \fDHNRM Dynamic Topic Modeling H \u00b5\u2032 2 \u00b52 \u00b5\u2032 m \u00b5\u2032 n \u00b5\u2032 1 \u00b51 \u00b5m \u00b5n T(Sq(\u00b5\u2032 1)) T(Sq(\u00b5\u2032 2)) T(Sq(\u00b5\u2032 m)) \u00b51j \u03b81ji x1ji \u00b52j \u03b82ji x2ji \u00b5mj \u03b8mji xmji \u00b5nj \u03b8nji xnji W1 N1 W2 Wm Wn N2 Nm Nn t1 t2 tm tn \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 H \u00b5\u2032 2 \u02dc \u00b52 \u00b5\u2032 m \u00b5\u2032 n \u00b5\u2032 1 \u02dc \u00b5\u2032 2 \u02dc \u00b51 \u02dc \u00b5\u2032 1 \u02dc \u00b5m \u02dc \u00b5\u2032 m \u02dc \u00b5n \u02dc \u00b5\u2032 n T(Sq(\u02dc \u00b51)) T(Sq(\u02dc \u00b52)) T(Sq(\u02dc \u00b5m)) \u00b51j \u03b81ji x1ji \u00b52j \u03b82ji x2ji \u00b5mj \u03b8mji xmji \u00b5nj \u03b8nji xnji W1 N1 W2 Wm Wn N2 Nm Nn t1 t2 tm tn \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 Figure 1. The time dependent topic model. The left plot corresponds to directly manipulating on normalized random measures (9), the right one corresponds to manipulating on unnormalized random measures (10). T: Point transition; Sq: Subsampling with acceptance rate q; \u2295: Superposition. Here m = n \u22121 in the \ufb01gures. Point transitions can be done in di\ufb00erent ways with di\ufb00erent transition kernels T(\u00b7). In this paper, following (Lin et al., 2010), when inheriting from NRM \u00b5, we draw atoms \u03b8\u2032 k from the base measure as \u00b5 conditioned on its current statistics. Other ways of constructing transition kernels are left for further research. 3.2. Properties of the dependency operators The three dependency operators on the NRMs inherit some of the nice properties from the underlying Poisson process. It not only enables quantitatively controlling dependencies introduced after and before the operations, as is shown in (Section 4 Chen et al., 2012), but also maintains a nice equivalence relation between the NRM\u2019s and the corresponding CRM\u2019s. In the following theorem, we will use \u02dc \u2295, \u02dc Sq(\u00b7) and \u02dc T(\u00b7) to denote the three operations on their corresponding CRM\u2019s3. Theorem 2 The following time dependent random measures (9) and (10) are equivalent: \u2022 Manipulate the normalized random measures: \u00b5\u2032 m \u223cT(Sq(\u00b5\u2032 m\u22121)) \u2295\u00b5m, m > 1. (9) \u2022 Manipulate the completely random measures: \u02dc \u00b5\u2032 m \u223c\u02dc T( \u02dc Sq(\u02dc \u00b5\u2032 m\u22121)) \u2295\u02dc \u00b5m, m > 1. \u00b5\u2032 m = \u02dc \u00b5\u2032 m \u02dc \u00b5\u2032 m(X), (10) 3The de\ufb01nitions of \u02dc \u2295, \u02dc Sq(\u00b7) and \u02dc T(\u00b7) are similar to the NRMs\u2019, see (Section 1.5.1 Chen et al., 2012) for details. Furthermore, the resulting NRMs \u00b5\u2032 m\u2019s give the following: \u00b5\u2032 m = m X j=1 \u0000qm\u2212j \u02dc \u00b5j \u0001 (X) Pm j\u2032=1 (qm\u2212j\u2032 \u02dc \u00b5j\u2032) (X)T m\u2212j(\u00b5j), m > 1 where qm\u2212j \u02dc \u00b5 is the random measure with L\u00b4 evy measure qm\u2212j\u03bd(dt, dx) (\u03bd(dt, dx) is the L\u00b4 evy measure of \u02dc \u00b5). T m\u2212j(\u00b5) denotes point transition on \u00b5 for (m\u2212j) times . 3.3. Reformulation of the proposed model Theorem 2 in the last section allows us to \ufb01rst take superposition, subsampling, and point transition on the completely random measures \u02dc \u00b5g\u2019s and then do the normalization. Therefore, we make use of Theorem 2 to obtain the dynamic topic model in Figure 1(right) by expanding the recusive formula in (10), which is equivalent to the left one. The generating process of the new model is: \u2022 Generating independent CRM\u2019s \u02dc \u00b5m for time frame m = 1, \u00b7 \u00b7 \u00b7 , n, following (1). \u2022 Generating \u00b5\u2032 m for time frame m > 1, following (10). \u2022 Generating hierarchical NRM mixtures (\u00b5mj, \u03b8mji, xmji) following (7). The reason for this reformulation is because the inference on the model in Figure 1(left) appears to be infeasible. In general, the posterior of an NRM introduce \fDHNRM Dynamic Topic Modeling complex dependencies between jumps, thus sampling is unclear after taking the three dependency operators. On the other hand, the model in Figure 1(right) is more amenable to computation because the NRMs and the three operators are decoupled. It allows us to \ufb01rst generate the dependent CRM\u2019s, then use the slice sampler introduced in Section 2.2 to sample the posterior of the corresponding NRMs. From now on, we will focus on the model in Figure 1(right). In the next section, we discuss its sampling procedure. 4. Sampling To introduce our sampling method we use the familiar Chinese restaurant metaphor (e.g. (Teh et al., 2006)) to explain key statistics. In this model customers for the variable \u00b5mj correspond to words in a document, restaurants to documents, and dishes to topics. In time frame m, \u2022 xmji: the customer i in the jth restaurant. \u2022 smji: the dish that xmji is eating. \u2022 nmjk: nmjk = P i \u03b4smji=k, the number of customers in \u00b5mj eating dish k. \u2022 tmjr: the table r in the jth restaurant. \u2022 \u03c8mjr: the dish that the table tmjr is serving. \u2022 n\u2032 mk: n\u2032 mk = P j P r \u03b4\u03c8mjr=k, the number of customers4 in \u00b5\u2032 m eating dish k. \u2022 \u02dc n\u2032 mk: \u02dc n\u2032 mk = n\u2032 mk, the number of customers in \u02dc \u00b5\u2032 m eating dish k. \u2022 \u02dc nmk: \u02dc nmk = P m\u2032\u2265m \u02dc n\u2032 m\u2032k, the number of customers in \u02dc \u00b5m eating dish k. We will do the sampling by marginalizing out \u00b5mj\u2019s. As it turns out, the remaining random variables that require sampling are smji, n\u2032 mk, as well as \u02dc \u00b5m = X k Jmk\u03b4\u03b8k, \u02dc \u00b5\u2032 m = X k J\u2032 mk\u03b4\u03b8k Note the tmjr and \u03c8mjr are not sampled as we sample the n\u2032 mk directly. Thus our sampler deals with the following latent statistics and variables: smji, n\u2032 mk, Jmk, J\u2032 mk and some auxiliary variables are sampled to support these. 4the customers in \u00b5\u2032 m corresponds to the tables in \u00b5mj. For convenient, we also regard a CRM as a restaurant. Sampling Jmk. Given \u02dc nmk, we use the slice sampler introduced in (Gri\ufb03n & Walker, 2011) to sample these jumps, with the posterior given in (3). Note that the mass Mm\u2019s are also sampled, see (Sec.1.3 Chen et al., 2012). The resulting {Jmk} are those jumps that exceed a threshold de\ufb01ned in the slice sampler, thus the number of jumps is \ufb01nite. Sampling J\u2032 mk. J\u2032 mk is obtained by subsampling of {Jm\u2032k}m\u2032\u2264m5. By using a Bernoulli variable zmk, J\u2032 mk = ( Jm\u2032k if zmk = 1 0 if zmk = 0. We compute the posterior p(zmk = 1|\u02dc \u00b5m, {\u02dc n\u2032 mk}) to decide whether to inherit this jump to \u02dc \u00b5\u2032 m or not. These posteriors are given in (Corollary 3 Chen et al., 2012). In practice, we found it mixes faster if we integrate out zmk\u2019s. (Lemma 9 Chen et al., 2012) shows that q-subsampling of a CRM with L\u00b4 evy measure \u03bd(\u00b7) results in another CRM with L\u00b4 evy measure q\u03bd(\u00b7), thus the jump sizes in the resultant CRM are scaled by q, meaning that J\u2032 mk = qm\u2212m\u2032Jm\u2032k. After the sampling of {J\u2032 mk}, we normalize it and obtain the NRM \u00b5\u2032 m, \u00b5\u2032 m = P k rmk\u03b4\u03b8k where rmk = J\u2032 mk/ P k\u2032 J\u2032 mk\u2032 Sampling smji, n\u2032 mk. The following procedures are similar to sampling an HDP. The only di\ufb00erence is that \u00b5mj and \u00b5\u2032 m are NRMs instead of DPs. The sampling method goes as follows: \u2022 Sampling smji: We use a similar strategy as the sampling by direct assignment algorithm for the HDP (Teh et al., 2006), the conditional posterior of smji is: p(smji = k|\u00b7) \u221d(\u03c9k + \u03c90Mmrmk)g0(xmji|\u03b8k) where \u03c90 and \u03c9k depend on the corresponding L\u00b4 evy measure of \u00b5mj (see (Theorem 2 James et al., 2009)). When \u00b5mj is a DP, then \u03c9k \u221dnmjk and \u03c90 \u221d1. When \u00b5mj is a NGG, \u03c9k \u221dnmjk \u2212a and \u03c90 \u221da(b + vmj)a, where vmj is the introduced auxiliary variables which can be sampled by an adaptive-rejection sampler using the posterior given in (Proposition 1 James et al., 2009). \u2022 Sampling n\u2032 mk: Using the similar strategy as in (Teh et al., 2006), we sample n\u2032 mk by simulating the (generalized) Chinese Restaurant Process, following the prediction rule (the probabilities of 5 Since all the atoms across {\u02dc \u00b5m\u2032} are unique, J\u2032 mk is inherited from only one of {Jm\u2032k}. \fDHNRM Dynamic Topic Modeling generating a new table or sitting on existing tables) of \u00b5mk in (Proposition 2 James et al., 2009). 5. Experiments 5.1. Power-law in the NGG We \ufb01rst investigate the power-law phenomena in the NGG, we sample it using the scheme of (James et al., 2009) and compare it with the DP in Figure 2. #data in each cluster #clusters NGG (a = 0.5, M = 100) 100 101 102 103 100 101 102 103 0 2000 4000 6000 8000 10000 0 100 200 300 #data (customers) #clusters (tables)     DP (M = 10) NGG (a = 0.2, M = 10) Figure 2. Power-law phenomena in NGG. The \ufb01rst plot shows the #data VS. #clusters, the second shows the size s of each cluster VS. total number of clusters with size s. 5.2. Datasets We tested our time dependent dynamic topic model on 9 datasets, removing stop-words and words appearing less than 5 times. ICML, JMLR, TPAMI are crawled from their websites and the abstracts are parsed. The preprocessed NIPS dataset is from (Globerson et al., 2007). The Person dataset is extracted from Reuters RCV1 using the query \u201cperson\u201d under Lucene. The Twitter datasets are updates from three sports twitter accounts: ESPN FirstTake (Twitter1), sportsguy33 (Twitter2) and SportsNation (Twitter3) obtained with the TweetStream API (http://pypi.python.org/pypi/tweetstream) to collect the last 3200 updates from each. The Daily Kos blogs (BDT) were pre-processed by (Yano et al., 2009). Statistics for the data sets are given in Table 1. Illustration: Figure 3 gives an example of topic evolutions in the Twitter2 dataset. We can clearly see that the three popular sports in the USA, i.e., basketball, football and baseball, evolve reasonably with time. For example, MLB starts in April each year, showing a peak in baseball topic, and then slowly evolves with decreasing topic proportions. Also, in August one footdataset vocab docs words epochs ICML 2k 765 44k 2007\u20132011 JMLR 2.4k 818 60k 12 vols TPAMI 3k 1108 91k 2006\u20132011 NIPS 14k 2483 3.28M 1987-2003 Person 60k 8616 1.55M 08/96\u201308/97 Twitter1 6k 3200 16k 14 months Twitter2 6k 3200 31k 16 months Twitter3 6k 3200 25k 29 months BDT 8k 2649 234k 11/07\u201304/08 Table 1. Data statistics 10/10 12/10 02/11 04/11 06/11 08/11 10/11 12/11 0 10 20 30 40 50 60 NBA game story NBA game top list time time NBA madness March sports facebook favorite game stars fun facebook NBA final tonight tweet set classic coming win award buy weekend live list 2011 ruin sports game NBA NBA game texans Celts losing minutes sports NFL picks week report week sports NFL picks report week supercontest NFL playoff report Mega chat trade super Jacoby breaking report Jalenrose talk report Mega Welts playoff Jacoby report report week NFL sports coming mustache NFL week report week NFL report podcast report round playoff sox playoff game heat finals win final game sox run front game play fight sox left front game playoff Vegas sox game sox start red kings pessimistic home fight game quater win field game jump penalty bruins ready vegas supercontest week picks NFL picks palmer round NFL picks vegas picks supercontest NFL picks round NFL playoff football1 basketball baseball football2 knicks president picks report crowd wins cup series Dallas Rodgers playoffs team game game power mailbag sports Lakers record sports column Lakers sports west Celts games tonight Figure 3. Topic evolution on Twitter. Words in red have increased, and blue decreased. ball topic is born, indicating a new season begins. Figure 4 gives an example of the word probability change in a single topic for the JMLR. 5.3. Quantitative Evaluations Comparisons We \ufb01rst compare our model with two popular dynamic topic models where the author\u2019s own code was available for our use: (1) the dynamic topic model by Blei and La\ufb00erty (Blei & La\ufb00erty, 2006) and (2) the hierarchical Dirichlet process, where we used a three level HDP, with the middle level DP\u2019s representing the base topic distribution for the documents in a particular time. For fair comparison, similar to (Blei & La\ufb00erty, 2006), we held out the data in previous time but used their statistics to help the training of the current time data, this is implemented in the HDP code by Teh. Furthermore, we also tested the proposed model without power-law, which is to use a DP instead of an NGG. We tested our model on the 9 datasets, for each dataset we used 80% for training and held out 20% for testing. The hyperparameters for DHNGG is set to a = 0.2 in this set of experiments with subsampling rate being 0.9, which is found to work well in practice. The topic-word distributions are symmetric Dirichlet with prior set to 0.3. Table 2 shows the test \fDHNRM Dynamic Topic Modeling Figure 4. Topic evolution on JMLR. Shows a late developing topic on software, before during and after the start of MLOSS.org in 2008. log-likelihoods for all these methods, which are calculated by \ufb01rst removing the test words from the topics and adding them back one by one and collecting the add-in probabilities as the testing likelihood (Teh et al., 2006). For all the methods we ran 2000 burn in iterations, followed by 200 iterations to collect samples. The results are averages over these samples. From Table 2 we see the proposed model DHNGG works best, with an improvement of 1%-3% in test log-likelihoods over the HDP model. In contrast the time dependent model iDTM of Ahmed & Xing (2010) only showed a 0.1% improvement over HDP on NIPS, implying the superiority of DHNRM over iDTM. Hyperparameter sensitivity In NGG, there are hyperparameters a and b, where a controls the behavior of the power-law. In this section we study the in\ufb02uences of these two hyperparameters to the model. We varied a among (0.1, 0.2, 0.3, 0.5, 0.7, 0.9) while \ufb01xed the subsampling rate to 0.9 in this experiment. We run these settings on all these datasets, the training likelihoods are shown in Figure 5. From these results we consider a = 0.2 to be a good choice in practice. In\ufb02uence of the subsampling rate One of the distinct features of our model compared to other time dependent topic models is that the dependency comes partially from subsampling the previous time random measures, thus it is interesting to study the impact of subsampling rates to this model. In this experiment, we \ufb01xed a = 0.2, and varied the subsampling rate q among (0.1, 0.2, 0.3, 0.5, 0.7, 0.9, 1.0). The results are shown in Figure 6. From Figure 6, it is interesting to see that on the academic datasets, e.g., ICML,JMLR, the best results are achieved when q is approximately equal to 1; these datasets have higher correlations. While for the Twitter datasets, the best results are 0.1 0.2 0.3 0.5 0.7 0.9 \u22122.28 \u22122.275 \u22122.27 \u22122.265x 10 5 a 0.1 0.2 0.3 0.5 0.7 0.9 \u22123.15 \u22123.14 \u22123.13 x 10 5 a 0.1 0.2 0.3 0.5 0.7 0.9 \u22124.8 \u22124.78 \u22124.76 \u22124.74x 10 5 a 0.1 0.2 0.3 0.5 0.7 0.9 \u22121.038 \u22121.036 \u22121.034 x 10 7 a 0.1 0.2 0.3 0.5 0.7 0.9 \u22121.47 \u22121.465 \u22121.46 x 10 6 a Figure 5. Training log-likelihoods in\ufb02uenced by hyperparameters a. From left to right (top-down) are the results on ICML, JMLR, TPAMI, Person and BDT. achieved when q is equal to 0.5 \u223c0.7, indicating that people tend to discuss more changing topics in these datasets. 0.1 0.3 0.5 0.7 0.9 1 \u22122.29 \u22122.28 \u22122.27 x 10 5 0.1 0.3 0.5 0.7 0.9 1   \u22123.2 \u22123.15 \u22123.1x 10 5 0.1 0.3 0.5 0.7 0.9 1   \u22124.82 \u22124.78 \u22124.74 x 10 5 0.1 0.3 0.5 0.7 0.9 1   \u22121.04 \u22121.035 \u22121.03 x 10 7 0.1 0.2 0.3 0.5 0.7 0.9 1   \u22121.05 \u22121.04 \u22121.03 x 10 5 0.1 0.2 0.3 0.5 0.7 0.9 1   \u22122.19 \u22122.18 \u22122.17 x 10 5 0.1 0.2 0.3 0.5 0.7 0.9 1   \u22121.6 \u22121.58 \u22121.56x 10 5 0.1 0.2 0.3 0.5 0.7 0.9 1   \u22121.48 \u22121.47 \u22121.46 x 10 6 Figure 6. Training log-likelihoods in\ufb02uenced by the subsampling rate q(\u00b7). The x-axes represent q, the y-axes represent training log-likelihoods. From top-down, left to right are the results on ICML, JMLR, TPAMI, Person, Twitter1, Twitter2, Twitter3 and BDT datasets, respectively. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1205.4159v2",
+                    "title": "Theory of Dependent Hierarchical Normalized Random Measures",
+                    "abstract": "This paper presents theory for Normalized Random Measures (NRMs), Normalized\nGeneralized Gammas (NGGs), a particular kind of NRM, and Dependent Hierarchical\nNRMs which allow networks of dependent NRMs to be analysed. These have been\nused, for instance, for time-dependent topic modelling. In this paper, we first\nintroduce some mathematical background of completely random measures (CRMs) and\ntheir construction from Poisson processes, and then introduce NRMs and NGGs.\nSlice sampling is also introduced for posterior inference. The dependency\noperators in Poisson processes and for the corresponding CRMs and NRMs is then\nintroduced and Posterior inference for the NGG presented. Finally, we give\ndependency and composition results when applying these operators to NRMs so\nthey can be used in a network with hierarchical and dependent relations.",
+                    "authors": "Changyou Chen, Wray Buntine, Nan Ding",
+                    "published": "2012-05-18",
+                    "updated": "2012-05-25",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "math.ST",
+                        "stat.ML",
+                        "stat.TH"
+                    ],
+                    "main_content": "Introduction 4 2 Background 4 2.1 Constructing Completely Random Measures from Poisson processes . 5 2.2 Normalized random measures . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Slice sampling normalized random measure mixtures . . . . . . . . . 11 2.3.1 Sampling: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Operations 13 3.1 Operations on Poisson processes . . . . . . . . . . . . . . . . . . . . . 14 3.2 Operations on random measures . . . . . . . . . . . . . . . . . . . . . 15 3.2.1 Operations on CRMs . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2 Operations on NRMs . . . . . . . . . . . . . . . . . . . . . . . 15 4 Posteriors for the NGG 16 4.1 Simple Posterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Conditional Posterior . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Dependencies and Properties of Operations 20 5.1 Dependencies between NRMs via Operations . . . . . . . . . . . . . . 20 5.2 Properties of the three dependency operations . . . . . . . . . . . . . 22 A Proofs 28 \f4 Normalized Random Measures 1 Introduction This paper presents theory for Normalized Random Measures (NRMs), Normalized Generalized Gammas (NGGs), a particular kind of NRM, and Dependent Hierarchical NRMs which allow networks of dependent NRMs to be analysed. These have been used, for instance, for time-dependent topic modelling [CDB12]. Dependency models are getting more and more popular in machine learning recently due to the fact of correlated data we are facing at, e.g., real data is always correlated with each other rather than independent. The pioneer work of MachEachern [Mac99, Mac00] treats the jumps and atoms to be stochastic between dependent models. While there are many ways of constructing dependent nonparametric models, e.g., from a stick-breaking construction [GS09], or from a hierarchical construction [TJBB06], in this paper, following the idea of [LGF10], we construct dependency normalized random measures from the underlying Poisson processes of the corresponding completely random measures [Kin67]. This construction is intuitive and allow \ufb02exibly controlling of the dependencies. A related construction in the statistical literature is by Lijoi et al. [A. 12] that deals with modeling two groups of data. In this paper, we \ufb01rst introduce in Section 2 some mathematical background of completely random measures (CRMs) and their construction from Poisson processes, and then introduce NRMs and NGGs. Slice sampling is also introduced to do the posterior sampling of NRMs using techniques from [GW11]. The dependency operators in Poisson processes and for the corresponding CRMs and NRMs is then introduced in Section 3 following the work of [Kin93, LGF10]. Posterior inference for the NGG are then developed in Section 4 based on the results of [JLP09]. Then we give the dependency and composition results when applying these operators to NRMs in Section 5. Proofs are given in the Appendix, Section A. 2 Background In this section we brie\ufb02y introduce background of Poisson processes, the corresponding completely random measures, dependency operations on these random measures, and normalized random measures. Section 2.1 explains how to construct completely random measures from Poisson processes. Section 3.1 introduces operations on Poisson processes to construct dependent Poisson processes. Section 3.2 adapts these operations to the corresponding completely random measures (CRMs). Constructing normalized random measures (NRMs) from CRMs is discussed in Section 2.2 along with details of the Normalized Generalized Gamma (NGG), a particular kind of NRM for which the details have been worked out. A slice sampler for sampling an NRM is described in Section 2.3. We \ufb01rst give an illustration of the basic construction for an NRM. for a target domain X. The Poisson process is used to create a countable (and usually) in\ufb01nite set points in a product space of R+ with the target domain X, as shown in the left of \fChen & Buntine & Ding 5 Figure 1. The distribution is then a discrete one on these points. The distribution can be pictured by dropping lines from each point (t, x) down to (0, x), and then normalizing all these lines so their sum is one. The resulting picture shows the set of weighted impulses that make up the constructed NRM on the target domain. X R+ Counting process: N(\u00b7) = P k \u03b4(Jk,xk)(\u00b7) \u2192 X R+ Completely random measure: \u02dc \u00b5(\u00b7) = P k Jk\u03b4xk(\u00b7) Figure 1: Constructing a completely random measure from a counting process N(\u00b7) with points at (Jk, xk). 2.1 Constructing Completely Random Measures from Poisson processes In contrast to the general class of completely random measure (CRM) [Kin67], which admits a unique decomposition as the summation over there parts: a deterministic measure, a purely atomic measure with \ufb01xed atom locations and a measure1 with random jumps and atoms, in this paper, we restrict it to the class of pure jump processes [FK72], which has the following form \u02dc \u00b5 = \u221e X k=1 Jk\u03b4xk, (1) where J1, J2, \u00b7 \u00b7 \u00b7 > 0 are called the jumps of the process, and x1, x2, \u00b7 \u00b7 \u00b7 are a sequence of independent random variables drawn from a base measurable space (X, B(X))2. It is shown that these kinds of CRMs can be constructed from Poisson processes with speci\ufb01c mean measures \u03bd(\u00b7). We will start from some de\ufb01nitions. Poisson Distributions: A random variable X taking values in N = {0, 1, \u00b7 \u00b7 \u00b7 , \u221e} is said to have the Poisson distribution with mean c in (0, \u221e) if p(X = k|c) = e\u2212cck k! , k \u2208N, (2) 1can be continuous or discrete. 2B(X) means the \u03c3-algebra of X, we sometimes omit this and use X to denote the measurable space. \f6 Normalized Random Measures then X < \u221ealmost surely and E[X] = Var[X] = c. Poisson Processes: Let (S, S) be a measure space where S is the \u03c3-algebra of S. Let \u03bd(\u00b7) be a measure on it. A Poisson process on S is de\ufb01ned to be a random subset \u03a0 \u2208S such that if N(A) is the number of points of \u03a0 in the measurable subset A \u2286S, then a) N(A) is a random variable having the Poisson distribution with mean \u03bd(A), and b) whenever A1, \u00b7 \u00b7 \u00b7 , An are in S and disjoint, the random variables N(A1), \u00b7 \u00b7 \u00b7 , N(An) are independent. The integer-value random measure N(\u00b7) is called a Poisson random measure and the Poisson process is denoted as \u03a0 \u223cPoissonP(\u03bd), where \u03bd is called the mean measure of the Poisson process. Completely Random Measure: In this paper, we de\ufb01ne a random measure on (X, B(X)) to be a linear functional of the Poisson random measure N(\u00b7), whose mean measure \u03bd(dt, dx) de\ufb01ned on a product space S = R+ \u00d7 X: \u02dc \u00b5(B) = Z R+\u00d7B tN(dt, dx), \u2200B \u2208B(X). (3) The mean measure \u03bd(dt, dx) is called the L\u00b4 evy measure of \u02dc \u00b5. The general treatment of constructing random measures from Poisson random measures can be found in [Jam05]. Note that the random measure \u02dc \u00b5 in construction (3) has the same form as Equation (1) because N(\u00b7) is composed of a countable number of points. It can be proven to be a completely random measure [Kin67] on X, meaning that for arbitrary disjoint subsets {Ai \u2208X} of the measurable space, the random variables {\u02dc \u00b5(Ai)} are independent. For the completely random measure de\ufb01ned above to always be \ufb01nite, it is necessary that R R+\u00d7X t \u03bd(dt, dx) be \ufb01nite, and therefore for every z > 0, \u03bd([z, \u221e)\u00d7X) = R \u221e z R X \u03bd(dt, dx) is \ufb01nite [Kin93]. It follows that there will always be a \ufb01nite number of points with jumps Jk > z for that z > 0. Therefore in the bounded product space [z, \u221e)\u00d7X the measure \u03bd(dt, dx) is \ufb01nite. So it is meaningful to sample those points (Jk, xk) with Jk > z by \ufb01rst getting the count of points K sampled from a Poisson with (\ufb01nite) mean \u03bd([z, \u221e) \u00d7 X), and then to sample the K points according to the distribution of \u03bd(dt,dx) \u03bd([z,\u221e)\u00d7X). Without loss of generality, the L\u00b4 evy measure of Equation (3) can be represented as \u03bd(dt, dx) = M\u03c1\u03b7(dt|x)H(dx), where \u03b7 denotes the hyper-parameters if any of a measure on t, H(dx) is a probability measure so H(X) = 1, and M is called the mass of the L\u00b4 evy measure. Note the total measure of \u03c1\u03b7(dt|x) is not standardized in any way so in principle some mass could also appear in \u03c1\u03b7(dt|x). The mass is used as a concentration parameter for the random measure. \fChen & Buntine & Ding 7 A realization of \u02dc \u00b5 on X can be constructed by sampling from the underlying Poisson process in a number of ways, either in rounds for decreasing bounds z using the logic just given, or by explicitly sampling the jumps in order. The later goes as follows [FK72]: Lemma 1 (Sampling a CRM) Sample a CRM \u02dc \u00b5 with L\u00b4 evy measure \u03bd(dt, dx) = M\u03c1\u03b7(dt|x)H(dx) as follows. \u2022 Draw i.i.d. samples xi from the base measure H(dx). \u2022 Draw the corresponding weights Ji for these i.i.d. samples in decreasing order, which goes as: \u2013 Draw the largest jump J1 from the cumulative distribution function P(J1 \u2264j1) = exp n \u2212M R \u221e j1 \u03c1\u03bd(dt|xi) o . \u2013 Draw the second largest jump J2 from the cumulative distribution function P(J2 \u2264j2) = exp n \u2212M R j1 j2 \u03c1\u03bd(dt|x2) o . \u2013 \u00b7 \u00b7 \u00b7 \u2022 The random measure \u02dc \u00b5 then can now be realized as \u02dc \u00b5 = P i Ji\u03b4xi. As a random variable is uniquely determined by its Laplace transformation, the random measure \u02dc \u00b5 is uniquely characterized by its Laplace functional through the L\u00b4 evy-Khintchine representation of a L\u00b4 evy process [C \u00b810]. That is, for any measurable function f : X \u2192R+, we have E \u0014 exp \u001a \u2212 Z X f(x)\u02dc \u00b5(dx) \u001b\u0015 = exp \u001a \u2212 Z R+\u00d7X [1 \u2212exp {\u2212tf(x)}] \u03bd(dt, dx) \u001b , (4) Now instead of dealing with \u02dc \u00b5 itself, we deal with \u03bd(dt, dx), which is called the L\u00b4 evy measure of \u02dc \u00b5, whose role in generating the measure via a Poisson process was explained above. In the case where the measure on the jumps is not dependent on the data x, so \u03c1\u03b7(dt|x) = \u03c1\u03b7(dt), then \u02dc \u00b5 is called homogeneous, which is the case considered in this paper. When f does not depend on x, (4) simpli\ufb01es to E [exp {\u2212f \u02dc \u00b5(B)}] = exp \u001a \u2212M p(B) Z R+ [1 \u2212exp {\u2212tf}] \u03c1\u03b7(dt) \u001b . (5) Note the term inside the exponential plays an important role in subsequent theory, so it is given a name. Laplace exponent: The Laplace exponent, denoted as \u03c8\u03b7(f) for a CRM with parameters \u03b7 is given by \u03c8\u03b7(f) = Z R+\u00d7X [1 \u2212exp {\u2212tf}] \u03bd(dt, dx) = M Z R+ [1 \u2212exp {\u2212tf}] \u03c1\u03b7(dt) (homogeneous case) . (6) \f8 Normalized Random Measures Note that to guarantee the positiveness of jumps in the random measure, \u03c1(dt) in the L\u00b4 evy measure should satisfy R \u221e 0 \u03c1\u03b7(dt) = +\u221e[C \u00b810], which leads to the following equations: \u03c8\u03b7(0) = 0, \u03c8\u03b7(+\u221e) = +\u221e. (7) That \u03c8\u03b7(f) is \ufb01nite for \ufb01nite positive f implies (or is a consequence of) R \u221e 0 t\u03c1\u03b7(dt) being \ufb01nite. Remark There are thus four di\ufb00erent ways to de\ufb01ne or interpret a CRM: 1. via the linear functional of Equation (3), 2. through the L\u00b4 evy-Khintchine representation of Equation (4) using the Laplace exponent, 3. sampling in order of decreasing jumps using Lemma 1, and 4. sampling in blocks of decreasing jump values as discussed before Lemma 1. 2.2 Normalized random measures Normalized Random Measures (NRM) Based on (3), a normalized random measure on (X, B(X)) is de\ufb01ned as3 \u00b5 = \u02dc \u00b5 \u02dc \u00b5(X). (8) The original idea of constructing random probabilities by normalizing completely random measures on R, namely increasing additive processes, can be found in [ELP03], where it is termed normalized random measures with independent increment (NRMI) and the existence of such random measures is proved. This idea can be easily generalized from R to any parameter space X, e.g., X being the Dirichlet distribution space in topic modeling. Also note that the idea of normalized random measures can be taken as doing a transformation Tr(\u00b7) on completely random measures, that is \u00b5 = Tr(\u02dc \u00b5). In the normalized random measure case, Tr(\u00b7) is a transformation such that Tr(\u02dc \u00b5(X)) = 1. A concise survey of other kinds of transformations can be found in [LP10]. Taking di\ufb00erent L\u00b4 evy measures \u03bd(dt, dx) of (4), we can obtain di\ufb00erent NRMs. We use NRM(\u03b7, M, H(\u00b7)) to denote the normalized random measure, where M is the total mass, which usually needs to be sampled in the model, and H(\u00b7) is the base probability measure, \u03b7 is the set of other hyper-parameters to the measure on the jumps, depending on the speci\ufb01c NRMs. In this paper, we are interested in a class of NRMs called normalized generalized Gamma processes: 3In this paper, we use \u00b5 to denote a normalized random measure, while use \u02dc \u00b5 to denote its unnormalized counterpart. \fChen & Buntine & Ding 9 Normalized Generalized Gamma Processes Generalized Gamma processes are random measures proposed by Brix [Bri99] for constructing shot noise Cox processes. They have the L\u00b4 evy measures as \u03bd(dt, dx) = e\u2212bt t1+aH(dx), b > 0, 0 < a < 1. (9) By normalizing the generalized Gamma process as in (8), we obtain the normalized generalized Gamma process (NGG). For ease of representation and sampling, we convert the NGG into a di\ufb00erent form using the following lemma. Lemma 2 Let a normalised random measure be de\ufb01ned using L\u00b4 evy density \u03bd(dx, dt). Then scaling t by \u03bb > 0 yields an equivalent NRM up to a factor. That is, the normalised measure obtained using \u03bd(dx, dt/\u03bb) is equivalent to the normalised measure obtained using \u03bb \u03bd(dx, dt). By this lemma, without loss of generality, we can instead represent the NGG by eliminating the parameter b above. Normalized Generalized Gamma: The NGG with shape parameter a, total mass (or concentration) parameter M and base distribution H(\u00b7), denoted NGG(a, M, H(\u00b7)), has L\u00b4 evy density M \u03c1a(dt)H(dx) where \u03c1a(t) = a \u0393(1 \u2212a) e\u2212t t1+a . Note that similar to the two parameter Poisson-Dirichlet process [PY97], the normalized generalized Gamma process with a \u0338= 0 can also produce power-law phenomenon, making it di\ufb00erent from the Dirichlet process and suitable to model real data. Proposition 1 ([LMP07]) Let Kn be the number of components induced by the NGG with parameter a and mass M or the Dirichlet process with total mass M. Then for the NGG, Kn/na \u2192Sa,M almost surely, where Sa,M is a strictly positive random variable parameterized by a and M. For the DP, Kn/ log(n) \u2192M. Figure 2 demonstrates the power law phenomena in the NGG compared to the Dirichlet process (DP). We sample it using the generalized Blackwell-MacQueen sampling scheme [JLP09]. Each data to be sampled can choose an existing cluster or create a new cluster, resulting in K clusters with N data points in total. Many familiar stochastic processes are special/limiting cases of normalized generalized Gamma processes, e.g., Dirichlet processes arise when a \u21920. Normalized inverse-Gaussian processes (N-IG) arise when a = 1 2 and b = 1 2. If b \u21920, we get the \u03c3-stable process, and if a \u21920 and b depends on x, we get the extended Gamma process. \f10 Normalized Random Measures 0 2000 4000 6000 8000 10000 0 50 100 150 200 #data (customers) #clusters (tables)     DP (M = 10) NGG(a = 0.2, M = 10) #data in each cluster #clusters NGG (a = 0.5, M = 10) 100 101 102 103 100 101 102 103 Figure 2: Power-law phenomena in NGG. The \ufb01rst plot shows the #data versus #clusters compared with DP, the second plot shows the size s of each cluster versus total number of clusters with size s. Remark For the NGG, key formula used subsequently are as follows: \u03c8a(v) = M ((1 + v)a \u22121) Z \u221e L \u03c1a(dt) = |Q(\u2212a, L)| Z \u221e L e\u2212vx\u03c1a(dt) = (1 + v)a|Q(\u2212a, L(1 + v))| Z L 0 \u00001 \u2212e\u2212vt\u0001 \u03c1a(t)dt = ((1 + v)a \u22121) + (1 + v)a|Q(\u2212a, L(1 + v))| \u2212|Q(\u2212a, L)| where Q(x, y) = \u0393(x, y)/\u0393(x) is the regularized upper incomplete Gamma function. Some mathematical libraries provide it for a negative \ufb01rst argument, or it can be evaluated using Q(\u2212a, z) = Q(1 \u2212a, z) \u2212 1 \u0393(1 \u2212a)z\u2212ae\u2212z, using an upper incomplete Gamma function de\ufb01ned only for positive arguments. Finally, because probabilities for a NRM necessarily have the divisor \u02dc \u00b5(X ) = P\u221e k=1 Jk, and thus likelihoods of the NRM should involve powers of \u02dc \u00b5(X ), a trick is widely used to eliminate these terms. Latent relative mass: Consider the case where N data are observed. By introducing the auxiliary variable, called latent relative mass, UN = \u0393N/\u02dc \u00b5(X ) where \u0393N \u223c\u0393(1, N), then it follows that 1 \u02dc \u00b5(X )N p(\u0393N)d\u0393N = exp ( \u2212UN \u221e X k=1 Jk ) dUN \fChen & Buntine & Ding 11 Thus the N-th power of the normaliser can be replaced by an exponential term in the jumps which factorizes, at the expense of introducing the new latent variable v. To the best of our knowledge, the idea of this latent variable originals from [Jam05] and is future explicitly studied in [JLP06, JLP09, GW11], etc.. 2.3 Slice sampling normalized random measure mixtures Slice sampling an NRM has been discussed in several papers, here we follow the method in [GW11], to brie\ufb02y introduce the ideas behind it. It deals with the normalized random measure mixture of the type \u00b5 = \u221e X k=1 \u03c9k\u03b4\u03b8k, \u03b8si \u223c\u00b5, xi \u223cg0(\u00b7|\u03b8si) (10) where \u03c9k = Jk/ P\u221e l=1 Jl, J1, J2, \u00b7 \u00b7 \u00b7 are the jumps of the corresponding CRM de\ufb01ned in (3), \u03b8k\u2019s are the components of the mixture model drawn i.i.d. from a parameter space H(\u00b7), si denotes the component that xi belongs to, and g0(\u00b7|\u03b8k) is the density function to generate data on component k. Given the observations \u20d7 x, we introduce a slice latent variable ui for each xi so that we only consider those components whose jump sizes Jk\u2019s are larger than the corresponding ui\u2019s. Furthermore, the auxiliary variable UN (latent relative mass) is introduced to decouple each individual jump Jk and their in\ufb01nite sum of the jumps P\u221e l=1 Jl appeared in the denominators of \u03c9k\u2019s. For clari\ufb01cation, we list the notation and its description in Table 1. Based on [GW11], we have the following posterior Lemma. Lemma 3 The posterior of the in\ufb01nite mixture model (10) with the above auxiliary variables is proportional to p(\u20d7 \u03b8, J1, \u00b7 \u00b7 \u00b7 , JK, K, \u20d7 u, L,\u20d7 s, UN|\u20d7 x, NRM(\u03b7, M, H(\u00b7))) \u221d exp ( \u2212UN K X k=1 Jk ) exp \u001a \u2212M Z L 0 \u00001 \u2212e\u2212UNt\u0001 \u03c1\u03b7(t)dt \u001b UN\u22121 N p(J1, \u00b7 \u00b7 \u00b7 , JK) K Y k=1 h(\u03b8k) N Y i=1 1(Jsi > ui)g0(xi|\u03b8si), (11) where 1(a) is a indicator function returning 1 if a is true and 0 otherwise, h(\u00b7) is the density of H(\u00b7), J\u2217= P\u221e k=K+1 Jk, L = min{\u20d7 u}, and p(J1, \u00b7 \u00b7 \u00b7 , JK) = QK k=1 \u03c1\u03b7(Jk) R \u221e L \u03c1\u03b7(t)dt is the jump (large than L) distribution derived from the underlying Poisson process (actually, \u20d7 J follows a compound Poisson process, meaning that it has K \u223c Poisson(M R \u221e L \u03c1\u03b7(dt)) jumps, while each jump has density \u03c1\u03b7(Jk) R \u221e L \u03c1\u03b7(s)dt, here Poisson(x) means Poisson distribution with mean x). The expressions for the NGG needed to work with this lemma were given in the remark at the end of Section 2.2. Thus the integral term in Equation (11) can be turned into an expression involving incomplete Gamma functions. \f12 Normalized Random Measures Table 1: List of notation. Notation Description K #components with jump sizes larger than a threshold L \u03b8k, k = 1, \u00b7 \u00b7 \u00b7 , K Components in the mixture model M Total mass of the random measure Jk, k = 1, \u00b7 \u00b7 \u00b7 , K Jump sizes of the random measure with all Jk > L J\u2217 Sum of the remaining jump sizes, J\u2217= P\u221e k=K+1 Jk, Jk < L yi, i = 1, \u00b7 \u00b7 \u00b7 , N Observed data nk, k = 1, \u00b7 \u00b7 \u00b7 , K #data attached to each component N total number of data points si, i = 1, \u00b7 \u00b7 \u00b7 , N Variables indicating which component yi belongs to ui, i = 1, \u00b7 \u00b7 \u00b7 , N slice variable uniformly distributed in (0, Jsi] for yi L L = min{\u20d7 u} UN An auxiliary variable introduced to make the sampling feasible g0(\u00b7|\u03b8k) Density function to generate data on component \u03b8k h(\u03b8k) Density of H(\u03b8k) pM(M) Prior for M \u03bd(dt, dx) L\u00b4 evy measure of the random measure with decomposition \u03bd(dt, dx) = \u03c1\u03b7(dt)H(dx) considered in this paper 2.3.1 Sampling: First, we denote the parameter set as C = n \u20d7 \u03b8, J1, \u00b7 \u00b7 \u00b7 , JK, K, \u20d7 u, L,\u20d7 s, UN, M o , then the sampling goes as \u2022 Sampling \u20d7 s: From (11) we get p(si = k|C\\{si}) \u221d1(Jk > ui)g0(yi|\u03b8k) (12) \u2022 Sampling UN: Similarly p(UN|C\\{UN}) \u221d UN\u22121 N exp ( \u2212UN K X k=1 Jk ) exp \u001a \u2212M Z L 0 [1 \u2212exp {\u2212UNt}] \u03c1\u03b7(dt) \u001b , (13) \fChen & Buntine & Ding 13 which can be sampled using rejection sampling from a proposal distribution Ga \u0010 n, PK k=1 Jk \u0011 , here Ga(a, b) means a Gamma distribution with shape parameter a and scale parameter b. \u2022 Sampling \u03b8: The posterior of \u03b8k with prior density h(\u03b8k) is p(\u03b8k|C\\{\u03b8k}) \u221dh(\u03b8) Y i|si=k g0(yi|\u03b8k). (14) \u2022 Sampling K, {J1, \u00b7 \u00b7 \u00b7 , JK}: Sampling for Jk can be done separately for those associated with data points (\ufb01xed points) and for those that are not. Based on [JLP09], when integrating out \u20d7 u in (11), the posterior of the jump Jk with data attached (nk > 0) is proportional to Jnk k exp {\u2212UNJk} \u03c1\u03b7(Jk), (15) While for those without data attached (nk = 0), based on [GW11], conditional on UN, the number of these jumps follows a Poisson distribution with mean M Z \u221e L exp{\u2212UNt}\u03c1\u03b7(dt), while their lengths t have densities proportional to exp{\u2212UNt}\u03c1\u03b7(dt)1(t > L). \u2022 Sampling \u20d7 u: \u20d7 u are uniformly distributed in the interval (0, Jsi] for each i. After sampling the \u20d7 u, L is set to L = min{\u20d7 u}. \u2022 Sampling M: The posterior of M with prior pM(M) is p(M|C\\{M}) \u221dpM(M)MK exp \u001a \u2212M \u0014Z \u221e L \u03c1\u03b7(dt) + Z L 0 [1 \u2212exp {\u2212UNt}] \u03c1\u03b7(dt) \u0015\u001b . (16) pM(M) is usually taken to be Gamma distributed, so the posterior of M can be sampled conveniently. 3 Operations This section introduces the dependency operations used. These are developed for Poisson processes, CRMs and NRMs. \f14 Normalized Random Measures 3.1 Operations on Poisson processes We review three operations that transform Poisson processes in order to construct dependent completely random measures. For details, refer to [Kin93, LGF10]. Superposition of Poisson processes Given a set of Poisson processes \u03a01, \u03a02, \u00b7 \u00b7 \u00b7 , \u03a0n, the superposition of these Poisson processes is de\ufb01ned as the union of the points in these Poisson processes: \u03a0 := n [ i=1 \u03a0i. (17) Lemma 4 (Superposition Theorem) Let \u03a01, \u00b7 \u00b7 \u00b7 , \u03a0n be n independent Poisson processes on S with \u03a0k \u223cPoissonP(\u03bdk), then the superposition of these n Poisson processes is still a Poisson process with \u03a0 \u223cPoissonP(P i \u03bdi). Subsampling of Poisson processes Subsampling of a Poisson process with sampling rate q(\u03b8) is de\ufb01ned to be selecting the points of the Poisson process via independent Bernoulli trials with acceptance rate q(\u03b8). Lemma 5 (Subsampling Theorem) Let \u03a0 \u223cPoissonP(\u03bd) be a Poisson process on the space S and q : S \u2192[0, 1] be a measurable function. If we independently draw z\u03b8 \u2208{0, 1} for each \u03b8 \u2208\u03a0 with P(z\u03b8 = 1) = q(\u03b8), and let \u03a0k = {\u03b8 \u2208\u03a0 : z\u03b8 = k} for k = 0, 1, then \u03a00, \u03a01 are independent Poisson processes on S with S1\u2212q(\u03a0) := \u03a00 \u223c PoissonP((1 \u2212q)\u03bd) and Sq(\u03a0) := \u03a01 \u223cPoissonP(q\u03bd). Point transition of Poisson processes Point transition of a Poisson process \u03a0 on space (S, S), denoted as T(\u03a0), is de\ufb01ned as moving each point of the Poisson process independently to other locations following a probabilistic transition kernel T, which is de\ufb01ned to be a function T : S \u00d7 S \u2192[0, 1] 4 such that for each \u03b8 \u2208S, T (\u03b8, \u00b7) is a probability measure on E that describes the distribution of where the point \u03b8 moves, and for each A \u2208S, T (\u00b7, A) is integrable. Thus, T(\u03a0) := {\u03b8\u2032 : \u03b8\u2032 \u223c T (\u03b8, \u00b7)|\u03b8 \u2208\u03a0}. With a little abuse of notation, we use T (\u03b8) to denote a sample from T (\u03b8, \u00b7) in this paper. Thus T (\u03b8) is a stochastic function. Lemma 6 (Transition Theorem) Let \u03a0 \u223cPoissonP(\u03bd) be a Poisson process on space (S, S), T a probability transition kernel, then T(\u03a0) \u223cPoissonP(T \u03bd). (18) where T \u03bd can be considered as a transformation of measures over S de\ufb01ned as (T \u03bd)(A) := R S T (\u03b8, A)\u03bd(d\u03b8) for A \u2208S. 4In the following we will use T (\u00b7) to denote the point transition operation, while use T (\u00b7, \u00b7) to denote the corresponding transition kernel. \fChen & Buntine & Ding 15 3.2 Operations on random measures 3.2.1 Operations on CRMs The dependency operations de\ufb01ned on Poisson processes in Section 3.1 can be naturally generalized to the completely random measures given the construction in (3). Formally, we have Superposition of CRMs Given n independent CRMs \u02dc \u00b51, \u00b7 \u00b7 \u00b7 , \u02dc \u00b5n on X, the superposition (\u02dc \u2295) is de\ufb01ned as: \u02dc \u00b51 \u02dc \u2295\u02dc \u00b52 \u02dc \u2295\u00b7 \u00b7 \u00b7 \u02dc \u2295\u02dc \u00b5n := \u00b51 + \u00b52 + \u00b7 \u00b7 \u00b7 + \u00b5n . Subsampling of CRMs Given a CRM \u02dc \u00b5 = P\u221e k=1 Jk\u03b4\u03b8k on X, and a measurable function q : X \u2192[0, 1]. If we independently draw z(\u03b8) \u2208{0, 1} for each \u03b8 \u2208X with p(z(\u03b8) = 1) = q(\u03b8), the subsampling of \u02dc \u00b5, is de\ufb01ned as \u02dc Sq(\u02dc \u00b5) := X k z(\u03b8k)Jk\u03b4\u03b8k, (19) Point transition of CRMs Given a CRM \u02dc \u00b5 = P\u221e k=1 Jk\u03b4\u03b8k on X, the point transition of \u02dc \u00b5, is to draw atoms \u03b8\u2032 k from a transformed base measure to yield a new random measure as \u02dc T(\u02dc \u00b5) := \u221e X k=1 Jk\u03b4\u03b8\u2032 k. 3.2.2 Operations on NRMs The operations on NRMs can be naturally generalized from those on CRMs: Superposition of NRMs Given n independent NRMs \u00b51, \u00b7 \u00b7 \u00b7 , \u00b5n on X, the superposition (\u2295) is: \u00b51 \u2295\u00b52 \u2295\u00b7 \u00b7 \u00b7 \u2295\u00b5n := c1\u00b51 + c2\u00b52 + \u00b7 \u00b7 \u00b7 + cn\u00b5n . where the weights cm = \u02dc \u00b5m(X) P j \u02dc \u00b5j(X) and \u02dc \u00b5m is the unnormalized random measures corresponding to \u00b5m. Subsampling of NRMs Given a NRM \u00b5 = P\u221e k=1 rk\u03b4\u03b8k on X, and a measurable function q : X \u2192[0, 1]. If we independently draw z(\u03b8) \u2208{0, 1} for each \u03b8 \u2208X with p(z(\u03b8) = 1) = q(\u03b8), the subsampling of \u00b5, is de\ufb01ned as Sq(\u00b5) := X k:z(\u03b8k)=1 rk P j z(\u03b8j)rj \u03b4\u03b8k, (20) \f16 Normalized Random Measures Point transition of NRMs Given a NRM \u00b5 = P\u221e k=1 rk\u03b4\u03b8k on X, the point transition of \u00b5, is to draw atoms \u03b8\u2032 k from a transformed base measure to yield a new NRM as T(\u00b5) := \u221e X k=1 rk\u03b4\u03b8\u2032 k . The de\ufb01nitions are constructed so the following simple lemma holds. Lemma 7 Superposition, subsampling or point transition of NRMs is equivalent to superposition, subsampling or point transition of their underlying CRMs. Thus one does not need to distinguish between whether these operations are on CRMs or NRMs. 4 Posteriors for the NGG This section develops posteriors for the single NGG, for a standard version p \u0010 \u20d7 X|NGG(a, M, H(\u00b7) \u0011 and a version conditioned on the latent relative mass UN, p \u0010 \u20d7 X|UN, NGG(a, M, H(\u00b7) \u0011 . The second version is done because, as shown, the \ufb01rst version requires computing a complex recursive function. 4.1 Simple Posterior James et al. [JLP09] develop posterior analysis as follows. This theorem simpli\ufb01es their results and specialises them to the NGG. Theorem 2 (Posterior Analysis for the NGG) Consider the NGG(a, M, H(\u00b7)). For a data vector \u20d7 X of length N there are K distinct values X\u2217 1, ..., X\u2217 K with counts n1, ..., nK respectively. The posterior marginal is given by p \u0010 \u20d7 X|NGG(a, M, H(\u00b7) \u0011 = eMaK\u22121T N,K a,M \u0393(N) K Y k=1 (1 \u2212a)nk\u22121h(X\u2217 k) . (21) where T N,K a,M = Z \u221e M   1 \u2212 \u0012M t \u00131/a!N\u22121 tK\u22121e\u2212tdt . (22) Moreover, the predictive posterior is given by: p(XN+1 \u2208dx| \u20d7 X, NGG (a, M, H(\u00b7)) = \u03c90H(dx) + K X k=1 \u03c9k\u03b4X\u2217 k(dx) \fChen & Buntine & Ding 17 where the weights sum to 1 (PK k=0 \u03c9k = 1) are derived as \u03c90 \u221d aT N+1,K+1 a,M T N+1,K a,M \u03c9k \u221d (nk \u2212a) (23) Note that an alternative de\ufb01nition of T N,K a,M is T N,K a,M = aMK eM Z R+ uN\u22121 (1 + u)N\u2212KaeM\u2212M(1+u)adu , and various scaled versions of this integral are presented in the literature. Introducing a \u0393(b/a, 1) prior on M and then marginalising out M makes the term in eM\u2212M(1+u)a disappear since the integral over M can be carried inside the integral over u. Corollary 1 Let \u20d7 \u00b5 \u223cNGG (a, M, H(\u00b7)) and suppose M \u223c\u0393(b/a, 1) then it follows that \u20d7 \u00b5 \u223cPDP(a, b, H(\u00b7)) For computation, the issue here will be computing the terms T N,K a,M . Therefore we present some results for this. Lemma 8 (Evaluating T N,K a,M :) Have T N,K a,M de\ufb01ned as in Theorem 2. Then the following formula hold: T N,K a,M \u2264 \u0393(K, M) , (24) T N,K a,M = N\u22121 X n=0 \u0012N \u22121 n \u0013 \u0000\u2212M1/a\u0001n \u0393 \u0000K \u2212n a \u0001 \u0393 \u00001 \u2212n a \u0001 \u0393 \u0010 1 \u2212n a, M \u0011 \u2200N, K \u2208N+ , (25) T N,K+2 a,M = K T N+1,K a,M + N \u22121 a \u0010 T N,K a,M \u2212T N,K+1 a,M \u0011 , \u2200N > 2, K \u2208N+ (26) where \u0393(x, y) is the upper incomplete gamma function, de\ufb01ned for y > 0 and x \u0338= 0, \u22121, \u22122, .... Moreover, for Equation (25), ka cannot be integral for k = 1, ..., K\u22121. Another recursion is needed when a = 1/R for some R \u2208N+, R > 1. Then T N+1,K a,M = T N,K a,M \u2212M1/aT N,K\u22121/a a,M , \u2200K > R, N, K \u2208N+ , (27) It can be seen there are two di\ufb00erent situations. When a = 1/R for some R \u2208 N+, R > 1, then one can recurse down on N. But otherwise, one recurses down on K. Moreover, T N,K a,M is a strictly decreasing function of N and M, but an increasing function of K and a. For computation, Equation (25) can be used to compute T N,1 a,M and T N,2 a,M in terms of \u0393 \u00001 \u2212N a \u0001 . This equation may not be usable for K > 2 and may be unstable. Thereafter, for K > 2 in T N,K a,M the recursion of Equation (26) can be applied. \f18 Normalized Random Measures Remark The Poisson-Dirichlet Process and Dirichlet Process are well known for their ease of use in a hierarchical context [TJBB06, CDB11, BH12]. The NGG has the same general form. The major issue with this posterior theory is that one needs to precompute the terms T N,K a,M . While the Poisson-Dirichlet Process has a similar style, it has a generalised Stirling number dependent only on the discount a [BH12]. The di\ufb00erence is that for the PDP we can tabulate these terms for a given discount parameter a and still vary the concentration parameter (b above, but corresponding to M) easily. For the NGG, any tables of T N,K a,M would need to be recomputed with every change in mass parameter M. This might represent a signi\ufb01cant computational burden. 4.2 Conditional Posterior James et al. [JLP09] also develop conditional posterior analysis as follows. This theorem simpli\ufb01es their results and specialises them to the NGG. Theorem 3 (Conditional Posterior Analysis for the NGG) Consider the NGGa,M and the situation of Theorem 2. The conditional posterior marginal, conditioned on the auxiliary variable UN, is given by p \u0010 \u20d7 X|UN = u, NGG (a, M, H(\u00b7)) , N \u0011 = (Ma (1 + u)a)K PN k=1 SN k,a (Ma (1 + u)a)k K Y k=1 (1\u2212a)nk\u22121h(X\u2217 k) . (28) Moreover, the predictive posterior is given by: p \u0010 XN+1 \u2208dx| \u20d7 X, UN = u, NGG (a, M, H(\u00b7)) , N \u0011 = \u03c90H(dx) + K X k=1 \u03c9k\u03b4X\u2217 k(dx) where the weights sum to 1 (PK k=0 \u03c9k = 1) are derived as \u03c90 \u221d Ma (1 + u)a \u03c9k \u221d nk \u2212a . (29) The posterior for UN is given by: p \u0010 UN = u| \u20d7 X, NGG (a, M, H(\u00b7)) , N \u0011 = aMK T N,K a,M uN\u22121 (1 + u)N\u2212Kae\u2212M(1+u)a . (30) A posterior distribution is also presented by James et al. as their major result of Theorem 1 [JLP09]. We adapt it here to the NGG. Theorem 4 In the context of Theorem 3 the conditional posterior of the normalised random measure \u02dc \u00b5 given data \u20d7 X of length N and latent relative mass UN = u is given by \u20d7 \u00b5 = T T + J+ \u20d7 \u00b5\u2032 + J+ T + J+ K X k=1 pk\u03b4X\u2217 k \fChen & Buntine & Ding 19 where \u20d7 \u00b5\u2032 \u223c NGG \u0012 a, M 1 + u, H(\u00b7) \u0013 , T \u223c fT(t) where L\u00b4 evy measure of fT(t) = Ma \u0393(1 \u2212a)s\u2212a\u22121e\u2212(1+u)s , J+ \u223c \u0393(N \u2212Ka, 1 + u) , \u20d7 p \u223c DirichletK (\u20d7 n \u2212a) . Here, \u20d7 \u00b5\u2032, J+ and \u20d7 p are jointly independent and T, J+ and \u20d7 p are jointly independent. Note in particular the densities given for \u20d7 \u00b5\u2032 and T are not independent from each other. While an explicit density is not given for T, its expected value is easily computed via the Laplace transform as Ma(1 + u)a\u22121. Gri\ufb03n et al. [GW11] present an alternative technique for obtaining the conditional posterior. The following is adapted from their main sampler after integrating out the slice variables. Theorem 5 (Sampling Posterior) Consider a bound 0 < L < \u221ewhich is su\ufb03ciently small so that it is less than the jumps Jk associated with all the observed data. For an NRM given by NRM (\u03b7, M, H(\u00b7)), the number of jumps KL with value Jk > L is a random variable as well as their values J1, ..., JKL. The resultant posterior is as follows: p \u0010 \u20d7 X, UN = u | KL, J1, ..., JKL, N, NRM (\u03b7, M, H(\u00b7)) \u0011 = uN\u22121e\u2212M R L 0 (1\u2212e\u2212us)\u03c1\u03b7(s)ds KL Y k=1 e\u2212(1+u)JkJnk k h(X\u2217 k)1nk>0 where X\u2217 k are the unique data values (from \u20d7 X) and nk are the count of data from \u20d7 X having the value X\u2217 k. The expressions for the NGG needed to work with this lemma were given in the remark at the end of Section 2.2. We further simplify this by marginalising out jumps Jk and then taking the limit as L \u21920. Note we have renumbered indexes so that nk > 0 for all k = 1, ..., K where K \u2264KL. This matches the conditionals of Theorem 3 so is seen to be correct. Corollary 2 (Reduced Sampling Posterior) In the context of Theorem 5, assume there are K jumps with attached data such that nk > 0. The resultant posterior is as follows: p \u0010 \u20d7 X, UN = u, K | N, NGG (a, M, H(\u00b7)) \u0011 = uN\u22121 (1 + u)N\u2212Ka (Ma)K eM\u2212M(1+u)a K Y k=1 (1 \u2212a)nk\u22121h(X\u2217 k) . (31) \f20 Normalized Random Measures Moreover, p \u0010 \u20d7 X, UN = u, K, J1, ..., JK | N, NGG (a, M, H(\u00b7)) \u0011 = uN\u22121 \u0012 Ma \u0393(1 \u2212a) \u0013K eM\u2212M(1+u)a K Y k=1 Jnk\u2212a\u22121 k e\u2212(1+u)Jkh(X\u2217 k) . (32) Remark With the use of the latent relative mass UN, the NGG lends itself to hierarchical reasoning without a need to compute the recursive series T N,K a,M . This can be done with either the jumps integrated out, or the jumps retained. 5 Dependencies and Properties of Operations This section presents a number of results to do with the operations applied to the NRMs. First dependencies such as covariances are presented. Then some further properties are developed for when the operations are used in a network. 5.1 Dependencies between NRMs via Operations Properties of the NRMs here are given in terms of the Laplace exponent and its derivatives. In the Dirichlet process case, we have \u03c8(v) = M log(1 + v), while in the normalized generalized Gamma process case, we have \u03c8a(v) = M ((1 + v)a \u22121). Because the dependencies involve the total masses signi\ufb01cantly, we use a modi\ufb01ed version of the Laplace exponent in all these results. De\ufb01ne \u02dc \u03c8\u03b7(v) = 1 M \u03c8\u03b7(v), which has the mass removed. Di\ufb00erent from the Dirichlet process, the total masses M are no longer independent from their normalized jumps in general normalized random measures. However, we can still derive the correlations between di\ufb00erent NRMs. The following Theorems summarize these results. Lemma 9 (Mean and Variance of an NRM) Given a normalized random measure \u00b5 on X with the underlying L\u00b4 evy measure \u03bd(dt, dx) = M\u03c1\u03b7(dt)P(dx), for \u2200B \u2208B(X). The mean of this NRM is given by E[\u00b5(B)] = P(B) . (33) The variance of this NRM is given by Var(\u00b5(B)) = P(B)(P(B) \u22121)M Z \u221e 0 v \u02dc \u03c8\u2032\u2032 \u03b7(v) exp n \u2212M \u02dc \u03c8\u03b7(v) o dv . (34) \fChen & Buntine & Ding 21 Remark For DP, the corresponding variances are: VarDP(\u00b5(B)) = P(B)(1 \u2212P(B)) M + 1 . For NGG, it is VarNGG(\u00b5(B)) = P(B)(1 \u2212P(B))1 \u2212a a eMM 1 a|\u0393(\u22121 a, M)|. For large M the upper incomplete gamma function used here has the property that eMM1+ 1 a|\u0393(\u22121 a, M)| \u21921 and so we get for large M VarNGG(\u00b5(B)) \u2192P(B)(1 \u2212P(B))1 \u2212a Ma . Theorem 6 (Dependency via superposition) Suppose \u00b5i, i = 1, \u00b7 \u00b7 \u00b7 , n are n independent normalized random measures on X with the underlying L\u00b4 evy measures \u03bdi(dt, dx) = Mi\u03c1\u03b7(dt)P(dx), let \u00b5 = \u00b51 \u2295\u00b7 \u00b7 \u00b7 \u2295\u00b5n, B \u2208B(X), then the covariance between \u00b5k(k < n) and \u00b5 is Cov (\u00b5k(B), \u00b5(B)) = P(B)Mk Z \u221e 0 \u03b3(Mk, P(B), v) exp ( \u2212( X j\u0338=k Mj) \u02dc \u03c8\u03b7(v) ) dv +P(B)2   1 \u2212 2 P i\u0338=k Mi P j Mj ! . (35) where \u03b3(Mk, P(B), v) = (36) Z v 0 \u0010 P(B)Mk \u02dc \u03c8\u2032 \u03b7(v1)2 \u2212\u02dc \u03c8\u2032\u2032 \u03b7(v1) \u0011 exp n \u2212Mk \u02dc \u03c8\u03b7(v1) o dv1 Theorem 7 (Dependency via subsampling) Let \u02dc \u00b5 be a completely random measure on X with L\u00b4 evy measure \u03bd(dt, dx) = M\u03c1\u03b7(dt)P(dx), \u00b5 = \u02dc \u00b5 \u02dc \u00b5(X). The covariance between \u00b5 and its subsampling version Sq(\u00b5), denoted as \u00b5q, with sampling rate q(\u00b7) on B \u2208B(X) is Cov (\u00b5q(B), \u00b5(B)) = P(B)Mq Z \u221e 0 \u03b3(Mq, P(B), v) exp n \u2212(M \u2212Mq) \u02dc \u03c8\u03b7(v) o dv + P(B)2 \u00122Mq \u2212M M \u0013 , (37) where Mq := (q\u02dc \u00b5)(X) = R X q(x)\u02dc \u00b5(x)dx. \f22 Normalized Random Measures Theorem 8 (Dependency via point transition) Let \u02dc \u00b5 be a random measure on X with L\u00b4 evy measure \u03bd(dt, dx) = M\u03c1\u03b7(dt)P(dx), \u00b5 = \u02dc \u00b5 \u02dc \u00b5(X). Let B \u2208B(X), A = T (B) := {x : x \u223cT (y, \u00b7), y \u2208B} be the set of points obtained after the point transition on B, thus P(A) = R B P(T (x))dx. Suppose A and B are disjoint (which is usually the case when the transition operator T is appropriately de\ufb01ned), the covariance between \u00b5 and its point transition version T(\u00b5) on B \u2208B(X) is Cov (\u00b5(B), (T\u00b5)(B)) = P(A)P(B) (38) \u0012 M2 Z \u221e 0 Z v1 0 \u02dc \u03c8\u2032 \u03b7(v2)2 exp n \u2212M \u02dc \u03c8\u03b7(v2) o dv2dv1 \u22121 \u0013 5.2 Properties of the three dependency operations We \ufb01rst prove the following two Lemmas about superposition and subsampling of CRMs. A straightforward extension of Theorem 1 [JLP09] leads to the following Lemma about the posterior of CRMs under superposition. Lemma 10 (Posterior of CRMs under superposition) Let \u02dc \u00b51, \u02dc \u00b52, \u00b7 \u00b7 \u00b7 , \u02dc \u00b5n be n independent CRMs de\ufb01ned on space X, with L\u00b4 evy measures \u03bdi(dt, dx) for i = 1, \u00b7 \u00b7 \u00b7 , n. Let \u02dc \u00b5 = \u2295n i=1\u02dc \u00b5i. (39) Then given observed data X = {Xi} (we use X\u2217 k to denote the distinct values among X) and a latent relative mass Un, the posterior of \u02dc \u00b5 is given by (we use x|(y) to denote the variable x conditioned on y) \u02dc \u00b5|(Un, X) = \u02dc \u00b5|(Un) + K X k=1 Jk\u03b4X\u2217 k, (40) where 1. \u02dc \u00b5|(Un) is a CRM with L\u00b4 evy measure \u03bd(dt, dx) = e\u2212ut  n X i=1 \u03bdi(dt, dx) ! , 2. X\u2217 k (k = 1, \u00b7 \u00b7 \u00b7 , K) are the \ufb01xed points of discontinuity and Jk\u2019s are the corresponding jumps with densities proportional to tnke\u2212ut  n X i=1 \u03bdi(dt, dx) ! , where nk is the number of data attached at jump Jk. \fChen & Buntine & Ding 23 3. \u02dc \u00b5|(Un) and Jk\u2019s are independent. By subsampling, we can prove the following formula of the L\u00b4 evy measure under subsampling. Lemma 11 (L\u00b4 evy measure under subsampling) Let \u02dc \u00b5 = P\u221e k=1 Jk\u03b4X\u2217 k be a CRM with L\u00b4 evy measure \u03bd(dt, dx). Let Sq(\u02dc \u00b5) be its subsampling version with acceptance rate q(\u00b7), then Sq(\u02dc \u00b5) has the L\u00b4 evy measure of q(dx)\u03bd(dt, dx). Now we give some properties about compositions of of the dependency operations which follow simply. Lemma 12 (Composition of dependency operators) Given CRMs \u02dc \u00b5, \u02dc \u00b5\u2032 and \u02dc \u00b5\u2032\u2032, the following hold: \u2022 Two subsampling operations are commutative. So with acceptance rates q(\u00b7) and q\u2032(\u00b7), then Sq\u2032(Sq(\u02dc \u00b5)) = Sq(Sq\u2032(\u02dc \u00b5)). Both are equal to Sq\u2032q(\u02dc \u00b5)). \u2022 A constant subsampling operation commutes with a point transition operation. Thus Sq(T(\u02dc \u00b5)) = T(Sq(\u02dc \u00b5)) where the acceptance rate q is indepenent of the data space. \u2022 Subsampling and point transition operations distribute over superposition. Thus for acceptance rate q(\u00b7) and point transition T(\u00b7), Sq(\u02dc \u00b5 \u2295\u02dc \u00b5\u2032) = Sq(\u02dc \u00b5) \u2295Sq(\u02dc \u00b5\u2032) , T(\u02dc \u00b5 \u2295\u02dc \u00b5\u2032) = T(\u02dc \u00b5) \u2295T(\u02dc \u00b5\u2032) . \u2022 Superposition is commutative and associative. Thus \u02dc \u00b5 \u2295\u02dc \u00b5\u2032 = \u02dc \u00b5\u2032 \u2295\u02dc \u00b5 and (\u02dc \u00b5 \u2295\u02dc \u00b5\u2032) \u2295\u02dc \u00b5\u2032\u2032 = \u02dc \u00b5 \u2295(\u02dc \u00b5\u2032 \u2295\u02dc \u00b5\u2032\u2032). Thus when subsampling operations are all constant, a composition of subsampling, point transition and superposition operations admits a normal form where all the subsampling operations are applied \ufb01rst, then the transition operations and lastly the superposition operations. Lemma 13 (Normal form for compositions) Assume subsampling operations all have a constant acceptance rate. A normal form for a composition of subsampling, point transition and superposition operations is obtained by applying the following rules until no further can apply. Sq(Sq\u2032(\u02dc \u00b5)) \u2192 Sqq\u2032(\u02dc \u00b5)) , Sq(T(\u02dc \u00b5)) \u2192 T(Sq(\u02dc \u00b5)) , Sq(\u02dc \u00b5 \u2295\u02dc \u00b5\u2032) \u2192 Sq(\u02dc \u00b5) \u2295Sq(\u02dc \u00b5\u2032) , T(\u02dc \u00b5 \u2295\u02dc \u00b5\u2032) \u2192 T(\u02dc \u00b5) \u2295T(\u02dc \u00b5\u2032) . The remaining top level set of superpositions are then \ufb02attened out by removing any precedence ordering. \f24 Normalized Random Measures Note that Lemmas 10, 11, 12 and 13 all apply to NRMs as well due to Lemma 7. Now it is ready to state the main theorem about the relation between the CRM and the corresponding NRM under the three dependency operations. Theorem 9 (Equivalence Theorem) Assume the subsampling rates q(\u00b7) are independent (constant)5 for each point of the corresponding Poisson process, the following dependent random measures (41) and (42) are equivalent: \u2022 Manipulate the normalized random measures: \u00b5\u2032 m \u223cT(Sq(\u00b5\u2032 m\u22121)) \u2295\u00b5m, for m > 1. (41) \u2022 Manipulate the completely random measures: \u02dc \u00b5\u2032 m \u223c\u02dc T( \u02dc Sq(\u02dc \u00b5\u2032 m\u22121)) \u2295\u02dc \u00b5m, for m > 1. \u00b5\u2032 m = \u02dc \u00b5\u2032 m \u02dc \u00b5\u2032 m(X), (42) Furthermore, both resulting NRMs \u00b5\u2032 m\u2019s are equal to: \u00b5\u2032 m = m X j=1 (qm\u2212j \u02dc \u00b5j) (X) Pm j\u2032=1 (qm\u2212j\u2032 \u02dc \u00b5j\u2032) (X)T m\u2212j(\u00b5j), for m > 1 where qm\u2212j \u02dc \u00b5 is the random measure with L\u00b4 evy measure qm\u2212j(dx)\u03bd(dt, dx), and \u03bd(dt, dx) is the L\u00b4 evy measure of \u02dc \u00b5. T m\u2212j(\u00b5) denotes point transition on \u00b5 for (m \u2212j) times . In the posterior sampling for subsampling operation, we can prove the following posterior of the Bernoulli variables. Theorem 10 (Posterior acceptance rates for subsampling) Let \u02dc \u00b5\u2032 = P k Jk\u03b8k be a completely random measure on X, \u02dc \u00b5 = Sq(\u02dc \u00b5\u2032) := P k zkJk\u03b4k be its subsampling version, where zk\u2019s are independent Bernoulli random variables with acceptance rate q. Further de\ufb01ne \u00b5 = \u02dc \u00b5 \u02dc \u00b5(X). Given n = P k nk observed data in \u00b5, the posterior of zk is: p(zk = 1|\u02dc \u00b5, n) = ( 1 if nk > 0, q/J q/J+(1\u2212q)/J\u2212k if nk = 0. (43) where J = (P k\u2032 zk\u2032Jk\u2032)n, J\u2212k = \u0010P k\u2032\u0338=k zk\u2032Jk\u2032 \u0011n . 5This assumption is to deal with the case when considering point transition, meaning we can drop this assumption if no point transition operation is considered. \fChen & Buntine & Ding 25 Corollary 3 (Posterior acceptance rates in sampling J\u2032 mk in Section 4 [CDB12]) Using the terminology as in Section 4 [CDB12], the posterior p(zmk = 1|\u02dc \u00b5m, {\u02dc n\u2032 mk}) is computed as: \u2022 If \u02dc n\u2032 mk > 0, then p(zmk = 1|\u02dc \u00b5m, {\u02dc n\u2032 mk}) = 1. \u2022 Otherwise, p(zmk = 1|\u02dc \u00b5m, {\u02dc n\u2032 mk}) = qm\u2212m\u2032/Jm qm\u2212m\u2032/Jm + (1 \u2212qm\u2212m\u2032)/J\u2212k m , where Jm = \u0000P m\u2032\u2264m P k\u2032 zm\u2032k\u2032Jm\u2032k\u2032\u0001\u02dc n\u2032 m\u00b7, J\u2212k m = \u0010P m\u2032\u2264m P k\u2032\u0338=k zm\u2032k\u2032Jm\u2032k\u2032 \u0011\u02dc n\u2032 m\u00b7, and \u02dc n\u2032 m\u00b7 = P k\u2032 \u02dc n\u2032 mk\u2032. Acknowledgements. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program."
+                }
+            ],
+            "Ruiyi Zhang": [
+                {
+                    "url": "http://arxiv.org/abs/2403.09113v2",
+                    "title": "AutoLoRA: Automatically Tuning Matrix Ranks in Low-Rank Adaptation Based on Meta Learning",
+                    "abstract": "Large-scale pretraining followed by task-specific finetuning has achieved\ngreat success in various NLP tasks. Since finetuning all parameters of large\npretrained models poses substantial computational and memory challenges,\nseveral efficient finetuning methods have been developed. Among them, low-rank\nadaptation (LoRA), which finetunes low-rank incremental update matrices on top\nof frozen pretrained weights, has proven particularly effective. Nonetheless,\nLoRA's uniform rank assignment across all layers, along with its reliance on an\nexhaustive search to find the best rank, leads to high computation costs and\nsuboptimal finetuning performance. To address these limitations, we introduce\nAutoLoRA, a meta learning based framework for automatically identifying the\noptimal rank of each LoRA layer. AutoLoRA associates each rank-1 matrix in a\nlow-rank update matrix with a selection variable, which determines whether the\nrank-1 matrix should be discarded. A meta learning based method is developed to\nlearn these selection variables. The optimal rank is determined by thresholding\nthe values of these variables. Our comprehensive experiments on natural\nlanguage understanding, generation, and sequence labeling demonstrate the\neffectiveness of AutoLoRA.",
+                    "authors": "Ruiyi Zhang, Rushi Qiang, Sai Ashish Somayajula, Pengtao Xie",
+                    "published": "2024-03-14",
+                    "updated": "2024-03-17",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Large Language Models (LLMs) (Radford et al., 2019; Brown et al., 2020) have demonstrated stateof-the-art performance across a variety of NLP tasks, spanning from Natural Language Understanding (NLU) (Wang et al., 2018) to Natural Language Generation (NLG) (Radev et al., 2020), a trajectory highlighted by the success of models like ChatGPT (OpenAI, 2023). Their success largely *Equal contribution. stems from a two-stage process: initial pretraining on vast amounts of unlabeled texts, followed by finetuning on specific downstream tasks. However, as models scale up, for instance transitioning from RoBERTa-large\u2019s 355 million parameters (Liu et al., 2019) to GPT-3\u2019s staggering 175 billion parameters (Brown et al., 2020), finetuning becomes highly expensive in computation. To address this challenge, many efficient finetuning methods (Houlsby et al., 2019) have been developed. For instance, the Adapters method (Houlsby et al., 2019) inserts lightweight layers (called adapters) into pretrained networks. During finetuning, only these adapters are updated while the pretrained layers are kept frozen. One limitation of this method is that the adapters incur additional computation overhead during inference. Another approach, prefix tuning (Lester et al., 2021), introduces trainable prefix parameters which are prepended to the input sequence while making the pretrained model parameters frozen. Nevertheless, determining the optimal length of the prefix can be tricky. A prefix that is too short cannot capture enough information, while an overlong prefix may largely reduce the maximum length of the input sequence. To address these limitations, LoRA (Hu et al., 2022) proposes to add low-rank incremental update matrices to pretrained weight matrices. During finetuning, only the incremental matrices are trained while the pretrained ones are frozen. The low-rank parameterization significantly reduces the number of finetuning parameters. While achieving parameter-efficient finetuning without increasing inference costs, LoRA has two limitations. First, the update matrices at different layers share the same rank, without considering the varying properties across layers. Different layers in a pretrained model have varying importance to a downstream task and should be adapted differently, which requires the number of trainable parameters to be layer-specific. EmarXiv:2403.09113v2  [cs.CL]  17 Mar 2024 \fOptimize Update Matrices Low-Rank Matrix \ud835\udc48 Low-Rank Matrix \ud835\udc49 Selection Variables Optimize Selection Variables Low-Rank Matrix \ud835\udc48 Low-Rank Matrix \ud835\udc49 Selection Variables Iterate until convergence Meta Learning of Selection Variables Thresholding Selection Variables Retrain Update Matrix with Optimal Rank Determine Matrix Rank Figure 1: An overview of AutoLoRA. In the meta learning process, AutoLoRA learns selection variables with two iterative steps. Firstly, the weights in the update matrices are optimized on the training dataset. Secondly, the selection variables are updated on the validation dataset. These two steps are iterated until convergence is achieved. Upon acquiring the optimal values of the selection variables, AutoLoRA determines the optimal matrix ranks by thresholding these values. Subsequently, the ranks of update matrices in the LoRA layers are set to the learned optimal ranks and retrained on the combination of training and validation data. ploying a uniform rank across all layers compromises this purpose, which renders some layers to be under-parameterized (leading to suboptimal finetuning performance) while others unnecessarily over-parameterized (leading to computation inefficiency). Second, obtaining the optimal rank in LoRA typically involves an extensive manual hyperparameter search, which is time-consuming and poses scalability issues. To address the aforementioned limitations of LoRA, we introduce the AutoLoRA framework to automatically determine the optimal rank for each LoRA layer. In AutoLoRA, we first decompose an update matrix into the product of two low-rank matrices (with rank k), in alignment with the LoRA methodology. This product can be expressed as the summation of k rank-1 matrices. For each rank-1 matrix, we assign a continuous trainable selection variable \u03b1 \u2208[0, 1] indicating the matrix\u2019s relative importance in the summation. After learning, if \u03b1 is close to zero, the corresponding rank-1 matrix is removed from the summation. These selection variables effectively control the rank of an update matrix. Learning \u03b1 directly on a training dataset together with the update matrices can result in overfitting, and the network learned in this way lacks generalization ability. To mitigate this problem, we formulate the search process of \u03b1 as a meta learning (Finn et al., 2017) problem. First, we finetune the weights in the rank-1 matrices on a training dataset. Second, we optimize the \u03b1 values by minimizing the loss on a validation dataset. These two steps iterate until convergence. Subsequently, we derive the optimal rank of each LoRA layer by thresholding the learned \u03b1 values. Once the optimal rank is identified for each layer, the weights in the low-rank update matrices are retrained on the combination of training and validation data. An overview of our proposed method is illustrated in Figure 1. The major contributions of this paper are summarized as follows. \u2022 We propose AutoLoRA, a meta learning based approach that can automatically determine the optimal and layer-specific ranks of update matrices, alleviating the burden of manually tuning them as in LoRA. \u2022 Extensive experiments on natural language understanding and generation tasks demonstrate the effectiveness of AutoLoRA. 2 Related Works 2.1 Parameter Efficient Finetuning Methods Various methods have been developed for efficiently finetuning pretrained models. These methods update only a small subset of the weights in large pretrained models, leaving the majority of the parameters frozen. According to Aghajanyan et al. (2021), weight matrices in large pretrained models tend to have a small intrinsic dimension, offering \ftheoretical intuitions for finetuning pretrained models with low-dimensional reparameterization. Impressively, these methods can sometimes surpass the performance of full finetuning, particularly in downstream tasks with limited training data. Some efficient finetuning methods finetune the pretrained model by updating trainable prompts while leaving its pretrained parameters frozen. For example, Prompt-tuning (Lester et al., 2021) learns \u201csoft prompts\u201d for language models to perform specific downstream tasks. Prefix-tuning (Li and Liang, 2021) optimizes a sequence of continuous task-specific vectors for natural language generation tasks. P-tuning (Liu et al., 2023) optimizes a small neural network which generates continuous prompt embeddings to finetune GPT models for natural language understanding tasks. LLaMAAdapter (Zhang et al., 2023b) learns trainable prompts for the LLaMA (Touvron et al., 2023a) model. However, selecting appropriate prompt length can be challenging, as short prompts cannot capture sufficient information while overlong prompts significantly reduce the input sequences\u2019 length. Another line of research involves finetuning the pretrained model by inserting trainable modules into the model while keeping pretrained parameters frozen. For example, Adapter (Houlsby et al., 2019) proposes to inject additional trainable adapter layers into pretrained Transformer (Vaswani et al., 2017) models. IA3 (Liu et al., 2022) multiplies the output of activation functions in the pretrained model with trainable vectors. Compacter (mahabadi et al., 2021) inserts hypercomplex multiplication layers (Zhang et al., 2021) to the pretrained model, offering more efficiency than those in Adapters. These methods incur additional inference overhead due to computing the inserted modules. AdaLoRA (Zhang et al., 2023a) aims to overcome the problem that LoRA evenly distributes the budget of updates across all LoRA layers by adaptively allocating the budget according to their importance scores. However, since both the importance score and update matrices are learned on the same training dataset, there is an increased risk of overfitting. 2.2 Meta Learning Various meta learning methods have been proposed for better adaptation of models to new tasks with minimal training data. For instance, ModelAgnostic Meta-Learning (MAML) (Finn et al., 2017) is a gradient based meta learning method, aiming to train model weights for fast adaptation to new tasks with small amounts of data in a few gradient descent steps. Meta-SGD is an extension of MAML (Li et al., 2018). It not only learns model weights, but also optimizes learning rates for fast adaptation to new tasks. Reptile (Nichol et al., 2018) is a first-order meta learning algorithm, which serves as a simpler alternative to MAML. Reptile repeatedly moves the initialization of meta parameters towards the model weights trained on a specific task, sidestepping second-order gradient computation. Orthogonal to these previous methods, our meta learning based method is used for tuning matrix ranks in LoRA. 3 Preliminaries In LoRA (Hu et al., 2022), a weight matrix Wl \u2208 Rml\u00d7nl at layer l in a downstream model is parameterized as Wl = f Wl + \u2206l, where f Wl is the weight matrix at layer l in a pretrained model and \u2206l is an incremental update matrix. \u2206l is parameterized as the product of two low-rank matrices: \u2206l = UlVl, where Ul \u2208Rml\u00d7kl and Vl \u2208Rkl\u00d7nl. kl, which is much smaller than ml and nl, is the rank of \u2206l. Equivalently, \u2206l can be written as the summation of kl rank-1 matrices: \u2206l = kl X j=1 \u2206j l , (1) where \u2206j l is the outer-product between the j-th column of Ul and the j-th row of Vl. 4 Method 4.1 Overview In AutoLoRA, we aim to automatically determine the rank kl in Eq.(1), instead of manually specifying it as in LoRA. To achieve this goal, we associate each rank-1 matrix in an update matrix with a selection variable and reparameterize the update matrix as a weighted sum of rank-1 matrices. A meta learning based approach is developed to learn these selection variables. After learning, if the value of a selection variable is close to zero, its corresponding rank-1 matrix is removed. In this way, we can determine the optimal rank for each update matrix based on the selection variables. An overview of the AutoLoRA algorithm is shown in Algorithm 1. \fAlgorithm 1 AutoLoRA: Automatically Search for Optimal Rank Initialize selection variables A = {\u03b1l}1\u2264l\u2264M and update matrices {\u2206l = Pkl j=1 \u03b1j l \u2206j l }1\u2264l\u2264M while not converged do 1. Update weight parameters W by descending \u2207W Ltr(W, Dtr). 2. Update selection variables A by descending \u2207ALval(W \u2212\u03b7\u2207W Ltr(W, Dtr), Dval). end while Derive the best rank k\u2217 l = |{\u03b1\u2217 l,j|1 \u2264j \u2264kl, \u03b1\u2217 l,j \u2265\u03bb}| for LoRA layer l from optimal values \u03b1\u2217 l . 4.2 Reparameterize Update Matrices We associate each rank-1 matrix \u2206j l in Eq.(1) with a selection variable \u03b1j l \u2208[0, 1] and reparameterize \u2206l as a weighted sum of rank-1 matrices: \u2206l = kl X j=1 \u03b1j l \u2206j l . (2) \u03b1j l can be interpreted as the importance of \u2206j l . If \u03b1j l is close to 0, \u2206j l will be removed from \u2206l, which effectively reduces the rank of \u2206l by one. In other words, the rank of \u2206l is equivalent to the number of non-zero values in {\u03b1j l }kl j=1. By learning these selection variables based on their fitness to data, we can automatically determine the rank of \u2206l. We add a constraint that the sum of {\u03b1j l }kl j=1 is equal to one: Pkl j=1 \u03b1j l = 1. This constraint renders the optimization of {\u03b1j l }kl j=1 difficult. To address this problem, instead of optimizing {\u03b1j l }kl j=1 directly, we parameterize them using softmax: \u03b1j l = exp(\u03b2j l ) Pkl i=1 exp(\u03b2i l) , (3) and learn the unconstrained variables {\u03b2j l }kl j=1. 4.3 Learn Selection Variables Let A = {\u03b1j l |1 \u2264j \u2264kl, 1 \u2264l \u2264M} denote all selection variables, where M is the number of layers in the pretrained model. We propose a meta learning based approach to learn A. Let Ltr denote the downstream task\u2019s training loss defined on a training dataset Dtr. Given the weight parameters Wl = f Wl + \u2206l at layer l in the downstream model, we first perform a one-step gradient descent update of Wl: c Wl = Wl \u2212\u03b7\u2207WlLtr({Wl}M l=1, Dtr), (4) where \u03b7 is a learning rate. Then we evaluate {c Wl}M l=1 on a validation dataset Dval. The validation loss Lval({c Wl}M l=1, Dval) is a function of A since Lval depends on {c Wl}M l=1 which depends on A. We optimize A by minimizing the validation loss: minA Lval({c Wl}M l=1, Dval). (5) We use an approximate gradient-based algorithm (Choe et al., 2023) to solve this problem. The updates of W and A in Eq.(4) and Eq.(5) are iteratively performed until convergence. 4.4 Determine Matrix Rank Given the optimally learned selection variables A\u2217, we determine the rank of each update matrix based on A\u2217. For each layer l, we count the number of entries in {\u03b1j l }kl j=1 that satisfy \u03b1j l \u2265\u03bb, where \u03bb denotes a threshold. This number would be the optimal rank for \u2206l. We set \u03bb to be \u03bb = 1/kl. This threshold guarantees the automatically determined rank is at least one. 4.5 Retrain Update Matrices The thresholding operations in Section 4.4 incurs a discrepancy: when training the update matrices in Section 4.3, all rank-1 matrices are used to make predictions; however, after thresholding, some rank-1 matrices are dropped, which may hurt performance. To bridge this discrepancy, we retrain the update matrices. Specifically, for each update matrix, we set its rank to be the optimal value determined in Section 4.4, then train them by minimizing the finetuning loss on the combination of training and validation datasets. 5 Experiments 5.1 Experimental Setup The baseline methods used in this work include Adapter (Houlsby et al., 2019), LoRA (Hu et al., 2022), and AdaLoRA (Zhang et al., 2023a). We examine the efficacy of AutoLoRA by finetuning a RoBERTa-base model (Liu et al., 2019), a RoBERTa-large model, and a GPT2-medium model (Radford et al., 2019) on natural language \fMethod Params CoLA SST-2 MRPC QQP MNLI QNLI RTE STS-B Avg. Full FT 125.0M 61.6 94.8 89.3 90.3 86.7 92.8 76.9 91.2 85.5 Adapter 0.9M 58.8 94.0 88.4 89.1 86.5 92.5 71.2 89.9 83.8 LoRA 0.3M 59.0 94.5 89.1 89.6 86.9 92.9 75.8 91.1 84.9 AdaLoRA 0.3M 58.8 94.0 89.4 89.9 87.0 93.0 75.9 90.6 85.0 AutoLoRA 0.3M 61.3 94.9 89.4 90.3 87.0 92.9 77.0 90.8 85.5 Table 1: Performance and the number of parameters (Params) of AutoLoRA and baseline methods for finetuning RoBERTa-base on the GLUE benchmark. Higher value is better for all metrics. The best results are shown in bold. We also provide the performance of full finetuning (Full FT) for reference. Figure 2: Results of finetuning the RoBERTa-large model on the MRPC, QQP, and SST-2 datasets. Y-axis represents accuracy on GLUE development sets. understanding (NLU), natural language generation (NLG), and sequence labeling datasets. We include detailed comparison of these two pretrained models in Appendix B. A Transformer (Vaswani et al., 2017) model consists of several stacked Transformer blocks (layers), and each block contains a multi-head attention (MHA) module and a fully-connected neural network. Each head in an MHA module includes a query projection layer, a key projection layer, and a value projection layer. In adherence to the standard setting in LoRA, we select only the query and value projection layers as trainable LoRA layers, leaving other layers frozen. Both RoBERTa-base and GPT2-medium possess 12 Transformer layers, which results in 24 trainable LoRA layers. The RoBERTa-large model, with 24 Transformer layers, has 48 trainable LoRA layers. We set the initial dimension of selection variables \u03b1l to be 8 at each layer, i.e., kl = 8. The rank for each layer in LoRA baselines is set as 4, resulting in a similar number of trainable parameters as that in AutoLoRA. We use AdamW (Loshchilov and Hutter, 2019) as the optimizer for both AutoLoRA and baseline methods. We set the batch size as 16 for NLU and NLG tasks, and 32 for the sequence labeling task. We set the learning rate for optimizing weight parameters W in Eq.(4) to be 1e \u22124, and the learning rate for optimizing selection variables A in Eq.(5) to be 1e \u22123. All experiments were conducted on NVIDIA A100 GPUs. Our implementation is based on Pytorch (Paszke et al., 2019), HuggingFace Transformers (Wolf et al., 2020), and the Betty library (Choe et al., 2023). 5.2 Experiments on Natural Language Understanding Tasks We conduct extensive experiments on eight datasets from the General Language Understanding Evaluation (GLUE) benchmark (Wang et al., 2018) to evaluate the performance of AutoLoRA on NLU tasks. The GLUE benchmark contains single sentence classification, sentence pair classification, and regression tasks for language acceptability evaluation, sentiment analysis, sentence similarity measurement, and natural language inference. We use accuracy as the evaluation metrics for the SST-2, MRPC, QQP, MNLI, QNLI, and RTE tasks. We use Matthew\u2019s correlation for the CoLA task and Spearman\u2019s correlation for the STS-B task. Since the test sets of the GLUE benchmark are not publicly available, following previous studies (Zhang et al., 2022), we use the AutoLoRA framework to finetune a RoBERTa-base model on the GLUE training set and evaluate it on the GLUE development set. We split the original training set into a new training set and a validation set with a ratio of 1:1, which are used as Dtr and Dval in Eq.(4) and Eq.(5) respectively. Please note that baselines methods are trained on the original training set and our method does not unfairly use more data than baselines. Table 1 shows the performance of AutoLoRA \fE2E WebNLG Method Param BLEU NIST MET ROUGE-L CIDEr BLEU MET TER Full FT 354.9M 68.0 8.61 46.1 69.0 2.38 46.5 38.0 0.53 Adapter 11.1M 67.0 8.50 45.2 66.9 2.31 50.2 38.0 0.46 LoRA 0.3M 67.1 8.54 45.7 68.0 2.33 50.7 39.5 0.46 AdaLoRA 0.3M 67.0 8.55 45.5 68.1 2.32 50.6 39.4 0.44 AutoLoRA 0.3M 67.9 8.68 46.0 68.9 2.37 50.8 39.6 0.44 Table 2: Performance and the number of parameters of AutoLoRA and baseline methods for finetuning GPTmedium on the E2E and WebNLG datasets. Higher value is better for all metrics except TER. Lower TER value indicates better performance. on the GLUE development sets, compared with baseline methods. AutoLoRA achieves the best performance on 6 out of 8 datasets, and obtains an average performance of 85.5, outperforming all baseline methods. As AutoLoRA outperforms LoRA on average, we can conclude that the optimal ranks learned by AutoLoRA are better than the manually tuned ranks in LoRA. The reasons are two-fold. First, AutoLoRA allows different layers to have distinct ranks, sufficiently accounting for the fact that different layers have varying properties and need to have layer-specific amounts of tunable parameters. In contrast, LoRA uniformly uses the same rank for all layers, without considering the difference across layers. Second, AutoLoRA learns the continuous selection variables (which determine the ranks) by maximizing the finetuning performance on validation data via gradient descent. The search space is continuous, which allows more comprehensive exploration of rank configurations. In contrast, LoRA performs manual tuning of ranks in a discrete space, where the number of rank configurations is relatively limited. Furthermore, AutoLoRA outperforms the AdaLoRA baseline on average. The reason is that AdaLoRA uses a single dataset to simultaneously learn rank-1 matrices and their importance scores, which can easily lead to overfitting. In contrast, our method splits the training dataset into two disjoint sets, learns rank-1 matrices on one set, and optimizes selection variables on the other set, which is more resilient to overfitting. In addition, we present the results of fully finetuning a RoBERTa-base model. Results indicate that AutoLoRA attains performance on par with the full finetuning method, while utilizing significantly fewer parameters. We further examine the efficacy of AutoLoRA with larger pretrained models. Specifically, we applied AutoLoRA to finetune a RoBERTa-large model (Liu et al., 2019) on the MRPC, QQP, and SST-2 datasets. The RoBERTa-large model comprises 355 million parameters, in contrast to the RoBERTa-base, which only contains 125 million. As shown in Figure 2, the performance of AutoLoRA surpasses both baseline methods across all three datasets, demonstrating AutoLoRA\u2019s robust effectiveness in finetuning pretrained models with various sizes. 5.3 Experiments on Natural Language Generation Tasks In addition to NLU tasks, we also evaluate the effectiveness of AutoLoRA in NLG tasks. The experiments were conducted on two datasets: E2E (Novikova et al., 2017) and WebNLG (Gardent et al., 2017). The E2E dataset contains around 50,000 data-sentence pairs in the restaurant domain. Given the data record of a restaurant, the task is to generate a text description for the restaurant. The WebNLG dataset contains more than 10,000 datasentence pairs extracted from DBpedia. The data contains triples with a format of (subject, property, object), and the task is to generate a text as a verbalisation of these triples. We use BLEU (Papineni et al., 2002), NIST (Lin and Och, 2004), METEOR (Banerjee and Lavie, 2005), ROUGEL (Lin and Hovy, 2004), and CIDEr (Vedantam et al., 2015) as evaluation metrics for the E2E dataset. For the WebNLG dataset, we use BLEU, METEOR, and TER (Snover et al., 2006) as evaluation metrics. AutoLoRA was applied to finetune a GPT-medium model. Table 2 shows the performance of AutoLoRA on the E2E test set and WebNLG test set. AutoLoRA achieves the best performance in terms of all five metrics on the E2E dataset. It outperforms or is on par with baseline methods on the WebNLG dataset \fMethod Param Precision F1 Full FT 125.0M 70.3 74.9 Adapter 0.9M 66.9 71.3 LoRA 0.3M 68.5 72.2 AdaLoRA 0.3M 69.4 73.0 AutoLoRA 0.3M 70.1 74.2 Table 3: Performance and the number of parameters of AutoLoRA and baseline methods for finetuning a RoBERTa-base model on the BioNLP dataset. in terms of all three metrics. This demonstrates AutoLoRA\u2019s effectiveness in finetuning pretrained models for NLG tasks. The analysis of reasons that AutoLoRA outperforms LoRA and AdaLoRA is similar to that in Section 5.2. Moreover, the performance of AutoLoRA is on par with that of the full finetuning method, while the number of parameters in AutoLoRA is substantially less. 5.4 Experiments on Sequence Labeling In this section, we evaluate AutoLoRA on a sequence labeling task. Different from the GLUE tasks which perform classification on an entire sentence (focusing on capturing global semantics), sequence labeling performs classification on each token in a sentence (emphasizing capturing local context). The experiments were conducted on the BioNLP (Collier and Kim, 2004) dataset, which is a Named Entity Recognition dataset containing biological entities such as DNA, RNA, and protein. F1 is used as the evaluation metric. AutoLoRA was applied to finetune a RoBERTa-base model for this task. Table 3 shows the performance of AutoLoRA on the BioNLP test set, compared with baseline methods. AutoLoRA outperforms all baseline methods in terms of F1 score. The analysis of reasons is similar to that in Section 5.2. In line with our previous findings on NLU and NLG tasks, AutoLoRA can effectively finetune pretrained models for sequence labeling. 5.5 Ablation Studies In this section, we perform ablation studies to investigate the effectiveness of individual modules in our method. The studies were performed on the GLUE benchmark. No Constraints. We examine the effectiveness of the sum-to-one constraint in Section 4.2 by removing the constraints from AutoLoRA. Specifically, we directly use a threshold \u03bb = Pkl j=1 \u03b1j l /kl to obtain the optimal discrete ranks without any constraints in the meta learning process (AutoLoRA w/o cst.). Results in Table 4 show that AutoLoRA outperforms this ablation setting on average, indicating the effectiveness of this sum-to-one constraint. The reason is that adding such a constraint can make the selection variables better represent the relative importance of rank-1 matrices, which facilitate accurate pruning of less important rank-1 matrices. Element-wise Sigmoid. We further examine the effectiveness of the sum-to-one constraint in Section 4.2 by comparing AutoLoRA with an ablation setting that applies element-wise sigmoid operations on selection variables. Specifically, for each \u03b1j l , we use sigmoid to constrain its value into [0, 1] in the meta learning process, and a threshold of 0.5 is used to obtain discrete ranks (AutoLoRA sigmoid). Results in Table 4 show that AutoLoRA outperforms this ablation setting on average. In this ablation setting, \u03b1j l no longer directly indicates the relative importance of rank-1 matrices, making it challenging to select an appropriate threshold. Meta Learning. We examine the effectiveness of the meta learning framework by setting \u03b7 = 0 in Algorithm 1. This ablation setting can be interpreted as an alternative learning method where two optimization steps are carried out alternatively on two different splits of the training dataset. Results in Table 4 show that AutoLoRA outperforms AutoLoRA (\u03b7 = 0) on average, demonstrating the efficacy of the meta learning strategy. 5.6 Qualitative Analysis Figure 3 presents the optimal rank determined by AutoLoRA for the QQP, MNLI, and E2E datasets. For the QQP and MNLI datasets, we utilized a RoBERTa-base backbone, while a GPT2-medium backbone was employed for the E2E dataset. In this figure, column i corresponds to the i-th Transformer block in the pretrained model. Each row corresponds to a dataset and a layer type (query projection and value projection layer). As can be seen, the optimal ranks learned by AutoLoRA for different layers have varying values. This is aligned with the hypothesis discussed in Section 1 that different layers need different matrix ranks. Vanilla LoRA ignores this difference and uniformly uses the same \fMethod CoLA SST-2 MRPC QQP MNLI QNLI RTE STS-B Avg. AutoLoRA (w/o cst.) 61.0 93.7 88.5 90.0 87.2 92.1 77.5 90.5 85.1 AutoLoRA (sigmoid) 59.7 94.1 88.3 89.8 86.9 92.6 75.7 90.7 84.7 AutoLoRA (\u03b7 = 0) 61.2 94.8 89.3 90.1 87.1 92.8 77.3 90.5 85.2 AutoLoRA 61.3 94.9 89.4 90.3 87.0 92.9 77.0 90.8 85.5 Table 4: Ablation studies. Performance comparison of AutoLoRA, AutoLoRA without constraint (AutoLoRA w/o cst.), AutoLoRA with element-wise sigmoid (AutoLoRA sigmoid), and AutoLoRA with \u03b7 = 0. We evaluated all methods on the development sets of the GLUE benchmark. Figure 3: Optimal ranks of the LoRA layers obtained by AutoLoRA on the QQP, MNLI, and E2E datasets. We finetuned RoBERTa-base on QQP and MNLI, and GPT2-medium on E2E. Both RoBERTa-base and GPT2-medium consist of 12 Transformer layers. We only search for the ranks in the query and value projection layers. Method AdaLoRA LoRA+Grid Search AutoLoRA Cost x1 x14.29 x1.91 Table 5: Comparison of average training cost between AutoLoRA and baseline methods on the SST-2, MNLI, and QQP datasets. We normalize the average training time of AdaLoRA as 1. rank across layers, which leads to inferior performance. Our method provides a computationally efficient mechanism to learn these layer-specific ranks, which takes much less time than grid search (as shown in Section 5.7). 5.7 Computation Costs Table 5 shows the average training cost of AutoLoRA and two baseline methods on the SST2, MNLI, and QQP datasets. We normalize the average training time of AdaLoRA as 1 for reference. In LoRA, we use grid search to tune the ranks, with 16 configurations. As can be seen, our method is much more efficient than performing grid search of ranks in LoRA. Grid search is conducted in a discrete space. For each configuration of ranks, LoRA needs to run from scratch, which is very time-consuming. In contrast, our method performs the search in a continuous space via gradient method, which can efficiently explore many configurations without restarting. Compared with AdaLoRA, our method has significantly better performance as shown in Tables 1, 2, and 3, without substantially increasing the computation costs. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2005.01618v1",
+                    "title": "Reward Constrained Interactive Recommendation with Natural Language Feedback",
+                    "abstract": "Text-based interactive recommendation provides richer user feedback and has\ndemonstrated advantages over traditional interactive recommender systems.\nHowever, recommendations can easily violate preferences of users from their\npast natural-language feedback, since the recommender needs to explore new\nitems for further improvement. To alleviate this issue, we propose a novel\nconstraint-augmented reinforcement learning (RL) framework to efficiently\nincorporate user preferences over time. Specifically, we leverage a\ndiscriminator to detect recommendations violating user historical preference,\nwhich is incorporated into the standard RL objective of maximizing expected\ncumulative future rewards. Our proposed framework is general and is further\nextended to the task of constrained text generation. Empirical results show\nthat the proposed method yields consistent improvement relative to standard RL\nmethods.",
+                    "authors": "Ruiyi Zhang, Tong Yu, Yilin Shen, Hongxia Jin, Changyou Chen, Lawrence Carin",
+                    "published": "2020-05-04",
+                    "updated": "2020-05-04",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.IR",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Traditional recommender systems depend heavily on user history. However, these approaches, when implemented in an of\ufb02ine manner, cannot provide satisfactory performance due to sparse history data and unseen dynamic new items (e.g., new products, recent movies, etc.). Recent work on recommender systems has sought to interact with users, to adapt to user preferences over time. Most existing interactive recommender systems are designed based on simple user feedback, such as clicking data or updated ratings [6, 30, 33]. However, this type of feedback contains little information to re\ufb02ect complex user attitude towards various aspects of an item. For example, a user may like the graphic of a dress but not its color. A click or numeric rating is typically not suf\ufb01cient to express such a preference, and thus it may lead to poor recommendations. By contrast, allowing a recommender system to use natural-language feedback provides richer information for future recommendation, especially for visual item recommendation [20, 21]. With natural-language feedback, a user can describe features of desired items that are lacking in the current recommended items. The system can then incorporate feedback and subsequently recommend more suitable items. This type of recommendation is referred to as text-based interactive recommendation. Flexible feedback with natural language may still induce undesired issues. For example, a system may ignore the previous interactions and keep recommending similar items, for which a user has expressed the preference before. To tackle these issues, we propose a reward constrained recommendation (RCR) framework, where one sequentially incorporates constraints from previous feedback into the recommendation. Speci\ufb01cally, we formulate the text-based interactive recommendation as a constraint-augmented reinforcement learning (RL) problem. Compared to standard constraintaugmented RL, there are no explicit constraints in text-based interactive recommendation. To this end, we use a discriminator to detect violations of user preferences in an adversarial manner. To further 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada. \u2217Equal contribution. Work done while RZ was a research intern at Samsung Research America. arXiv:2005.01618v1  [cs.CL]  4 May 2020 \fvalidate our proposed RCR framework, we extend it to constrained text generation to discourage undesired text generation. The main contributions of this paper are summarized as follows. (i) A novel reward constrained recommendation framework is developed for text-based interactive recommendation, where constraints work as a dynamically updated critic to penalize the recommender. (ii) A novel way of de\ufb01ning constraints is proposed, in an adversarial manner, with better generalization. (iii) Extensive empirical evaluations are performed on text-based interactive recommendation and constrained text generation tasks, demonstrating consistent performance improvement over existing approaches. 2 Background 2.1 Reinforcement Learning Reinforcement learning aims to learn an optimal policy for an agent interacting with an unknown (and often highly complex) environment. A policy is modeled as a conditional distribution \u03c0(a|s), specifying the probability of choosing action a \u2208A when in state s \u2208S. Formally, an RL problem is characterized by a Markov decision process (MDP) [39], M = \u27e8S, A, P, R\u27e9. In this work, we consider recommendation for \ufb01nite-horizon environments with the average reward criterion. If the agent chooses action a \u2208A at state s \u2208S, then the agent will receive an immediate reward r(s, a), and the state will transit to s\u2032 \u2208S with probability P(s\u2032|s, a). The expected total reward of a policy \u03c0 is de\ufb01ned as [43]: JR(\u03c0) = \u221e X t=1 EP,\u03c0 [r(st, at)] . (1) In (1) the sum is over in\ufb01nite time steps, but in practice we will be interested in \ufb01nite horizons. The goal of an agent is to learn an optimal policy that maximizes JR(\u03c0). A constrained Markov decision process (CMDP) [3] extends the MDP framework by introducing the constraint C(s, a) (mapping a state-action pair to costs, similar to the usual reward) 2 and a threshold \u03b1 \u2208[0, 1]. Denoting the expectation over the constraint C(s, a) as JC(\u03c0) = P\u221e t=1 EP,\u03c0[C(st, at)], the constrained policy optimization thus becomes [1]: max \u03c0\u2208\u03a0 JR(\u03c0), s.t. JC(\u03c0) \u2264\u03b1 . (2) 2.2 Text-based Interactive Recommendation as Reinforcement Learning We employ an RL-based formulation for sequential recommendation of items to users, utilizing user feedback in natural language. Denote st \u2208S as the state of the recommendation environment at time t and at \u2208A as the recommender-de\ufb01ned items from the candidate items set A. In the context of a recommendation system, as discussed further below, the state st corresponds to the state of sequential recommender, implemented via a LSTM [24] state tracker. At time t, the system recommends item at based on the current state st at time t. After viewing item at, a user may comment on the recommendation in natural language (a sequence of natural-language text) xt, as feedback. The recommender then receives a reward rt and perceives the new state st+1. Accordingly, we can model the recommendation-feedback loop as an MDP M = \u27e8S, A, P, R\u27e9, where P : S \u00d7 A \u00d7 S 7\u2192R is the environment dynamic of recommendation and R : S \u00d7 A 7\u2192R is the reward function used to evaluate recommended items. The recommender seeks to learn a policy parameterized by \u03b8, i.e., \u03c0\u03b8(a|s), that corresponds to the distribution of items conditioned on the current state of the recommender. The recommender is represented as an optimal policy that maximizes the expected reward as JR(\u03c0) = P t EP,\u03c0 [r(st, at)]. At each time step, the recommender sequentially selects potential desired items at each time step via at = arg maxa\u2208A \u03c0\u03b8(a|st). 3 Proposed Method In text-based interactive recommendation, users provide natural-language-based feedback. We consider the recommendation of visual items [20, 21]. As shown in Figure 2, the system recommends an item to the user, with its visual appearance. The user then views the recommended item and gives feedback in natural language, describing the desired aspects that the current recommended item 2For simplicity, we here only introduce one constraint function; in practice, there may be many constraint functions. 2 \fRecommended Image at <latexit sha1_base64=\"4kX/AjHNVjqY1TH I4qakymbL4Uc=\">AB+nicbVC7TsMwFHXKq5RXCiOLRYXEVCWlEoyVWBiLRB9SG0WO47ZWnTi yb0BV6KewMIAQK1/Cxt/gtBmg5UiWj865Vz4+QSK4Bsf5tkobm1vbO+Xdyt7+weGRXT3uapkqy jpUCqn6AdFM8Jh1gINg/UQxEgWC9YLpTe73HpjSXMb3MEuYF5FxzEecEjCSb1ezYSBFqGeRuT CZ+DbNafuLIDXiVuQGirQ9u2vYShpGrEYqCBaD1wnAS8jCjgVbF4ZpolhE7JmA0MjUnEtJct os/xuVFCPJLKnBjwQv29kZFI5+HMZERgole9XPzPG6QwuvYyHicpsJguHxqlAoPEeQ845IpRE DNDCFXcZMV0QhShYNqmBLc1S+vk26j7l7WG3fNWqtZ1FGp+gMXSAXaEWukVt1EUPaJn9Ir erCfrxXq3PpajJavYOUF/YH3+AJdJlCo=</latexit> User I prefer high heel\u0011\u0003 Image Database  Recommender Visual Encoder Image Features MLP Matching Predicted Features Next Recommendation                        Feature Extractor Textual Encoder MLP User Rewards  Discriminator History Comments Visual Encoder LSTM Visual Encoder MLP at+1 <latexit sha1_base64=\"ogx9g3nh+OUXWUC A+riW+fRDnE=\">AB/nicbVDNS8MwHE3n15xfVfHkJTgEQRjtHOhx4MXjBPcBWylpm5haVK SVBil4L/ixYMiXv07vPnfmG496OaDkMd7vx95eUHCqNKO821V1tY3Nreq27Wd3b39A/vwqKdEK jHpYsGEHARIEUY56WqGRkqA4YKQfTG8Lv/9IpKCP+hZQrwYjTmNKEbaSL59ko0CwUI1i8 0FUe5n+tLNfbvuNJw54CpxS1IHJTq+/TUKBU5jwjVmSKmh6yTay5DUFDOS10apIgnCUzQmQ0M5 ionysn8HJ4bJYSRkOZwDefq740MxaoIaCZjpCdq2SvE/7xhqMbL6M8STXhePFQlDKoBSy6g CGVBGs2MwRhSU1WiCdIqxNYzVTgrv85VXSazbcq0bzvlVvt8o6quAUnIEL4IJr0AZ3oAO6AIM MPINX8GY9WS/Wu/WxGK1Y5c4x+APr8wdM2ZWm</latexit> Constraint Penalty st <latexit sha1_base64=\"Eqhcd0WCWa7epyYJfJSROl6Cci0=\">AB9XicbVDLSsNAFL3xWe ur6tLNYBFclaSKuiy4cVnBPqCNZTKdtEMnD2ZulBLyH25cKOLWf3Hn3zhps9DWAwOHc+7lnjleLIVG2/62VlbX1jc2S1vl7Z3dvf3KwWFbR4livMUiGamuRzWXIuQtFCh5N1acBp7kHW9yk/udR 60iMJ7nMbcDegoFL5gFI30A8oj0/1dkgxWxQqdo1ewayTJyCVKFAc1D56g8jlgQ8RCap1j3HjtFNqULBJM/K/UTzmLIJHfGeoSENuHbTWeqMnBplSPxImRcimam/N1IaD0NPDOZp9SLXi7+5 /US9K/dVIRxgjxk80N+IglGJK+ADIXiDOXUEMqUMFkJG1NFGZqiyqYEZ/HLy6RdrzntfrdRbVxWdRgmM4gTNw4AoacAtNaAEDBc/wCm/Wk/VivVsf89EVq9g5gj+wPn8AUomTAg=</latexi t> Figure 1: Overview of the reward constrained recommender model. When receiving the recommended images, the user gives natural-language feedback, and this feedback will be used for the next item recommendation, as well as preventing future violations. lacks. The system then incorporates the user feedback and recommends (ideally) more-suitable items, until the desired item is found. While users provide natural-language feedback on recommendations, standard RL methods may overlook the information from the feedback and recommend items that violate the user\u2019s previous feedback. To better understand this issue, consider the example in Figure 2. In round 3, the system forgets, and recommends an item that violates previous user preference on the \u2018ankle boots\u2019. I prefer ankle boots. I prefer shoes with  suede texture. I prefer ankle boots. I prefer shoes with  moc toe. Round 1 Round 2 Round 3 Round 4 Figure 2: An example of text-based interactive recommendation. To alleviate this issue, we consider using feedback from users as constraints, and formulate text-based interactive recommendation as a constrained policy optimization problem. The difference between the investigated problem and conventional constrained policy optimization [3, 5] is that constraints are added sequentially, affecting the search space of a policy in a different manner. Our model is illustrated in Figure 1. 3.1 Recommendation as Constrained Policy Optimization We consider an RL environment with a large number of discrete actions, deterministic transitions, and deterministic terminal returns. Suppose we have the user preference as constraints JC(\u03c0\u03b8) when making recommendations. The objective of learning a recommender is de\ufb01ned as: JR(\u03c0\u03b8) = \u221e X t=1 EP,\u03c0\u03b8 [r(st, at)] , s.t. JC(\u03c0\u03b8) \u2264\u03b1 . (3) If one naively augments previous user preferences as a hard constraint, i.e., exactly attributes matching, it usually leads to a sub-optimal solution. To alleviate this issue, we propose to use a learned constraint function based on the visual and textual information. Constraint Functions In text-based interactive recommendation, we explicitly use the user preference as constraints. Speci\ufb01cally, we exploit user feedback and put it as sequentially added constraints. To generalize well on the constraints, we learn a discriminator C\u03c6 parameterized by \u03c6 as the constraint function. We de\ufb01ne two distributions on feedback-recommendation pairs, i.e., non-violation distribution pr, and violation distribution pf (details provided in Appendix A.2). The objective of the discriminator is to minimize the following objective: L(\u03c6) = \u2212E(s,a)\u223cpf [log(C\u03c6(s, a))] \u2212E(s,a)\u223cpr [log(1 \u2212C\u03c6(s, a))] . (4) With the discriminator as the constraint i.e., JC\u03c6(\u03c0\u03b8) \u225cP\u221e t=1 EP,\u03c0\u03b8[C\u03c6(st, at)], the constraint is imposed. However, directly solving the constrained-optimization problem in (3) is dif\ufb01cult, and we 3 \femploy the Lagrange relaxation technique [4] to transform the original objective to an equivalent problem as: min \u03bb\u22650 max \u03b8 L(\u03bb, \u03b8, \u03c6) = min \u03bb\u22650 max \u03b8 \u0002 JR(\u03c0\u03b8) \u2212\u03bb \u00b7 (JC\u03c6(\u03c0\u03b8) \u2212\u03b1) \u0003 , (5) where \u03bb \u22650 is a Lagrange multiplier. Note that as \u03bb increases, the solution to (5) converges to that of (3). The goal is to \ufb01nd a saddle point (\u03b8\u2217(\u03bb\u2217), \u03bb\u2217) of (5), that can be achieved approximately by alternating gradient descent/ascent. Speci\ufb01cally, the gradient of (5) can be estimated using policy gradient [43] as: \u2207\u03b8L(\u03b8, \u03bb, \u03c6) = EP,\u03c0[(r(st, at) \u2212\u03bbC\u03c6(st, at)) \u2207\u03b8 log \u03c0\u03b8(st, at)] , (6) \u2207\u03bbL(\u03b8, \u03bb, \u03c6) = \u2212(EP,\u03c0[C\u03c6(st, at)] \u2212\u03b1) , (7) where C\u03c6(st, at) is the general constraint, speci\ufb01ed in the following. Penalized Reward Functions Note that the update in (6) is similar to the actor-critic method [43]. While the original use of a critic in reinforcement learning was for variance reduction [43], here we use it to penalize the policy for constraint violations. In order to ensure the constraints, \u03bb is also optimized using policy gradient via (7). The optimization proceeds intuitively as: i) when a violation happens (i.e., C\u03c6(s, a) > \u03b1), \u03bb will increase to penalize the policy. ii) If there is no violation (i.e., C\u03c6(s, a) < \u03b1), \u03bb will decrease to give the policy more reward. Model Training We alternatively update the constraint function, i.e., the discriminator and the recommender \u03c0\u03b8, similar to the Generative Adversarial Network (GAN) [16]. Speci\ufb01cally, the parameters are updated via the following rules: \u03b8k+1 = \u0393\u03b8[\u03b8k + \u03b71(k)\u2207\u03b8L(\u03bbk, \u03b8k, \u03c6k)] , (8) \u03c6k+1 = \u03c6k + \u03b72(k)\u2207\u03c6L(\u03bbk, \u03b8k, \u03c6k) , (9) \u03bbk+1 = \u0393\u03bb[\u03bbk \u2212\u03b73(k)\u2207\u03bbL(\u03bbk, \u03b8k, \u03c6k)] , (10) Algorithm 1 Reward Constrained Recommendation Input: constraint C(\u00b7), threshold \u03b1, learning rates \u03b71(k) > \u03b72(k) > \u03b73(k) Initialize recommender and discriminator parameters with pretrained ones, Lagrange multipliers \u03bb0 = 0 repeat for t = 0, 1, ..., T \u22121 do Sample action at \u223c\u03c0, observe next state st+1, reward rt and penalties ct \u02c6 Rt = rt \u2212\u03bbkct Recommender update with (8) end for Discriminator update with (9) Lagrange multiplier update with (10) until Model converges return recommender (policy) parameters \u03b8 where \u0393\u03b8 is a projection operator, which keeps the stability as the parameters are updated within a trust region; \u0393\u03bb projects \u03bb into the range [0, \u03bbmax]. We denote a three-timescale Reward Constrained Recommendation process, i.e., the three parts are updated with different frequency and step sizes: the recommender aims to maximize the expected reward with less violations following (8). As described in the Algorithm 1, the discriminator is updated following (9) to detect new violations, and \u03bb is updated following (10). 3.2 Model Details We discuss details on model design when applying the proposed framework in a textbased recommender system. Feature Extractor Our feature extractor consists of the encoders of text and visual inputs. Similar to [21], we consider the case where the visual attributes are available. We encode the raw images of the items by ResNet50 [22] and an attribute network, i.e., the visual feature cvis t of the item at is the concatenation of ResNet(at) and AttrNet(at). The input of the attribute network is an item\u2019s encoding by ResNet50 and the attribute network outputs this items\u2019 attribute values. We further encode the user comments in texts by an embedding layer, a LSTM and a linear mapping. Given a user comment xt, the \ufb01nal output of textual context is denoted as ctxt t . The encoded image and comment are further concatenated as the input to an MLP, and then the recommender component. 4 \fRecommender With the visual feature cvis t and textual feature ctxt t , the recommender perceives the state in an auto-regressive manner. At time t, the state is st = f(g([cvis t , ctxt t ]), st\u22121), where g is an MLP for textual and visual matching, and f is the LSTM unit [24]. Since our goal in each user session is to \ufb01nd items with a set of desired attribute values, we use the policy \u03c0\u03b8 with multi-discrete action spaces [13, 23]. For each attribute, the desired attribute value by the user is sampled from a categorical distribution. Given the state st, the probability of choosing a particular attribute value is output by a three-layer fully connected neural network with a softmax activation function. The recommender samples the values of different attributes from \u03c0\u03b8. If K items are recommended at each time, we select the items that are top K closest to the sampled attribute values under Euclidean distance in the visual attribute space. Discriminator The discriminator is designed to discriminate whether a recommended item at time t violates previous user comments in the current session. That is, given the visual feature of current image cvis t , and textual features {ctxt j }t\u22121 j=1, the discriminator outputs whether the image violates the user comment. In practice, this discriminator is a three-layer fully connected neural network and trained on-the-\ufb02y to incrementally learn the multimodal matching between the user comments and item visual features. Following Algorithm 1, we update the discriminator after each user session, where a user interacts with the system for several time steps, or quits. To further enhance the results, when making recommendations, we reject some items based on this discriminator. If an item at sampled by the recommender has high probability of violating the previous comments {xi}t\u22121 i=1, we ignore this item and sample another item to recommend. 3.3 Extension to Constrained Text Generation In this section, we describe how to extend our framework for constrained text generation. Encoder Decoder Discriminator (Constraint) z <latexit sha1_base64=\"TRovG9Ka/9/Hu2EkE4jI87rMPVM=\">AB+HicbVC7Ts MwFL0pr1IeDTCyWFRITFVSEDBWYmEsEn1IbVQ5jtNadZzIdpDaqF/CwgBCrHwKG3+D02aAliNZPjrnXvn4+AlnSjvOt1Xa2Nza3invVvb2Dw6r9tFxR8WpJLRNYh7Lno8V5 UzQtma014iKY58Trv+5C73u09UKhaLRz1NqBfhkWAhI1gbaWhXs4Ef80BNI3Oh2Xxo15y6swBaJ25BalCgNbS/BkFM0ogKThWqu86ifYyLDUjnM4rg1TRBJMJHtG+oQJHV HnZIvgcnRslQGEszREaLdTfGxmOVB7NTEZYj9Wql4v/ef1Uh7dexkSairI8qEw5UjHKG8BUxSovnUEwkM1kRGWOJiTZdVUwJ7uqX10mnUXcv642Hq1rzuqijDKdwBhfg wg04R5a0AYCKTzDK7xZM+vFerc+lqMlq9g5gT+wPn8AJ06TXg=</latexit> X <latexit sha1_base64=\"8bo0KUKzPB21U/r5/1MHPAduQk0=\">AB+HicbVDLSgMxFL3js9Z Hqy7dBIvgqsxUZcFNy4r2Ae0Q8lkMm1oJhmSjFCHfokbF4q49VPc+Tdm2lo64GQwzn3kpMTJxp47rfztr6xubWdmnvLu3f1CpHh51tEwVoW0iuVS9AGvKmaBtwynvURHAecdoPJbe53H6nS TIoHM02oH+ORYBEj2FhpWK1kg0DyUE9je6HebFituXV3DrRKvILUoEBrWP0ahJKkMRWGcKx13MT42dYGUY4nZUHqaYJhM8on1LBY6p9rN58Bk6s0qIqnsEQbN1d8bGY51Hs1OxtiM9bKXi/95/d REN37GRJIaKsjioSjlyEiUt4BCpigxfGoJorZrIiMscLE2K7KtgRv+curpNOoexf1xv1lrXlV1FGCEziFc/DgGpwBy1oA4EUnuEV3pwn58V5dz4Wo2tOsXMf+B8/gDzlZM8</latexit> Discriminator (General) \u02c6 X <latexit sha1_base64=\"Y5k7GBEfunzN0PKl6RCkOjpKRXE=\">AB/nicbVDLSsNAFL3xWes rKq7cDBbBVUmqMuCG5cV7AOaUCaTSTt0kgkzE6GEgr/ixoUibv0Od/6NkzYLbT0wzOGce5kzJ0g5U9pxvq2V1bX1jc3KVnV7Z3dv3z47CiRSULbRHAhewFWlLOEtjXTnPZSXEcNoNxreF32k UjGRPOhJSv0YDxMWMYK1kQb2sTfCOs+9QPBQTWJzod50OrBrTt2ZAS0TtyQ1KNEa2F9eKEgW0QTjpXqu06q/RxLzQin06qXKZpiMsZD2jc0wTFVfj6LP0VnRglRJKQ5iUYz9fdGjmNVZDOTMdYjte gV4n9eP9PRjZ+zJM0Tcj8oSjSAtUdIFCJinRfGIJpKZrIiMsMREm8aqpgR38cvLpNOouxf1xv1lrXlV1lGBEziFc3DhGpwBy1oA4EcnuEV3qwn68V6tz7moytWuXMEf2B9/gDhnZYJ</latex it> Lae <latexit sha1_base64=\"XcLvoZwJOgSKB1xWDQUMfce92TM=\">AB+XicbVDLSsNAFJ34rPU VdelmsAiuSlJFXRbcuHBRwT6gDWEyvWmHTiZhZlIoIX/ixoUibv0Td/6NkzYLbT0wcDjnXu6ZEyScKe0439ba+sbm1nZlp7q7t39waB8d1ScSgptGvNY9gKigDMBbc0h14igUQBh24wuSv87hSk YrF40rMEvIiMBAsZJdpIvm0PIqLHlPDsIfczArlv15y6MwdeJW5JaqhEy7e/BsOYphEITlRqu86ifYyIjWjHPLqIFWQEDohI+gbKkgEysvmyXN8bpQhDmNpntB4rv7eyEik1CwKzGSRUy17hfif10 91eOtlTCSpBkEXh8KUYx3jogY8ZBKo5jNDCJXMZMV0TCSh2pRVNSW4y19eJZ1G3b2sNx6vas3rso4KOkVn6AK56AY10T1qoTaiaIqe0St6szLrxXq3Phaja1a5c4L+wPr8Af4dk9o=</latexit> Figure 3: Overview of the constrained textgeneration model: Lae is the reconstruction term from the VAE in pretraining. The constraint will give a penalty when generated text violates the constraint discriminator. We consider text generation with speci\ufb01c constraints. Speci\ufb01cally, we consider the scenario of controlling for negative sentiments. For example, a generator may generate some offensive or negative words, which will affect the user experience in some situations, such as with an online chatbot for helping consumers. To alleviate this issue, we applied the proposed RCR methods for text generation. We assume each sentence is generated from a latent vector z \u223cp(z), where p(z) is the distribution of a latent code. Text generation is then formulated as the learning of a distribution: p(X) = R zx p(X|zx)q(zx|X)dzx, where p corresponds to a decoder and q to an encoder model, within the encoder-decoder framework; z is the latent code containing content information. The generator learns a policy \u03c0\u03b8 to generate a sequence Y = (y1, . . . , yT ) of length T. Here each yt is a token from vocabulary A. The objective is to maximize the expected reward with less constraint violations, de\ufb01ned as: L(\u03b8, \u03bb, \u03c6) = min \u03bb\u22650 max \u03b8 EY \u223c\u03c0\u03b8 [r(Y ) \u2212\u03bb(C\u03c6(Y ) \u2212\u03b1)] , (11) where r is the reward function, that can be a metric reward (e.g., BLEU) or a learned reward function with general discriminator [53]; C\u03c6(\u00b7) is the constraint discriminator for the generation. In practice, we pretrain our generator \u03c0\u03b8 with a variational autoencoder (VAE) [26], and we only use the decoder as our generator. More details about the pretrained model are provided in Appendix A.1. There is a constraint for the generation, and the framework is illustrated in Figure 3. The general discriminator can be a language model [49], and the constraint is a learned function parameterized by a neural network. During inference, the model generates text based on draws from an isotropic Gaussian distribution, i.e., z \u223cN(0, I). Here we only consider the static constraint with non-zero \ufb01nal deterministic reward. 4 Related Work Constrained Policy Optimization Constrained Markov Decision Processes [3] are employed in a wide range of applications, including analysis of electric grids [27] and in robotics [9, 18]. Lagrange multipliers are widely used to solve the CMDP problem [5, 44], as adopted in our proposed framework. 5 \fOther solutions of CMDP include use of a trust region [1], and integrating prior knowledge [12]. Additionally, some previous work manually selects the penalty coef\ufb01cient [14, 32, 38]. In contrast with standard methods, our constraint functions are: (i) sequentially added via natural-language feedback; (ii) parameterized by a dynamically updated neural network with better generalization. Text-Based Recommender System Communications between a user and recommendation system have been leveraged to understand user preference and provide recommendations. Entropy-based methods and bandits have been studied in question selection [11, 35]. Deep learning and reinforcement learning models have been proposed to understand user conversations and make recommendations [2, 10, 17, 31, 42, 54, 57]. Similar to [11, 31, 42, 54], the items are associated with a set of attributes in our recommendation setting. In the existing works, the content of the conversation serves as the constraint when a system makes recommendations. However, in most existing works, constraints from the conversations are not explicitly modeled. By contrast, this paper proposes a novel constrained reinforcement learning framework to emphasize the constraints when making recommendations. Interactive Image Retrieval Leveraging user feedback on images to improve image retrieval has been studied extensively [46]. Depending on the feedback format, previous works can be categorized into relevance feedback [40, 48] and relative-attributes feedback [28, 37, 50]. In these works, the attributes to describe the images are pre-de\ufb01ned and \ufb01xed. To achieve more \ufb02exible and precise representation of the image attributes, Guo, et al. [20] proposes an end-to-end approach, without pre-de\ufb01ning a set of attributes. Their goal is to improve the ranking of the target item, while we focus on recommending items that do not violate the users\u2019 previous comments in the iterative recommendation. Thus, we develop a different evaluation simulator as detailed in Section 5.1. In [54], it is assumed that an accurate discriminator pretrained on huge-amount of\ufb02ine data is available at the beginning, which is usually impractical. Instead, our novel RCR framework learns the discriminator from scratch and dynamically updates the model \u03c6 and its weight \u03bb by (9) and (10) online. Constrained Text Generation Adversarial text generation [7, 15, 34, 36, 53, 55] use reinforcement learning (RL) algorithms for text generation. They use the REINFORCE algorithm to provide an unbiased gradient estimator for the generator, and apply the roll-out policy to obtain the reward from the discriminator. LeakGAN [19] adopts a hierarchical RL framework to improve text generation. GSGAN [29] and TextGAN [25, 56] use the Gumbel-softmax and soft-argmax representation, respectively, to deal with discrete data. Wang, et al. [47] put topic-aware priors on the latent codes to generate text on speci\ufb01c topics. All these works consider generating sentences with better quality and diversity, without explicit constraints. 5 Experiments We apply the proposed methods in two applications: text-based interactive recommendation and constrained text generation, to demonstrate the effectiveness of our proposed RCR framework. 5.1 Text-Based Interactive Recommendation Dataset and Setup Our approaches are evaluated on the UT-Zappos50K dataset [51, 52]. UTZappos50K is a shoe dataset consisting of 50,025 shoe images. This dataset provides rich attribute data and we focus on shoes category, shoes subcategory, heel height, closure, gender and toe style in our evaluation. Among all the images, 40,020 images are randomly sampled as training data and the rest are used as test data. To validate the generalization ability of our approach, we compare the performance on seen items and unseen items. The seen items are the items in the training data where the item visual attributes are carefully labeled. The unseen items are the items in the test data. We assume the unseen items are newly collected and have no labeled visual attributes. We train the attribute network on the training data, under the cross-entropy loss. The ResNet50 is pretrained on ImageNet and is \ufb01xed subsequently. When we report the results on seen and unseen items, their attribute values are predicted by the attribute network. We pretrain the textual encoder, where the labels are the described attribute values, under the cross-entropy loss. The training data consists of the comments collected by annotators as detailed later in this section. In reinforcement learning, we use Adam [26] as the optimizer. We set \u03b1 = 0.5 and \u03bbmax = 1. We de\ufb01ne the reward as the visual similarity between the recommended and desired items. Similar to [21], in our task both images and their visual attributes are available to measure the similarity. It is desired that the recommended item becomes more similar to the desired item with more user interactions. Thus, at time t, given the recommended item at and the desired item a\u2217, we 6 \f0 10000 20000 30000 40000 Training iterations 15 20 25 30 35 40 45 NI RCR RL 0 10000 20000 30000 40000 Training iterations 20 40 60 80 100 120 140 NV RCR RL 0 10000 20000 30000 40000 Training iterations 20 40 60 80 SR@30 RCR RL 0 2000 4000 6000 8000 10000 Number of Samples 0.0 0.2 0.4 0.6 0.8 Figure 4: Number of Interactions (NI), Number of Violations (NV), Success Rate@30 (SR@30) with respect to training iterations and the values of \u03bb in RCR with respect to number of samples. The RL method converges much slower than the RCR. SR@10 \u2191 SR@20 \u2191 SR@30 \u2191 NI \u2193 NV \u2193 RL (Unseen) 19% 44% 63% 26.75 \u00b1 1.67 70.02 \u00b1 6.20 RL + Naive (Unseen) 52% 83% 94% 12.72 \u00b1 0.93 16.47 \u00b1 2.75 RCR (Unseen) 74% 86% 94% 10.91 \u00b1 1.06 11.32 \u00b1 1.98 RCR (Seen) 78% 91% 92% 10.34 \u00b1 1.18 12.25 \u00b1 2.99 Table 1: Comparisons between different approaches. Except the row of RCR (seen) reporting results on training data, all the results are on the test data with unseen items. want to minimize their visual difference. In detail, we maximize the following visual reward rt = \u2212||ResNet(at) \u2212ResNet(a\u2217)||2 \u2212\u03bbatt||AttrNet(at) \u2212AttrNet(a\u2217)||0, where || \u00b7 ||2 is the L2 norm, || \u00b7 ||0 is the L0 norm, and we set \u03bbatt = 0.5 to ensure the scales of the two distances are similar. If the system is not able to \ufb01nd the desired item before 50 interactions, we will terminate this user session and the system will receive an extra reward \u22123 (i.e., a penalty). Online Evaluation We cannot directly detect the violations with existing text-based interactive recommendation dataset [20], since there are no attribute labels for the images. A recent relevant fashion dataset provides the attribute labels 3 derived from the text metadata [21]. Unfortunately, we observe that the user\u2019s comments are usually unrelated to the attribute labels. Therefore, we need to collect the user\u2019s comments relevant to attributes with groundtruth, for our evaluation purpose. Further, evaluating the proposed system requires the ability to get access to all user reactions to any possible items at each time step. For the evaluation on the UT-Zappos50K dataset, we use a similar simulator to Guo, et al. [20]. This simulator acts as a surrogate for real human users by generating their comments in natural language. The generated comments describe the prominent visual attribute differences between any pair of desired and candidate items. To achieve this, we collect user comments relevant to the attributes with groundtruth and train a user simulator. A training dataset is collected for 10,000 pairs of images with visual attributes. These pairs are prepared such that in each pair there is a recommended item and a desired item. Given a pair of images, one user comment is collected. The data are collected in a scenario in which the customer talks with the shopping assistant to get the desired items. The annotators act as the customers to express the desired attribute values of items. For the evaluation purpose, we adopt a simpli\ufb01ed setting and instruct the annotators to describe the comments related to a \ufb01xed set of visual attributes. Thus, the comments in our evaluation are relatively simpler compared to the real-world sentences. Considering this, we further augment the collected user comment data as follows. From the real-world sentences collected from annotators, we derive several sentence templates. Then, we generate 20,000 labeled sentences by \ufb01lling these templates with the groundtruth attribute label. On the augmented user comment data, we train the user simulator. Our user simulator is implemented via a sequence-to-sequence model. The inputs of the user simulator are the differences on one attribute value between the candidate and desired items. Given the inputs, the user simulator generates a sentence describing the visual attribute difference between the candidate item and the desired item. We use two LSTMs as the encoder and decoder. The dimensionality of the latent code is set as 256. We use Adam as the optimizer, where the initial learning is set as 0.001 with batch size of 64. Note that for evaluating how the current recommended item\u2019s visual attributes satisfy the user\u2019s previous feedback, our user simulator on UT-Zappos50K only generates simple comments on the visual attribute difference between the candidate image and the desired image: we can calculate how many attributes violate the users\u2019 previous feedback based on the visual attribute groundtruth available in UT-Zappos50K. 3Available at https://github.com/hongwang600/image_tag_dataset/tree/master/tags. 7 \fWe de\ufb01ne four evaluation metrics: i) task success rate (SR@K), which is the success rate after after K interactions; ii) number of user interactions before success (NI); and iii) number of violated attributes (NV). In each user session, we assume the user aims to \ufb01nd items with a set of desired attribute values sampled from the dataset. We report results averaged over 100 sessions with standard error. We develop an RL baseline approach by ignoring the constraints (i.e., discriminator) in RCR. A major difference between our RL baseline approach and Guo, et al. [20] is that we consider the attributes in the model learning, while the attributes are ignored in [20]. We compare RCR with the RL without constraints, as well as RL methods with naive constraints, i.e., naively using hard constraints. That is, we track all the visual attributes previously described by the user in this session, and make further recommendations based on the matching between them and the items in dataset. I want boots. I prefer knee  high. Show me shoes  with round toe. Good! I prefer ankle. I want the shoes  with pull-on closure. I prefer ankle. I prefer the ones  for women. (fail to meet the  constraint of ankle)  Show me more  shoes with round toe. I prefer elastic  gore. I prefer shoes for  women. Good! (a) (b) (c) Round 1 Round 2 Round 3 Round 4 Figure 5: Three use cases, from logged experimental results. (a) and (b) are successful use cases by RCR. (c) is not successful by RL, which demonstrate the common challenge of failing to meet the constraint in recommendation. Analysis All models are trained for 100,000 iterations (user sessions), and the results with standard errors under different metrics are shown in Table 1. The proposed RCR framework shows consistent improvements on most metrics, compared with the baselines. The gap between RL with naive constraints and RCR demonstrate the learned constraint (discriminator) has better generalization. Figure 4 shows the metrics with standard errors of RL and the proposed RCR in the \ufb01rst 40,000 iterations. RCR shows much faster convergence than RL. The last sub\ufb01gure shows the values of \u03bb with different number of samples. It is interesting to see that \u03bb increases at the initial stage because of too many violations. Then, with less violations, \u03bb decreases to a relatively small value as \u03bb = 0.04 and then remains stable, which is the automatically learned weight of the discriminator. Some examples in Figure 5 show how the constraint improves the recommendation. 5.2 Constrained Text Generation Experimental Setup We use the Yelp review dataset [41] to validate the proposed methods. We split the data as 444,000, 63,500, and 127,000 sentences in the training, validation and test sets, respectively. The generator is trained on the Yelp dataset to generate reviews without sentiment labels. We de\ufb01ne the reward of the generated sentence as the probability of being real and the constraint is to generate positive reviews, i.e., the generator will receive a penalty if it generates negative reviews. The constraint is a neural network with a classi\ufb01cation accuracy of 97.4% on the validation set, trained on sentences with the sentiment labels. We follow the strategy in [19, 53] and adopt the BLEU score, referenced by test set with only positive reviews (test-BLEU) and themselves (self-BLEU) to evaluate the quality of generated samples. We also report the violation rate (VR), the percentage of generated negative reviews violating the constraint. Test-BLEU-2 3 4 5 Self-BLEU-2 3 4 VR RL 0.807 0.622 0.469 0.376 0.658 0.315 0.098 40.36% RCR (ours) 0.840 0.651 0.492 0.392 0.683 0.348 0.151 10.49% Table 2: Comparison between RCR and standard RL for constrained text generation on Yelp. RL without Constraints RCR the ceiling is low , the place smells awful , gambling sucked . i have been here a few times and each time has been great ! bad food , bad service , takes too much time . food was good , but overall it was a very bad dining experience . my entree was a sea bass which was well prepared and tasty . the food is delicious and very consistently so . the waitress was horrible and came by maybe once every hour . every dish was so absolutely delicious and seasoned perfectly . he is the most compassionate vet i have ever met . compared to other us cities , this place ranks very generous in my book . then you already know what this tastes like . thank you my friends for letting us know this \ufb01nest dining place in lv . great service and the food was excellent . the lines can get out of hand sometimes but it goes pretty quick . Table 3: Randomly selected examples of text generation by two methods. Analysis As illustrated in Table 2, RCR achieves better test-BLEU scores than standard RL, demonstrating high-quality generated sentences. Further, RCR shows a little higher but reasonable 8 \fself-BLEU scores, since we only generate sentences with positive sentiments, leading to lower diversity. Our proposed method shows much lower violation rate, demonstrating the effectiveness of RCR. Some randomly generated examples are shown in Table 3. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2005.01279v1",
+                    "title": "Improving Adversarial Text Generation by Modeling the Distant Future",
+                    "abstract": "Auto-regressive text generation models usually focus on local fluency, and\nmay cause inconsistent semantic meaning in long text generation. Further,\nautomatically generating words with similar semantics is challenging, and\nhand-crafted linguistic rules are difficult to apply. We consider a text\nplanning scheme and present a model-based imitation-learning approach to\nalleviate the aforementioned issues. Specifically, we propose a novel guider\nnetwork to focus on the generative process over a longer horizon, which can\nassist next-word prediction and provide intermediate rewards for generator\noptimization. Extensive experiments demonstrate that the proposed method leads\nto improved performance.",
+                    "authors": "Ruiyi Zhang, Changyou Chen, Zhe Gan, Wenlin Wang, Dinghan Shen, Guoyin Wang, Zheng Wen, Lawrence Carin",
+                    "published": "2020-05-04",
+                    "updated": "2020-05-04",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Text generation is an important area of investigation within machine learning. Recent work has shown excellent performance on a number of tasks, by combining reinforcement learning (RL) and generative models. Example applications include image captioning (Ren et al., 2017; Rennie et al., 2016), text summarization (Li et al., 2018b; Paulus et al., 2017; Rush et al., 2015), and adversarial text generation (Guo et al., 2017; Lin et al., 2017; Yu et al., 2017; Zhang et al., 2017; Zhu et al., 2018). The sequence-to-sequence framework (Seq2Seq) (Sutskever et al., 2014) is a popular technique for text generation. However, models from such a setup are typically trained to predict the next token given previous ground-truth tokens as input, causing what is termed exposure bias (Ranzato et al., 2016). By contrast, sequence-level training with RL provides an effective means of solving this challenge, by treating text generation as a sequential decision-making problem. By directly optimizing an evaluation score (cumulative rewards) (Ranzato et al., 2016), state-of-the-art results have been obtained in many text-generation tasks (Paulus et al., 2017; Rennie et al., 2016). However, one problem in such a framework is that rewards in RL training are particularly sparse, since a scalar reward is typically only available after an entire sequence has been generated. Furthermore, the recurrent models focus more on local \ufb02uency, and may cause inconsistent semantic meanings for long text generation. For RL-based text generation, most existing works rely on a model-free framework, which has been criticized for its high variance and poor sample ef\ufb01ciency (Sutton and Barto, 1998). On the other hand, while model-based RL methods do not suffer from these issues, they are usually dif\ufb01cult to train in complex environments. Further, a learned policy is usually restricted by the capacity of an environment model. Recent developments on model-based RL (Gu et al., 2016; Kurutach et al., 2018; Nagabandi et al., 2017) combine the advantages of these two approaches, and have achieved improved performance by learning a model-free policy, assisted by an environment model. In addition, model-based RL has been employed recently to solve problems with extremely sparse rewards, with curiosity-driven methods (Pathak et al., 2017). In this paper, we propose a model-based imitation-learning method to overcome the aforementioned issues in text-generation tasks. Our main idea is to employ an explicit guider network to model the generation environment in the feature space of sentence tokens, used to emit intermediate rewards by matching the predicted features from the guider network and features from generated sentences. The guider network is trained to encode global structural information of training sentences, and thus is useful to guide next-token prediction in the generative process. Within the proposed framework, to assist the guider network, we also develop a new type of self-attention mechanism to provide high-level planning-ahead information arXiv:2005.01279v1  [cs.CL]  4 May 2020 \fand maintain consistent semantic meaning. Our experimental results demonstrate the effectiveness of proposed methods. 2 Background Text Generation Model Text generation models learn to generate a sentence Y = (y1, . . . , yT ) of length T, possibly conditioned on some context X. Here each yt is a token from vocabulary A. Starting from the initial state s0, a recurrent neural network (RNN) produces a sequence of states (s1, . . . , sT ) given an input sentence-feature representation (e(y1), . . . , e(yT )), where e(\u00b7) denotes a word embedding function mapping a token to its ddimensional feature representation. The states are recursively updated with a function known as the cell: st = h(st\u22121, e(yt)). One typically assigns the following probability to an observation y at location t: p(y|Y<t) = [softmax(g(st))]y. Together (g, h) speci\ufb01es a probabilistic model \u03c0, i.e., log \u03c0(Y ) = X t log p(yt|Y<t). (1) To train the model \u03c0, one typically uses maximum likelihood estimation (MLE), via minimizing the cross-entropy loss, i.e., JMLE(\u03c0) = \u2212E[log \u03c0(Y )]. In order to generate sentence Y s from a (trained) model, one iteratively applies the following operations: ys t+1 \u223cMulti(1, softmax(g(st))), (2) st = h(st\u22121, e(ys t )) , (3) where Multi(1, \u00b7) denotes one draw from a multinomial distribution. Model-Based Imitation Learning Text generation can be considered as an RL problem with a large number of discrete actions, deterministic transitions, and deterministic terminal rewards. It can be formulated as a Markov decision process (MDP) M = \u27e8S, A, P, r, \u03b3\u27e9, where S is the state space, A is the action space, P is the deterministic environment dynamics, r(s, y) is a reward function, and \u03b3 \u2208(0, 1) is the discrete-time discount factor. The policy \u03c0\u03c6, parameterized by \u03c6, maps each state s \u2208S to a probability distribution over A. The objective is to maximize the expected reward: J(\u03c0) = \u221e X t=1 EP,\u03c0 \u0002 \u03b3t\u22121 \u00b7 r(st, yt) \u0003 . (4) In model-based imitation learning (Baram et al., 2017; Cheng et al., 2019), a model is built to make predictions for future state st+\u25b3t conditioned on the current state1, which can be used for action selection, e.g., next-token generation. This model is typically a discrete-time system, taking the current state-action pair (st, yt) as input, and outputting an estimate of the future state st+\u25b3t at time t + \u25b3t. At each step t, yt is chosen based on the model, and the model will re-plan with the updated information from the dynamics. This control scheme is different from a standard model-based method, and is referred to as model-predictive control (MPC) (Nagabandi et al., 2017). Note that in our setting, the state in RL typically corresponds to the current generated sentences Y1,...,t instead of the RNN state of generator (decoder). 3 Proposed Model The model is illustrated in Figure 1, with an autoeocoder (AE) structure for sentence feature extraction and generation. The encoder is shared for sentences from both training data and generated data, as explained in detail below. Overall, text generation can be formulated as an imitationlearning problem. At each timestep t, the agent, also called a generator (which corresponds to the LSTM decoder), takes the current LSTM state as input, denoted as st. The policy \u03c0\u03c6(\u00b7|st) parameterized by \u03c6 is a conditional generator, to generate the next token (action) given st, the observation representing the current generated sentence. The objective of text generation is to maximize the total reward as in (4). We detail the components for our proposed model in the following subsections. 3.1 The Guider Network The guider network, implemented as an RNN with LSTM units, is adopted to model environment dynamics to assist text generation. The idea is to train a guider network such that its predicted sentence features at each time step are used to assist next-word generation and construct intermediate rewards, which in turn are used to optimize the sentence generator. Denote the guider network as G\u03c8(sG t\u22121, f t), with parameters \u03c8 and input arguments (sG t\u22121, f t) at time t, to explicitly write out the dependency on the guider network latent state sG t\u22121 from the previous time step. 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Solid lines mean gradients are backpropagated in training; dash lines mean gradients are not backpropagated. CNN is the feature extractor, and MLP outputs the parameters of the Gaussian density which is compatible with the initial state of the LSTM Guider and Decoder. by an encoder network. Speci\ufb01cally, let the current generated sentence be Y1...t (encouraged to be the same as parts of a training sentence in training), with f t calculated as: f t = Enc(Y1...t). The initial state of the guider network is the encoded feature of a true input sentence by the same convolutional neural network (CNN), i.e., sG 0 = Enc(X), where Enc(\u00b7) denotes the encoder transformation, implemented with a CNN (Zhang et al., 2017). Importantly, the input to the guider network, at each time point, is de\ufb01ned by features from the entire sentence generated to that point. This provides an important \u201cguide\u201d to the LSTM decoder, accounting for the global properties of the generated text. Text Generation with Planning We \ufb01rst explain how one uses the guider network to guide next-word generation for the generator (the LSTM decoder in Figure 1). Our framework is inspired by the MPC method (Nagabandi et al., 2017), and can be regarded as a type of plan-ahead attention mechanism. Given the feature f t at time t from the current input sentence, the guider network produces a prediction G\u03c8(sG t\u22121, f t) as a future feature representation, by feeding f t into the LSTM guider. Since the training of the guider network is based on real data (detailed in the next paragraph), the predicted feature contains global-structure information of the training sentences. To utilize such information to predict the next word, we combine the predicted feature with the output of the decoder by constructing an attention-like mechanism. Speci\ufb01cally, we \ufb01rst apply a linear transformation \u03d5 on the predicted feature G\u03c8(sG t\u22121, f t), forming a weight vector wt \u225c\u03d5 \u0000G\u03c8(sG t\u22121, f t) \u0001 . The weight wt is applied to the output Ot of the LSTM decoder by an element-wise multiplication operation. The result is then fed into a softmax layer to generate the next token yt. Formally, the generative process is written as: Ot = g(st\u22121), wt = \u03d5(G\u03c8(sG t\u22121, f t)), (5) yt \u223cMulti(1, softmax(Ot \u00b7 wt)), (6) sG t = hG(sG t\u22121, f t), st = h(st\u22121, e(yt)) . (7) Guider Network Training Given a sentence of feature representations (f 1, f 2, . . . f T ) for a training sentence, we seek to update the guider network such that it is able to predict f t+c given f t, where c > 0 is the number of steps that are looked ahead. We implement this by forcing the predicted feature, G\u03c8(sG t , f t), to match both the sentence feature f t+c (\ufb01rst term in (8)) and the corresponding feature-changing direction (second term in (8)). This is formalized by maximizing an objective function of the following form at time t: J\u03c8 G = Dcos \u0010 f t+c, G\u03c8(sG t\u22121, f t) \u0011 (8) + Dcos \u0010 f t+c \u2212f t, G\u03c8(sG t\u22121, f t) \u2212f t \u0011 , where Dcos(\u00b7, \u00b7) denotes the cosine similarity2. By maximizing (8), an ideal guider network should be able to predict the true next words conditioned on the current word in a sentence. As a result, the prediction is used to construct an intermediate reward, used to update the generator (the LSTM decoder), as described further below. 3.2 Feature-Matching Rewards and Generator Optimization As in many RL-based text-generation methods, such as SeqGAN (Yu et al., 2017) and LeakGAN (Guo et al., 2017), the generator is updated based on policy-gradient methods. As a result, collecting rewards in the generation process is critical. 2We found that the cosine similarity worked better than the l2-norm. \fThough SeqGAN (Yu et al., 2017) has proposed to use rollout to get rewards for each generated word, the variance of the rewards is typically too high to be useful practically. In addition, the computational cost may be too high for practical use. We below describe how to use the proposed guider network to de\ufb01ne intermediate rewards, leading to a de\ufb01nition of feature-matching reward. Feature-Matching Rewards We \ufb01rst de\ufb01ne an intermediate reward to generate a particular word. The idea is to match the ground-truth features from the CNN encoder in Figure 1 with those generated from the guider network. Equation (8) indicates that the further the generated feature is from the true feature, the smaller the reward should be. To this end, for each time t, we de\ufb01ne the intermediate reward for generating the current word as: rg t = 1 2c c X i=1 (Dcos(f t, \u02c6 f t)+ Dcos(f t \u2212f t\u2212i, \u02c6 f t \u2212f t\u2212i)) , where \u02c6 f t = G\u03c8(sG t\u2212c\u22121, f t\u2212c) is the predicted feature. Intuitively, f t \u2212f t\u2212i measures the difference between the generated sentences in feature space; the reward is high if it matches the predicted feature transition \u02c6 f t \u2212f t\u2212i from the guider network. At the last step of text generation, i.e., t = T, the corresponding reward measures the quality of the whole generated sentence, thus it is called a \ufb01nal reward. The \ufb01nal reward is de\ufb01ned differently from the intermediate reward, discussed below for both the unconditionaland conditional-generation cases. Note that a token generated at time t will in\ufb02uence not only the rewards received at that time but also the rewards at subsequent time steps. Thus we propose to de\ufb01ne the cumulative reward, PT i=t \u03b3irg i with \u03b3 a discount factor, as a featurematching reward. Intuitively, this encourages the generator to focus on achieving higher long-term rewards. Finally, in order to apply policy gradient to update the generator, we combine the featurematching reward with the problem-speci\ufb01c \ufb01nal reward, to form a Q-value reward speci\ufb01ed below. Similar to SeqGAN, the \ufb01nal reward is de\ufb01ned as the output of a discriminator, evaluating the quality of the whole generated sentence, i.e., the smaller the output, the less likely the generation is a true sentence. As a result, we combine the adversarial reward rf \u2208[0, 1] by the discriminator (Yu et al., Algorithm 1 Model-based Imitation Learning for Text Generation Require: generator policy \u03c0\u03c6; guider network G\u03c8; a sequence dataset {X1...T } by some expert policy. 1: Initialize G\u03c8, D\u03b8 with random weights. 2: while Imitation Learning phase do 3: Update generator \u03c0\u03c6, guider G\u03c8 with MLE loss. 4: end while 5: while Reinforcement Learning phase do 6: Generate a sequence Y1...T \u223c\u03c0\u03c6. 7: Compute Qt, and update \u03c0\u03c6. 8: end while 2017) with the guider-matching rewards, to de\ufb01ne a Q-value reward as Qt = (PT i=t \u03b3irg i ) \u00d7 rf. Generator Optimization The generator is initialized by pre-training on sentences with an autoencoder structure, based on MLE training. After that, the \ufb01nal Q-value reward Qt is used as a reward for each time t, with standard policy gradient optimization methods to update the generator. Speci\ufb01cally, the policy gradient is \u2207\u03c6J = E(st\u22121,yt)\u223c\u03c1\u03c0 [Qt\u2207\u03c6 log p(yt|st\u22121; \u03c6, \u03d5)] , \u2207\u03d5J = E(st\u22121,yt)\u223c\u03c1\u03c0 [Qt\u2207\u03d5 log p(yt|st\u22121; \u03c6, \u03d5)] , where p(yt|st\u22121; \u03c6, \u03d5) is the probability of generating yt given st\u22121 in the generator. Algorithm 1 describes the proposed model-based imitation learning framework for text generation. Model-based or Model-free Text generation seeks to generate the next word (action) given the current (sub-)sentence (state). The generator is considered as an agent that learns a policy to predict the next word given its current state. In previous work (Ranzato et al., 2016), a metric reward is given and the generator is trained to only maximize the metric reward by trial, thus this is model-free learning. In the proposed method, the guider network models the environment dynamics, and is trained by minimizing the cosine similarity between the prediction and the ground truth on real text. For generator training, it maximizes the reward which is determined by the metric and guider network, and thus is model-free learning with model-based boosting (Gu et al., 2016). 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styles. To make the guider network focus on the guidance of styles, we assign the label l as the initial state sG 0 of the guider network. Speci\ufb01cally, at each step t, we feed the current sentence representation f t and label l into the guider network: Ot = g(st\u22121), wt = \u03d5(G\u03c8(sG t\u22121, [f t, l])), (9) yt \u223cMulti(1, softmax(Ot \u00b7 wt)). (10) For the generator, we put an adversarial regularizer on the encoded latent s0(X) and penalize it if it contains the sentiment information, by maximizing the entropy, i.e., max P l p(l| s0(X)) log p(l| s0(X)), where p is a pre-trained classi\ufb01er. Intuitively, the generator gives candidate words represented by Ot, while the guider makes a choice implicitly by wt based on the sentiment information. The sentiment information is contained in wt, while the content of the original sentence is represented by Ot. To achieve styletransfer, one feeds the original sentence X with the target style label l to get the transferred sentence Y with style l. Following previous work (Hu et al., 2017; Yang et al., 2018; Cheng et al., 2020), we adopt a classi\ufb01er as the discriminator and the soft-argmax approach (Kusner and Miguel, 2016) for the update of generator instead of policy gradient (Sutton and Barto, 1998). 5 Related Work We \ufb01rst review related works that combine RL and GAN for text generation. As one of the most representative models in this direction, SeqGAN (Yu et al., 2017) adopts Monte-Carlo search to calculate rewards. However, such a method introduces high variance in policy optimization. There are a number of works proposed subsequently to improve the reward-generation process. For example, RankGAN (Lin et al., 2017) proposes to replace the reward from the GAN discriminator with a rankingbased reward, MaliGAN (Che et al., 2017) modi\ufb01es the GAN objective and proposes techniques to reduce gradient variance, MaskGAN (Fedus et al., 2018) uses a \ufb01lling technique to de\ufb01ne a Q-value reward for sentence completion, RelGAN (Nie et al., 2019) uses a relational memory based generator for the long-distance dependency modeling, FMGAN (Chen et al., 2018) uses a feature mover distance to match features of real and generated sentences inspired by optimal transport (Chen et al., 2019; Zhang et al., 2018), and LeakGAN (Guo et al., 2017) tries to address the sparse-reward issue for long-text generation with hierarchical RL by utilizing the leaked information from a GAN discriminator. One problem of LeakGAN is that it tends to over\ufb01t the training data, yielding generated sentences that are often not diverse. By contrast, by relying on a model-based imitation learning approach, our method learns global-structure information, which generates more-diverse sentences, and can be extended to conditional text generation. Zhang et al. (2020) designed a differentiable nested Wasserstein distance for semantic matching, which can be applied for further improvement. RL techniques can also be used in other ways for text generation (Bachman and Precup, 2015). For example, Ranzato et al. (2016) trained a Seq2Seq model by directly optimizing the BLEU/ROUGE scores with the REINFORCE algorithm. To reduce variance of the vanilla REINFORCE, Bahdanau et al. (2017) adopted the actor-critic framework for sequence prediction. Furthermore, Rennie et al. (2016) trained a baseline algorithm with a greedy decoding scheme for the REINFORCE method. Note that all these methods can only obtain reward after a whole sentence is generated. Planning techniques in RL have also been explored to improve text generation (Gulcehre et al., 2017; Serdyuk et al., 2018). Zhang et al. (2020) introduced the selfimitation scheme to exploit historical high-quality sentences for enhanced exploration. Compared to these related works, the proposed guider network can provide a planning mechanism and intermediate rewards. \fMethod Test-BLEU-2 3 4 5 Self-BLEU-2 3 4 SeqGAN (Yu et al., 2017) 0.820 0.604 0.361 0.211 0.807 0.577 0.278 RankGAN (Lin et al., 2017) 0.852 0.637 0.389 0.248 0.822 0.592 0.230 GSGAN (Kusner and Miguel, 2016) 0.810 0.566 0.335 0.197 0.785 0.522 0.230 TextGAN (Zhang et al., 2017) 0.910 0.728 0.484 0.306 0.806 0.548 0.217 LeakGAN (Guo et al., 2017) 0.922 0.797 0.602 0.416 0.912 0.825 0.689 MLE (Caccia et al., 2018) 0.902 0.706 0.470 0.392 0.787 0.646 0.485 GMGAN (ours) 0.949 0.823 0.635 0.421 0.746 0.511 0.319 Table 1: Test-BLEU (\u2191) and Self-BLEU (\u2193) scores on Image COCO. Method Test-BLEU-2 3 4 5 Self-BLEU-2 3 4 SeqGAN (Yu et al., 2017) 0.630 0.354 0.164 0.087 0.728 0.411 0.139 RankGAN (Lin et al., 2017) 0.723 0.440 0.210 0.107 0.672 0.346 0.119 GSGAN (Kusner and Miguel, 2016) 0.723 0.440 0.210 0.107 0.807 0.680 0.450 TextGAN (Zhang et al., 2017) 0.777 0.529 0.305 0.161 0.806 0.662 0.448 LeakGAN (Guo et al., 2017) 0.923 0.757 0.546 0.335 0.837 0.683 0.513 MLE (Caccia et al., 2018) 0.902 0.706 0.470 0.392 0.787 0.646 0.485 GMGAN (ours) 0.923 0.727 0.491 0.303 0.814 0.576 0.328 Table 2: Test-BLEU (\u2191) and Self-BLEU (\u2193) scores on EMNLP2017 WMT News. 6 Experiments We test the proposed framework on unconditional and conditional text generation tasks, and analyze the results to understand the performance gained by the guider network. We also perform an ablation investigation on the improvements brought by each part of our proposed method, and consider non-parallel style transfer. All experiments are conducted on a single Tesla P100 GPU and implemented with TensorFlow and Theano. Details of the datasets, the experimental setup and model architectures are provided in the Appendix. 6.1 Implementation Details Encoder as the feature extractor For unconditional generation, the feature extractor that generates inputs for the guider network shares the CNN part of the encoder. We stop gradients from the guider network to the encoder CNN in the training process. For conditional generation, we use a pretrained feature extractor, trained similarly to the unconditional generation. Training procedure As with many imitationlearning models (Bahdanau et al., 2017; Rennie et al., 2016; Sutskever et al., 2014), we \ufb01rst train the encoder-decoder part based on the off-policy data with an MLE loss. Then we use RL training to \ufb01ne-tune the trained generator. We adaptively transfer the training from MLE loss to RL loss, similar to (Paulus et al., 2017; Ranzato et al., 2016). Initial states We use the same initial state for both the generator and the guider networks. For conditional generation, the initial state is the encoded latent code of the conditional information for both training and testing. For unconditional generation, the initial state is the encoded latent code of a target sentence in training and random noise in testing. 6.2 Adversarial Text Generation We focus on adversarial text generation, and compare our approach with a number of related works (Guo et al., 2017; Lin et al., 2017; Yu et al., 2017; Zhang et al., 2017; Zhu et al., 2018). In this setting, a discriminator in the GAN framework is added to the model in Figure 1 to guide the generator to generate high-quality sentences. This is implemented by de\ufb01ning the \ufb01nal reward to be the output of the discriminator. All baseline experiments are implemented on the texygen platform (Zhu et al., 2018). We adopt the BLEU score, referenced by the test set (test-BLEU, higher value implies better quality) and itself (self-BLEU, lower value implies better diversity) (Zhu et al., 2018) to evaluate quality of generated samples, where test-BLEU evaluates the reality of generated samples, and self-BLEU measures the diversity. A good generator should achieve both a high test-BLEU score and a low self-BLEU score. In practice, we use \u25b3t = c = 4 and \u03b3 = 0.25. We call the proposed method guidermatching GAN (GMGAN) for unconditional text \fgeneration. More details of GMGAN are provided in Appendix D. Short Text Generation: COCO Image Captions We use the COCO Image Captions Dataset, in which most sentences have a length of about 10 words. Since we consider unconditional text generation, only image captions are used as the training data. After preprocessing, we use 120,000 random sample sentences as the training set, and 10,000 as the test set. The BLEU scores with different methods are listed in Table 1. We observe that GMGAN performs signi\ufb01cantly better than the baseline models. Speci\ufb01cally, besides achieving higher test-BLEU scores, the proposed method also generates samples with very good diversity in terms of self-BLEU scores. LeakGAN represents the state-of-the-art in adversarial text generation, however, its diversity measurement is relatively poor (Zhu et al., 2018). We suspect that the high BLEU score achieved by LeakGAN is due to its mode collapse on some good samples, resulting in high self-BLEU scores. Other baselines achieve lower self-BLEU scores since they cannot generate reasonable sentences. Long Text Generation: EMNLP2017 WMT Following (Zhu et al., 2018), we use the News section in the EMNLP2017 WMT4 Dataset as our training data. The dataset consists of 646,459 words and 397,726 sentences. After preprocessing, the training dataset contains 5,728 words and 278,686 sentences. The BLEU scores with different methods are provided in Table 2. Compared with other methods, LeakGAN and GMGAN achieve comparable test-BLEU scores, demonstrating high-quality generated sentences. Again, LeakGAN tends to over-\ufb01t on training data, leading to much higher (worse) self-BLEU scores. Our proposed GMGAN shows good diversity of long text generation with lower self-BLEU scores. Other baselines obtain both low self-BLEU and testBLEU scores, leading to more random generations. Human Evaluation Simply relying on the above metrics is not suf\ufb01cient to evaluate the proposed method (Caccia et al., 2018). Following previous work (Guo et al., 2017), we perform human evaluations using Amazon Mechnical Turk, evaluating the text quality based on readability and meaningfulness (whether sentences make sense) on the EMNNLP2017 WMT News dataset. We ask the worker to rate the input sentence with scores scalScores Criteria 5 (Best) It is consistent, informative, grammatically correct. 4 It is grammatically correct and makes sense. 3 It is mostly meaningful and with small grammatical error. 2 It needs some time to understand and has grammatical errors. 1 (Worst) Meaningless, not readable. Table 3: Human evaluation rating criteria. Methods MLE SeqGAN RankGAN GSGAN Human scores 2.45\u00b10.14 2.57\u00b10.15 2.91\u00b10.17 2.48\u00b10.14 Methods textGAN LeakGAN GMGAN Real Human scores 3.11\u00b10.16 3.47\u00b10.15 3.89\u00b10.15 4.21\u00b10.14 Table 4: Results of human evaluation with different methods on EMNLP2017 WMT dataset. ing from 1 to 5, with 1 as the worst score and 5 as the best. The detailed criteria is listed in Table 3. We require all the workers to be native English speakers, with approval rate higher than 90% and at least 100 assignments completed. We randomly sample 100 sentences generated by each model. Ten native English speakers on Amazon Mechanical Turk are asked to rate each sentence. The average human rating scores are shown in Table 4, indicating GMGAN achieves higher human scores compared to other methods. As examples, Table 5 illustrates some generated samples by GMGAN and its baselines. The performance on the two datasets indicates that the generated sentences of GMGAN are of higher global consistency and better readability than SeqGAN and LeakGAN. More generated examples are provided in the Appendix. Ablation Study We conduct ablation studies on long text generation to investigate the improvements brought by each part of our proposed method. We \ufb01rst test the bene\ufb01ts of using the guider network. Among the methods compared, Guider is the standard MLE model with the guider network. We further compare RL training with i) only \ufb01nal rewards , ii) only feature-matching rewards, and iii) combining both rewards, namely GMGAN. The results are shown in Table 6. We observe that guider network plays an important role in improving the performance. RL training with \ufb01nal rewards given by a discriminator typically damages the generation quality, but feature-matching reward produces sentences with much better diversity due to the ability of exploration. \fMethod COCO Image Captions EMNLP2017 WMT News SeqGAN (1) A person and black wooden table. (2) A closeup of a window at night. (1) She added on a page where it was made clear more old but public got said. (2) I think she\u2019re guys in four years , and more after it played well enough. LeakGAN (1) A bathroom with a black sink and a white toilet next to a tub. (2) A man throws a Frisbee across the grass covered yard. (1)\"I\u2019m a fan of all the game, I think if that\u2019s something that I\u2019ve not,\" she said, adding that he would not be decided. (2) The UK is Google\u2019 s largest non-US market, he has added \"20, before the best team is amount of fewer than one or the closest home or two years ago. GMGAN (1) Bicycles are parked near a row of large trees near a sidewalk. (2) A married couple posing in front of a piece of birthday cake. (1) \"Sometimes decisions are big, but they\u2019re easy to make,\" he told The Sunday Times in the New Year. (2) A BBC star has been questioned by police on suspicion of sexual assault against a 23-year-old man , it was reported last night. Table 5: Examples of generated samples with different methods on COCO and EMNLP datasets. Methods MLE Guider Final Stepwise GMGAN Test-BLEU-2 0.761 0.920 0.843 0.914 0.923 BLEU-3 0.468 0.723 0.623 0.704 0.727 BLEU-4 0.230 0.489 0.390 0.457 0.491 BLEU-5 0.116 0.289 0.221 0.276 0.303 Self-BLEU-2 0.664 0.812 0.778 0.798 0.814 BLEU-3 0.338 0.589 0.525 0.563 0.576 BLEU-4 0.113 0.360 0.273 0.331 0.328 Table 6: Ablation study on EMNLP2017 WMT. (a) (b) Figure 3: Guider-Matching Rewards Illustrations. Case Study of Guider-Matching Rewards Figure 3(a) illustrates the feature-matching rewards in the generation. Figure 3(a) shows an example of failure generation in the training stage, when two sentences are combined by the word \u2018was\u2019. It is grammatically wrong to select \u2018was\u2019 for the generator, thus the guider network gives a small reward. We can see that the rewards become lower with more time steps, which is consistent with the exposure bias. Figure 3(b) shows a successful generation, where the rewards given by the guider are relatively high (larger than 0.5). These observations validate that: (i) exposure bias exists in MLE training. (ii) RL training with exploration can help reduce the effects of exposure bias. (iii) Our proposed feature-matching rewards can provide meaningful guidance to maintain sentence structure and \ufb02uency. Model Acc(%) BLEU BLEU-ref CVAE (Shen et al., 2017) 73.9 20.7 7.8 Controllable (Hu et al., 2017) 86.7 58.4 BackTrans (Prabhumoye et al., 2018) 91.2 2.8 2.0 DeleteAndRetrieval (Li et al., 2018a) 88.9 36.8 14.7 Guider (Ours) 92.7 52.1 25.4 Table 7: Non-parallel text style transfer results on the test set with human references. 6.3 Non-parallel Text-style Transfer We test the proposed framework on the non-parallel text-style-transfer task, where the goal is to transfer one sentence in one style (e.g., positive) to a similar sentence but with a different style (e.g., negative). Pair-wise information should be inferred from the training data, which becomes more challenging. For a fair comparison, we use the same data and its split method as in (Shen et al., 2017). Speci\ufb01cally, there are 444,000, 63,500, and 127,000 sentences with either positive or negative sentiments in the training, validation and test sets, respectively. To measure whether the original sentences (in the test set) have been transferred to the desired sentiment, we follow the settings of (Shen et al., 2017) and employ a pretrained CNN classi\ufb01er, which achieves an accuracy of 97.4% on the validation set, to evaluate the transferred sentences. We also report the BLEU scores with original sentences (BLEU) and human references (BLEU-ref) (Li et al., 2018a), to evaluate the content preservation of transferred sentences. Results are summarized in Table 7. Our proposed model exhibits higher transfer accuracy and better content preservation, indicating the guider network provides good sentiment guidance to better preserve the content information. \fFrom positive to negative Original: all the employees are friendly and helpful . Transferred: all the employees are rude and unfriendly . Original: i \u2019m so lucky to have found this place ! Transferred: i \u2019m so embarrassed that i picked this place . From negative to positive Original: the service was slow . Transferred: the service was fast and friendly . Original: i would never eat there again and would probably not stay there either . Transferred: i would de\ufb01nitely eat this place and i would recommend them . Table 8: Generated samples of guided style transfer. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2002.09072v1",
+                    "title": "GenDICE: Generalized Offline Estimation of Stationary Values",
+                    "abstract": "An important problem that arises in reinforcement learning and Monte Carlo\nmethods is estimating quantities defined by the stationary distribution of a\nMarkov chain. In many real-world applications, access to the underlying\ntransition operator is limited to a fixed set of data that has already been\ncollected, without additional interaction with the environment being available.\nWe show that consistent estimation remains possible in this challenging\nscenario, and that effective estimation can still be achieved in important\napplications. Our approach is based on estimating a ratio that corrects for the\ndiscrepancy between the stationary and empirical distributions, derived from\nfundamental properties of the stationary distribution, and exploiting\nconstraint reformulations based on variational divergence minimization. The\nresulting algorithm, GenDICE, is straightforward and effective. We prove its\nconsistency under general conditions, provide an error analysis, and\ndemonstrate strong empirical performance on benchmark problems, including\noff-line PageRank and off-policy policy evaluation.",
+                    "authors": "Ruiyi Zhang, Bo Dai, Lihong Li, Dale Schuurmans",
+                    "published": "2020-02-21",
+                    "updated": "2020-02-21",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Estimation of quantities de\ufb01ned by the stationary distribution of a Markov chain lies at the heart of many scienti\ufb01c and engineering problems. Famously, the steady-state distribution of a random walk on the World Wide Web provides the foundation of the PageRank algorithm (Langville & Meyer, 2004). In many areas of machine learning, Markov chain Monte Carlo (MCMC) methods are used to conduct approximate Bayesian inference by considering Markov chains whose equilibrium distribution is a desired posterior (Andrieu et al., 2002). An example from engineering is queueing theory, where the queue lengths and waiting time under the limiting distribution have been extensively studied (Gross et al., 2018). As we will also see below, stationary distribution quantities are of fundamental importance in reinforcement learning (RL) (e.g., Tsitsiklis & Van Roy, 1997). Classical algorithms for estimating stationary distribution quantities rely on the ability to sample next states from the current state by directly interacting with the environment (as in on-line RL or MCMC), or even require the transition probability distribution to be given explicitly (as in PageRank). Unfortunately, these classical approaches are inapplicable when direct access to the environment is not available, which is often the case in practice. There are many practical scenarios where a collection of sampled trajectories is available, having been collected off-line by an external mechanism that chose states and recorded the subsequent next states. Given such data, we still wish to estimate a stationary quantity. One important example is off-policy policy evaluation in RL, where we wish to estimate the value of a policy different from that used to collect experience. Another example is off-line PageRank (OPR), where we seek to estimate the relative importance of webpages given a sample of the web graph. Motivated by the importance of these off-line scenarios, and by the inapplicability of classical methods, we study the problem of off-line estimation of stationary values via a stationary distribution corrector. Instead of having access to the transition probabilities or a next-state sampler, we assume only access to a \ufb01xed sample of state transitions, where states have been sampled from an unknown distribution and next-states are sampled according to the Markov chain\u2019s transition operator. The \u2217Equal contribution. \u2020Work done while interning at Google. 1 arXiv:2002.09072v1  [stat.ML]  21 Feb 2020 \fPublished as a conference paper at ICLR 2020 off-line setting is indeed more challenging than its more traditional on-line counterpart, given that one must infer an asymptotic quantity from \ufb01nite data. Nevertheless, we develop techniques that still allow consistent estimation under general conditions, and provide effective estimates in practice. The main contributions of this work are: \u2022 We formalize the problem of off-line estimation of stationary quantities, which captures a wide range of practical applications. \u2022 We propose a novel stationary distribution estimator, GenDICE, for this task. The resulting algorithm is based on a new dual embedding formulation for divergence minimization, with a carefully designed mechanism that explicitly eliminates degenerate solutions. \u2022 We theoretically establish consistency and other statistical properties of GenDICE, and empirically demonstrate that it achieves signi\ufb01cant improvements on several behavior-agnostic offpolicy evaluation benchmarks and an off-line version of PageRank. The methods we develop in this paper fundamentally extend recent work in off-policy policy evaluation (Liu et al., 2018; Nachum et al., 2019) by introducing a new formulation that leads to a more general, and as we will show, more effective estimation method. 2 BACKGROUND We \ufb01rst introduce off-line PageRank (OPR) and off-policy policy evaluation (OPE) as two motivating domains, where the goal is to estimate stationary quantities given only off-line access to a set of sampled transitions from an environment. Off-line PageRank (OPR) The celebrated PageRank algorithm (Page et al., 1999) de\ufb01nes the ranking of a web page in terms of its asymptotic visitation probability under a random walk on the (augmented) directed graph speci\ufb01ed by the hyperlinks. If we denote the World Wide Web by a directed graph G = (V, E) with vertices (web pages) v \u2208V and edges (hyperlinks) (v, u) \u2208E, PageRank considers the random walk de\ufb01ned by the Markov transition operator v \u2192u: P (u|v) = (1\u2212\u03b7) |v| 1(v,u)\u2208E + \u03b7 |V | , (1) where |v| denotes the out-degree of vertex v and \u03b7 \u2208[0, 1) is a probability of \u201cteleporting\u201d to any page uniformly. De\ufb01ne dt (v) := P (st = v|s0 \u223c\u00b50, \u2200i < t, si+1 \u223cP(\u00b7|si)), where \u00b50 is the initial distribution over vertices, then the original PageRank algorithm explicitly iterates for the limit d (v) := ( limt\u2192\u221edt (v) if \u03b3 = 1 (1 \u2212\u03b3) P\u221e t=0 \u03b3tdt (v) if \u03b3 \u2208(0, 1) . (2) The classical version of this problem is solved by tabular methods that simulate Equation 1. However, we are interested in a more scalable off-line version of the problem where the transition model is not explicitly given. Instead, consider estimating the rank of a particular web page v\u2032 from a large web graph, given only a sample D = {(v, u)i}N i=1 from a random walk on G as speci\ufb01ed above. We would still like to estimate d(v\u2032) based on this data. First, note that if one knew the distribution p by which any vertex v appeared in D, the target quantity could be re-expressed by a simple importance ratio d (v\u2032) = Ev\u223cp h d(v) p(v)1v=v\u2032 i . Therefore, if one had the correction ratio function \u03c4 (v) = d(v) p(v), an estimate of d (v\u2032) can easily be recovered via d (v\u2032) \u2248\u02c6 p (v\u2032) \u03c4 (v\u2032), where \u02c6 p (v\u2032) is the empirical probability of v\u2032 estimated from D. The main attack on the problem we investigate is to recover a good estimate of the ratio function \u03c4. Policy Evaluation An important generalization of this stationary value estimation problem arises in RL in the form of policy evaluation. Consider a Markov Decision Process (MDP) M = \u27e8S, A, P, R, \u03b3, \u00b50\u27e9(Puterman, 2014), where S is a state space, A is an action space, P (s\u2032|s, a) denotes the transition dynamics, R is a reward function, \u03b3 \u2208(0, 1] is a discounted factor, and \u00b50 is the initial state distribution. Given a policy, which chooses actions in any state s according to the probability distribution \u03c0(\u00b7|s), a trajectory \u03b2 = (s0, a0, r0, s1, a1, r1, . . .) is generated by \ufb01rst sampling the initial state s0 \u223c\u00b50, and then for t \u22650, at \u223c\u03c0(\u00b7|st), rt \u223cR(st, at), and st+1 \u223cP(\u00b7|st, at). The value of a policy \u03c0 is the expected per-step reward de\ufb01ned as: Average: R(\u03c0) := limT \u2192\u221e 1 T +1E hPT t=0 rt i , Discounted: R\u03b3(\u03c0) := (1\u2212\u03b3)E [P\u221e t=0 \u03b3trt] . (3) In the above, the expectation is taken with respect to the randomness in the state-action pair P (s\u2032|s, a) \u03c0 (a\u2032|s\u2032) and the reward R (st, at). Without loss of generality, we assume the limit exists for the average case, and hence R(\u03c0) is \ufb01nite. 2 \fPublished as a conference paper at ICLR 2020 Behavior-agnostic Off-Policy Evaluation (OPE) An important setting of policy evaluation that often arises in practice is to estimate R\u03b3 (\u03c0) or R (\u03c0) given a \ufb01xed dataset D = {(s, a, r, s\u2032)i}N i=1 \u223c P (s\u2032|s, a) p (s, a), where p (s, a) is an unknown distribution induced by multiple unknown behavior policies. This problem is different from the classical form of OPE, where it is assumed that a known behavior policy \u03c0b is used to collect transitions. In the behavior-agnostic scenario, however, typical importance sampling (IS) estimators (e.g., Precup et al., 2000) do not apply. Even if one can assume D consists of trajectories where the behavior policy can be estimated from data, it is known that that straightforward IS estimators suffer a variance exponential in the trajectory length, known as the \u201ccurse of horizon\u201d (Jiang & Li, 2016; Liu et al., 2018). Let d\u03c0 t (s, a) = P (st = s, at = a|s0 \u223c\u00b50, \u2200i < t, ai \u223c\u03c0 (\u00b7|si) , si+1 \u223cP(\u00b7|si, ai)). The stationary distribution can then be de\ufb01ned as \u00b5\u03c0 \u03b3 (s, a) := ( limT \u2192\u221e 1 T +1 PT t=0 d\u03c0 t (s, a) = limt\u2192\u221ed\u03c0 t (s, a) if \u03b3 = 1 (1 \u2212\u03b3) P\u221e t=0 \u03b3td\u03c0 t (s, a) if \u03b3 \u2208(0, 1) . (4) With this de\ufb01nition, R(\u03c0) and R\u03b3 (\u03c0) can be equivalently re-expressed as R\u03b3(\u03c0) := E\u00b5\u03c0 \u03b3 [R (s, a)] = Ep h \u00b5\u03c0 \u03b3 (s,a) p(s,a) R (s, a) i . (5) Here we see once again that if we had the correction ratio function \u03c4 (s, a) = \u00b5\u03c0 \u03b3 (s,a) p(s,a) a straightforward estimate of R\u03b3(\u03c0) could be recovered via R\u03b3(\u03c0) \u2248E\u02c6 p [\u03c4 (s, a) R (s, a)], where \u02c6 p (s, a) is an empirical estimate of p (s, a). In this way, the behavior-agnostic OPE problem can be reduced to estimating the correction ratio function \u03c4, as above. We note that Liu et al. (2018) and Nachum et al. (2019) also exploit Equation 5 to reduce OPE to stationary distribution correction, but these prior works are distinct from the current proposal in different ways. First, the inverse propensity score (IPS) method of Liu et al. (2018) assumes the transitions are sampled from a single behavior policy, which must be known beforehand; hence that approach is not applicable in behavior-agnostic OPE setting. Second, the recent DualDICE algorithm (Nachum et al., 2019) is also a behavior-agnostic OPE estimator, but its derivation relies on a change-of-variable trick that is only valid for \u03b3 < 1. This previous formulation becomes unstable when \u03b3 \u21921, as shown in Section 6 and Appendix E. The behavior-agnostic OPE estimator we derive below in Section 3 is applicable both when \u03b3 = 1 and \u03b3 \u2208(0, 1). This connection is why we name the new estimator GenDICE, for GENeralized stationary DIstribution Correction Estimation. 3 GENDICE As noted, there are important estimation problems in the Markov chain and MDP settings that can be recast as estimating a stationary distribution correction ratio. We \ufb01rst outline the conditions that characterize the correction ratio function \u03c4, upon which we construct the objective for the GenDICE estimator, and design ef\ufb01cient algorithm for optimization. We will develop our approach for the more general MDP setting, with the understanding that all the methods and results can be easily specialized to the Markov chain setting. 3.1 ESTIMATING STATIONARY DISTRIBUTION CORRECTION The stationary distribution \u00b5\u03c0 \u03b3 de\ufb01ned in Equation 4 can also be characterized via \u00b5 (s\u2032, a\u2032) = (1 \u2212\u03b3) \u00b50 (s\u2032) \u03c0 (a\u2032|s\u2032) + \u03b3 Z \u03c0 (a\u2032|s\u2032) P (s\u2032|s, a) \u00b5 (s, a) ds da | {z } (T \u25e6\u00b5)(s\u2032,a\u2032) , \u2200(s\u2032, a\u2032) \u2208S \u00d7 A. (6) At \ufb01rst glance, this equation shares a super\ufb01cial similarity to the Bellman equation, but there is a fundamental difference. The Bellman operator recursively integrates out future (s\u2032, a\u2032) pairs to characterize a current pair (s, a) value, whereas the distribution operator T de\ufb01ned in Equation 6 operates in the reverse temporal direction. When \u03b3 < 1, Equation 6 always has a \ufb01xed-point solution. For \u03b3 = 1, in the discrete case, the \ufb01xedpoint exists as long as T is ergodic; in the continuous case, the conditions for \ufb01xed-point existence become more complicated (Meyn & Tweedie, 2012) and beyond the scope of this paper. The development below is based on a divergence D and the following default assumption. Assumption 1 (Markov chain regularity) For the given target policy \u03c0, the resulting state-action transition operator T has a unique stationary distribution \u00b5 that satis\ufb01es D(T \u25e6\u00b5\u2225\u00b5) = 0. 3 \fPublished as a conference paper at ICLR 2020 In the behavior-agnostic setting we consider, one does not have direct access to P for element-wise evaluation or sampling, but instead is given a \ufb01xed set of samples from P (s\u2032|s, a) p (s, a) with respect to some distribution p (s, a) over S \u00d7 A. De\ufb01ne T p \u03b3,\u00b50 to be a mixture of \u00b50\u03c0 and Tp; i.e., let T p \u03b3,\u00b50 ((s\u2032, a\u2032) , (s, a)) := (1 \u2212\u03b3) \u00b50 (s\u2032) \u03c0 (a\u2032|s\u2032) + \u03b3 \u03c0 (a\u2032|s\u2032) P (s\u2032|s, a) p (s, a) | {z } Tp((s\u2032,a\u2032),(s,a)) . (7) Obviously, conditioning on (s, a, s\u2032) one could easily sample a\u2032 \u223c\u03c0 (a\u2032|s\u2032) to form (s, a, s\u2032, a\u2032) \u223c Tp ((s\u2032, a\u2032) , (s, a)); similarly, a sample (s\u2032, a\u2032) \u223c\u00b50 (s\u2032) \u03c0 (a\u2032|s\u2032) could be formed from s\u2032. Mixing such samples with probability \u03b3 and 1 \u2212\u03b3 respectively yields a sample (s, a, s\u2032, a\u2032) \u223c T p \u03b3,\u00b50 ((s\u2032, a\u2032) , (s, a)). Based on these observations, the stationary condition for the ratio from Equation 6 can be re-expressed in terms of T p \u03b3,\u00b50 as p (s\u2032, a\u2032) \u03c4 \u2217(s\u2032, a\u2032) = (1 \u2212\u03b3) \u00b50 (s\u2032) \u03c0 (a\u2032|s\u2032) + \u03b3 Z \u03c0 (a\u2032|s\u2032) P (s\u2032|s, a) p (s, a) \u03c4 \u2217(s, a) ds da | {z } (T p \u03b3,\u00b50\u25e6\u03c4 \u2217)(s\u2032,a\u2032) , (8) where \u03c4 \u2217(s, a) := \u00b5(s,a) p(s,a) is the correction ratio function we seek to estimate. One natural approach to estimating \u03c4 \u2217is to match the LHS and RHS of Equation 8 with respect to some divergence D (\u00b7\u2225\u00b7) over the empirical samples. That is, we consider estimating \u03c4 \u2217by solving the optimization problem min \u03c4\u22650 D \u0000T p \u03b3,\u00b50 \u25e6\u03c4\u2225p \u00b7 \u03c4 \u0001 . (9) Although this forms the basis of our approach, there are two severe issues with this naive formulation that \ufb01rst need to be recti\ufb01ed: i) Degenerate solutions: When \u03b3 = 1, the operator T p \u03b3=1,\u00b50 is invariant to constant rescaling: if \u03c4 \u2217= T p \u03b3=1,\u00b50 \u25e6\u03c4 \u2217then c\u03c4 \u2217= T p \u03b3=1,\u00b50 \u25e6(c\u03c4 \u2217) for any c \u22650. Therefore, simply minimizing the divergence D \u0000T p \u03b3=1,\u00b50 \u25e6\u03c4\u2225p \u00b7 \u03c4 \u0001 cannot provide a desirable estimate of \u03c4 \u2217. In fact, in this case the trivial solution \u03c4 \u2217(s, a) = 0 cannot be eliminated. ii) Intractable objective: The divergence D \u0000T p \u03b3,\u00b50 \u25e6\u03c4\u2225p \u00b7 \u03c4 \u0001 involves the computation of T p \u03b3,\u00b50 \u25e6 \u03c4, which in general involves an intractable integral. Thus, evaluation of the exact objective is intractable, and neglects the assumption that we only have access to samples from T p \u03b3,\u00b50 and are not able to evaluate it at arbitrary points. We address each of these two issues in a principled manner. 3.2 ELIMINATING DEGENERATE SOLUTIONS To avoid degenerate solutions when \u03b3 = 1, we ensure that the solution is a proper density ratio; that is, the property \u03c4 \u2208\u039e := {\u03c4 (\u00b7) \u22650, Ep [\u03c4] = 1} must be true of any \u03c4 that is a ratio of some density to p. This provides an additional constraint that we add to the optimization formulation min \u03c4\u22650 D \u0000T p \u03b3,\u00b50 \u25e6\u03c4\u2225p \u00b7 \u03c4 \u0001 , s.t., Ep [\u03c4] = 1. (10) With this additional constraint, it is obvious that the trivial solution \u03c4 (s, a) = 0 is eliminated as an infeasible point of Eqn (10), along with other degenerate solutions \u03c4 (s, a) = c\u03c4 \u2217(s, a) with c \u0338= 1. Unfortunately, exactly solving an optimization with expectation constraints is very complicated in general (Lan & Zhou, 2016), particularly given a nonlinear parameterization for \u03c4. The penalty method (Luenberger & Ye, 2015) provides a much simpler alternative, where a sequence of regularized problems are solved min \u03c4\u22650 J (\u03c4) := D \u0000T p \u03b3,\u00b50 \u25e6\u03c4\u2225p \u00b7 \u03c4 \u0001 + \u03bb 2 (Ep [\u03c4] \u22121)2 , (11) with \u03bb increasing. The drawback of the penalty method is that it generally requires \u03bb \u2192\u221eto ensure the strict feasibility, which is still impractical, especially in stochastic gradient descent. The in\ufb01nite \u03bb may induce unbounded variance in the gradient estimator, and thus, divergence in optimization. However, by exploiting the special structure of the solution sets to Equation 11, we can show that, remarkably, it is unnecessary to increase \u03bb. Theorem 1 For \u03b3 \u2208(0, 1] and any \u03bb > 0, the solution to Equation 11 is given by \u03c4 \u2217(s, a) = u(s,a) p(s,a) . The detailed proof for Theorem 1 is given in Appendix A.1. By Theorem 1, we can estimate the desired correction ratio function \u03c4 \u2217by solving only one optimization with an arbitrary \u03bb > 0. 4 \fPublished as a conference paper at ICLR 2020 3.3 EXPLOITING DUAL EMBEDDING The optimization in Equation 11 involves the integrals \u0000T p \u03b3,\u00b50 \u25e6\u03c4 \u0001 and Ep [\u03c4] inside nonlinear loss functions, hence appears dif\ufb01cult to solve. Moreover, obtaining unbiased gradients with a naive approach requires double sampling (Baird, 1995). Instead, we bypass both dif\ufb01culties by applying a dual embedding technique (Dai et al., 2016; 2018). In particular, we assume the divergence D is in the form of an f-divergence (Nowozin et al., 2016) D\u03c6 \u0000\u0000T p \u03b3,\u00b50 \u25e6\u03c4 \u0001 \u2225p \u00b7 \u03c4 \u0001 := R p \u00b7 \u03c4 (s, a) \u03c6 \u0012 (T p \u03b3,\u00b50\u25e6\u03c4)(s,a) p\u00b7\u03c4(s,a) \u0013 ds da where \u03c6 (\u00b7) : R+ \u2192R is a convex, lower-semicontinuous function with \u03c6 (1) = 0. Plugging this into J (\u03c4) in Equation 11 we can easily check the convexity of the objective Theorem 2 For an f-divergence with valid \u03c6 de\ufb01ning D\u03c6, the objective J (\u03c4) is convex w.r.t. \u03c4. The detailed proof is provided in Appendix A.2. Recall that a suitable convex function can be represented as \u03c6 (x) = maxf x\u00b7f \u2212\u03c6\u2217(f), where \u03c6\u2217is the Fenchel conjugate of \u03c6 (\u00b7). In particular, we have the representation 1 2x2 = maxu ux \u22121 2u2, which allows us to re-express the objective as J (\u03c4) = R p \u00b7 \u03c4 (s\u2032, a\u2032) \u001a maxf \u0014 (T p \u03b3,\u00b50\u25e6\u03c4)(s\u2032,a\u2032) p\u00b7\u03c4(s\u2032,a\u2032) f \u2212\u03c6\u2217(f) \u0015\u001b ds\u2032da\u2032 + \u03bb n maxu h u (Ep [\u03c4] \u22121) \u2212u2 2 io . (12) Applying the interchangeability principle (Shapiro et al., 2014; Dai et al., 2016), one can replace the inner max in the \ufb01rst term over scalar f to maximize over a function f (\u00b7, \u00b7) : S \u00d7 A \u2192R min \u03c4\u22650 max f:S\u00d7A\u2192R,u\u2208R J (\u03c4, u, f) = (1 \u2212\u03b3) E\u00b50\u03c0 [f (s, a)] + \u03b3ETp [\u03c4 (s, a) f (s\u2032, a\u2032)] \u2212Ep [\u03c4 (s, a) \u03c6\u2217(f (s, a))] + \u03bb \u0010 Ep [u\u03c4 (s, a) \u2212u] \u2212u2 2 \u0011 . (13) This yields the main optimization formulation, which avoids the aforementioned dif\ufb01culties and is well-suited for practical optimization as discussed in Section 3.4. Remark (Other divergences): In addition to f-divergence, the proposed estimator Equation 11 is compatible with other divergences, such as the integral probability metrics (IPM) (M\u00a8 uller, 1997; Sriperumbudur et al., 2009), while retaining consistency. Based on the de\ufb01nition of the IPM, these divergences directly lead to min-max optimizations similar to Equation 13 with the identity function as \u03c6\u2217(\u00b7) and different feasible sets for the dual functions. Speci\ufb01cally, maximum mean discrepancy (MMD) (Smola et al., 2006) requires \u2225f\u2225Hk \u22641 where Hk denotes the RKHS with kernel k; the Dudley metric (Dudley, 2002) requires \u2225f\u2225BL \u22641 where \u2225f\u2225BL := \u2225f\u2225\u221e+ \u2225\u2207f\u22252; and Wasserstein distance (Arjovsky et al., 2017) requires \u2225\u2207f\u22252 \u22641. These additional requirements on the dual function might incur some extra dif\ufb01culty in practice. For example, with Wasserstein distance and the Dudley metric, we might need to include an extra gradient penalty (Gulrajani et al., 2017), which requires additional computation to take the gradient through a gradient. Meanwhile, the consistency of the surrogate loss under regularization is not clear. For MMD, we can obtain a closed-form solution for the dual function, which saves the cost of the inner optimization (Gretton et al., 2012), but with the tradeoff of requiring two independent samples in each outer optimization update. Moreover, MMD relies on the condition that the dual function lies in some RKHS, which introduces additional kernel parameters to be tuned and in practice may not be suf\ufb01ciently \ufb02exible compared to neural networks. 3.4 A PRACTICAL ALGORITHM We have derived a consistent stationary distribution correction estimator in the form of a min-max saddle point optimization Equation 13. Here, we present a practical instantiation of GenDICE with a concrete objective and parametrization. We choose the \u03c72-divergence, which is an f-divergence with \u03c6 (x) = (x \u22121)2 and \u03c6\u2217(y) = y+ y2 4 . The objective becomes J\u03c72 (\u03c4, u, f) = (1 \u2212\u03b3) E\u00b50\u03c0 [f (s, a)] + \u03b3ETp [\u03c4 (s, a) f (s\u2032, a\u2032)] \u2212Ep \u0002 \u03c4 (s, a) \u0000f (s, a) + 1 4f 2 (s, a) \u0001\u0003 + \u03bb \u0010 Ep [u\u03c4 (s, a) \u2212u] \u2212u2 2 \u0011 . (14) There two major reasons for adopting \u03c72-divergence: i) In the behavior-agnostic OPE problem, we mainly use the ratio correction function for estimating b Ep [\u02c6 \u03c4 (s, a) R (s, a)], which is an expectation. Recall that the error between the estimate and ground-truth can then be bounded by total variation, which is a lower bound of \u03c72-divergence. 5 \fPublished as a conference paper at ICLR 2020 ii) For the alternative divergences, the conjugate of the KL-divergence involves exp (\u00b7), which may lead to instability in optimization; while the IPM variants introduce extra constraints on dual function, which may be dif\ufb01cult to be optimized. The conjugate function of \u03c72-divergence enjoys suitable numerical properties and provides squared regularization. We have provided an empirical ablation study that investigates the alternative divergences in Section 6.3. To parameterize the correction ratio \u03c4 and dual function f we use neural networks, \u03c4 (s, a) = nnw\u03c4 (s, a) and f (s, a) = nnwf (s, a), where w\u03c4 and wf denotes the parameters of \u03c4 and f respectively. Since the optimization requires \u03c4 to be non-negative, we add an extra positive neuron, such as exp (\u00b7), log (1 + exp (\u00b7)) or (\u00b7)2 at the \ufb01nal layer of nnw\u03c4 (s, a). We empirically compare the different positive neurons in Section 6.3. For these representations, and unbiased gradient estimator \u2207(w\u03c4 ,u,wf )J (\u03c4, u, f) can be obtained straightforwardly, as shown in Appendix B. This allows us to apply stochastic gradient descent to solve the saddle-point problem Equation 14 in a scalable manner, as illustrated in Algorithm 1. 4 THEORETICAL ANALYSIS We provide a theoretical analysis for the proposed GenDICE algorithm, following a similar learning setting and assumptions to (Nachum et al., 2019). Assumption 2 The target stationary correction are bounded, \u2225\u03c4 \u2217\u2225\u221e\u2264C < \u221e. The main result is summarized in the following theorem. A formal statement, together with the proof, is given in Appendix C. Theorem 3 (Informal) Under mild conditions, with learnable F and H, the error in the objective between the GenDICE estimate, \u02c6 \u03c4, to the solution \u03c4 \u2217(s, a) = u(s,a) p(s,a) is bounded by E [J (\u02c6 \u03c4) \u2212J (\u03c4 \u2217)] = e O \u0010 \u03f5approx (F, H) + 1 \u221a N + \u03f5opt \u0011 , where E [\u00b7] is w.r.t. the randomness in D and in the optimization algorithms, \u03f5opt is the optimization error, and \u03f5approx (F, H) is the approximation induced by (F, H) for parametrization of (\u03c4, f). The theorem shows that the suboptimality of GenDICE\u2019s solution, measured in terms of the objective function value, can be decomposed into three terms: (1) the approximation error \u03f5approx, which is controlled by the representation \ufb02exibility of function classes; (2) the estimation error due to sample randomness, which decays at the order of 1/ \u221a N; and (3) the optimization error, which arises from the suboptimality of the solution found by the optimization algorithm. As discussed in Appendix C, in special cases, this suboptimality can be bounded below by a divergence between \u02c6 \u03c4 and \u03c4 \u2217, and therefore directly bounds the error in the estimated policy value. There is also a tradeoff between these three error terms. With more \ufb02exible function classes (e.g., neural networks) for F and H, the approximation error \u03f5approx becomes smaller. However, it may increase the estimation error (through the constant in front of 1/ \u221a N) and the optimization error (by solving a harder optimization problem). On the other hand, if F and H are linearly parameterized, estimation and optimization errors tend to be smaller and can often be upper-bounded explicitly in Appendix C.3. However, the corresponding approximation error will be larger. 5 RELATED WORK Off-policy Policy Evaluation Off-policy policy evaluation with importance sampling (IS) has has been explored in the contextual bandits (Strehl et al., 2010; Dud\u00b4 \u0131k et al., 2011; Wang et al., 2017), and episodic RL settings (Murphy et al., 2001; Precup et al., 2001), achieving many empirical successes (e.g., Strehl et al., 2010; Dud\u00b4 \u0131k et al., 2011; Bottou et al., 2013). Unfortunately, ISbased methods suffer from exponential variance in long-horizon problems, known as the \u201ccurse of horizon\u201d (Liu et al., 2018). A few variance-reduction techniques have been introduced, but still cannot eliminate this fundamental issue (Jiang & Li, 2016; Thomas & Brunskill, 2016; Guo et al., 2017). By rewriting the accumulated reward as an expectation w.r.t. a stationary distribution, Liu et al. (2018); Gelada & Bellemare (2019) recast OPE as estimating a correction ratio function, which signi\ufb01cantly alleviates variance. However, these methods still require the off-policy data to be collected by a single and known behavior policy, which restricts their practical applicability. The only published algorithm in the literature, to the best of our knowledge, that solves agnostic-behavior off-policy evaluation is DualDICE (Nachum et al., 2019). However, DualDICE was developed for 6 \fPublished as a conference paper at ICLR 2020 discounted problems and its results become unstable when the discount factor approaches 1 (see below). By contrast, GenDICE can cope with the more challenging problem of undiscounted reward estimation in the general behavior-agnostic setting. Note that standard model-based methods (Sutton & Barto, 1998), which estimate the transition and reward models directly then calculate the expected reward based on the learned model, are also applicable to the behavior-agnostic setting considered here. Unfortunately, model-based methods typically rely heavily on modeling assumptions about rewards and transition dynamics. In practice, these assumptions do not always hold, and the evaluation results can become unreliable. Markov Chain Monte Carlo Classical MCMC (Brooks et al., 2011; Gelman et al., 2013) aims at sampling from \u00b5\u03c0 by iteratively simulting from the transition operator. It requires continuous interaction with the transition operator and heavy computational cost to update many particles. Amortized SVGD (Wang & Liu, 2016) and Adversarial MCMC (Song et al., 2017; Li et al., 2019) alleviate this issue via combining with neural network, but they still interact with the transition operator directly, i.e., in an on-policy setting. The major difference of our GenDICE is the learning setting: we only access the off-policy dataset, and cannot sample from the transition operator. The proposed GenDICE leverages stationary density ratio estimation for approximating the stationary quantities, which distinct it from classical methods. Density Ratio Estimation Density ratio estimation is a fundamental tool in machine learning and much related work exists. Classical density ratio estimation includes moment matching (Gretton et al., 2008), probabilistic classi\ufb01cation (Bickel et al., 2007), and ratio matching (Nguyen et al., 2008; Sugiyama et al., 2008; Kanamori et al., 2009). These classical methods focus on estimating the ratio between two distributions with samples from both of them, while GenDICE estimates the density ratio to a stationary distribution of a transition operator, from which even one sample is dif\ufb01cult to obtain. PageRank Yao & Schuurmans (2013) developed a reverse-time RL framework for PageRank via solving a reverse Bellman equation, which is less sensitive to graph topology and shows faster adaptation with graph change. However, Yao & Schuurmans (2013) still considers the online manner, which is different with our OPR setting. 6 EXPERIMENTS In this section, we evaluate GenDICE on OPE and OPR problems. For OPE, we use one or multiple behavior policies to collect a \ufb01xed number of trajectories at some \ufb01xed trajectory length. This data is used to recover a correction ratio function for a target policy \u03c0 that is then used to estimate the average reward in two different settings: i) average reward; and ii) discounted reward. In both settings, we compare with a model-based approach and step-wise weighted IS (Precup et al., 2000). We also compare to Liu et al. (2018) (referred to as \u201cIPS\u201d here) in the Taxi domain with a learned behavior policy1. We speci\ufb01cally compare to DualDICE (Nachum et al., 2019) in the discounted reward setting, which is a direct and current state-of-the-art baseline. For OPR, the main comparison is with the model-based method, where the transition operator is empirically estimated and stationary distribution recovered via an exact solver. We validate GenDICE in both tabular and continuous cases, and perform an ablation study to further demonstrate its effectiveness. All results are based on 20 random seeds, with mean and standard deviation plotted. Our code is publicly available at https://github.com/zhangry868/GenDICE. 6.1 TABULAR CASE Of\ufb02ine PageRank on Graphs One direct application of GenDICE is off-line PageRank (OPR). We test GenDICE on a Barabasi-Albert (BA) graph (synthetic), and two real-world graphs, Cora and Citeseer. Details of the graphs are given in Appendix D. We use the log KL-divergence between estimated stationary distribution and the ground truth as the evaluation metric, with the ground truth computed by an exact solver based on the exact transition operator of the graphs. We compared GenDICE with model-based methods in terms of the sample ef\ufb01ciency. From the results in Figure 1, GenDICE outperforms the model-based method when limited data is given. Even with 20k samples for a BA graph with 100 nodes, where a transition matrix has 10k entries, GenDICE still 1We used the released implementation of IPS (Liu et al., 2018) from https://github.com/zt95/ infinite-horizon-off-policy-estimation. 7 \fPublished as a conference paper at ICLR 2020 shows better performance in the of\ufb02ine setting. This is reasonable since GenDICE directly estimates the stationary distribution vector or ratio, while the model-based method needs to learn an entire transition matrix that has many more parameters. 0.5k 1k 2k 5k 10k 20k # of Samples -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 log KL-Divergence BA Graph (100 Nodes) Model-Based GenDICE (ours) 1k 2k 5k 10k 20k 50k # of Samples -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 BA Graph (500 Nodes) 2k 5k 10k 20k 50k 100k # of Samples -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 log KL-Divergence Citeseer 2k 5k 10k 20k 50k 100k # of Samples -1.6 -1.4 -1.2 -1.0 Cora Figure 1: Stationary Distribution Estimation on BA and real-world graphs. Each plot shows the log KLdivergence of GenDICE and modelbased method towards the number of samples. Off-Policy Evaluation with Taxi We use a similar taxi domain as in Liu et al. (2018), where a grid size of 5 \u00d7 5 yields 2000 states in total (25 \u00d7 16 \u00d7 5, corresponding to 25 taxi locations, 16 passenger appearance status and 5 taxi status). We set the target policy to a \ufb01nal policy \u03c0 after running tabular Q-learning for 1000 iterations, and set another policy \u03c0+ after 950 iterations as the base policy. The behavior policy is a mixture controlled by \u03b1 as \u03c0b = (1 \u2212\u03b1)\u03c0 + \u03b1\u03c0+. For the model-based method, we use a tabular representation for the reward and transition functions, whose entries are estimated from behavior data. For IS and IPS, we \ufb01t a policy via behavior cloning to estimate the policy ratio. In this speci\ufb01c setting, our methods achieve better results compared to IS, IPS and the model-based method. Interestingly, with longer horizons, IS cannot improve as much as other methods even with more data, while GenDICE consistently improve and achieves much better results than the baselines. DualDICE only works with \u03b3 < 1. GenDICE is more stable than DualDICE when \u03b3 becomes larger (close to 1), while still showing competitive performance for smaller discount factors \u03b3. 100 200 400 800 Truncated Length -10 -9 -8 -7 -6 log MSE =1.0 100 200 400 800 Truncated Length -10 -9 -8 -7 -6 =0.999 0.99 0.995 0.999 0.9999 1.0 Discount Factor  -10 -9 -8 -7 =0.0 0.0 0.2 0.4 0.6 0.8 Mixing Ratio  -11 -10 -9 -8 -7 =1.0 0.0 0.2 0.4 0.6 0.8 Mixing Ratio  -10 -9 -8 -7 =0.999 Model-Based Importance Sampling DualDICE IPS GenDICE (ours) Figure 2: Results on Taxi Domain. The plots show log MSE of the tabular estimator across different trajectory lengths, different discount factors and different behavior policies (x-axis). 6.2 CONTINUOUS CASE We further test our method for OPE on three control tasks: a discrete-control task Cartpole and two continuous-control tasks Reacher and HalfCheetah. In these tasks, observations (or states) are continuous, thus we use neural network function approximators and stochastic optimization. Since DualDICE (Nachum et al., 2019) has shown the state-of-the-art performance on discounted OPE, we mainly compare with it in the discounted reward case. We also compare to IS with a learned policy via behavior cloning and a neural model-based method, similar to the tabular case, but with neural network as the function approximator. All neural networks are feed-forward with two hidden layers of dimension 64 and tanh activations. More details can be found in Appendix D. Due to limited space, we put the discrete control results in Appendix E and focus on the more challenging continuous control tasks. Here, the good performance of IS and model-based methods in Section 6.1 quickly deteriorates as the environment becomes complex, i.e., with a continuous action space. Note that GenDICE is able to maintain good performance in this scenario, even when using function approximation and stochastic optimization. This is reasonable because of the dif\ufb01culty of \ufb01tting to the coupled policy-environment dynamics with a continuous action space. Here we also empirically validate GenDICE with off-policy data collected by multiple policies. As illustrated in Figure 3, all methods perform better with longer trajectory length or more trajectories. When \u03b1 becomes larger, i.e., the behavior policies are closer to the target policy, all methods performs better, as expected. Here, GenDICE demonstrates good performance both on averagereward and discounted reward cases in different settings. The right two \ufb01gures in each row show the log MSE curve versus optimization steps, where GenDICE achieves the smallest loss. In the discounted reward case, GenDICE shows signi\ufb01cantly better and more stable performance than the 8 \fPublished as a conference paper at ICLR 2020 strong baseline, DualDICE. Figure 4 also shows better performance of GenDICE than all baselines in the more challenging HalfCheetah domain. Figure 3: Results on Reacher. The left three plots in the \ufb01rst row show the log MSE of estimated average per-step reward over different numbers of trajectories, truncated lengths, and behavior policies (M1 and M2 mean off-policy set collected by multiple behavior policies with \u03b1 = [0.0, 0.33] and \u03b1 = [0.0, 0.33, 0.66]). The right two \ufb01gures show the loss curves towards the optimization steps. Each plot in the second row shows the average reward case. 20 40 60 100 200 Truncated Length -4 -2 0 2 log MSE =1.0 20 40 60 100 200 Truncated Length -6 -4 -2 0 2 =0.99 100 200 400 1000 5000 # of Trajectories -4 -3 -2 -1 0 log MSE =0.99 0.0 0.33 0.66 M1 M2 Different Behaviour Policies -5 -4 -3 -2 -1 =1.0 0.0 0.33 0.66 M1 M2 Different Behaviour Policies -4 -3 -2 -1 =0.99 Model-Based Importance Sampling DualDICE GenDICE (ours) Figure 4: Results on HalfCheetah. Plots from left to the right show the log MSE of estimated average per-step reward over different truncated lengths, numbers of trajectories, and behavior policies in discounted and average reward cases. 6.3 ABLATION STUDY Finally, we conduct an ablation study on GenDICE to study its robustness and implementation sensitivities. We investigate the effects of learning rate, activation function, discount factor, and the speci\ufb01cally designed ratio constraint. We further demonstrate the effect of the choice of divergences and the penalty weight. Effects of the Learning Rate Since we are using neural network as the function approximator, and stochastic optimization, it is necessary to show sensitivity to the learning rate with {0.0001, 0.0003, 0.001, 0.003}, with results in Figure 5. When \u03b1 = 0.33, i.e., the OPE tasks are relatively easier and GenDICE obtains better results at all learning rate settings. However, when \u03b1 = 0.0, i.e., the estimation becomes more dif\ufb01cult and only GenDICE only obtains reasonable results with the larger learning rate. Generally, this ablation study shows that the proposed method is not sensitive to the learning rate, and is easy to train. Activation Function of Ratio Estimator We further investigate the effects of the activation function on the last layer, which ensure the non-negative outputs required for the ratio. To better understand which activation function will lead to stable trainig for the neural correction estimator, we empirically compare using i) (\u00b7)2; ii) log(1 + exp(\u00b7)); and iii) exp(\u00b7). In practice, we use the (\u00b7)2 since it achieves low variance and better performance in most cases, as shown in Figure 5. Effects of Discount Factors We vary \u03b3 \u2208{0.95, 0.99, 0.995, 0.999, 1.0} to probe the sensitivity of GenDICE. Speci\ufb01cally, we compare to DualDICE, and \ufb01nd that GenDICE is stable, while DualDICE becomes unstable when the \u03b3 becomes large, as shown in Figure 6. GenDICE is also more general than DualDICE, as it can be applied to both the average and discounted reward cases. Effects of Ratio Constraint In Section 3, we highlighted the importance of the ratio constraint. Here we investigate the trivial solution issue without the constraint. The results in Figure 6 demonstrate the necessity of adding the constraint penalty, since a trivial solution prevents an accurate corrector from being recovered (green line in left two \ufb01gures). 9 \fPublished as a conference paper at ICLR 2020 Figure 5: Results of ablation study with different learning rates and activation functions. The plots show the log MSE of estimated average per-step reward over training and different behavior policies. Figure 6: Results of ablation study with constraint penalty and discount factors. The left two \ufb01gures show the effect of ratio constraint on estimating average per-step reward. The right three \ufb01gures show the log MSE for average per-step reward over training and different discount factor \u03b3. -4.8 -4.6 -4.4 -4.2 -4.0 -3.8 Log KL-Divergence JS 2 Wasserstein Hellinger MMD KL -4.800 -4.775 -4.750 -4.725 -4.700 -4.675 -4.650 -4.625 -4.600 Log KL-Divergence 0.1 0.5 1.0 2.0 4.0 5.0 (a) (b) Figure 7: Results of ablation study with (a) different divergence and (b) weight of penalty \u03bb. The plots show the log KL-Divergence of OPR on Barabasi-Albert graph. Effects of the Choice of Divergences We empirically test the GenDICE with several other alternative divergences, e.g., Wasserstein-1 distance, Jensen-Shannon divergence, KL-divergence, Hellinger divergence, and MMD. To avoid the effects of other factors in the estimator, e.g., function parametrization, we focus on the of\ufb02ine PageRank task on BA graph with 100 nodes and 10k of\ufb02ine samples. All the experiments are evaluated with 20 random trials. To ensure the dual function to be 1-Lipchitz, we add the gradient penalty. Besides, we use a learned Gaussian kernel in MMD, similar to Li et al. (2017). As we can see in Figure 7(a), the GenDICE estimator is compatible with many different divergences. Most of the divergences, with appropriate extra techniques to handle the dif\ufb01culties in optimization and carefully tuning for extra parameters, can achieve similar performances, consistent with phenomena in the variants of GANs (Lucic et al., 2018). However, KL-divergence is an outlier, performing noticeably worse, which might be caused by the ill-behaved exp (\u00b7) in its conjugate function. The \u03c72divergence and JS-divergence are better, which achieve good performances with fewer parameters to be tuned. Effects of the Penalty Weight The results of different penalty weights \u03bb are illustrated in Figure 7(b). We vary the \u03bb \u2208[0.1, 5] with \u03c72-divergence. Within a large range of \u03bb, the performances of the proposed GenDICE are quite consistent, which justi\ufb01es Theorem 1. The penalty multiplies with \u03bb. Therefore, with \u03bb increases, the variance of the stochastic gradient estimator also increases, which explains the variance increasing in large \u03bb in Figure 7(b). In practice, \u03bb = 1 is a reasonable choice for general cases. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2001.06944v1",
+                    "title": "Nested-Wasserstein Self-Imitation Learning for Sequence Generation",
+                    "abstract": "Reinforcement learning (RL) has been widely studied for improving\nsequence-generation models. However, the conventional rewards used for RL\ntraining typically cannot capture sufficient semantic information and therefore\nrender model bias. Further, the sparse and delayed rewards make RL exploration\ninefficient. To alleviate these issues, we propose the concept of\nnested-Wasserstein distance for distributional semantic matching. To further\nexploit it, a novel nested-Wasserstein self-imitation learning framework is\ndeveloped, encouraging the model to exploit historical high-rewarded sequences\nfor enhanced exploration and better semantic matching. Our solution can be\nunderstood as approximately executing proximal policy optimization with\nWasserstein trust-regions. Experiments on a variety of unconditional and\nconditional sequence-generation tasks demonstrate the proposed approach\nconsistently leads to improved performance.",
+                    "authors": "Ruiyi Zhang, Changyou Chen, Zhe Gan, Zheng Wen, Wenlin Wang, Lawrence Carin",
+                    "published": "2020-01-20",
+                    "updated": "2020-01-20",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Sequence generation is an important research topic in machine learning, covering a wide range of applications, including machine translation [Bahdanau et al., 2015, Cho et al., 2014, Sutskever et al., 2014], image captioning [Anderson et al., 2017, Vinyals et al., 2015, Xu et al., 2015], and text summarization [Paulus et al., 2017, Rush et al., 2015]. Standard sequence generation follows an auto-regressive model design under maximum likelihood estimation (MLE) learning [Husz\u00e1r, 2015, Sutskever et al., 2014, Wiseman and Rush, 2016]. That is, models are trained to maximize the expected Proceedings of the 23rdInternational Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2020, Palermo, Italy. PMLR: Volume 108. Copyright 2020 by the author(s). log-likelihood of the next word conditioned on its preceding ground-truth partial sentence. However, when testing, the generated partial sequence is fed to the generator to draw the next token. Such a discrepancy between training and testing, commonly known as exposure bias, leads to accumulated approximation errors along the sequence-generation trajectory [Bengio et al., 2015, Ranzato et al., 2016]. To address exposure bias, reinforcement learning (RL) techniques have been introduced [Ranzato et al., 2016]. Unlike MLE, which only leverages training examples, RL can also exploit samples drawn from the current policy. Improvements are gained from reinforcing the training towards more-plausible generations, typically based on a user-speci\ufb01ed reward function [Ranzato et al., 2016, Yu et al., 2017]. However, the manually designed rewards often target speci\ufb01c desirable properties in sequence generation (e.g., matching n-gram overlap between generated sequences and ground-truth references), which unintentionally induces extra bias and is often criticized as a bad proxy for human evaluation [Wang et al., 2018a, Hu et al., 2019]. Concerns have also been raised w.r.t. e\ufb03cient exploration in sequence generation. In existing RL-based methods for sequence generation [Bahdanau et al., 2017, Ranzato et al., 2016, Rennie et al., 2016], all experiences are treated as equivalent. However, merely relying on policy samples to explore often leads to forgetting a high-reward trajectory, unless it can be re-sampled frequently [Liang et al., 2018]. This problem becomes severe in the sparse-reward setting in sequence generation, i.e., the reward is only available after the whole sentence is generated. Motivated by the above observations, we present a novel nested-Wasserstein Self-Imitation Learning (WSIL) framework for sequence generation. Speci\ufb01cally, we propose the nested-Wasserstein distance, a generalization of the Wasserstein distance, and exploit it to measure distance between the behavior policy and the arti\ufb01cial policy de\ufb01ned by the replay bu\ufb00er to encourage self-imitation. The nested-Wasserstein distance is well suited for distributional semantic matching bearXiv:2001.06944v1  [cs.CL]  20 Jan 2020 \fNested-Wasserstein Self-Imitation Learning for Sequence Generation tween two (sequence) distributions whose samples are still discrete distributions, as in the case of sequence generation. The proposed WSIL is inspired by and derived from the policy optimization with Wasserstein trust-regions [Zhang et al., 2018b]. It provides a novel reward function to match the generated sequences with the high-reward sequences in the replay bu\ufb00er, encouraging distributional semantic matching rather than simple n-gram overlapping. The main contributions of this paper are summarized as follows. (i) A novel nested-Wasserstein self-imitation learning framework is developed for sequence generation, exploiting historical good explorations for better future exploration. (ii) A novel nested-Wasserstein distance is introduced for sequence generation via distributional semantic matching, e\ufb00ectively alleviating the model training bias imposed by conventional rewards. (iii) Extensive empirical evaluation is performed on both unconditional and conditional text generation tasks, demonstrating consistent performance improvement over existing state-of-the-art approaches. 2 Background Sequence-generation model We consider the problem of discrete sequence generation, which learns to generate a sequence Y = (y1, . . . , yT ) \u2208Y of length T, possibly conditioned on context X. Here each yt is a token from vocabulary A. Pairs (X, Y ) are used for training a sequence-generation model. We are particularly interested in applications to text generation, where Y is a sentence and each yt is a word. Starting from the initial state s0, a recurrent neural network (RNN) produces a sequence of states (s1, . . . , sT ) given an input sequence-feature representation (e(y1), . . . , e(yT )), where e(\u00b7) denotes a word embedding mapping a token to its d-dimensional feature representation. The states are recursively updated with a function known as the cell: st = h\u03b8(st\u22121, e(yt)), where \u03b8 denotes the model parameters. Popular implementations include Long Short-Term Memory (LSTM) [Hochreiter and Schmidhuber, 1997] and the Gated Recurrent Unit (GRU) [Cho et al., 2014]. In order to generate sequence Y s from a (trained) model, one iteratively applies the following operations: ys t+1 \u223cMulti(softmax(g(st))) , (1) st = h(st\u22121, e(ys t )) , (2) where Multi(\u00b7) denotes a multinomial distribution. In conditional generation, s0 is initialized with Enc(X), where Enc(\u00b7) encodes the relevant information from the context [Bahdanau et al., 2017, Cho et al., 2014]. For unconditional generation, one typically draws s0 from a standard Gaussian distribution. Sequence generation as an RL problem Sequence generation can be considered as an RL problem with deterministic state transition and sparse reward. It can be formulated as a Markov decision process (MDP) M = \u27e8S, A, P, r\u27e9, where S is the state space, A is the action space, P is the deterministic environment dynamics and r(s, y) is a reward function. The policy \u03c0\u03b8, parameterized by \u03b8, maps each state s \u2208S to a probability distribution over A. The objective is to maximize the expected reward, de\ufb01ned as: J(\u03c0\u03b8) = EY \u223c\u03c0\u03b8 [r(Y )] = T X t=1 E(st,yt)\u223c\u03c0\u03b8 [r(st, yt)] , (3) where Y \u225c(s1, y1, \u00b7 \u00b7 \u00b7 , sT , yT ) is a trajectory from policy \u03c0\u03b8 with yt \u2208A, and r(Y ) represents the reward for a sentence Y , and r(st, yt) is the step-wise reward. RL seeks to learn an optimal policy, that maximizes the expected total reward J(\u03c0\u03b8). Optimal transport on discrete domains The optimal transport (OT) distance Wc(\u00b5, \u03bd) is a discrepancy score that measures the distance between two probability distributions \u00b5(\u00b7) and \u03bd(\u00b7) w.r.t. a cost function c(\u00b7, \u00b7). Speci\ufb01cally, we consider two discrete distributions \u00b5 \u225cPn i=1 ui\u03b4zi and \u03bd \u225cPm j=1 vj\u03b4z\u2032 j with \u03b4z the Dirac delta function centered on z. The weight vectors u = {ui}n i=1 \u2208\u2206n and v = {vj}m j=1 \u2208\u2206m respectively belong to the n and m-dimensional simplex, i.e., Pn i=1 ui = Pm j=1 vj = 1. Accordingly, Wasserstein distance is equivalent to solving the following minimization problem: Wc(\u00b5, \u03bd) = min T\u2208\u0393(\u00b5,\u03bd) m X i=1 n X j=1 Tij \u00b7 c(zi, z\u2032 j) = min T\u2208\u0393(\u00b5,\u03bd) \u27e8T, C\u27e9, (4) where Pn j=1 Tij = 1 m and Pm i=1 Tij = 1 n are the constraints, \u27e8\u00b7, \u00b7\u27e9represents the Frobenius dot-product, and C is the cost matrix de\ufb01ned by Cij = c(zi, z\u2032 j). Intuitively, the OT distance is the minimal cost of transporting mass from \u00b5 to \u03bd. 3 Distributional Semantic Matching We \ufb01rst consider evaluating the sentence from syntactic and semantic perspectives. Conventional metric rewards (e.g., BLEU) can capture the syntactic structure better, where the exact matching of words (or short phases) to the reference sequences is encouraged, which induces strong bias in many cases. As such, we focus on the semantic matching and propose the nestedWasserstein distance, which de\ufb01nes the distance between two sequence distributions. Nested-Wasserstein distance provides a natural way to manifest semantic matching compared with the conventional rewards used in existing RL-based sequence models. Alternatively, we can train a discriminator to learn the reward model, but empirically it only rewards high-quality generations, even though they may be characterized by mode \fRuiyi Zhang1 Changyou Chen2 Zhe Gan3 Zheng Wen4 Wenlin Wang1 Lawrence Carin1  Reference:             There are six freshmen playing football . Candidate 1 (C1):   There are six freshmen reading papers . Candidate 2 (C2):    Six freshmen are playing soccer . BLEU ROUGE-L CIDEr Naive Wasserstein C1 36.8 50.0 163.7 84.1 76.3 C2 0.0 35.8 55.9 42.5 80.1 Table 1: Comparison of di\ufb00erent rewards in terms of the sequence-level (higher is better). The top \ufb01gure illustrates the Wasserstein reward of comparing two candidate sentences with a reference sentence, which will automatically match semantically similar words. Dominant edges are shown in dark blue, determined by the optimal transport matrix T. collapse [He et al., 2019]. This undermines diversity, an important aspect in evaluation. To better understand the issue, consider the example on sequence matching in Table 1. One can also use a naive way of semantic matching, i.e., measuring a distance between average word embeddings. It is clear that while the \ufb01rst candidate sentence has a similar syntactic structure to the reference, the second candidate sentence is more semantically consistent with the reference. However, popular hard-matching metrics [Papineni et al., 2002, Vedantam et al., 2015] and the naive method consistently indicate the \ufb01rst candidate is a better match to the reference. The above contradiction can be alleviated if the reward metric is more semantic-aware. So motivated, the remainder of this section is devoted to a discussion of design and implementation of Wasserstein rewards. The general idea is to match the semantic features via minimizing the Wasserstein distance between hypothesis sentences and their references in the semantic space. A nested version of the Wasserstein distance arises when integrating the distributional semantic matching into the objective of sequence distribution matching. De\ufb01nition 1 (Wasserstein Distance between Sequence Pairs) Consider sequence Y = (y1, . . . , yT ) as a discrete distribution pY = 1 T P t \u03b4e(yt) in the semantic space, with the length-normalized point mass placed at the word embedding, i.e., zt = e(yt) of each token yt from the sequence Y . Given a hypothesis sequence Y w.r.t. a reference sequence Y \u2032, we de\ufb01ne the Wasserstein distance as Wc(pY , pY \u2032) \u225cminT\u27e8T, C\u27e9 between pY and pY \u2032 with cost c(z, z\u2032). When the cosine distance ccos(z, z\u2032) = 1 \u2212 z\u22baz\u2032 \u2225z\u22252\u2225z\u2032\u22252 is used as our cost, we de\ufb01ne the Wasserstein reward as rs(Y, Y \u2032) \u225c \u27e8T\u2217, 1 \u2212C\u27e9, where T\u2217is the optimal transport matrix. Word-level Sequence-level Distribution-level (Y, Wc) <latexit sha1_base64=\"kihwQJy8uvgG1ufWusiVIeoNki4=\">A AB+3icbVDLSsNAFL3xWesr1qWbwSJUkJLUgi4LblxWsA9pQ5hMJ+3QyYOZiVhCfsWNC0Xc+iPu/BsnbRbaemDgcM693DPHizmTyrK+jb X1jc2t7dJOeXdv/+DQPKp0ZQIQjsk4pHoe1hSzkLaUx2o8FxYHac+b3uR+75EKyaLwXs1i6gR4HDKfEay05JqV2jDAakIwTx+yC9R zyblrVq26NQdaJXZBqlCg7Zpfw1FEkoCGinAs5cC2YuWkWChGOM3Kw0TSGJMpHtOBpiEOqHTSefYMnWlhPxI6BcqNFd/b6Q4kHIWeHoy DyqXvVz8zxskyr92UhbGiaIhWRzyE45UhPIi0IgJShSfaYKJYDorIhMsMFG6rIuwV7+8irpNur2Zb1x16y2mkUdJTiBU6iBDVfQgltoQ wcIPMEzvMKbkRkvxrvxsRhdM4qdY/gD4/MHxkqTkg=</latexit> (PY , Wnc) <latexit sha1_base64=\"g7CsEuJzg6A6KsEVQ9W3AUQF3lk=\">A ACDXicbVDLSsNAFJ3UV62vqEs3g1WoICWpBV0W3LisYB/ShDCZTNqhk0yYmQgl5Afc+CtuXCji1r07/8ZJ7EKrF4Y5nHsv95zjJ4xKZV mfRmVpeWV1rbpe29jc2t4xd/f6kqcCkx7mjIuhjyRhNCY9RUjw0QFPmMDPzpZdEf3BEhKY9v1CwhboTGMQ0pRkpTnUcCKkJr6fdXM vc3zOAjmL9Adv81M48LIY5yeWbeaVlnwL7DnoA7m1fXMDyfgOI1IrDBDUo5sK1FuhoSimJG85qSJAhP0ZiMNIxRKSblW5yeKyZAIZc 6BcrWLI/NzIUyUKjniyky8VeQf7XG6UqvHAzGiepItpWeShMGVQcFtHAgAqCFZtpgLCgWivEyQVjrAmg7BXrT8F/RbTfus2bpu1zvte RxVcAOQPY4Bx0wBXogh7A4B48gmfwYjwYT8ar8fY9WjHmO/vgVxnvXzs2m5o=</latexit> (A, ccos) <latexit sha1_base64=\"t3tjJvbaDbOdcVOM9hp4Wk0LMPU=\">A ACAHicbVDLSsNAFJ34rPUVdeHCzWARKkhJakGXFTcuK9gHNCFMptN26GQmzEyErLxV9y4UMStn+HOv3HSZqGtBy4czrmXe+8JY0aVdp xva2V1bX1js7RV3t7Z3du3Dw47SiQSkzYWTMheiBRhlJO2pqRXiwJikJGuHkNve7j0QqKviDnsbEj9CI0yHFSBspsI+rXoT0GCOW3mQ XEAeph4XKzgO74tScGeAycQtSAQVagf3lDQROIsI1ZkipvuvE2k+R1BQzkpW9RJEY4Qkakb6hHEVE+ensgQyeGWUAh0Ka4hrO1N8TKYqU mkah6cyvVYteLv7n9RM9vPZTyuNE47ni4YJg1rAPA04oJgzaGICypuRXiMZIa5NZ2YTgLr68TDr1mntZq983Ks1GEUcJnIBTUAUu AJNcAdaoA0wyMAzeAVv1pP1Yr1bH/PWFauYOQJ/YH3+AOw3le4=</latexit> Figure 1: Illustration of nested-Wasserstein distance (Wnc) over distributions of sequences (PY ), showing how the distance is de\ufb01ned in a nested manner to measure distance of sequence distributions. ccos is the word ground metric; Wc is the sequence ground metric. Nested-Wasserstein distance Our ultimate goal is to measure distance between two policy distributions instead of sequence pairs. Given two sets of sequences from two policies, one aims to incorporate the semantic information between sequences into the distance measure. To this end, we propose the nested-Wasserstein distance in De\ufb01nition 2. Figure 1 illustrates the nestedWasserstein, considering both wordand sequence-level matching with Wasserstein distance. De\ufb01nition 2 (Nested-Wasserstein Distance) Consider two sets of sequences Y = {Yi}K i=1 and Y \u2032 = {Y \u2032 j }K\u2032 j=1 drawn from two sequence distributions PY and PY \u2032, where K and K\u2032 are the number of sequences in Y and Y \u2032. The nested-Wasserstein distance, denoted as Wnc(PY , PY \u2032), is a metric measuring the distance between PY and PY \u2032 de\ufb01ned in a nested manner: Wnc(PY , PY \u2032) \u225cmin T s K X i=1 K\u2032 X j=1 T s ijWc(pYi, pY \u2032 j ) , (5) where T \u2032 ij \u22650 satis\ufb01es P i T s ij = 1 K and P j T s ij = 1 K\u2032 ; and Wc(\u00b7, \u00b7) denotes the c-Wasserstein distance de\ufb01ned in (4). Remark 1 The word \u201cnested\u201d comes from the de\ufb01nition in (5), which essentially consists of two nested levels of Wasserstein distances. The proposed nestedWasserstein distance brings in the semantic information via the distance measure Wc in the \ufb01rst level distance. Note that we have omitted the expectation over samples in (5) for simplicity, as we essentially use a single set of samples to approximate Wnc(\u00b7, \u00b7) in algorithms. Sample-based estimation of nested-Wasserstein distance Computing the exact Wasserstein distance is computationally intractable [Arjovsky et al., 2017, Genevay et al., 2018, Salimans et al., 2018], let alone the proposed nested-Wasserstein distance. Fortunately, we can employ the recently proposed IPOT algorithm [Xie et al., 2018] to obtain an e\ufb03cient approximation. Speci\ufb01cally, IPOT considers the following proximal gradient descent to solve the optimal transport matrix T via iterative optimization, i.e., T(t+1) = arg minT\u2208\u03a0(\u00b5,\u03bd) \b \u27e8T, C\u27e9+ \u03b3 \u00b7 DKL(T, T(t)) \t , where \fNested-Wasserstein Self-Imitation Learning for Sequence Generation 1/\u03b3 > 0 is the generalized step size and the generalized KL-divergence DKL(T, T(t)) = P i,j Tij log Tij T(t) ij \u2212P i,j Tij + P i,j T(t) ij is used as the proximity metric. Standard Sinkhorn iterations [Cuturi, 2013] are used to solve the above sub-problem. The IPOT was designed to approximately calculate the standard Wasserstein distance. Here we extend it to calculate the nested-Wasserstein distance by applying IPOT twice in a nested manner, i.e., in the sequence and distribution levels, respectively. The full approach of IPOT is summarized as Algorithm 2 in Appendix B. 4 Nested-Wasserstein Self-Imitation Learning Purely adopting the nested-Wasserstein distance as the reward in a standard policy-gradient method is not e\ufb00ective, because the syntactic information is missing. Speci\ufb01cally, we consider sequences generated from a conditional behavior policy \u03c0\u03b8,X, parameterized by \u03b8 with the conditional variable X. For example, in image captioning, each sequence is generated conditioned on a given image. For unconditional generation, the conditional variable is empty. Instead of combining the rewards with di\ufb00erent weights [Liu et al., 2017, Pasunuru et al., 2017], we present the nested-Wasserstein Self-Imitation Learning (WSIL) framework, which provides a novel way of leveraging both syntactic (metric) and semantic (Wasserstein) information. The overall idea of the proposed nested-Wasserstein selfimitation learning is to de\ufb01ne a Wasserstein trust-region between the current policy (a.k.a. behavior policy) and the arti\ufb01cial policy de\ufb01ned by the replay bu\ufb00er. Intuitively, the Wasserstein trust-region encourages the self-imitation of historical high-reward sequences, which provides semantic signals to guide training, in addition to the stabilizing e\ufb00ect from trust-region optimization. Furthermore, a replay bu\ufb00er is used to store highreward historical sequences, whose induced conditional policy is denoted \u03c0B,X. Our new objective function with a Wasserstein trust-region is de\ufb01ned as: J(\u03c0\u03b8) = EX\u223cpd{EY s\u223c\u03c0\u03b8,X [r(Y s)] \u2212\u03bb \u00b7 Wnc(\u03c0\u03b8,X, \u03c0B,X)} , (6) where Wnc is the nested-Wasserstein distance de\ufb01ned in De\ufb01nition 2, and r(\u00b7) can be a metric reward between Y s and the ground-truth references Y . With a little abuse of notation, but for conciseness, we use \u03c0\u03b8 to denote both the policy and the distribution over the sequences. Distinct from classic trust-region policy optimization, which de\ufb01nes the trust region based on KL-divergence [Schulman et al., 2015], WSIL de\ufb01nes the trust region based on the nested-Wasserstein Generator Wasserstein Rewards Generated Sequences   Standard Rewards Sequences from  Replay Bu\ufb00er Replay  Bu\ufb00er High-Rewards Sequences Figure 2: Illustration of the proposed nestedWasserstein Self-Imitation Learning (WSIL) framework, where Wasserstein self-imitation rewards are de\ufb01ned to encourage the generator to imitate samples from the replay bu\ufb00er. The standard RL framework is given in the gray dotted box. distance between the behavior policy \u03c0\u03b8,X and the arti\ufb01cial policy \u03c0B,X. Note when K = K\u2032 = 1, the nested Wasserstein distance degenerates to the de\ufb01nition of Wasserstein distance between two sequences. Remark 2 Unconditional Generation: By considering samples (features) themselves as discrete distributions, we replace the mean square di\ufb00erence over features of sequence pairs, i.e., Euclidean norm, with the Wasserstein distance. Then for the distributions of sequences, we again adopt the Wasserstein distance as in WGAN [Arjovsky et al., 2017] but in the discrete domain. Thus, the Wasserstein distance is de\ufb01ned in a nested manner. Remark 3 Conditional Generation: We replace the exact matching of sequence pairs with metric rewards in RL training, with the Wasserstein distance. In this case, we are matching two conditional distributions with Wasserstein distance, instead of matching the generated sentence with all reference sentences by average. This is a more suitable way as a generated sentence does not necessarily need to match all the references. For simplicity, we sometimes omit the \ufb01rst expectation EX\u223cpd. With the proposed nested Wasserstein distance, we propose the Wasserstein self-imitation scheme in (6), as illustrated in Figure 2. We seek to use historical highreward sequences to de\ufb01ne a \u201cself-imitation\u201d reward function, which is then combined with the original reward function to update the generator with policy gradient methods. Intuitively, higher self-imitation rewards are achieved when the generated sequences are close to historical high-reward sequences. Thus the generator is guided to perform self imitation and we call this method indirect nested-Wasserstein self-imitation learning (WSIL-I). The word \u201cindirect\u201d comes from the mechanism that historical sequences interact with the policy indirectly via the self-imitation reward. WSIL-I incorporates a self-imitation reward, denoted as rs(Y s, Y b), into the objective function. Here Y b \fRuiyi Zhang1 Changyou Chen2 Zhe Gan3 Zheng Wen4 Wenlin Wang1 Lawrence Carin1 denotes a sample from the replay bu\ufb00er and Y s denotes a sample from the current policy. To this end, we replace the Wasserstein distance Wc in the nestedWasserstein distance with rs(Y s, Y b) in the general objective (6). Speci\ufb01cally, we de\ufb01ne the two sets of sample sequences from \u03c0\u03b8,X and \u03c0B,X to be Y s \u225c{Y s i }K i=1 and Y b \u225c{Y b j }K\u2032 j=1, with sizes of K and K\u2032, respectively. Here Y s i \u223c\u03c0\u03b8,X and Y b j \u223c\u03c0B,X, \u2200j. {Y s i }K i=1 and Y b will be used in calculating the nested-Wasserstein distance. Let rns(Y s i , Y b) \u225cP j T s ijrs(Y s i , Y b j ) be the nested-Wasserstein reward, with Ts = {T s ij} the optimal weights in distribution-level. Based on (6), the objective of WSIL-I is adapted to be: JI(\u03c0\u03b8) \u225cEX\u223cpdEY s\u223c\u03c0\u03b8,X \u0002 r(Y s) + \u03bbrns(Y s, Y b) \u0003 , (7) where r is the original RL reward; rns is the nestedWasserstein reward. Since not all historically explored samples are helpful for updating the current policy, we only consider a subset of the high-reward sequences when performing self-imitation. Using K trajectories sampled i.i.d. from \u03c0\u03b8 and introducing a baseline b, the gradient estimate of WSIL-I is expressed as: \u2207\u03b8JI(\u03c0\u03b8) \u2248\u2212 K X k=1 [(r(Y s k ) \u2212b)\u2207\u03b8 log \u03c0\u03b8(Y s k ) + \u03bbrns(Y s k , Y b)\u2207\u03b8 log \u03c0\u03b8(Y s k )] . (8) In practice, I \u0002 r(Y b) > r(Y s) \u0003 will be combined with the nested-Wasserstein rewards, where I(\u00b7) = 1 if the condition is satis\ufb01ed, and 0 otherwise; b is the baseline to stabilize training. If the reward of a historical highreward sequence is greater than the current one (i.e., r(Y b) > r(Y s)), the generator learns to imitate this high-reward sequence. Otherwise, the update based on the historical sequence is not performed due to the I(\u00b7) operator. This encourages the agent to only imitate its good historical explorations. We have also developed another way to implement (direct) WSIL (WSIL-D) as discussed in the Appendix A. Algorithm 1 describes the general implementation procedure of the WSIL. Exploration E\ufb03ciency The exploration space of MLE is the examples in the training set [Tan et al., 2018], i.e., no exploration is performed in supervised training. In contrast, standard policy optimization [Ranzato et al., 2016] basically allows the whole exploration space. However, the exploration may become ine\ufb03cient since it may be too \ufb02exible, and some good sequences observed in history tend to be less explored and imitated due to the sparse rewards. Our proposed WSIL aims to provide more e\ufb03cient and systematic exploration. It allows the whole-space exploration, but re-weights the exploration space to focus more on the exploration that may provide better performance with the Wasserstein trust-region. Algorithm 1 Nested-Wasserstein Self-Imitation. Require: Generator policy \u03c0\u03b8; a sequence dataset D = {Y1...T }N 1 ; a possibly empty condition X = {X}N 1 . Initialize \u03c0\u03b8 and replay bu\ufb00er B. Pretrain generator \u03c0\u03b8 with MLE. repeat Generate K sequences Y s = {Y s k }K k=1, where Y s k \u223c\u03c0\u03b8. Update replay bu\ufb00er B using Y s. if Self-Imitation then Sample K\u2032 sequences Y b = {Y b j }K\u2032 j=1, where Y b j \u223c\u03c0B. Estimate the OT matrix T and Ts via IPOT Compute rns(Y s k , Y b) and update \u03c0\u03b8 with (8). else Update the generator \u03c0\u03b8 with (3) using Y s. end if until Algorithm converges MLE RL WSIL Figure 3: Exploration space of di\ufb00erent methods. Circle: ground truth; Star: high-reward sequences. Increasing Self-Imitation According to the theory of Wasserstein gradient \ufb02ows [Villani, 2008], 1/\u03bb can be interpreted as a generalized decaying learning rate. With more explorations, \u03bb becomes larger, and the algorithm should focus more on the self-imitation learning, providing a guideline to balance the standard RL training and self-imitation learning. More details are provided in Appendix B. Practically, nested-Wasserstein provides weak supervision focusing on semantic matching, which is reasonable since the historical high-reward sequences contain some noises. 5 Related Work Optimal transport Kusner et al. [2015] proposed the word mover\u2019s distance (WMD) and \ufb01rst applied optimal transport (OT) to NLP; OT has also been employed to improve topic modeling [Huang et al., 2016]. The transportation cost is usually de\ufb01ned as Euclidean distance, and OT distance is approximated by solving a Kantorovich-Rubinstein dual [Gulrajani et al., 2017] or a less-accurate lower bound [Kusner et al., 2015]. Yurochkin et al. [2019] proposed a hierarchical OT representation for document, but the hierarchy was in wordand topic-level based on the WMD. Our work considers nested-Wasserstein distance, presenting an e\ufb03cient IPOT-based implementation for OT distance approximation [Xie et al., 2018], successfully using it to guide sequence generation. \fNested-Wasserstein Self-Imitation Learning for Sequence Generation Self-Imitation Learning Experience replay has been widely considered in RL. Deterministic policy gradient [Silver et al., 2014, Lillicrap et al., 2016] performs experience replay, but is limited to continuous control. Actor-critic approaches [Konda and Tsitsiklis, 2000] can also utilize a replay bu\ufb00er to improve performance. Prioritized experience replay [Schaul et al., 2015] samples trajectories based on the time-di\ufb00erence error, and we adopt it in our implementation. These approaches indiscriminately bu\ufb00er all experiences, while the approach proposed here only bu\ufb00ers high-reward experience. Further, episodic control [Lengyel and Dayan, 2008] can be regarded as an extreme way of exploiting past experience, trying to reproduce its best past decisions, but retrieving states leads to poor e\ufb03ciency and generalization in testing. Self-imitation learning was \ufb01rst applied in Atari games and Mujoco [Oh et al., 2018, Gangwani et al., 2018], reporting performance improvement w.r.t. sparse rewards. Compared with that work, our solution considers a novel self-imitation learning scheme in the context of sequence generation. RL for Sequence Generation RL techniques have been explored in detail for sequence generation. For example, a Seq2Seq model can be trained by directly optimizing BLEU/ROUGE scores via policy gradient [Ranzato et al., 2016, Bahdanau et al., 2017]. Furthermore, Rennie et al. [2016] baselines the actor with the reward of a greedy-decoding sequence for the REINFORCE method. Model-based RL and hierarchical RL have also been studied for sequence generation [Zhang et al., 2018a, Huang et al., 2019]. Further, a learned discriminator (or, critic) can also be used to provide sequencelevel guidance. By constructing di\ufb00erent objectives, previous work [Yu et al., 2017, Lin et al., 2017, Guo et al., 2017, Fedus et al., 2018] combines the policygradient algorithm with the original GAN training procedure. However, mode-collapse problems make the training of these methods challenging. By contrast, we propose the use of self-imitation learning, and maintain a replay bu\ufb00er to exploit past good explorations. 6 Experiments We evaluate the proposed method on both unconditional and conditional text-generation tasks, considering standard benchmark datasets. Our approach achieves state-of-the-art results on unconditional text generation and video captioning. We also observed improved performance on image captioning though relying on much simpler features compared to prior state-ofthe-art methods. We also perform ablation studies to understand the improvements brought by self-imitation and Wasserstein rewards individually. Details of the datasets, experimental setup and model architectures are provided in Appendix C. Figure 4: Demonstration of nested-Wasserstein distance in word-level (left) and sentence-level (right). Figure 5: An example of image captioning. The right generated sentence is better but given a lower CIDEr. Implementation Details A few key techniques are required for successful model training. (i) The reward from a greedy-decoding sentence is used as the baseline [Rennie et al., 2016] in conditional text generation; in unconditional text generation, a constant baseline is used. (ii) A single large replay bu\ufb00er is maintained for unconditional generation, and multiple replay bu\ufb00ers are maintained for di\ufb00erent conditions in conditional generation. (iii) For each pair of sentences, the shorter one should be padded to the same length as the longer one for a balanced optimal transport, which is a key implementation technique. Demonstration of nested-Wasserstein Figure 4 shows the optimal matching in word-level (T) and sentence-level (Ts). It is interesting to see that all similar words (e.g., bike and cycle) are matched with each other (higher weights), which cannot be achieved via exact hard-matching metrics. At the distributionlevel, we show an example in captioning tasks, where we have \ufb01ve reference and hypothesis sentences. Traditional methods will match a hypothesis sentence to each of the references and average over them; while our method performs distributional semantic matching, i.e., only matching similar references instead of all of them. For example, the third hypothesis is almost matched with the \ufb01fth reference, because they are more similar. This is reasonable, because the references are usually very di\ufb00erent, and equivalently matching with all of them is confusing for the generator. As shown in Figure 5, CIDEr focuses more on the locality \ufb02uency and equivalent matching with all references, while nestedWasserstein performs distributional semantic matching. More examples are provided in the Appendix. \fRuiyi Zhang1 Changyou Chen2 Zhe Gan3 Zheng Wen4 Wenlin Wang1 Lawrence Carin1 Method Test-BLEU-2 3 4 5 Self-BLEU-2 3 4 MLE [Caccia et al., 2018] 0.902 0.706 0.470 0.392 0.787 0.646 0.485 SeqGAN [Yu et al., 2017] 0.820 0.604 0.361 0.211 0.807 0.577 0.278 RankGAN [Lin et al., 2017] 0.852 0.637 0.389 0.248 0.822 0.592 0.230 TextGAN [Zhang et al., 2017] 0.910 0.728 0.484 0.306 0.806 0.548 0.217 FMGAN [Chen et al., 2018] 0.911 0.782 0.584 0.382 0.834 0.643 0.405 LeakGAN [Guo et al., 2017] 0.922 0.797 0.602 0.416 0.912 0.825 0.689 WSIL-D (ours) 0.917 0.774 0.576 0.393 0.797 0.569 0.284 WSIL-I (ours) 0.922 0.778 0.576 0.396 0.813 0.600 0.326 Table 2: Test-BLEU (\u2191) and Self-BLEU (\u2193) scores on Image COCO. Method Test-BLEU-2 3 4 5 Self-BLEU-2 3 4 MLE [Caccia et al., 2018] 0.905 0.701 0.464 0.278 0.764 0.522 0.295 SeqGAN [Yu et al., 2017] 0.630 0.354 0.164 0.087 0.728 0.411 0.139 RankGAN [Lin et al., 2017] 0.723 0.440 0.210 0.107 0.672 0.346 0.119 TextGAN [Zhang et al., 2017] 0.777 0.529 0.305 0.161 0.806 0.662 0.448 FMGAN [Chen et al., 2018] 0.913 0.751 0.512 0.315 0.830 0.682 0.427 LeakGAN [Guo et al., 2017] 0.923 0.757 0.546 0.335 0.837 0.683 0.513 SIL-D (ours) 0.875 0.634 0.401 0.243 0.724 0.466 0.256 SIL-I (ours) 0.869 0.633 0.399 0.242 0.710 0.455 0.263 WSIL-D (ours) 0.931 0.736 0.503 0.317 0.795 0.553 0.299 WSIL-I (ours) 0.926 0.726 0.492 0.307 0.815 0.595 0.380 Table 3: Test-BLEU (\u2191) and Self-BLEU (\u2193) scores on EMNLP2017 WMT News. 6.1 Unconditional Text Generation We compare our approach with a number of related RL-based GAN models for unconditional text generation [Guo et al., 2017, Lin et al., 2017, Yu et al., 2017, Zhang et al., 2017]. Our implementation is developed based on the LeakGAN model, by incorporating Wasserstein self-imitation learning. All baseline experiments are performed on the texygen platform [Zhu et al., 2018]. The corpus-level BLEU score is employed to evaluate the generated sentences. Speci\ufb01cally, we follow the strategy in Yu et al. [2017], Guo et al. [2017] and adopt the BLEU score, referenced by test set (testBLEU) and themselves (self-BLEU) to evaluate the quality of generated samples. Test-BLEU evaluates the goodness of generated samples, and self-BLEU measures their diversity. The BLEU scores for 1000 generated sentences are averaged to obtain the \ufb01nal score for each model. A good generator should achieve both a high test-BLEU score and a low self-BLEU score. Following previous work [Guo et al., 2017], we test the proposed method on the short and long text generation on Image COCO and EMNLP2017 WMT News datasets. The BLEU scores with di\ufb00erent methods are provided in Tables 2 and 3. Analysis Compared with other methods, LeakGAN, WSIL-D and WSIL-I achieve comparable test-BLEU scores, demonstrating high-quality generated sentences. However, LeakGAN tends to over-\ufb01t on training data, leading to much higher (worse) self-BLEU scores. Our proposed methods, by contrast, show good diversity of the generated text with lower self-BLEU scores. Other baselines obtain both low self-BLEU and test-BLEU scores, leading to more random generations. Ablation Study We conduct ablation studies on EMNLP2017 WMT news to investigate the improvements brought by each part of WSIL. First, we test the bene\ufb01ts of using two types of self-imitation schemes. We compare RL training with (i) self-imitation (SIL-D and SIL-I), where only a replay bu\ufb00er and conventional matching (features extracted from a neural network) are employed; and (ii) Wasserstein self-imitation (WSIL-D and WSIL-I). Results are shown in Table 3. We observe that the self-imitation strategy, with speci\ufb01c replay bu\ufb00er construction, can alleviate the discrepancies between reward model bias and conventional rewards (e.g., self-BLEU). Without Wasserstein rewards, we achieve lower self-BLEU at the sacri\ufb01ce of test-BLEU. When combining with Wasserstein rewards, WSIL-D and WSIL-I show superior performance relative to the baselines. The random generated samples in Appendix D and human evaluations further validate this. Sweep the Temperature To better evaluate the proposed method, we follow Caccia et al. [2018] to evaluate the trade-o\ufb00between the quality and diversity. We use the F1-BLEU score as a metric, which considers both quality and diversity, and is de\ufb01ned as the geometry average of BLEU score and 1\u2212Self-BLEU: F1-BLEU = 2 \u00d7 BLEU \u00d7 (1-Self-BLEU) BLEU + (1-Self-BLEU) . (9) Figure 6 indicates that WSIL is consistently better than the MLE model on the F1-BLEU-4 score. \fNested-Wasserstein Self-Imitation Learning for Sequence Generation Method BLEU-4 METEOR ROUGE-L CIDEr ED-LG [Yao et al., 2015] 35.2 25.2 SA-LSTM [Xu et al., 2016] 36.6 25.9 SCST [Pasunuru et al., 2017] 40.5 28.4 61.4 51.7 MBP [Wang et al., 2018b] 41.3 28.7 61.7 48.0 Our Implementations MLE 39.2 27.8 59.8 46.6 MIXER [Ranzato et al., 2016] 40.2 27.9 60.8 50.3 SCST [Rennie et al., 2016] 40.7 27.9 61.6 51.3 WSIL-D 42.5 29.0 62.4 52.1 WSIL-I 41.6 28.4 62.0 52.2 Table 4: Video captioning results on MSR-VTT. Method BLEU-4 METEOR ROUGE-L CIDEr S & T [Vinyals et al., 2015] 27.7 23.7 85.5 OT [Chen et al., 2019] 31.0 24.6 94.7 Adaptive [Lu et al., 2017] 33.2 26.6 108.5 TD [Anderson et al., 2017] 33.3 26.3 55.3 111.4 Our Implementations MLE 28.8 24.4 52.0 91.3 MIXER [Ranzato et al., 2016] 30.8 24.7 52.9 101.2 SCST [Rennie et al., 2016] 32.1 25.4 53.9 105.5 WSIL-D 31.8 25.7 54.0 107.4 WSIL-I 32.0 25.6 53.9 107.6 Table 5: Image captioning results on COCO. Methods MLE LeakGAN SIL-D SIL-I Human scores 2.97\u00b10.05 2.63\u00b10.05 2.54\u00b10.05 2.55\u00b10.05 Methods Real WSIL-D WSIL-I Human scores 4.11\u00b10.04 3.49\u00b10.05 3.41\u00b10.05 Table 6: Results of human evaluation. 1.0 1.1 1.2 1.3 1.4 1.5 Reverse Temprature 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 F1-BLEU4 MLE WSIL-D WSIL-I 0 5 10 15 20 25 30 Epochs 0.47 0.48 0.49 0.50 0.51 0.52 0.53 CIDEr on Validation Set SCST WSIL-D WSIL-I Figure 6: F1-BLEU-4 on sweeping temperature on unconditional generation; CIDEr scores of Video Captioning on validation set. Human Evaluation Simply relying on the above metrics is not su\ufb03cient to evaluate the proposed method [Caccia et al., 2018]. Following previous work [Guo et al., 2017], we performed additional human evaluation on the EMNNLP2017 WMT News dataset using Amazon Mechnical Turk. We require all the workers to be native English speakers, with approval rate higher than 95% and at least 100 assignments completed. Previous work has shown higher scores of LeakGAN compared with other baselines [Guo et al., 2017], therefore we mainly focus on the comparison of our methods with LeakGAN. We randomly sampled 200 sentences from each model, and asked 5 di\ufb00erent workers to score each sentence on a scale of 1 to 5, considering its readability and meaning. Results are shown in Table 6, which indicates better performance of the proposed WSIL. 6.2 Conditional Text Generation Video Captioning We conduct experiments on the MSR-VTT dataset [Xu et al., 2016] for video captioning. The MSR-VTT is a large-scale video dataset, consisting of 20 video categories. The dataset was split into 6513 and 3487 clips in the training and testing sets. Each video is annotated with about 20 captions. For each video, we sample at 3 fps and extract Inceptionv4 [Szegedy et al., 2017] features from these sampled frames. We report BLEU-4 [Papineni et al., 2002], CIDEr [Vedantam et al., 2015], and METEOR [Banerjee and Lavie, 2005] scores. Results are summarized in Table 4. Consistent improvements are observed with the WSIL framework. WSIL-D performs slightly better than WSIL-I, both yielding much higher optimized CIDEr and METEOR scores than SCST. This indicates that Wasserstein self-imitation can improve the semantic matching between generated sentences and their references, while achieving reasonable exactmatching-based metric scores. Image Captioning We consider image captioning using the COCO dataset [Lin et al., 2014], which contains 123,287 images in total, each of which is annotated with at least 5 captions. Following with Karpathy\u2019s split [Karpathy and Fei-Fei, 2015], 113,287 images are used for training and 5,000 images are used for validation and testing. We follow the implementation of the SCST approach [Rennie et al., 2016], and use extracted image tags [Gan et al., 2017] as image features (encoder). We report BLEU-k (k from 1 to 4) [Papineni et al., 2002], CIDEr [Vedantam et al., 2015], and METEOR [Banerjee and Lavie, 2005] scores. Results are summarized in Table 5. Compared with the MLE baseline, RL-based methods signi\ufb01cantly increase the overall performance under all evaluation metrics. We choose CIDEr as the optimizing metric, since it performs best [Rennie et al., 2016]. Our proposed WSIL shows improvement on most metrics compared with the SCST baseline. Examples of generated captions are provided in Appendix E. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1902.07239v1",
+                    "title": "Scalable Thompson Sampling via Optimal Transport",
+                    "abstract": "Thompson sampling (TS) is a class of algorithms for sequential\ndecision-making, which requires maintaining a posterior distribution over a\nmodel. However, calculating exact posterior distributions is intractable for\nall but the simplest models. Consequently, efficient computation of an\napproximate posterior distribution is a crucial problem for scalable TS with\ncomplex models, such as neural networks. In this paper, we use distribution\noptimization techniques to approximate the posterior distribution, solved via\nWasserstein gradient flows. Based on the framework, a principled\nparticle-optimization algorithm is developed for TS to approximate the\nposterior efficiently. Our approach is scalable and does not make explicit\ndistribution assumptions on posterior approximations. Extensive experiments on\nboth synthetic data and real large-scale data demonstrate the superior\nperformance of the proposed methods.",
+                    "authors": "Ruiyi Zhang, Zheng Wen, Changyou Chen, Lawrence Carin",
+                    "published": "2019-02-19",
+                    "updated": "2019-02-19",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction In many online sequential decision-making problems, such as contextual bandits [8] and reinforcement learning [24], an agent needs to learn to take a sequence of actions to maximize its expected cumulative reward, while repeatedly interacting with an unknown environment. Moreover, since in such problems the agent\u2019s actions affect both its rewards and its observations, it faces the well-known exploration-exploitation dilemma. Consequently, the exploration strategy is crucial for a learning algorithm: typically, under-exploration will make the algorithm stick at a sub-optimal strategy, while over-exploration tends to incur a huge exploration cost. Various exploration strategies have been proposed, including \u03f5-greedy (EG), Boltzmann exploration [23, 10], upper-con\ufb01dence bound (UCB) [1, 5] type exploration, and Thompson sampling (TS). Among them, TS [25, 22], which is also known as posterior sampling or probability matching, is a widely used exploration strategy with good practical performance [15, 11] and theoretical guarantees [21, 2, 22]. The vanilla TS is computationally ef\ufb01cient when there is a closed-form posterior, such as with Bernoulli or Gaussian rewards. For cases without a closed-form posterior, other variants of TS have also been developed [3, 11]. However, most such TS algorithms cannot be extended to cases with complex generalization models, such as neural networks, in a computationally ef\ufb01cient manner. In this paper, adopting ideas from Wasserstein-gradient-\ufb02ow literature, we propose a general particlebased distribution optimization framework for Thompson sampling. Our framework improves the recently proposed particle-based framework for TS [17]. Speci\ufb01cally, our framework employs a set of particles interacting with each other to approximate the posterior distribution (thus it is called particle-interactive Thompson sampling, or \u03c0-TS), while [17] treats particles independently. Speci\ufb01cally, we optimize the posterior distribution in TS based on the Wasserstein-gradient-\ufb02ow framework. In this setting, Bayesian sampling in TS becomes a convex optimization problem on the space of probability measures, thus the optimality of the learned distribution could be guaranteed. For tractability, the posterior distribution in TS is approximated by a set of particles (a.k.a.samples). We test our framework on a number of applications, \ufb01rst in a simulated dynamic scenario and then on large-scale real-world datasets. In all cases, our proposed particle-interactive Thompson sampling signi\ufb01cantly outperforms other baselines. Infer to Control: NIPS 2018 Workshop on Probabilistic Reinforcement Learning and Structured Control. arXiv:1902.07239v1  [stat.ML]  19 Feb 2019 \f2 Background 2.1 Contextual Bandits and Thompson Sampling We consider a contextual bandit characterized by a triple (X, A, P), where X \u2286\u211cd is a context (state) space with dimension d, A = {1, 2, . . . , K} is a \ufb01nite action space, and P encodes the reward distributions at all the context-action pairs. The agent is assumed to know X and A, but not P. The agent repeatedly interacts with the contextual bandit for T rounds. At each round t = 1, . . . , T, the agent \ufb01rst observes a context xt \u2208X, which is independently chosen by the environment. Then, the agent adaptively chooses an action at \u2208A, based on the current context xt and the agent\u2019s past observations. Finally, the agent observes and receives a reward rt, which is conditionally independently drawn from the reward distribution Pxt,at. The agent\u2019s objective is to learn to minimize its expected cumulative regret in the \ufb01rst T rounds, i.e., R(T) = PT t=1 E [maxa\u2208A[r(xt, a)] \u2212rt]. Many practical online decision-making problems that \ufb01t into the framework of contextual bandit have intractably large scale. Speci\ufb01cally, in such problems, at least one of X and A has unmanageably large cardinality (if it is discrete) or dimension (if it is continuous). One standard approach to develop scalable learning algorithms for such large-scale problems is to exploit generalization models. Speci\ufb01cally, in this paper, we assume that the learning agent has access to a generalization model m(x, a; \u03b8) for the mean reward function \u00af r(x, a), where \u03b8 is the model parameters. We assume that the generalization model m is \u201caccurate\u201d in the sense that there exists a speci\ufb01c model parameter vector \u03b8\u2217, with m(x, a; \u03b8\u2217) \u2248\u00af r(x, a), \u2200(x, a) \u2208X \u00d7 A. Note that m is a function of the contextaction pair (x, a) and the parameter vector \u03b8. One example of the above-mentioned generalization model is neural network and simpler generalization models, such as linear regression models and logistic regression models, can be viewed as special cases of it. Thompson sampling (TS) [25] is a widely used class of algorithms for sequential decision-making. For the contextual bandits with reward generalization, assuming the reward generalization is perfect, the vanilla version of TS proceeds as follows: Given a prior distribution p0(\u03b8) on the model parameters \u03b8 and the set of past observations Dt \u225c{(xi, ai, ri)}t\u22121 i=1, Thompson sampling [25] maintains a posterior distribution over \u03b8 as pt\u22121 \u225cp(\u03b8| Dt\u22121). Then, at each time t, it \ufb01rst samples a parameter vector \u02c6 \u03b8t from the current posterior pt\u22121; then, it chooses action at \u2208argmaxa m(xt, a; \u02c6 \u03b8t), and receives the reward rt; \ufb01nally it updates the posterior. The pseudocode is provided in Algorithm 2. 2.2 Wasserstein Gradient Flows Wasserstein gradient \ufb02ows (WGF) is a generalization of gradient \ufb02ows on Euclidean space (reviewed in Section B.2), by lifting the differential equation above onto the space of probability measures, denoted P(\u2126) with \u2126\u2282Rd. Formally, we \ufb01rst endow a Riemannian geometry [9] on P(\u2126). The geometry is characterized by the length between two elements (two distributions), de\ufb01ned by the second-order Wasserstein distance: W 2 2 (\u00b5, \u03bd) \u225cinf\u03b3 nR \u2126\u00d7\u2126\u2225\u03b8 \u2212\u03b8\u2032\u22252 2d\u03b3(\u03b8, \u03b8\u2032) : \u03b3 \u2208\u0393(\u00b5, \u03bd) o , where \u0393(\u00b5, \u03bd) is the set of joint distributions over (\u03b8, \u03b8\u2032) such that the two marginals equal \u00b5 and \u03bd, respectively. If \u00b5 is absolutely continuous w.r.t. the Lebesgue measure, there is a unique optimal transport plan from \u00b5 to \u03bd, i.e., a mapping T : Rd \u2192Rd pushing \u00b5 onto \u03bd satisfying T#\u00b5 = \u03bd. Here T#\u00b5 denotes the pushforward measure [27] of \u00b5. The Wasserstein distance thus can be equivalently reformulated as W 2 2 (\u00b5, \u03bd) \u225cinfT \bR \u2126\u2225\u03b8 \u2212T(\u03b8)\u22252 2d\u00b5(\u03b8) \t. Consider P(\u2126) with a Riemannian geometry endowed by the second-order Wasserstein metric. Let {\u00b5\u03c4}\u03c4\u2208[0,1] be an absolutely continuous curve in P(\u2126) with distance between \u00b5\u03c4 and \u00b5\u03c4+h measured by W 2 2 (\u00b5\u03c4, \u00b5\u03c4+h). We overload the de\ufb01nition of T to denote the underlying transformation from \u00b5\u03c4 to \u00b5\u03c4+h as \u03b8\u03c4+h = Th(\u03b8\u03c4). Motivated by the Euclidean-space case, if we de\ufb01ne v\u03c4(\u03b8) \u225climh\u21920 Th(\u03b8\u03c4 )\u2212\u03b8\u03c4 h as the velocity of the particle, a gradient \ufb02ow can be de\ufb01ned on P(\u2126) correspondingly in Lemma 1 [4]. Lemma 1 Let {\u00b5\u03c4}\u03c4\u2208[0,1] be an absolutely-continuous curve in P(\u2126) with \ufb01nite second-order moments. Then for a.e. \u03c4 \u2208[0, 1], the above vector \ufb01eld v\u03c4 de\ufb01nes a gradient \ufb02ow on P(\u2126) as \u2202\u03c4\u00b5\u03c4 + \u2207\u03b8 \u00b7 (v\u03c4 \u00b5\u03c4) = 0, where \u2207\u03b8 \u00b7 a \u225c\u2207\u22a4 \u03b8 a for a vector a. Function F is in the space of probability measures P(\u2126), mapping a probability measure \u00b5 to a real value, i.e., F : P(\u2126) \u2192R. Consequently, it can be shown that v\u03c4 in Lemma 1 has the form v\u03c4 = \u2212\u2207x \u03b4F \u03b4\u00b5\u03c4 (\u00b5\u03c4) [4], where \u03b4F \u03b4\u00b5\u03c4 is called the \ufb01rst variation of functional F at \u00b5\u03c4 [13]. Based on 2 \fthis, gradient \ufb02ows on P(\u2126) can be written in a form of partial differential equation (PDE) as \u2202\u03c4\u00b5\u03c4 = \u2212\u2207\u03b8 \u00b7 (v\u03c4 \u00b5\u03c4) = \u2207\u03b8 \u00b7 \u0012 \u00b5\u03c4\u2207\u03b8( \u03b4F \u03b4\u00b5\u03c4 (\u00b5\u03c4)) \u0013 . (1) 3 Thompson Sampling via Optimal Transport This section describes our proposed Particle-Interactive Thompson sampling (\u03c0-TS) framework. We \ufb01rst interpret Thompson sampling as a WGF problem, then propose an energy function to design a speci\ufb01c WGF, and \ufb01nally propose particle-approximation methods to solve the \u03c0-TS problem. The posterior distribution of \u03b8 in Thompson sampling is de\ufb01ned as pt\u22121 \u225cp(\u03b8| Dt\u22121) \u221deU(\u03b8), where the potential energy is de\ufb01ned as U(\u03b8) \u225clog p(D|\u03b8) + log p0(\u03b8) = t X i=1 \u0012 log p(ri| xi, ai, \u03b8) + 1 t log p0(\u03b8) \u0013 = N X i=1 Ui(\u03b8). (2) To apply WGFs for posterior approximation in Thompson sampling, a variational (posterior) distribution for \u03b8, denoted as \u00b5(\u03b8), is learned by solving an appropriate gradient-\ufb02ow problem. To make the stationary distribution of the WGF consistent with the target posterior distribution, we de\ufb01ne an energy functional characterizing the similarity between the current variational distribution and the true distribution pt induced by the rewards as: F(\u00b5) \u225c\u2212 Z U(\u03b8)\u00b5(\u03b8)d\u03b8 + Z \u00b5(\u03b8) log \u00b5(\u03b8)d\u03b8 = KL (\u00b5\u2225pt) . (3) The energy functional F(\u00b5) de\ufb01nes a landscape determined by the rewards, whose minimum is obtained at \u00b5 = pt. We investigate the discrete-gradient-\ufb02ow (DGF) method to solve (1). Algorithm 1 Particle-Interactive TS (\u03c0-TS) Require: D0 = \u2205; initialize particles \u03980 = {\u03b8i 0}M i=1; 1: for t = 1, 2, . . . , T do 2: Observe context xt 3: Draw \u02c6 \u03b8t uniformly from \u0398t 4: Select at \u2208arg maxa m(xt, a; \u02c6 \u03b8t) 5: Observe and receive reward rt 6: Dt+1 = Dt \u222a(xt, at, rt) 7: Update \u0398t+1, according to (5) 8: end for Discrete gradient \ufb02ows (DGFs) approximate (1) by discretizing the continuous curve \u00b5t into a piecewise linear curve, leading to an iterative optimization problem to solve the intermediate points denoted as {\u00b5h k}k, where k denotes the discrete points, and h is refered to as the stepsize parameter. The iterative optimization problem is also known as JordanKinderleher-Otto (JKO) scheme [14], where for iteration k, \u00b5(h) k+1 is obtained by solving the following optimization problem: \u00b5(h) k+1 = arg min \u00b5 KL (\u00b5\u2225p\u03b8) + W 2 2 (\u00b5, \u00b5(h) k ) 2h . (4) Following methods such as those in [12], we proposed to use particle approximation to approximate \u00b5 with M particles {\u03b8i}M i=1 as \u00b5(h) \u2248 1 M PM i=1 \u03b4\u03b8i, where \u03b4\u03b8k is a delta function with a spike at \u03b8k. Consequently, the evolution of distributions described by (1) can be approximated with gradient ascent on particles. Speci\ufb01cally, (4) can be decomposed as F1 \u225c\u2212E\u00b5[log p(\u03b8|D)] + \u03bb1E\u00b5[log \u00b5], F2 \u225c\u03bb2E\u00b5[log \u00b5] + 1 2hW 2 2 (\u00b5, \u00b5(h) k ). According to [16], the gradient of the \ufb01rst term can be easily approximated as, \u2202F1/\u2202\u03b8i k = PM j=1 h \u2212\u03ba(\u03b8j k, \u03b8i k)\u2207\u03b8j kU(\u03b8i k) + \u2207\u03b8j k\u03ba(\u03b8j k, \u03b8i k) i , where \u03ba is the kernel function, which typically is the RBF kernel de\ufb01ned as \u03ba(\u03b8, \u03b8\u2032) = exp(\u2212\u2225\u03b8 \u2212\u03b8\u2032\u22252 2/h). For the second term F2, we can solve the entropy-regularized Wasserstein distance by introducing Lagrangian multipliers as: \u2202F2/\u2202\u03b8i k \u2248\u2212 P j uivjcije \u2212 cij \u03bb2 \u2202\u03b8i k = P j 2uivj \u0010 cij \u03bb2 \u22121 \u0011 exp\u2212 cij \u03bb2 (\u03b8i k \u2212 \u03b8j k\u22121). where cij \u225c\u2225\u03b8i \u2212\u03b8j\u22252 2. Theoretically, we need to adaptively update Lagrangian multipliers {ui, vj} as well to ensure the constraints in (3). In practice, however, we use a \ufb01xed scaling factor \u03b3 to approximate uivj for the sake of simplicity. The entropy-regularized Wasserstein term F2 works as a complex force between particles in two ways: i) When cij \u03bb > 1, \u03b8i k is pulled close to previous particles {\u03b8j k\u22121}, with force proportional to ( cij \u03bb \u22121)e\u2212cij/\u03bb; ii) when \u03b8i is close enough to a previous particle \u03b8j k, i.e., cij \u03bb < 1, \u03b8i k is pushed away, preventing it from collapsing to \u03b8j k. Formally, in the k-th iteration, the particles are updated with: \u03b8i k+1 = \u03b8i k + h M M X j=1 \u0014 \u2212\u03ba(\u03b8j k, \u03b8i k)\u2207\u03b8i kU(\u03b8i k) + \u2207\u03b8j k\u03ba(\u03b8j k, \u03b8i k) + \u0012cij \u03bb2 \u22121 \u0013 exp \u2212 cij \u03bb2 (\u03b8i k \u2212\u03b8j k\u22121) \u0015 (5) 3 \fBy applying the methods above to solve the WGF for Thompson sampling, we arrive at the ParticleInteractive Thompson Sampling (\u03c0-TS) framework. The pseudocode of \u03c0-TS is described in Algorithm 1. In \u03c0-TS, the initial particles are drawn from the model prior p0(\u03b8), which are maintained updated iteratively via discrete gradient \ufb02ow to approximate the posterior distributions. Different from vanilla Thompson sampling, one approximate posterior sample is randomly selected from the particle set \u0398t\u22121 in each iteration of \u03c0-TS to make decisions at time t. 4 Experiments We conduct experiments to verify the performance of our proposed \u03c0-TS framework in both static scenarios and contextual-bandit problems. Our implementation is in TensorFlow and will be released upon publication. All computations were run on a single Tesla P100 GPU and all results are averaged over 50 realizations. 4.1 Linear and Sparse Linear Contextual Bandits We consider a contextual bandit scenario where uncertainty estimation is driven by sequential decision making. This is more challenging because in this case the observations D are no longer i.i.d., leading to larger accumulative error as time goes on. We test the proposed method in the linear setting [19]. We can see from the \ufb01gure that the proposed method, \u03c0-TS-DGF, performs almost as well as Lin-TS, the exact model; whereas other methods such as Neural-Linear and VI-TS receive much larger regrets. The gap is mostly caused by the approximation error between the exact posterior and approximate posterior. Especially, VI-TS shows a higher regret variance. 4.2 Deep Contextual Bandits Following the settings of [19], we evaluate the algorithms on a range of bandit problems created from real-world data: Statlog, Covertype, Adult, Census, Financial, and Mushroom datasets. We normalize the cumulative regrets relative to that of the Uniform action selection, and plot the box-plot of the \ufb01nal normalized regrets in Figure 1. In Figure 2 are shown the mean (dark curves) and standard derivation (light areas) of regrets, along with number of pulls over 50 realizations. To conclude, \u03c0-TS outperforms other methods; the performance of Lin-TS is not as good due to its poor representation. With more data observed, it becomes increasingly dif\ufb01cult to approximate the exact posterior with the Lin-TS. With feature extracted by a neural network, the Neural Linear improves the performance and generally outperforms Lin-TS. Nevertheless, there are some cases where valid features cannot be well extracted by neural networks, leading to poor performance of Neural Linear. Furthermore, VI-TS consistently performs poorly with very high variances. The main cause might be that the underestimated uncertainty would lead to poor exploration. Our proposed \u03c0-TS outperforms other methods, since it can provide better uncertainty estimation than VI-TS, and endows more representation power than Lin-TS. Importantly, the performance of \u03c0-TS for relatively large datasets is much better than that of other methods. Figure 1: Normalized Cumulative Regret on all settings. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1811.00696v1",
+                    "title": "Sequence Generation with Guider Network",
+                    "abstract": "Sequence generation with reinforcement learning (RL) has received significant\nattention recently. However, a challenge with such methods is the sparse-reward\nproblem in the RL training process, in which a scalar guiding signal is often\nonly available after an entire sequence has been generated. This type of sparse\nreward tends to ignore the global structural information of a sequence, causing\ngeneration of sequences that are semantically inconsistent. In this paper, we\npresent a model-based RL approach to overcome this issue. Specifically, we\npropose a novel guider network to model the sequence-generation environment,\nwhich can assist next-word prediction and provide intermediate rewards for\ngenerator optimization. Extensive experiments show that the proposed method\nleads to improved performance for both unconditional and conditional\nsequence-generation tasks.",
+                    "authors": "Ruiyi Zhang, Changyou Chen, Zhe Gan, Wenlin Wang, Liqun Chen, Dinghan Shen, Guoyin Wang, Lawrence Carin",
+                    "published": "2018-11-02",
+                    "updated": "2018-11-02",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL"
+                    ],
+                    "main_content": "Introduction Sequence generation is an important area of investigation within machine learning. Recent work has shown excellent performance on a number of tasks, by combining reinforcement learning (RL) and generative models. Example applications include image captioning (Ren et al., 2017; Rennie et al., 2016), text summarization (Li et al., 2018; Paulus et al., 2017; Rush et al., 2015), and adversarial text generation (Guo et al., 2017; Lin et al., 2017; Yu et al., 2017; Zhang et al., 2017; Zhu et al., 2018). The sequence-to-sequence framework (Seq2Seq) (Sutskever et al., 2014) is a popular technique for sequence generation. However, models from such a setup are typically trained to predict the next token given previous ground-truth tokens as input, causing what is termed exposure bias (Ranzato et al., 2016). By contrast, sequencelevel training with RL provides an effective way to solve this challenge by treating sequence generation as an RL problem. By directly optimizing an evaluation score (cumulative rewards) (Ranzato et al., 2016), state-of-the-art results have been obtained in many sequence-generation tasks (Paulus et al., 2017; Rennie et al., 2016). However, one problem in such a framework is that rewards in the RL training are particularly sparse, since a scalar reward is typically only available after an entire sequence has been generated. For RL-based sequence generation, most existing works rely on a model-free framework via recurrent policy gradient (Wierstra et al., 2010). However, these methods have been criticized for their high variance and poor sample ef\ufb01ciency (Sutton and Barto, 1998). On the other hand, modelbased RL methods do not suffer from these issues, but they are usually dif\ufb01cult to train in complex environments. Furthermore, a learned policy is usually restricted by the capacity of an environment model. Recent developments on model-based RL (Kurutach et al., 2018; Nagabandi et al., 2017) combine the advantages of these two approaches, and have achieved improved performance by learning a model-free policy, assisted by an environment model. In addition, model-based RL has been employed recently to solve problems with extremely sparse rewards, with curiosity-driven methods (Pathak et al., 2017). Inspired by the ideas in (Kurutach et al., 2018; Nagabandi et al., 2017; Pathak et al., 2017), we propose a model-based RL method to overcome the sparse-reward problem in sequence-generation tasks. Our main idea is to employ a new guider network to model the generation environment in the feature space of sequence tokens, which is used to emit intermediate rewards by matching the predicted features from the guider network and features from generated sequences. The guider network is trained to encode global structural inarXiv:1811.00696v1  [cs.CL]  2 Nov 2018 \fformation of training sequences, useful to guide next-token generation in the generative process. Within the proposed framework, we also propose a new type of self-attention mechanism, to assist the guider network to provide high-level planningahead information. The intermediate rewards are combined with a \ufb01nal scalar reward, e.g., an evaluation score in a Seq2Seq generation model or the discriminator loss in the generative-adversarial-net (GAN) framework, to train a sequence generator with policy-gradient methods. Extensive experiments show improved performance of our method on both unconditional and conditional sequencegeneration tasks, relative to existing state-of-the-art methods. 2 Background 2.1 Sequence-to-Sequence Model A sequence-generation model learns to generate a sequence Y = (y1, . . . , yT ) conditioned on a possibly empty object X from a different feature space. Here yt \u2208A with A the alphabet set of output tokens. The pairs (X, Y ) are used for training a sequence-generation model. We use T to denote the length of an output sequence, and Y1,...,t to indicate a subsequence of the form (y1, . . . , yt). The output of a trained generator is denoted \u02c6 Y , which is typically not used during training. Since we focus on text generation in this paper, we will use token and word interchangeably to denote an element of a (text) sequence. Starting from the initial hidden state s0, a recurrent neural network (RNN) produces a sequence of states (s1, . . . , sT ) given an input sequencefeature representation (e(y1), . . . , e(yT )), where e(\u00b7) denotes a function mapping a token to its feature representation. Let et \u225ce(yt). The states are generated by applying a transition function h : st = h(st\u22121, et) for T times. The transition function h is implemented by a cell of an RNN, with popular choices being the Long Short-Term Memory (LSTM) (Hochreiter and Schmidhuber, 1997) and the Gated Recurrent Unit (GRU) (Cho et al., 2014). We will use the LSTM for our model. To generate a token \u02c6 yt \u2208A, a stochastic output layer is applied on the current state st: \u02c6 yt \u223cMulti(1, softmax(g(st\u22121))), (1) st = h(st\u22121, e(\u02c6 yt)) , (2) where Multi(1, \u00b7) denotes one draw from a multinomial distribution, and g(\u00b7) represents a linear transformation. Since the generated sequence Y is conditioned on X, one can simply start with an initial state encoded from X: s0 = s0(X) (Bahdanau et al., 2017; Cho et al., 2014). Finally, a conditional RNN can be trained for sequence generation with gradient ascent by maximizing the log-likelihood of a generative model. 2.2 Model-Based Reinforcement Learning Reinforcement learning is the problem of \ufb01nding an optimal policy for an agent interacting with an unknown environment, collecting a reward per action. A policy is de\ufb01ned as a conditional distribution, \u03c0(a|s), de\ufb01ning the probability over an action a \u2208A conditioned on a state variable s \u2208S. Formally, the problem can be described as a Markov decision process (MDP), M = \u27e8S, A, Ps, r, \u03b3\u27e9, where Ps(s\u2032|s, a) is the transition probability from state s to s\u2032 given action a; r(s, a) is an unknown reward function immediately following the action a performed at state s; \u03b3 \u2208[0, 1] is a discount factor regularizing future rewards. At each time step t, conditioned on the current state st, the agent chooses an action at \u223c\u03c0(\u00b7|st) and receives a reward signal r(st, at). The environment, as seen by the agent, then updates its state as st+1 \u223cPs(\u00b7|st, at). The goal is to learn an optimal policy such that one obtains the maximum expected total reward, e.g., by maximizing the total rewards, J(\u03c0) = \u221e X t=1 EPs,\u03c0 \u0002 \u03b3tr(st, at) \u0003 (3) = Est\u223c\u03c1\u03c0,at\u223c\u03c0 [r(st, at)] , (4) where \u03c1\u03c0 \u225cP\u221e t=1 \u03b3t\u22121Pr(s = st), and Pr(s) denotes the state marginal distribution induced by \u03c0. In model-based RL, a model of the dynamics Ps is built to make predictions for future states conditioned on the current state, which can be used for action selection, e.g., next-token generation. In practice, the model of dynamics is typically implemented as a discrete-time function, taking the current state-action pair (st, at) as input, and outputing an estimate of the next state st+\u25b3t at time t + \u25b3t. At each step, the best next action is chosen based on the current policy, and the model will re-plan with the updated information from the dynamics. This control scheme is referred to as model-predictive control (MPC) (Nagabandi et al., 2017). 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5OJH</latexit> Figure 1: Model Overview: Sequence generation with a guider network. Solid lines mean gradients are backpropagated in training; dash lines mean gradients are not backpropagated. typically corresponds to the current generated sentences Y1,...,t in sequence generation, thus we use the same notation f to denote both a state variable in RL and sequence generation. 3 Proposed Model To make the discussion explicit, we describe our model in the context of text generation, where tokens in a sequence are represented as words. The model is illustrated in Figure 1, with the \ufb01rst building block an Autoeocoder (AE) structure (the Encoder-Decoder in Figure 1) for sentence feature extraction and generation. The encoder is shared for sentences from both training data and generated data, as explained in detail below. Overall, text generation can be formulated as a sequential decision-making problem. At each timestep t, the agent, also called a generator (which corresponds to the LSTM Decoder in Figure 1), takes the current LSTM state as the environment state, denoted as st. The policy \u03c0\u03c6(\u00b7|st) parameterized by \u03c6 is a conditional generator, to generate the next token (action) given the state st of current generated sequence. At each time step, an immediate reward rt is also revealed, which is calculated based on the output of the guider network and used to update the sequence generator described below. The objective of sequence generation is to maximize the total rewards, as in (3). We detail the components of our proposed model in the following subsections. 3.1 Guider Network for Environment Dynamics The guider network, implemented as an RNN with LSTM units, is adopted to model environment dynamics to better assist sequence generation. The idea is to train a guider network such that its predicted sequence features at each time step are used to construct intermediate rewards in an RL setting, which in turn are used to optimize the sentence generator. Denote the guider network as G\u03c8(sG t\u22121, f t), with parameters \u03c8 and input arguments (sG t\u22121, f t) at time t to explicitly write out the dependency on the guider network latent state sG t\u22121 from the previous time step. Here f t is the input to the LSTM, which represents the feature of the current generated sentence after an encoder network. Speci\ufb01cally, let the current generated sentence be Y1...t (forced to be the same as parts of a training sequence in training), with f t calculated as: f t = Enc(Y1...t), where Enc(\u00b7) denotes the encoder transformation, implemented with a convolutional neural network (CNN) (Zhang et al., 2017); see Figure 1. The initial state of the guider network is the encoded feature of a true input sequence by the same CNN, i.e., sG 0 = Enc(X). Sequence-to-Sequence Generation with Planning We \ufb01rst explain how one uses the guider network to guide next-word generation for the generator (the LSTM decoder in Figure 1). Our framework is inspired by the MPC method (Nagabandi et al., 2017), and can be regarded as a type of planahead attention mechanism. Given the feature f t at time t from the current input sequence, the guider network produces a prediction G\u03c8(sG t\u22121, f t) as a future feature representation, by feeding f t into the LSTM. Since the training of the guider network is based on real data (detailed in the next paragraph), the predicted feature contains global-structure information of the training sequences. To utilize such information to predict the next word, we combine the predicted feature with the output of the decoder by constructing an attention-like mechanism. Speci\ufb01cally, we \ufb01rst apply a linear transformation \u03d5 on the predicted feature G\u03c8(sG t\u22121, f t), forming a weight vector wt \u225c\u03d5 \u0000G\u03c8(sG t\u22121, f t) \u0001 . Next, the weight wt is applied to the output Ot of the LSTM decoder by an element-wise multiplication operation. The result is then fed into a softmax layer to generate the next token yt. Formally, the generative process is written as Ot = g(st\u22121), (5) wt = \u03d5(G\u03c8(sG t\u22121, f t)), (6) yt \u223cMulti(1, softmax(Ot \u00b7 wt)), (7) sG t = hG(sG t\u22121, f t), st = h(st\u22121, e(yt)) . (8) \fGuider Network Optimization In training, given a sequence of feature representations (f 1, f 2, ... f T ) for a training sentence, we seek to update the guider network such that it is able to predict f t+c given f t, where c > 0 is the number of steps looked ahead. We implement this by forcing the predicted feature, G\u03c8(sG t , f t), to match both the sentence feature f t+c (\ufb01rst term in (9)) and the corresponding feature-changing direction (second term in (9)). This is formalized by maximizing an objective function of the following form at time t: J\u03c8 G =Dcos \u0010 f t+c, G\u03c8(sG t\u22121, f t) \u0011 + (9) Dcos \u0010 f t+c \u2212f t, G\u03c8(sG t\u22121, f t) \u2212f t \u0011 , where Dcos(\u00b7, \u00b7) denotes the cosine similarity. By minimizing (9), an ideal guider network should be able to predict the true next words conditioned on the current word in a sequence. As a result, the prediction is used to construct an intermediate reward, which is then used to update the generator (the LSTM decoder), as described further below. 3.2 Feature-Matching Rewards and Generator Optimization As in many RL-based sequence-generation methods, such as SeqGAN (Yu et al., 2017) and LeakGAN (Guo et al., 2017), the generator is updated based on policy-gradient methods. As a result, collecting rewards in the generation process is critical. Though (Yu et al., 2017) has proposed to use rollout to get rewards for each generated word, the variance of the rewards is typically too high to be practically useful. In addition, the computational cost may be too expensive for practical use. We describe how to use the proposed guider network to de\ufb01ne intermediate rewards below, leading to a de\ufb01nition of feature-matching reward. Feature-matching rewards We \ufb01rst de\ufb01ne an intermediate reward to generate a particular word. The idea is to match the ground-truth features from the CNN encoder in Figure 1 with those generated from the guider network. Equation (9) indicates that the further the generated feature is from the true feature, the smaller the reward should be. To this end, for each time t, we de\ufb01ne the intermediate reward for generating the current word as: rg t = 1 2c c X i=1 (Dcos(f t, \u02c6 f t)+ (10) Dcos(f t \u2212f t\u2212i, \u02c6 f t \u2212f t\u2212i)) , where \u02c6 f t = G\u03c8(sG t\u2212c\u22121, f t\u2212c) is the predicted feature. Intuitively, f t \u2212f t\u2212i measures the difference between the generated sequences in feature space; the reward will be high if it matches the predicted feature transition \u02c6 f t \u2212f t\u2212i from the guider network. At the last step of sequence generation, i.e., t = T, the corresponding reward measures the quality of the whole generated sequence, thus it is called a \ufb01nal reward. The \ufb01nal reward is de\ufb01ned differently from the intermediate reward, which will be discussed below for both the unconditionaland conditional-generation cases. Note a token generated at time t will in\ufb02uence not only the rewards received at that time but also the rewards at subsequent time steps. Thus we propose to de\ufb01ne the cumulative reward, PT i=t \u03b3irg i with \u03b3 a discount factor, as a feature-matching reward. Intuitively, this encourages the generator to focus on achieving higher long-term rewards. Finally, in order to apply policy gradient to update the generator, we combine the feature-matching reward with the problem-speci\ufb01c \ufb01nal reward, to form a Q-value reward speci\ufb01ed below. We consider unconditionaland conditional-sequence generation in the following. Unconditional generation This case corresponds to generating sequences from scratch. Similar to SeqGAN, the \ufb01nal reward is de\ufb01ned as the output of a discriminator, evaluating the quality of the whole generated sequence, i.e., the smaller the output, the less likely the generation is a true sequence. As a result, we combine the discriminator loss, denoted as rf 1 with the feature-matching rewards as follows, to de\ufb01ne a \ufb01nal Q-value reward: Qt = ( T X i=t \u03b3irg i ) \u00d7 rf . (11) Conditional generation This case corresponds to generating sequences conditioned on some input features, such as image features in image captioning. Following ideas from self-critical sequence training (SCST) (Rennie et al., 2016), the \ufb01nal reward rf s (also called the self-critical reward) is constructed by constituting a baseline reward, denoted as \u02c6 rf(Y \u2032), for variance reduction: rf s = rf(Y ) \u2212\u02c6 rf(Y \u2032) , (12) 1This reward is given by the discriminator in GAN framework, and rf \u2208[0, 1] \fwhere rf(Y ) is the reward of a sampled sentence Y by the current generator, and \u02c6 rf(Y \u2032) is the reward of a sentence obtained by choosing the words with the highest probabilities at each step t, i.e., a greedy decoding. Finally, the Q-value reward is de\ufb01ned as: Qt = ( rf s PT i=t \u03b3irg i if rf s > 0 rf s PT i=t \u03b3i(1 \u2212rg i ) otherwise (13) Generator optimization The sequence generator is initialized by pre-training on training sequences with an autoencoder structure, based on MLE training. After that, the \ufb01nal Q-value reward Qt is used as a reward for each time t, with standard policy gradient optimization methods to update the generator. Speci\ufb01cally, the policy gradient is \u2207\u03c6J = Est\u22121\u223c\u03c1\u03c0,yt\u223c\u03c0 [Qt\u2207\u03c6 log p(yt|st\u22121; \u03c6t\u22121)] , (14) where p(yt|st\u22121; \u03c6t\u22121) is the probability of generating yt given st\u22121 in the generator. Discussion For unconditional generation, the feature-matching reward is typically good enough, since the task focuses more on sentence structure, which is re\ufb02ected by the feature-matching reward. For conditional generation, however, a \ufb01nal reward in terms of an evaluation score (e.g., the BLEU score) is more important because the semantic information of the conditioned variable is encoded into the score. This \ufb01nal reward thus guides the generator to generate semantically consistent sentence w.r.t. the conditioned variable. However, when most of the \ufb01nal rewards are negative, it is well-known that a policy gradient method would fail because of a lack of positive reward signals. 3.3 Other Training Details Encoder as a feature extractor For unconditional generation, the feature extractor of the generating inputs for the guider network share the CNN part of the encoder. We stop gradients from the guider network to the encoder CNN in the training process. For conditional generation, we use a pretrained feature extractor, trained similarly to the unconditional generation and \ufb01xed later on. Training procedure As with many RL-based models (Bahdanau et al., 2017; Rennie et al., 2016; Sutskever et al., 2014), warm starting with a pretrained model is important. Thus we \ufb01rst pre-train the encoder-decoder part based on the training data with an MLE loss. After pre-training, we use RL training to \ufb01ne-tune the pre-trained generator. We adaptively transfer the training from MLE loss to RL loss, similar to (Paulus et al., 2017; Ranzato et al., 2016). Initial states We use the same initial state for both the sequence generator and the guider network. For conditional generation, the initial state is the encoded latent code of the conditional information for both training and testing. For unconditional generation, it is the encoded latent code of a target sequence in training and random noise in testing. 4 Related Work We \ufb01rst review related works that combine RL and GAN for text generation. As one of the most representative models in this direction, SeqGAN (Yu et al., 2017) adopts Monte-Carlo search to calculate rewards. However, such a method introduces high variance in policy optimization. There were a number of works proposed subsequently to improve the reward-generation process. For example, RankGAN (Lin et al., 2017) proposes to replace the reward from the GAN discriminator with a rankingbased reward; MaliGAN (Cho et al., 2014) modi\ufb01es the GAN objective and proposes techniques to reduce gradient variance; MaskGAN (Fedus et al., 2018) uses a \ufb01lling technique to de\ufb01ne a Q-value reward for sentence completion; LeakGAN (Guo et al., 2017) tries to address the sparse-reward issue for long-text generation with hierarchical RL by utilizing the leaked information from a GAN discriminator. One problem of LeakGAN is that it tends to over\ufb01t on training data, yielding generated sentences that are often not diverse. By contrast, by relying on a model-based RL approach, our method learns global-structure information, which generates more-diverse sentences, and can be extended to conditional sequence generation. RL techniques can also be used in other ways for sequence generation. For example, (Ranzato et al., 2016) trains a Seq2Seq model by directly optimizing the BLEU/ROUGE scores with the REINFORCE algorithm. To reduce variance of the vanilla REINFORCE, (Bahdanau et al., 2017) adopts the actor-critic framework for sequence prediction. Furthermore, (Rennie et al., 2016) trains a baseline with a greedy decoding scheme for the REINFORCE method. Note all these methods can only obtain rewards after a whole sentence is gen\ferated. Finally, planning techniques in RL have also been explored to improve sequence generation (Gulcehre et al., 2017; Serdyuk et al., 2018). Compared to these related works, the proposed guider network can provide a type of planningahead mechanism and intermediate rewards for RL training. Also, we consider using Q-value as the reward to encourage the generator focusing on longterm rewards. 5 Experiments We test the proposed framework on unconditional and conditional sequence generation tasks, and analyze the results to understand the performance gained by the guider network. We also perform an ablation investigation on the improvements brought by each part of our proposed method. All experiments are conducted on a single Tesla P100 GPU and implemented with TensorFlow. 5.1 Unconditional Text Generation We focus on adversarial text generation, and compare our approach with a number of related works (Guo et al., 2017; Lin et al., 2017; Yu et al., 2017; Zhang et al., 2017; Zhu et al., 2018). In this setting, a discriminator in the GAN framework is added to the model in Figure 1 to guide the generator to generate high-quality sequences. This is implemented by de\ufb01ning the \ufb01nal reward to be the output of the discriminator. All baseline experiments are implemented on the texygen platform2 (Zhu et al., 2018). We adopt the BLEU score, referenced by test set (test-BLEU) and themselves (self-BLEU) (Zhu et al., 2018) to evaluate quality of generated samples, where test-BLEU evaluates the reality of generated samples, and self-BLEU measures the diversity. A good generator should achieve both a high test-BLEU score and a low selfBLEU score. We call the proposed method featurematching GAN (GMGAN) for unconditional text generation. A detailed description of GMGAN is provided in the Supplementary Material. Short Text Generation: COCO Image Captions For this task, we use the COCO Image Captions Dataset (Lin et al., 2014), in which most sentences are of length about 10. Since we consider unconditional text generation, only image captions are used as the training data. After preprocessing, the training dataset consists of 27,842 words and 417,126 2https://github.com/geek-ai/Texygen Method BLEU-2 BLEU-3 BLEU-4 BLEU-5 MLE 0.820 0.607 0.389 0.248 SeqGAN 0.820 0.604 0.361 0.211 RankGAN 0.852 0.637 0.389 0.248 GSGAN 0.810 0.566 0.335 0.197 LeakGAN 0.922 0.797 0.602 0.416 TextGAN 0.926 0.774 0.552 0.362 GMGAN (ours) 0.949 0.823 0.635 0.421 Table 1: Test-BLEU scores on COCO Image Captions. Method BLEU-2 BLEU-3 BLEU-4 MLE 0.754 0.511 0.232 SeqGAN 0.807 0.577 0.278 RankGAN 0.822 0.592 0.288 GSGAN 0.785 0.522 0.230 LeakGAN 0.912 0.825 0.689 TextGAN 0.830 0.597 0.284 GMGAN (ours) 0.746 0.511 0.319 Table 2: Self-BLEU scores on COCO Image Captions. sentences. We use 120,000 random sample sentences as the training set, and 10,000 as the test set. The BLEU scores with different methods are listed in Tables 1 and 2. We observe that GMGAN performs signi\ufb01cantly better than the baseline models. Speci\ufb01cally, besides achieving higher test-BLEU scores, the proposed method can also generate samples with very good diversity in terms of self-BLEU scores. LeakGAN represents the state-of-the-art in adversarial text generation, however, its diversity measurement is relatively poor (Zhu et al., 2018). We suspect the high BLEU score achieved by LeakGAN is due to its mode collapse on some good samples, resulting in high self-BLEU scores. Other baselines achieve lower self-BLEU scores since they cannot generate reasonable sentences. Long Text Generation: EMNLP2017 WMT Following (Zhu et al., 2018), we use the News section in the EMNLP2017 WMT4 Dataset as our training data. The dataset consists of 646,459 words and 397,726 sentences. After preprocessing, the training dataset contains 5,728 words and 278,686 sentences. The BLEU scores with different methods are provided in Tables 4 and 5. Compared with other methods, LeakGAN and GMGAN achieves comparable test-BLEU scores, demonstrating high quality of the generated sentences. Again, LeakGAN tends to over-\ufb01t on training data, leading to much higher (worse) self-BLEU scores. Our proposed GMGAN shows good diversity of long text generation with lower self-BLEU scores. Other baselines obtain both low self-BLEU and test-BLEU scores, leading to more random genera\fMethod COCO Image Captions EMNLP2017 WMT News SeqGAN (1) A person made and black wooden table. (2) A closeup of a window at night. (1) She added on a page where it was made clear more old but public got said. (2) I think she\u2019re guys in four years , and more after it played well enough. LeakGAN (1) A bathroom with a black sink and a white toilet next to a tub. (2) A man throws a Frisbee across the grass covered yard. (1)\"I\u2019m a fan of all the game, I think if that\u2019s something that I\u2019ve not,\" she said, adding that he would not be decided. (2) The UK is Google\u2019 s largest non-US market, he has added \"20, before the best team is amount of fewer than one or the closest home or two years ago. GMGAN (1) Bicycles are parked near a row of large trees near a sidewalk. (2) A married couple posing in front of a piece of birthday cake. (1) \"Sometimes decisions are big, but they\u2019re easy to make,\" he told The Sunday Times in the New Year. (2) A BBC star has been questioned by police on suspicion of sexual assault against a 23-year-old man , it was reported last night. Table 3: Examples of generated samples with different methods on COCO and EMNLP datasets. Method BLEU-2 BLEU-3 BLEU-4 BLEU-5 MLE 0.761 0.468 0.230 0.116 SeqGAN 0.630 0.354 0.164 0.087 RankGAN 0.774 0.484 0.249 0.131 GSGAN 0.723 0.440 0.210 0.107 LeakGAN 0.920 0.725 0.502 0.321 TextGAN 0.777 0.529 0.305 0.161 GMGAN (ours) 0.923 0.727 0.491 0.303 Table 4: Test-BLEU scores on EMNLP2017 WMT. Method BLEU-2 BLEU-3 BLEU-4 MLE 0.664 0.338 0.113 SeqGAN 0.728 0.411 0.139 RankGAN 0.672 0.346 0.119 GSGAN 0.807 0.680 0.450 LeakGAN 0.857 0.696 0.553 TextGAN 0.806 0.662 0.448 GMGAN (ours) 0.814 0.576 0.328 Table 5: Self-BLEU scores on EMNLP2017 WMT. tions. Human Evaluation Besides quantitatively evaluating the results using BLEU scores, we also conduct a human evaluation on WMT News dataset, using Amazon Mechanical Turk. In this regard, we randomly sample 100 sentences generated by each model. 10 human judges are asked to rate the generated texts in a scale from 0 to 5 in terms of their readability. The averaged human rating scores are shown in Table 6, indicating GMGAN achieves higher human scores compared to other methods. The averaged human rating scores are shown in Table 6 and the results of GMGAN outperforms other methods. As examples, Table 3 illustrates some generated samples by GMGAN and its baselines. The performance on the two datasets indicates that the generated sentences of GMGAN are of higher global consistency and better readability than SeMethod Human Score MLE 2.45 SeqGAN 2.57 RankGAN 2.91 GSGAN 2.48 LeakGAN 3.47 textGAN 3.11 GMGAN (ours) 3.89 Table 6: Human evaluation results on EMNLP2017 WMT. qGAN and LeakGAN. More generated examples are provided in the Supplementary Material. 5.2 Conditional Generation We conduct experiments on image captioning (Karpathy and Fei-Fei, 2015) and text style transfer (Shen et al., 2017). We investigate the bene\ufb01ts brought by the proposed method in (13). In image captioning, instead of using a discriminator to de\ufb01ne \ufb01nal rewards for generated sentence, we adopt evaluation metrics computed based on human references. The \ufb01nal rewards appear more important as they contain ground-truth information. Feature-matching rewards work as a regularizer to maintain the semantic consistency and sentence structure, preventing language-\ufb02uency damages caused by only focusing on evaluation metrics (\ufb01nal rewards). We call our model in this setting a feature-matching sequence training (GMST) model. An overview of GMST is provided in the Supplementary Material. Image Captioning We \ufb01rst test our proposed model for image captioning on the MS COCO dataset (Lin et al., 2014), which contains 123,287 images in total. Each image is annotated with at least 5 captions. Following Karpathy\u2019s split (Karpa\fMethod BLEU-1 BLEU-2 BLEU-3 BLEU-4 METEOR CIDEr DeepVS (Karpathy and Fei-Fei, 2015) 62.5 45.0 32.1 23.0 19.5 66.0 ATT-FCN (You et al., 2016) 70.9 53.7 40.2 30.4 24.3 Soft Attention (Xu et al., 2015) 70.7 49.2 34.4 24.3 23.9 Hard Attention (Xu et al., 2015) 71.8 50.4 35.7 25.0 23.0 Show & Tell (Vinyals et al., 2015) 27.7 23.7 85.5 MSM (Yao et al., 2017) 73.0 56.5 42.9 32.5 25.1 98.6 No attention, Greedy, Tag AE 70.9 53.6 39.4 28.8 24.4 91.3 AE-g 71.0 53.9 39.6 28.9 24.3 92.8 SCST (BLEU-4) 73.2 57.1 43.9 33.6 24.5 95.9 GMST (BLEU-4) 73.4 57.5 44.3 33.9 24.5 97.1 SCST (CIDEr) 75.8 58.6 43.6 32.1 25.4 105.5 GMST (CIDEr) 76.1 59.0 44.1 32.6 25.5 107.4 No attention, Greedy, Resnet-152 AE 69.5 51.7 37.2 26.5 23.1 83.9 SCST (BLEU-4) 71.1 54.8 41.6 31.6 23.1 87.5 GMST (BLEU-4) 71.2 54.8 41.8 32.1 23.4 87.9 SCST (CIDEr) 73.9 56.1 41.2 30.0 24.3 98.6 GMST (CIDEr) 73.8 56.3 41.3 30.3 24.4 100.1 Table 7: Results for image captioning on the MS COCO dataset; the higher the better for all metrics (BLEU 1 to 4, METEOR, and CIDEr). thy and Fei-Fei, 2015), 5,000 images are used for both validation and testing. We report BLEU-k (k from 1 to 4) (Papineni et al., 2002), CIDEr (Vedantam et al., 2015), and METEOR (Banerjee and Lavie, 2005) scores. We consider two settings: (i) using a pre-trained 152-layer ResNet (He et al., 2016) for feature extraction; (ii) using semantic tags detected from the image as features (Gan et al., 2017). We use an LSTM with 512 hidden units, and train the model with the Adam optimizer (Kingma and Ba, 2014). The results are summarized in Table 7. When comparing an AutoEncoder (AE) with a variant implemented by adding a guider network (AE-g), reasonable improvements are observed. Next, we compare the proposed GMST with SCST, one of the state-of-the-art methods. Note the main difference between GMST and SCST is that the former employs our proposed feature-matching reward, while the latter only considers the \ufb01nal reward provided by evaluation metrics. GMST achieves higher scores compared with SCST on its optimized metrics. Style Transfer We next test the proposed framework on the non-parallel text-style-transfer task, where the goal is to transfer one sentence in one style (e.g., positive) to a similar sentence but with a different style (e.g., negative). For a fair comparison, we use the same data and its split method as in (Shen et al., 2017). Speci\ufb01cally, there are 444,000, 63,500, and 127,000 sentences with either positive or negative sentiments in the training, valiMethod Accuracy VAE (Shen et al., 2017) 23.2% Cross-align (Shen et al., 2017) 78.4% CVAE (Hu et al., 2017) 84.5% Ours 92.7% Table 8: Sentiment accuracy of transferred sentences. From positive to negative Original: the food is amazing ! Transferred: the food is horrible ! Original: all the employees are friendly and helpful . Transferred: all the employees are rude and unfriendly . Original: i \u2019m so lucky to have found this place ! Transferred: i \u2019m so embarrassed that i picked this place . From negative to positive Original: the service was slow . Transferred: the service was fast and friendly . Original: i would never eat there again and would probably not stay there either . Transferred: i would de\ufb01nitely eat this place and i would recommend them . Original: this place is aweful , everything about it was horrible ! Transferred: this place is incredible , and incredibly good service . Table 9: Style transfer samples. The \ufb01rst line is the original sentence, the second is the generated sentences after style transfer. dation and test sets, respectively. The task on this dataset is sentiment transfer. Table 9 lists some sentiment-transfer examples. The proposed method can transfer sentiment while maintaining its original content. To measure whether the original sentences (in the test set) have been transferred to the desired sentiment, we follow the settings of (Shen et al., 2017) and employ a pretrained CNN classi\ufb01er, which achieves an accu\fracy of 97.4% on the validation set, to evaluate the transferred sentences. Results are shown in Table 8. It can be observed that our proposed model exhibits much higher transfer accuracy, indicating the guider network provides good sentiment guidance for the generator. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1808.03030v1",
+                    "title": "Policy Optimization as Wasserstein Gradient Flows",
+                    "abstract": "Policy optimization is a core component of reinforcement learning (RL), and\nmost existing RL methods directly optimize parameters of a policy based on\nmaximizing the expected total reward, or its surrogate. Though often achieving\nencouraging empirical success, its underlying mathematical principle on {\\em\npolicy-distribution} optimization is unclear. We place policy optimization into\nthe space of probability measures, and interpret it as Wasserstein gradient\nflows. On the probability-measure space, under specified circumstances, policy\noptimization becomes a convex problem in terms of distribution optimization. To\nmake optimization feasible, we develop efficient algorithms by numerically\nsolving the corresponding discrete gradient flows. Our technique is applicable\nto several RL settings, and is related to many state-of-the-art\npolicy-optimization algorithms. Empirical results verify the effectiveness of\nour framework, often obtaining better performance compared to related\nalgorithms.",
+                    "authors": "Ruiyi Zhang, Changyou Chen, Chunyuan Li, Lawrence Carin",
+                    "published": "2018-08-09",
+                    "updated": "2018-08-09",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction There is recent renewed interest in reinforcement learning (Sutton & Barto, 1998; Kaelbling et al., 1996), largely as a consequence of the success of deep reinforcement learning (deep RL) (Mnih et al., 2015; Li, 2017), which is applicable to complex environments and has obtained state-of-the-art performance on several challenging problems. In reinforcement learning an agent interacts with the environment, seeking to learn an optimal policy that yields the maximum expected reward during the interaction. Generally speaking, a policy de\ufb01nes a distribution over actions conditioned on the states. Learning an optimal policy corresponds to searching for an element (a conditional action distribution) on the space of distributions that yields the best expected feedback (reward) to the agent as it interacts sequentially 1Duke University 2SUNY at Buffalo. Correspondence to: Changyou Chen <cchangyou@gmail.com>. Proceedings of the 35 th International Conference on Machine Learning, Stockholm, Sweden, PMLR 80, 2018. Copyright 2018 by the author(s). with the environment. A standard technique for policy learning is the policygradient (PG) method (Sutton et al., 2000). In PG, a policy is represented in terms of parameters, typically optimized by stochastic gradient descent (SGD) to maximize the expected total reward. A similar idea has been applied for learning deterministic policies, termed deterministic policy gradient (DPG) (Silver et al., 2014). Signi\ufb01cant progress has been made on advancing policy learning since introduction of deep learning techniques. As examples, the deep deterministic policy gradient (DDPG) method combines DPG and Q-learning (Watkins & Dayan, 1992) to jointly learn a policy and a Q-function for continuous control problems (Lillicrap et al., 2016). Trust region policy optimization (TRPO) improves PG by preserving the monotonicpolicy-improvement property (Schulman et al., 2015), implemented by imposing a trust-region constraint, de\ufb01ned as the Kullback-Leibler (KL) divergence between consecutive policies. Later, (Schulman et al., 2017b) proposed proximal policy optimization (PPO) to improve TRPO by optimizing a \u201csurrogate\u201d objective with an adaptive KL penalty and reward-clipping mechanism. Though obtaining encouraging empirical success, many of the aforementioned algorithms optimize parameters directly, and appear to lack a rigorous interpretation in terms of distribution optimization, e.g., it is not mathematically clear how sequentially optimizing policy parameters based on an expected-total-reward objective corresponds to optimizing the distribution of policy itself. In this paper we introduce gradient \ufb02ows in the space of probability distributions, called Wasserstein gradient \ufb02ows (WGF), and formulate policy optimization in RL as a WGF problem. Essentially, WGF induces a geometry structure (manifold) in the distribution space characterized by an energy functional. The length between elements on the manifold is de\ufb01ned by the second-order Wasserstein distance. Thus, searching for an optimal distribution corresponds to traveling along a gradient \ufb02ow on the space until convergence. In the context of deep RL, the energy functional is characterized by the expected reward. Gradient \ufb02ow corresponds to a sequence of policy distributions converging to an optimal policy during an iterative optimization procedure. From this perspective, convergence behavior of the optimization can be better understood. arXiv:1808.03030v1  [cs.LG]  9 Aug 2018 \fPolicy Optimization as Wasserstein Gradient Flows Traditional stochastic policies are limited by their simple representation ability, such as using multinomial (Mnih et al., 2015) or Gaussian policy distributions (Schulman et al., 2015). To overcome this issue, the proposed WGFbased stochastic policies employ general energy-based representations. To optimize the stochastic policy, we de\ufb01ne WGFs for RL in two settings: i) indirect-policy learning, de\ufb01ned on a distribution space for parameters; ii) directpolicy learning, de\ufb01ned on a distribution space for policy distributions. These correspond to two variants of our algorithms. The original form of the WGF problem is hard to deal with, as it is generally infeasible to directly optimize over a distribution (an in\ufb01nite-dimensional object). To overcome this issue, based on the Jordan-Kinderlehrer-Otto (JKO) method (Jordan et al., 1998), we propose a particlebased algorithm to approximate a continuous density function with particles, and derive the corresponding gradient formulas for particle updates. Our method is conceptually simple and practically ef\ufb01cient, which also provides a theoretically sound way to use trust-region algorithms for RL. Empirical experiments show improved performance over related reinforcement learning algorithms. 2. Preliminaries We review concepts and numerical algorithms for gradient \ufb02ows. We start from gradient \ufb02ows on the Euclidean space, and then extend them on the space of probability measures. 2.1. Gradient \ufb02ows on the Euclidean space For a smooth function\u2217F : Rd \u2192R, and a starting point x0 \u2208Rd, the gradient \ufb02ow of F(x) is de\ufb01ned as the solution of the differential equation: d x d\u03c4 = \u2212\u2207F(x(\u03c4)), for time \u03c4 > 0 and initial condition x(0) = x0. This is a standard Cauchy problem (Rulla, 1996), endowed with a unique solution if \u2207F is Lipschitz continuous. When F is nondifferentiable, the gradient is replaced with its subgradient, which gives a similar de\ufb01nition, omitted for simplicity. Numerical solution An exact solution to the above gradient-\ufb02ow problem is typically intractable. A standard numerical method, called the Minimizing Movement Scheme (MMS) (Gobbino, 1999), evolves x iteratively for small steps along the gradient of F at the current point. Denoting the current point as xk, the next point is xk+1 = xk \u2212\u2207F(xk+1)h, with stepsize h. Note xk+1 is equivalent to solving optimization problem xk+1 = arg minx F(x)+ \u2225x \u2212xk \u22252 2 2h , where \u2225\u00b7\u22252 denotes the vector 2-norm. Convergence of the {xk} sequence to the exact solution has been established (Ambrosio et al., 2005). Refer to Section A.1 of the Supplementary Material (SM) for details. \u2217We will focus on the convex case, since this is the case for many gradient \ufb02ows on the space of probability measures, as detailed subsequently. 2.2. Gradient \ufb02ows on the probability-measure space By placing the optimization onto the space of probability measures, denoted P(\u2126) with \u2126\u2282Rd, we arrive at Wasserstein gradient \ufb02ows. For a formal de\ufb01nition, we \ufb01rst endow a Riemannian geometry on P(\u2126). The geometry is characterized by the length between two elements (two distributions), de\ufb01ned by the 2nd-order Wasserstein distance: W 2 2 (\u00b5, \u03bd) \u225cinf \u03b3 \u001aZ \u2126\u00d7\u2126 \u2225x \u2212y \u22252 2d\u03b3(x, y) : \u03b3 \u2208\u0393(\u00b5, \u03bd) \u001b , where \u0393(\u00b5, \u03bd) is the set of joint distributions over (x, y) such that the two marginals equal \u00b5 and \u03bd, respectively. This is an optimal-transport problem, where one wants to transform \u00b5 to \u03bd with minimum cost (Villani, 2008). Thus the term \u2225x \u2212y \u22252 2 represents the cost to transport x in \u00b5 to y in \u03bd, and can be replaced by a general metric c(x, y) in a metric space. If \u00b5 is absolutely continuous w.r.t. the Lebesgue measure, there is a unique optimal transport plan from \u00b5 to \u03bd, i.e., a mapping T : Rd \u2192Rd pushing \u00b5 onto \u03bd satisfying T#\u00b5 = \u03bd. Here T#\u00b5 denotes the pushforward measure of \u00b5. The Wasserstein distance is equivalently reformulated as W 2 2 (\u00b5, \u03bd) \u225cinf T \u001aZ \u2126 c(x, T(x))d\u00b5(x) \u001b . Let {\u00b5\u03c4}\u03c4\u2208[0,1] be an absolutely continuous curve in P(\u2126) with \ufb01nite second-order moments. Consider W 2 2 (\u00b5\u03c4, \u00b5\u03c4+h). Motivated by the Euclidean-space case, if we de\ufb01ne v\u03c4(x) \u225climh\u21920 T (x\u03c4 )\u2212x\u03c4 h as the velocity of the particle, a gradient \ufb02ow can be de\ufb01ned on P(\u2126) correspondingly (Ambrosio et al., 2005). Lemma 1 Let {\u00b5\u03c4}\u03c4\u2208[0,1] be an absolutely-continuous curve in P(\u2126) with \ufb01nite second-order moments. Then for a.e. \u03c4 \u2208[0, 1], the above vector \ufb01eld v\u03c4 de\ufb01nes a gradient \ufb02ow on P(\u2126) as \u2202\u03c4\u00b5\u03c4 + \u2207\u00b7 (v\u03c4 \u00b5\u03c4) = 0, where \u2207\u00b7 a \u225c\u2207\u22a4 x a for a vector a. Function F in Section 2.1 is lifted to be a functional in the space of probability measures, mapping a probability measure \u00b5 to a real value, i.e., F : P(\u2126) \u2192R. F is the energy functional of a gradient \ufb02ow on P(\u2126). Consequently, it can be shown that v\u03c4 in Lemma 1 has the form v\u03c4 = \u2212\u2207\u03b4F \u03b4\u00b5\u03c4 (\u00b5\u03c4) (Ambrosio et al., 2005), where \u03b4F \u03b4\u00b5\u03c4 is called the \ufb01rst variation of F at \u00b5\u03c4. Based on this, gradient \ufb02ows on P(\u2126) can be written \u2202\u03c4\u00b5\u03c4 = \u2212\u2207\u00b7 (v\u03c4 \u00b5\u03c4) = \u2207\u00b7 \u0012 \u00b5\u03c4\u2207( \u03b4F \u03b4\u00b5\u03c4 (\u00b5\u03c4)) \u0013 . (1) Remark 2 Intuitively, an energy functional F characterizes the landscape structure (appearance) of the corresponding manifold, and the gradient \ufb02ow (1) de\ufb01nes a solution path on this manifold. Usually, by choosing appropriate F, the landscape is convex, e.g., the It\u00f3-diffusion case de\ufb01ned below. This provides a theoretical guarantee of optimal convergence of a gradient \ufb02ow. \fPolicy Optimization as Wasserstein Gradient Flows It\u00f3 diffusions as WGFs It\u00f3 diffusion de\ufb01nes a stochastic mapping T : Rd \u00d7 R \u2192Rd such that we have T (x, 0) = x and T (T (x, \u03c4), s) = T (x, s + \u03c4), for all x \u2208Rd and s, \u03c4 \u2208R. A typical example of this family is de\ufb01ned as T (x, \u03c4) = x\u03c4, where x\u03c4 is driven by a diffusion of the form: d x\u03c4 = \u2207U(x\u03c4)d\u03c4 + \u03c3(x\u03c4)dW . (2) Here U : Rd \u2192Rd, \u03c3 : Rd \u2192Rd\u00d7d are called the drift and diffusion terms, respectively; W is the standard d-dimensional Brownian motion. In Bayesian inference, we seek to make the stationary distribution of x\u03c4 approach a particular distribution p(x), e.g., a posterior distribution. One solution for this is to set U(x\u03c4) = 1 2 log p(x) and \u03c3(x\u03c4) as the d \u00d7 d identity matrix. The resulting diffusion is called Langevin dynamics (Welling & Teh, 2011). Denoting the distribution of x\u03c4 as \u00b5\u03c4, it is well known (Risken, 1989) that \u00b5\u03c4 is characterized by the Fokker-Planck (FP) equation: \u2202\u03c4\u00b5\u03c4 = \u2207\u00b7 \u0000\u2212\u00b5\u03c4\u2207U + \u2207\u00b7 \u0000\u00b5\u03c4\u03c3\u03c3\u22a4\u0001\u0001 . (3) Note (3) is in the gradient-\ufb02ow form of (1), where the energy functional F is de\ufb01ned as\u2020: F(\u00b5) \u225c\u2212 Z U(x)\u00b5(x)d x | {z } F1 + Z \u00b5(x) log \u00b5(x)d x | {z } F2 (4) Note F2 is the energy functional of a pure Brownian motion (e.g., U(x) = 0 in (2)). To verify the FP equation with (1), the \ufb01rst variation of F1 and F2 is calculated as \u03b4F1 \u03b4\u00b5 = \u2212U, \u03b4F2 \u03b4\u00b5 = log \u00b5 + 1 . (5) Substituting (5) into (1) recovers the FP equation (3). Numerical methods Inspired by the Euclidean-space case, gradient \ufb02ow (1) can be approximately solved by discretizing time, leading to an iterative optimization problem, where for iteration k: \u00b5(h) k+1 \u2208arg min\u00b5 F(\u00b5)+ W 2 2 (\u00b5,\u00b5(h) k ) 2h . Speci\ufb01cally, for It\u00f3-diffusion with F de\ufb01ned in (4), the optimization problem becomes: \u00b5(h) k+1 = arg min \u00b5 KL (\u00b5\u2225p(x)) + W 2 2 (\u00b5, \u00b5(h) k ) 2h , (6) where p(x) \u225c1 Z eU(x) is the target distribution. This procedure is called the Jordan-Kinderlehrer-Otto (JKO) scheme. Remarkably, the convergence of (6) can be guaranteed (Jordan et al., 1998), as stated in Lemma 3. \u2020We assume \u03c3 to be independent of x, which is the case in Langevin dynamics whose stationary distribution is set to be proportional to e\u2212U(x). As a result, we drop \u03c3 in the following. Lemma 3 Assume that log p(x) \u2264C1 is in\ufb01nitely differentiable, and \u2225\u2207log p(x)\u2225\u2264C2 (1 + C1 \u2212log p(x)) (\u2200x) for some constants {C1, C2}. Let T = hK, \u00b50 \u225cq0(x), and {\u00b5(h) k }K k=1 be the solution of the functional optimization problem (6), which are restricted to the space with \ufb01nite second-order moments. Then i) the problem (6) is convex; and ii) \u00b5(h) K converges to \u00b5T in the limit of h \u21920, i.e., limh\u21920 \u00b5(h) K = \u00b5T , where \u00b5T is the solution of (3) at T. Remark 4 Since the stationary distribution of the FP equation (3) is proportional to eU(x), Lemma 3 suggests that limk\u2192\u221e,h\u21920 \u00b5(h) k = 1 Z eU, a useful property to guide design of energy functionals for RL, as discussed in Section 4. 3. Particle Approximation for WGFs We focus on solving It\u00f3 diffusions with scheme (6). Directly reformulating gradient \ufb02ows as a sequential optimization problem in (6) is infeasible, because {\u00b5(h) k } are in\ufb01nitedimensional objects. We propose to use particle approximation to solve (6), where particles continuously evolve over time. There exist particle-based algorithms for gradient-\ufb02ow approximations, for example, the stochastic and deterministic particle methods in (Cottet & Koumoutsakos, 2000; Russo, 1990; Carrillo et al., 2017). However, they did not target the JKO scheme, and thus are not applicable to our setting. Another advantage of the JKO scheme is that it allows direct application of gradient-based algorithms, once we get gradients of the particles; thus, it is particularly useful in deep-learning-based methods where parameters are updated by backpropagating gradients through a network. Following similar idea as in (Chen et al., 2018), in the k-th iteration of our algorithm, M particles {x(i) k }M i=1 are used to approximate \u00b5(h) k : \u00b5(h) k \u2248 1 M PM i=1 \u03b4x(i) k . Our goal is to evolve {x(i) k } such that the corresponding empirical measure, \u00b5(h) \u225c 1 M PM i=1 \u03b4x(i), minimizes (6). A standard method is to use gradient descent to update the particles, where gradients { \u2202KL(\u00b5,\u00b5(h) k ) \u2202x(i) , \u2202W 2 2 (\u00b5,\u00b5(h) k ) \u2202x(i) } are required according to (6). By assuming x(i) k to evolve in the form of x(i) k+1 = x(i) k +h\u03c6(x(i) k ), with function \u03c6 restricted to an RKHS with kernel K(\u00b7, \u00b7), the gradient of the KL term is calculated as (Liu & Wang, 2016): \u2202KL(\u00b5(h), \u00b5(h) k ) \u2202x(i) \u221d1 M M X j=1 h K(x(j), x(i))\u2207x(j) log p(x(j)) +\u2207x(j)K(x(j), x(i)) i . (7) The gradient for W 2 2 (\u00b5(h), \u00b5(h) k ) is more involved, as the distance does not have a closed form. The Wasserstein term arises due to the Brownian motion in the diffusion process (2), and the non-differentiability of a sample path from a \fPolicy Optimization as Wasserstein Gradient Flows Brownian motion is translated into the Wasserstein distance. We develop a simple yet effective method to overcome this issue below. First, using a particle approximation, W 2 2 (\u00b5(h), \u00b5(h) k ) is simpli\ufb01ed as W 2 2 (\u00b5, \u00b5(h) k ) = inf pi,j X i,j pijc(x(i), x(j) k ) (8) s.t. X j pij = 1 M , X i pij = 1 M , where c(x1, x2) \u225c\u2225x1 \u2212x2 \u22252 2. Our goal turns to solving for the optimal {pij}. Since W2 comes from the Brownian motion, the energy functional in its gradient \ufb02ow is de\ufb01ned as F2 in (4). Solving the gradient \ufb02ow with the JKO scheme, at each iteration we minimize \u03bbF2 + W 2 2 (\u00b5, \u00b5(h) k ) with \u03bb a regularization parameter. Substituting W 2 2 with (8), introducing Lagrangian multipliers {\u03b1i, \u03b2i} to deal with the constraints, and letting cij \u225cc(x(i), x(j) k ), the dual problem is, L({pij}, {\u03b1i}, {\u03b2i}) = \u03bb X i,j pij log pij + pijcij + X i \u03b1i( X j pij \u22121 M ) + X j \u03b2j( X i pij \u22121 M ) The optimal pij have forms of p\u2217 ij = uie\u2212cij/\u03bbvj, where ui \u225ce\u22121 2 \u2212\u03b1i \u03bb , vj = e\u22121 2 \u2212 \u03b2j \u03bb . Assuming {ui} and {vj} are independent of {x(i)} and {x(j) k }, \u2202W 2 2 (\u00b5, \u00b5(h) k ) \u2202x(i) \u221d P j cije\u2212cij/\u03bb \u2202x(i) = X j 2 \u0010 1 \u2212cij \u03bb \u0011 e\u2212cij/\u03bb(x(i) \u2212x(j) k ) . (9) The gradients of particles can be obtained by combining (7) and (9), which are then optimized using SGD. Intuitively, from (9), the Wasserstein term contributes in two ways: i) When cij \u03bb > 1, x(i) is pulled close to previous particles {x(j) k }, with force proportional to ( cij \u03bb \u22121)e\u2212cij/\u03bb; ii) when x(i) is close enough to a previous particle x(j) k , i.e., cij \u03bb < 1, x(i) is pushed away, preventing it from collapsing to x(j) k . 4. Policy Optimization as WGFs Reinforcement learning is the problem of \ufb01nding an optimal policy for an agent interacting with an unknown environment, collecting a reward per action. A policy is de\ufb01ned as a conditional distribution, \u03c0(a | s), de\ufb01ning the probability over an action a \u2208A conditioned on a state variable s \u2208S. Formally, the problem can be described as a Markov decision process (MDP), M = \u27e8S, A, Ps, r, \u03b3\u27e9, where Ps(s\u2032 | s, a) is the transition probability from state s to s\u2032 given action a; r(s, a) is an unknown reward function immediately following the action a performed at state s; \u03b3 \u2208[0, 1] is a discount factor regularizing future rewards. We denote these variables with a subscript t to indicate their time dependency. At each time step t, conditioned on the current state st, the agent chooses an action at \u223c\u03c0(\u00b7| st) and receives a reward signal\u2021 r(a, s). The environment as seen by the agent then updates its state as st+1 \u223cPs(\u00b7| st, at). The goal is to learn an optimal policy such that one obtains the maximum expected total reward, e.g., by maximizing J(\u03c0) = \u221e X t=1 EPs,\u03c0 \u0002 \u03b3tr(a, s) \u0003 = Es\u223c\u03c1\u03c0,a\u223c\u03c0 [r(s, a)] (10) where \u03c1\u03c0 \u225cP\u221e t=1 \u03b3t\u22121Pr(st = s), and Pr(s) denotes the state marginal distribution induced by \u03c0. Optimizing the objective in (10) with a maximum entropy constraint provides us with a framework for training stochastic policies, where speci\ufb01c forms of these policy distribution are required, restricting the representation power. To de\ufb01ne a more general class of distributions that can represent more complex and multimodal distributions, we adopt the general energy-based policies (Haarnoja et al., 2017), and transform it into the WGF framework. Speci\ufb01cally, in the WGF framework, policies form a Riemannian manifold on the space of probability measures. The manifold structure is determined by the expected total reward (10), and the geodesic length between two elements (policy distributions) is de\ufb01ned as the standard second-order Wasserstein distance. With convex energy functionals (de\ufb01ned below), searching for an optimal policy reduces to running SGD on the manifold of probability measures. In the following, we de\ufb01ne gradient \ufb02ows on both parameterdistribution space and policy-distribution space, leading to indirect-policy learning and direct-policy learning, respectively. In indirect-policy learning, a WGF is de\ufb01ned over policy parameters; whereas in direct-policy learning, a WGF is de\ufb01ned over actions. For both settings, different energy functionals are de\ufb01ned based on the expected total reward, as detailed below. We note that most existing deep RL algorithms cannot be included into the two settings, without the concept of WGF. However, their speci\ufb01c techniques could be applied as intermediate ingredients in our framework. 4.1. Indirect-policy learning With indirect-policy learning, we do not optimize the stochastic policy \u03c0 directly. Instead, we aim to describe uncertainty of a policy with parameter distributions (weight uncertainty). Thus we de\ufb01ne a gradient \ufb02ow on the parameters. Let a policy be parameterized by \u03b8, denoted as \u03c0\u03b8. If we treat \u03b8 as stochastic and learn its posterior distribution p(\u03b8) in response to the expected total reward, the policy is implicitly learned in the sense that uncertainty in the parameter is transferred into the policy distribution in prediction. Following (Houthooft et al., 2016; Liu et al., 2017), the objective function is de\ufb01ned as: max p {Ep(\u03b8)[J(\u03c0\u03b8)] \u2212\u03b1KL(p\u2225p0)} (11) \u2021We assume the reward function to be deterministic, for simplicity; stochastic rewards can be addressed similarly. \fPolicy Optimization as Wasserstein Gradient Flows where p0(\u03b8) is the prior of \u03b8; \u03b1 \u2208[0, +\u221e) is the temperature hyper-parameter to balance exploitation and exploration in the policy. If we use an uninformative prior, p0(\u03b8) = const, the KL term is simpli\ufb01ed to the entropy as maxp{Ep(\u03b8)[J(\u03c0\u03b8)] + \u03b1H(p)}. By taking the derivative of the objective function, the optimal distribution is shown to have a simple closed form of p(\u03b8) \u221dexp (J(\u03c0\u03b8)/\u03b1) (Liu et al., 2017). This formulation is equivalent to a Bayesian formulation of parameter \u03b8, where p(\u03b8) can be seen as the \u201cposterior\u201d distribution, and exp(J(\u03c0\u03b8)/\u03b1) is the \u201clikelihood\u201d function. A variational (posterior) distribution for \u03b8, denoted as \u00b5(\u03b8), is learned by solving an appropriate gradient-\ufb02ow problem. We de\ufb01ne an energy functional characterizing the similarity between the current parameter distribution and the true distribution induced by the total reward as F(\u00b5) \u225c\u2212 Z J(\u03c0\u03b8)\u00b5(\u03b8)d\u03b8 + Z \u00b5(\u03b8) log \u00b5(\u03b8)d\u03b8 = KL (\u00b5\u2225p\u03b8) , (12) The energy functional de\ufb01nes a landscape determined by the expected total reward, and obtains its minimum when \u00b5 = p\u03b8. Proposition 5 For the gradient \ufb02ow with energy functional de\ufb01ned in (12), \u00b5 converges to p\u03b8 in the in\ufb01nite-time limit. To solve the above gradient-\ufb02ow problem, one can apply the JKO scheme with a stepsize h (we follow previous notation to use subscript k to denote discrete-time solutions and superscript h to denote the stepsize): \u00b5(h) k+1 = arg min \u00b5 KL (\u00b5\u2225p\u03b8) + W 2 2 (\u00b5, \u00b5(h) k ) 2h . (13) The above problem can be directly solved with gradient descent by adopting the particle approximation described in Section 3. Speci\ufb01cally, let the current particles be (\u03b8(i))M i=1. When calculating \u2202KL(\u00b5\u2225p\u03b8) \u2202\u03b8(i) as in (7), we need to evaluate \u2207\u03b8(i)J(\u03c0\u03b8(i)). This can be approximated with REINFORCE (Williams, 1992) or advantage actor critic (Schulman et al., 2016). For example, with REINFORCE, \u2207\u03b8(i)J(\u03c0\u03b8(i)) \u22481 T T X t=1 \u03b3t\u22121\u2207\u03b8(i) log \u03c0\u03b8(i)(at | st) \u02c6 Q\u03c0(st, at) where T is a horizon parameter, and \u02c6 Q\u03c0(st, at) is the Qvalue function. We call this variant of our framework Indirect Policy learning with WGF (IP-WGF). Remark 6 Assume gradients \u2207\u03b8(i)J(\u03c0\u03b8(i)) and \u2207\u03b8(i)W 2 2 are unbiased. Under the limit of M \u2192\u221eand h \u21920, and based on the fact that F in (12) is convex, Lemma 3 suggests the particle approximation converges to the global minimum p\u03b8. The conclusion applies, in the next section, similarly in the direct-policy-learning case. Existing methods such as TRPO (Schulman et al., 2015) and PPO (Schulman et al., 2017b) do not have such an interpretation, thus understanding their underlying convergence is more challenging. Furthermore, these methods optimize parameters directly as \ufb01xed points, deteriorating their ability to explore when policy distributions are inappropriately de\ufb01ned, as stochasticity only comes from the policy distributions. 4.2. Direct-policy learning When the dimension of parameter space is high, as is often the case in practice, IP-WGF can suffer from computation and storage inef\ufb01ciencies. In direct-policy learning, a gradient \ufb02ow is de\ufb01ned for the distribution of policies, thus a policy is directly optimized during learning. This approach appears to be more ef\ufb01cient and \ufb02exible, and connects more directly to existing works compared with indirect-policy learning. Speci\ufb01cally, we consider a general energy-based policies of the form \u03c0(a | s) \u221dexp(\u2212\u03b5(s, a)/\u03b1) that is able to model more complex distributions (Haarnoja et al., 2017). We formulate the direct-policy learning as policydistribution-based gradient \ufb02ows. The energy functional is de\ufb01ned with respect to the learned policy \u03c0, thus it should depend on states. To this end, let \u03b5s,\u03c0(a) = \u2212Q(at, st), where Q(at, st) \u225cr(at = a, st = s) + E(st+1,at+1,\u00b7\u00b7\u00b7 )\u223c(\u03c1\u03c0,\u03c0) P\u221e l=1 \u03b3lr(st+l, at+l). Q(at, st) is seen to be a functional depending on the current st and at, as well as the policy \u03c0. Integrating out the action a, an energy functional characterizing the similarity of the current policy \u03c0 and the optimal policy, ps,\u03c0(a | s) \u221deQ(a,s), is readily de\ufb01ned as Fs(\u03c0) \u225c\u2212 Z Q(a, s)\u03c0(a | s)d a + Z \u03c0(a | s) log \u03c0(a | s)d a = KL (\u03c0\u2225ps,\u03c0) . (14) Remark 7 Soft Q-learning (Haarnoja et al., 2017) adopts Q(a, s)+H(\u03c0) as the objective function, where the entropy of \u03c0, H(\u03c0) \u225c\u2212E\u03c0[log \u03c0], is included to add stochasticity into the corresponding Q-function. By treating the problem as a WGF, the stochasticity is modeled directly in the policy distribution, thus we do not include the entropy term, though it is of no harm to add it in. Proposition 8 For a WGF with the energy functional de\ufb01ned in (14), \u03c0(a | s) converges to ps,\u03c0(a) \u221deQ(a,s) with Q(a, s) satisfying the following modi\ufb01ed Bellman equation: Q(at, st) = r(at, st) + \u03b3Est+1\u223c\u03c1\u03c0[V\u03c0(st+1) \u2212H(\u03c0(\u00b7| st+1))] where V\u03c0(st+1) \u225clog R A exp(Q(a, st+1))d a. To solve the corresponding WGF, we again adopt the JKO scheme as in (13) to optimize the policy \u03c0 by particle approximation, i.e., \u03c0 \u221d 1 M PM i=1 \u03b4a(i). One challenge is that when calculating \u2202KL(\u03c0\u2225ps,\u03c0) \u2202a(i) , from (7), one needs to evaluate ps,\u03c0(a(i)), which is dif\ufb01cult due to the in\ufb01nite time horizon and the unknown reward function r(s, a) when calculating Q(a(i), s). To address this, we approximate the soft Q-function, Q(\u00b7, s), with a deep neural network Q\u03b8 s(s, a) \fPolicy Optimization as Wasserstein Gradient Flows parametrized by \u03b8, i.e., ps,\u03c0(a) \u221deQ\u03b8 s (s,a). The neural network Q\u03b8 s naturally leads to a soft approximation of the standard Q-function according to Proposition 8. As a result, the learning can be done by alternating between the following two steps. 1) Optimizing the policy Given Q\u03b8 s, we could adopt the particle approximation with the JKO scheme to optimize the policy. However, since a policy is a conditional distribution, one needs to introduce a set of particles for each state. For large or continuous state space, this becomes intractable. To mitigate this problem, we propose to use a stochastic state-conditioned neural network f\u03c6 \u03c6 \u03c6 parametrized by \u03c6 \u03c6 \u03c6 to approximate the policy. We call such a network a sampling network. The input to f\u03c6 \u03c6 \u03c6 is a concatenation of a state s and a randomnoise sample \u03be drawn from a simple distribution, e.g., the standard normal distribution. To optimize the sampling network, note that the JKO scheme, with energy functional (14), is written as (we rewrite \u03c0 as \u03c0\u03c6 \u03c6 \u03c6 to explicitly indicate the dependence of \u03c0 on \u03c6 \u03c6 \u03c6): \u03c0\u03c6 \u03c6 \u03c6 k+1 = arg min \u03c0\u03c6 \u03c6 \u03c6 KL \u0010 \u03c0\u03c6 \u03c6 \u03c6\u2225ps,\u03c0 \u0011 + W 2 2 (\u03c0\u03c6 \u03c6 \u03c6, \u03c0\u03c6 \u03c6 \u03c6 k) 2h \u225cJ\u03c6 \u03c6 \u03c6 \u03c0 . (15) The outputs of f\u03c6 \u03c6 \u03c6({\u03bei}; st) are particles (a(i) t )M i=1. Using chain rule we calculate the gradient of \u03c6 \u03c6 \u03c6 as \u2202J\u03c6 \u03c6 \u03c6 \u03c0 \u2202\u03c6 \u03c6 \u03c6 = E{\u03bei} \" \u2202J\u03c6 \u03c6 \u03c6 \u03c0 \u2202a(i) t \u2202a(i) t \u2202\u03c6 \u03c6 \u03c6 # . Thus \u03c6 \u03c6 \u03c6 can be updated using standard SGD, where \u2202J\u03c6 \u03c6 \u03c6 \u03c0 /\u2202a(i) t represents particle gradients in the WGF, and is approximated using techniques from Section 3; \u2202a(i) t /\u2202\u03c6 \u03c6 \u03c6 can be calculated by standard backpropagation. 2) Optimizing the Q-network\u00b7 We optimize the Qnetwork using the Bellman error as in the soft-Q learning setting (Haarnoja et al., 2017). Speci\ufb01cally, in each iteration, we optimize the following objective function: JQ(\u03b8) \u225cEst\u223cqst ,at\u223cqat \u00141 2 \u0010 \u02c6 Q \u00af \u03b8 s(st, at) \u2212Q\u03b8 s(st, at) \u00112\u0015 , where qst and qat are arbitrary distributions with support on S and A, respectively; \u02c6 Q \u00af \u03b8 s(st, at) = r(st, at)+\u03b3Est+1\u223c\u03c1\u03c0[V \u00af \u03b8 s (st+1)] is the target Q-value, with V \u00af \u03b8 s (st+1) = log Eqa\u2032 [ exp(Q \u00af \u03b8 s (st+1,a\u2032)) qa\u2032(a\u2032) ] \u2212H(qa\u2032); \u00af \u03b8 represents the parameters of the target Q-network, as used in standard deep Q-learning (Mnih et al., 2013). qat can be set to the distribution induced by the sampling network f\u03c6 \u03c6 \u03c6 (Haarnoja et al., 2017). Alternatively, the form of qat can be explicitly de\ufb01ned, e.g., using isotropic Gaussian or mixture of Gaussian distributions. The full algorithm is given in Section G of the SM. We call this variant of our framework Direct Policy learning with WGF (DP-WGF). Reducing Variance Note that when optimizing the Q-network, one needs to calculate the V \u03b8 s -function. This includes an integration over qa\u2032, which endows the high variance associated with Monte Carlo integration. Consequently, we propose to learn a V -network to approximate V \u03b8 s , denoted as \u00af V\u03c8 \u03c8 \u03c8(s) with parameter \u03c8 \u03c8 \u03c8. To learn the V -network, similar to (Haarnoja et al., 2018), we use an explicit policy distribution. As a result, we replace the sampling network f\u03c6 \u03c6 \u03c6 with a BNN discussed in Section 4.1, whose induced policy distribution is denoted \u02dc \u03c0\u03c6 \u03c6 \u03c6. Intuitively, V \u03b8 s (s) can be considered an approximation to the log-normalizer of exp(Q\u03b8 s(s, a)) over a. From the de\ufb01nition, the objective is de\ufb01ned as: JV (\u03c8 \u03c8 \u03c8) \u225c Est\u223cqst \u0000 \u00af V\u03c8 \u03c8 \u03c8(st) \u2212log Eat\u223c\u02dc \u03c0\u03c6 \u03c6 \u03c6(st) exp \u0000Q\u03b8 s(st, at)/\u02dc \u03c0\u03c6 \u03c6 \u03c6(at | st) \u0001 \u2212Eat\u223c\u02dc \u03c0\u03c6 \u03c6 \u03c6(st) log \u02dc \u03c0\u03c6 \u03c6 \u03c6(at | st) \u00012. In our implementation, we \ufb01nd the following approximation works well: JV (\u03c8 \u03c8 \u03c8) \u225c Est\u223cqst \u0010 \u00af V\u03c8 \u03c8 \u03c8(st) \u2212Eat\u223c\u02dc \u03c0\u03c6 \u03c6 \u03c6(st)[Q\u03b8 s(st, at) \u2212log \u02dc \u03c0\u03c6 \u03c6 \u03c6(at | st)] \u00112, which is inspired by (Haarnoja et al., 2018). We call this variant Direct Policy learning with WGF and Variance reduction (DP-WGF-V). 5. Connections with Related Works Soft-Q learning Though motivated from different perspectives, our DP-WGF results in a similar algorithm as soft-Q learning with energy based policies (Haarnoja et al., 2017). However, DP-WGF is more general, in that we can de\ufb01ne different sampling networks, such as a BNN, which can be optimized with the proposed IP-WGF technique. Soft actor-critic (SAC) SAC (Haarnoja et al., 2018) is an improvement of soft-Q learning, introducing a similar V -network as ours and modeling the policy with a mixture of Gaussians. DP-WGF-V is related to SAC, but with a V -network from a different perspective (variance reduction). Importantly, we de\ufb01ne a gradient \ufb02ow for policy distributions, allowing optimization from a distribution perspective. Trust-region methods Trust-Region methods are known to stabilize policy optimization in RL (Schulman et al., 2015). Schulman et al. (2017a) illustrate the equivalence between soft Q-learning and policy gradient. In the original TRPO setting, an objective function is optimized subject to a constraint that the updated policy is not too far from the current policy, in terms of the KL divergence (see Section G for a more detailed descriptions). The theory of TRPO suggests adding a penalty to the objective instead of adopting a constraint, resulting in a similar form as our framework. However, we use the Wasserstein distance to penalize the previous and current policies, which is a weaker metric than the KL divergence, and potentially leads to more robust solutions. This is evidenced by the development of Wasserstein GAN (Arjovsky et al.). As a result, our framework can be regarded as a trust-region-based counterpart for solving the soft Q-learning (Haarnoja et al., 2017) and SVPG (Liu et al., \fPolicy Optimization as Wasserstein Gradient Flows Dataset PBP SVGD WGF Boston -2.57 \u00b1 0.09 -2.50\u00b10.03 \u22122.40 \u00b1 0.10 Concrete -3.16 \u00b1 0.02 -3.08\u00b10.02 \u22122.95 \u00b1 0.06 Energy -2.04 \u00b1 0.02 -1.77\u00b10.02 \u22120.73 \u00b1 0.08 Kin8nm 0.90 \u00b1 0.01 0.98 \u00b1 0.01 0.97 \u00b1 0.02 Naval 3.73 \u00b1 0.01 4.09\u00b10.01 4.11 \u00b1 0.02 CCPP -2.80 \u00b1 0.05 -2.82\u00b10.01 \u22122.78 \u00b1 0.01 Winequality -0.97 \u00b1 0.01 -0.93\u00b10.01 \u22120.87 \u00b1 0.04 Yacht -1.63 \u00b1 0.02 -1.23\u00b10.04 \u22120.99 \u00b1 0.15 Protein -2.97 \u00b1 0.00 -2.95\u00b10.00 \u22122.88 \u00b1 0.01 YearPredict -3.60\u00b1NA -3.58 \u00b1 NA \u22123.57 \u00b1 NA Table 1. Averaged predictions, with standard deviations, in terms of test log-likelihood. 2017). Similar arguments hold for other trust-region methods such as PPO (Schulman et al., 2017b) and Trust-PCL (Nachum et al., 2017), which improve TRPO with either different objective or trust-region constraints. Noisy exploration Adding noise to the parameters for noisy exploration (Fortunato et al., 2018; Plappert et al., 2018) can be interpreted as a special case of our IP-WGF framework with a single particle. Isotropic Gaussian noisy exploration corresponds to the maximum a posterior (MAP) solution with a Gaussian assumption on the posterior distributions of parameters, potentially leading to inferior solutions when the assumption is not met. By contrast, our method is endowed with the ability to explore multimodal distributions, by optimizing the parameter distribution directly. More details are provided in Section C of the SM. 6. Experiments We test the proposed WGF framework from two perspectives: i) the effectiveness of the proposed particleapproximation method for WGF, and ii) the advantages of the WGF framework for policy optimization. For i), a standard regression model to learn optimal parameter distributions, i.e., posterior distributions. For ii), we test our algorithms on several domains in OpenAI rllab and Gym (Duan et al., 2016). All experiments are conducted on a single Tesla P100. Detailed settings are given in the SM. 6.1. Regression We use a single-layer BNN as a regression model. The parameters of the BNN are treated probabilistically and optimized with our WGF framework. We compare WGF, SVGD (Liu & Wang, 2016), Bayesian Dropout (Gal & Ghahramani, 2016) and PBP (Hern\u00e1ndez-Lobato & Adams, 2015). The RMSprop optimizer is employed. Detailed experimental settings and datasets are described in Section D.2 of the SM. We adopt the root-mean-squared error (RMSE) and test log-likelihood as the evaluation criteria. The experimental results are shown in Table 1 (complete results are provided in Section D.2 of the SM). It is observed that our proposed WGF obtains better results in both metrics, partially due to the \ufb02exibility of our particle approximation algorithm, which solves the original WGF problem effectively. Figure 1. Learning curves by IP-WGF and SVPG with REINFORCE and A2C. 6.2. Indirect-policy learning For this group of experiments, we compare IP-WGF with SVPG (Liu et al., 2017), a state-of-the-art method for indirect-policy learning, considering three classical continuous-control tasks: Cartpole Swing-Up, Double Pendulum, and Cartpole. Only policy parameters are updated by IP-WGF or SVPG, while the critics are updated with TD-error. We train our agents for 100 iterations on the easier Cartpole domain and 1000 iterations on the other two domains. Following the settings in (Liu et al., 2017; Houthooft et al., 2016; Zhang et al., 2018), the policy is parameterized as a two-layer (25-16 hidden units) neural network with tanh activation function. The maximum horizon length is set to 500. A sample size of 5000 is used for policy gradient estimation. We use M = 16 particles to approximate parameter distributions, and h = 0.1 as the discretized stepsize. REINFORCE (Williams, 1992) and advantage actor critic (A2C) (Schulman et al., 2016) are used as strategies of policy learning. Figure 2 plots the mean (dark curves) and standard derivation (light areas) of rewards over 5 runs. It is clear that in all tasks IP-WGF consistently converges faster than SVPG, and \ufb01nally converges to higher average rewards. The results are comparable to (Houthooft et al., 2016). The experiments demonstrate that employing the Wasserstein gradient \ufb02ows on policy optimization improves the performance, as suggested by our theory. \fPolicy Optimization as Wasserstein Gradient Flows WGF-DP-V SAC TRPO-GAE DDPG Domain Threshold MaxReturn. Episodes MaxReturn Epsisodes MaxReturn Episodes MaxReturn Episodes Swimmer 100 181.60 76 180.83 112 110.58 433 49.57 N/A Walker 3000 4978.59 2289 4255.05 2388 3497.81 3020 2138.42 N/A Hopper 2000 3248.76 678 3146.51 736 2604 1749 1317 N/A Humanoid 2000 3077.84 18740 2212.51 26476 5411.15 32261 2230.60 34652 Table 2. WGF-DP-V, TRPO, SAC, and DDPG results showing the max average rewards attained and the episodes to cross speci\ufb01c reward thresholds. WGF-DP-V often learn more sample-ef\ufb01ciently than the baselines, and WGF-DP-V can solve dif\ufb01cult domains such as Humanoid better than DDPG. Figure 2. Average return in MuJoCo tasks by Soft-Q, SAC and DP-WGF-V (\ufb01rst row), and by DDPG, TRPO-GAE and DP-WGF-V (second row). From left to right, the tasks are: Swimmer, Hopper, Walker and Humanoid, respectively. 6.3. Direct-policy Learning We compare our DP-WGF and DP-WGF-V frameworks with existing off-policy and on-policy deep RL algorithms on several tasks in MuJoCo, e.g., SAC, Soft-Q, DDPG (offpolicy) and TRPO-GAE (on-policy). Our DP-WGF-V is considered to be an off-policy actor-critic method. For all methods, value function and policy are parameterized as two-layer (128-128 hidden units) neural networks with tanh as the activation function. The maximum horizon length is set to 1000 when simulating expected total rewards. Three easier tasks (Swimmer, Hopper and Walker) in MuJoCo can be solved by a wide range of algorithms; while the more complex benchmark, the 21-dimensional Humanoid, is known to be very dif\ufb01cult to solve with off-policy algorithms (Duan et al., 2016). Implementation details of the algorithms are speci\ufb01ed in Section E.3 of the SM. Effectiveness of the Wasserstein trust-region We evaluate DP-WGF-V against SAC, and DP-WGF against Soft-Q on four Mujoco tasks, as they are closely related to our algorithms. Figure 2 (\ufb01rst row) plots average returns over epochs on the tasks. Similarly, our WGF-based methods converge faster and better than their counterparts due to the introduction of WGFs. Furthermore, by variance reduction, DP-WGF-V signi\ufb01cantly outperforms DP-WGF on all tasks. Comparisons with popular baselines Finally we compare DP-WGF-V with TRPO-GAE (Schulman et al., 2016) and DDPG (Lillicrap et al., 2016) on the same Mujoco tasks. In general, TRPO-GAE has been a state-of-the-art method for policy optimization. Figure 2 (second row) plots average returns over episodes, and it is observed that DP-WGF-V consistently outperforms other algorithms. Table 2 summarizes some key statistics, including the best attained average rewards and the episodes to reach the reward thresholds. It is observed that DP-WGF-V consistently outperform TRPOGAE and DDPG in terms of sample complexity, and often achieves higher rewards than TRPO-GAE. A particularly notable case, on Humanoid, shows DP-WGF-V substantially outperforms TRPO-GAE in terms of sample ef\ufb01ciency, while DDPG cannot learn a good policy at all. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1801.00085v2",
+                    "title": "Learning Structural Weight Uncertainty for Sequential Decision-Making",
+                    "abstract": "Learning probability distributions on the weights of neural networks (NNs)\nhas recently proven beneficial in many applications. Bayesian methods, such as\nStein variational gradient descent (SVGD), offer an elegant framework to reason\nabout NN model uncertainty. However, by assuming independent Gaussian priors\nfor the individual NN weights (as often applied), SVGD does not impose prior\nknowledge that there is often structural information (dependence) among\nweights. We propose efficient posterior learning of structural weight\nuncertainty, within an SVGD framework, by employing matrix variate Gaussian\npriors on NN parameters. We further investigate the learned structural\nuncertainty in sequential decision-making problems, including contextual\nbandits and reinforcement learning. Experiments on several synthetic and real\ndatasets indicate the superiority of our model, compared with state-of-the-art\nmethods.",
+                    "authors": "Ruiyi Zhang, Chunyuan Li, Changyou Chen, Lawrence Carin",
+                    "published": "2017-12-30",
+                    "updated": "2018-04-02",
+                    "primary_cat": "stat.ML",
+                    "cats": [
+                        "stat.ML",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Deep learning has achieved state-of-the-art performance on a wide range of tasks, including image classi\ufb01cation [Krizhevsky et al., 2012], language modeling [Sutskever et al., 2014], and game playing [Silver et al., 2016]. One challenge in training deep neural networks (NNs) is that such models may over\ufb01t to the observed data, yielding over-con\ufb01dent decisions in learning tasks. This is partially because most NN learning only seeks a point estimate for the model paProceedings of the 21st International Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spain. PMLR: Volume 84. Copyright 2018 by the author(s). rameters, failing to quantify parameter uncertainty. A natural way to ameliorate these problems is to adopt a Bayesian neural network (BNN) formulation. By imposing priors on the weights, a BNN utilizes available data to infer an approximate posterior distribution on NN parameters [MacKay, 1992]. When making subsequent predictions (at test time), one performs model averaging over such learned uncertainty, e\ufb00ectively yielding a mixture of NN models [Gal and Ghahramani, 2016, Zhang et al., 2016, Liu and Wang, 2016, Li et al., 2016a, Chen and Zhang, 2017]. BNNs have shown improved performance on modern achitectures, including convolutional and recurrent networks [Li et al., 2016b, Gan et al., 2017, Fortunato et al., 2017]. For computational convenience, traditional BNN learning typically makes two assumptions on the weight distributions: independent isotropic Gaussian distributions as priors, and fully factorized Gaussian proposals as posterior approximation when adopting variational inference [Hern\u00e1ndez-Lobato and Adams, 2015, Blundell et al., 2015, Liu and Wang, 2016, Feng et al., 2018, Pu et al., 2017b]. By examining this procedure, we note two limitations: (i) the independent Gaussian priors can ignore the anticipated structural information between the weights, and (ii) the factorized Gaussian posteriors can lead to unreasonable approximation errors and underestimate model uncertainty (underestimate variances). Recent attempts have been made to overcome these two issues. For example, [Louizos and Welling, 2016, Sun et al., 2017] introduced structural priors with the matrix variate Gaussian (MVG) distribution [Gupta and Nagar, 1999] to impose dependency between weights within each layer of a BNN. Further, nonparametric variational inference methods, e.g., Stein variational gradient descent (SVGD) [Liu and Wang, 2016], iteratively transport a set of particles to approximate the target posterior distribution (without making explicit assumptions about the form of the posterior, avoiding the aforementioned factorization asarXiv:1801.00085v2  [stat.ML]  2 Apr 2018 \fLearning Structural Weight Uncertainty for Sequential Decision-Making sumption). SVGD represents the posterior approximately in terms of a set of particles (samples), and is endowed with guarantees on the approximation accuracy when the number of particles is exactly in\ufb01nity [Liu, 2017]. However, since the updates within SVGD learning involve kernel computation in the parameter space of interest, the algorithm can be computationally expensive in a high-dimensional space. This becomes even worse in the case of structural priors, where a large amount of additional parameters are introduced, rendering SVGD ine\ufb03cient when directly applied for posterior inference. We propose an e\ufb03cient learning scheme for accurate posterior approximation of NN weights, adopting the MVG structural prior. We provide a new perspective to unify previous structural weight uncertainty methods [Louizos and Welling, 2016, Sun et al., 2017] via the Householder \ufb02ow [Tomczak and Welling, 2016]. This perspective allows SVGD to approximate a target structural distribution in a lower-dimensional space, and thus is more e\ufb03cient in inference. We call the proposed algorithm Structural Stein Variational Gradient Descent (S2VGD). We investigate the use of our structural-weightuncertainty framework for learning policies in sequential decision problems, including contextual bandits and reinforcement learning. In these models, uncertainty is particularly important because greater uncertainty on the weights typically introduces more variability into a decision made by a policy network [Kolter and Ng, 2009], naturally leading the policy to explore. As more data are observed, the uncertainty decreases, allowing the decisions made by a policy network to become more deterministic as the environment is better understood (exploitation when the policy becomes more con\ufb01dent). In all these models, structural weight uncertainty is inferred by our proposed S2VGD. We conduct several experiments, \ufb01rst demonstrating that S2VGD yields e\ufb00ective performance on classic classi\ufb01cation/regression tasks. We then focus our experiments on the motivating applications, sequential decision problems, for which accounting for NN weight uncertainty is believed to be particularly bene\ufb01cial. In these applications the proposed method demonstrates particular empirical value, while also being computationally practical. The results show that structural weight uncertainty gives better expressive power to describe uncertainty driving better exploration. 2 Preliminaries 2.1 Matrix variate Gaussian distributions The matrix variate Gaussian (MVG) distribution [Gupta and Nagar, 1999] has three parameters, describing the probability of a random matrix W \u2208 R\u21131\u00d7\u21132: p(W) \u225cMN(W; M, U, V) = exp \u00121 2tr[V\u22121(W \u2212M)\u22a4U\u22121(W \u2212M)] \u0013 (2\u03c0)\u21131\u21132/2|V|\u21131/2|U|\u21132/2 (1) where M \u2208R\u21131\u00d7\u21132 is the mean of the distribution. U \u2208R\u21131\u00d7\u21131 and V \u2208R\u21132\u00d7\u21132 encode covariance information for the rows and columns of W respectively. The MVG is closely related to the multivariate Gaussian distribution. Lemma 1 (Golub and Van Loan [2012]). Assume W follows the MVG distribution in (1), then vec(W) \u223cN(vec(M), V \u2297U) (2) where vec(M) is the vectorization of M by stacking the columns of M, and \u2297denotes the standard Kronecker product [Golub and Van Loan, 2012]. Furthermore, a linear transformation of an MVG distribution is still an MVG distribution. Lemma 2 (Golub and Van Loan [2012]). Assume W follows the MVG distribution in (1), A \u2208R\u21132\u00d7\u21131, C \u2208 R\u21132\u00d7\u21131, then, B \u225cAW \u223cMN(B; AM, AUA\u22a4, V) B \u225cWC \u223cMN(B; MC, U, C\u22a4VC) (3) MVG priors for BNNs For classi\ufb01cation and regression tasks on data D = {d1, \u00b7 \u00b7 \u00b7 , dN}, where di = {xi, yi}, with input xi and output yi, an L-layer NN parameterizes the mapping {g\u2113}L \u2113=1, de\ufb01ning the prediction of yi for xi as: \u02c6 yi = f(xi) = gL \u25e6gL\u22121 \u25e6\u00b7 \u00b7 \u00b7 \u25e6g0(xi), \u2200i. (4) where \u25e6represents function composition, i.e., A \u25e6B means A is evaluated on the output of B. Each layer g\u2113represents a nonlinear transformation. For example, with the Recti\ufb01ed Linear Unit (ReLU) activation function [Nair and Hinton, 2010], g\u2113(xi) = ReLU(W\u22a4 \u2113xi + b\u2113), where ReLU(x) \u225cmax (0, x), W\u2113 is the weight matrix for the \u2113th-layer, and b\u2113the corresponding bias term. The MVG can be adopted as a prior for the weight matrix in each layer, to impose the prior belief that there are intra-layer weight correlations, W\u2113\u223cMN(W; 0, U\u2113, V\u2113), (5) where the covariances U\u2113, V\u2113have components drawn independently from Inv-Gamma(a0, b0). The parameters are \u03b8 \u225c{W\u2113, log U\u2113, log V\u2113}, and the above distributions represent the prior p(\u03b8). Our goal with BNNs is to learn layer-wise structured weight \fRuiyi Zhang1 Chunyuan Li1 Changyou Chen2 Lawrence Carin1 uncertainty, described by the posterior distribution p(\u03b8|D) \u221dp(\u03b8)p(D|\u03b8), represented below as p for simplicity. When U = \u03c3I and V = \u03c3I, we reduce to BNNs with independent isotropic Gaussian priors [Blundell et al., 2015]. 2.2 Stein Variational Gradient Descent SVGD considers a set of particles {\u03b8i}M i=1 drawn from distribution q, and transforms them to better match the target distribution p, by update: \u03b8i \u2190\u03b8i + \u03f5\u03c6(\u03b8i), \u03c6 = arg max \u03c6\u2208F \u001a \u2202 \u2202\u03f5KL(q[\u03f5\u03c6]||p) \u001b , (6) where q[\u03f5\u03c6] is the updated empirical distribution, with \u03f5 as the step size, and \u03c6 as a function perturbation direction chosen to minimize the KL divergence between q and p. SVGD considers F as the unit ball of a vectorvalued reproducing kernel Hilbert space (RKHS) H associated with a kernel \u03ba(\u03b8, \u03b8\u2032). The RBF kernel is usually used as default. It has been shown [Liu and Wang, 2016] that: \u2212\u2202 \u2202\u03f5KL(q[\u03f5\u03c6]\u2225p)|\u03f5=0 = E\u03b8\u223cq[\u0393p\u03c6(\u03b8)], (7) with \u0393p\u03c6(\u03b8) \u225c\u2207\u03b8 log p(\u03b8|D)\u22a4\u03c6(\u03b8) + \u2207\u03b8 \u00b7 \u03c6(\u03b8), where \u2207\u03b8 log p(\u03b8) denotes the derivative of the logdensity of p; \u0393p is the Stein operator. Assuming that the update function \u03c6(\u03b8) is in a RKHS with kernel \u03ba(\u00b7, \u00b7), it has been shown in [Liu and Wang, 2016] that (7) is maximized with: \u03c6(\u03b8) = E\u03b8\u223cq[\u03ba(\u03b8, \u03b8\u2032)\u2207\u03b8 log p(\u03b8|D) + \u2207\u03b8\u03ba(\u03b8, \u03b8\u2032)]. (8) The expectation E\u03b8\u223cq[\u00b7] can be approximated by an empirical averaging of particles {\u03b8i}M i=1, resulting in a practical SVGD procedure as: \u03b8i \u2190\u03b8i + \u03f5 M M X j=1 h \u03ba(\u03b8j, \u03b8i)\u2207\u03b8j log p(\u03b8j|D) +\u2207\u03b8j\u03ba(\u03b8j, \u03b8i) i . (9) The \ufb01rst term to the right of the summation in (9) drives the particles \u03b8i towards the high probability regions of p, with information sharing across similar particles. The second term repels the particles away from each other, encouraging coverage of the entire distribution. SVGD applies the updates in (9) repeatedly, and the samples move closer to the target distribution p in each iteration. When using state-of-the-art stochastic gradient-based algorithms, e.g., RMSProp [Hinton et al., 2012] or Adam [Kingma and Ba, 2015], SVGD becomes a highly e\ufb03cient and scalable Bayesian inference method. Computational Issues Applying SVGD with structured distributions for NNs has many challenges. For a weight matrix W of size \u21131 \u00d7 \u21132, the number of parameters in U and V are \u21132 1 and \u21132 2, respectively. Hence, the total number of parameters \u03b8 needed to describe the distribution is \u21131\u21132 + \u21132 1 + \u21132 2, compared to \u21131\u21132 + 1 in traditional BNNs that employ isotropic Gaussian priors and factorization. The increase of parameter dimension by \u21132 1 + \u21132 2 \u22121 leads to signi\ufb01cant computational overhead. The problem becomes even more severe in two aspects in calculating the kernels: (i) The computation increases quadratically by M(M \u22121)(\u21132 1 + \u21132 2 \u22121)/2, (ii) the approximation to the repelling term in (9) using limited particles can be inaccurate in high dimensions. Therefore, it is desirable to transform the MVG distribution to a lower-dimensional representation. 3 BNN & Imposition of Structure 3.1 Reparameterization of the MVG Since the covariance matrices U and V are positive de\ufb01nite, we can decompose them as U = P\u039b1\u039b1P\u22a4, V = Q\u039b2\u039b2Q\u22a4, where P and Q are the corresponding orthogonal matrices, \u039b1 and \u039b2 are diagonal matrices with positive diagonal elements. According to Lemma 2, we show the following reparameterization of MVG: Proposition 3. For a random matrix C following an independent Gaussian distribution: vec(C) \u223cN(\u00b7, P\u22a4\u039b\u22121 1 M\u039b\u22121 2 Q, I), (10) the corresponding full-covariance MVG W in (1) can be reparameterized as W = P\u039b1C\u039b2Q\u22a4. The proof is in Section A of the Supplementary Material. Therefore, W drawn from MVG can be decomposed as the linear product of \ufb01ve random matrices: \u2022 C as a standard weight matrix with an independent Gaussian distribution. \u2022 \u039b1 and \u039b2 as the diagonal matrices, encoding the structural information within each row and column, respectively. \u2022 P and Q as orthogonal matrices, which characterize the structural information of weights between rows and columns, respectively. 3.2 MVG as Householder Flows Based on Proposition 3, we propose a layer decomposition: the one-layer weight matrix W with an MVG prior can be decomposed into a linear product of \ufb01ve matrices, as illustrated in Figure 1(a). Our layer decomposition provides an interesting interpretation for the original MVG layer: it is equivalent to several \fLearning Structural Weight Uncertainty for Sequential Decision-Making C \u21e41 \u21e42 P W \u21e5 \u21e5 \u21e5 \u21e5 Q (a) Layer Decomposition Q P or \u00b7 \u00b7 \u00b7 \u21e5 \u21e5 \u21e5 \u21e5 H1 H2 HK\u22121 HK (b) Householder Flow Figure 1: Illustration of the two proposed techniques to reduce parameter size in learning the distribution of W: (a) decomposition of W as a linear product of \ufb01ve matrices, and (b) approximation of P or Q as a linear product of K Householder matrices. Note that each rectangle indicates a matrix, \u201c\u00d7\u201d indicates the matrix product, and H in (b) is constructed using (12). within-layer transformations. The representations imposed in the standard weight matrix C are rotated by P and Q, and re-weighted by \u039b1 and \u039b2. Note that the layer decomposition maintains similar computational complexity as the original MVG layer. To reduce the computation bottleneck in the layer decomposition, we further propose to represent P and Q using Householder \ufb02ows [Tomczak and Welling, 2016]. Formally, a Householder transformation is a linear transformation that describes a re\ufb02ection about a hyperplane containing the origin. Householder \ufb02ow is a series of Householder transformations. Lemma 4 ([Sun and Bischof, 1995]). Any orthogonal matrix M of degree K can be expressed as a Householder \ufb02ow, i.e., a product of exactly K nontrivial Householder matrices., i.e., M = HKHK\u22121 \u00b7 \u00b7 \u00b7 H1. (11) Importantly, each Householder matrix H is constructed from a Householder vector v (which is orthogonal to the hyperplane): H = I \u22122 vv\u22a4 \u2225v\u22252 . (12) According to Lemma 4, P and Q can be represented as a product of Householder matrices {H(p) k } and {H(q) k }: P = H(p) K H(p) K\u22121 \u00b7 \u00b7 \u00b7 H(p) 1 Q = H(q) K H(q) K\u22121 \u00b7 \u00b7 \u00b7 H(q) 1 H(p) k = I \u22122v(p) k v(p)\u22a4 k / \u0010 v(p)\u22a4 k v(p) k \u0011 H(q) k = I \u22122v(q) k v(q)\u22a4 k / \u0010 v(q)\u22a4 k v(q) k \u0011 , (13) where v(p) k and v(q) k are the kth Householder vector for P and Q, respectively. This is illustrated in Figure 1 (b). Note that the degree K \u2264min{\u21131, \u21132}, with proof in Section B of Supplementary Material. In practice, K is a trade-o\ufb00hyperparameter, balancing the approximation accuracy and computation trade-o\ufb00. Since Householder \ufb02ows allow one to represent P or Q as K Householder vectors, the parameter sizes reduce from \u21132 1 and \u21132 2 to K\u21131 and K\u21132. Overall, we can model W with structured weight priors using only (K + 1)(\u21131 + \u21132) + \u21131\u21132 parameters. Therefore, we can e\ufb03ciently capture the structure information with only a slight increase of computational cost (i.e., (K + 1)(\u21131 + \u21132)). Interestingly, our method provides a unifying perspective of previous methods on learning structured weight uncertainty. In terms of prior distributions, when K = 0 (i.e., P = Q = I), our reparameterization reduces to [Louizos and Welling, 2016]. When K = 1, and \u039b1 = \u039b2 = I, our reparameterization reduces to [Sun et al., 2017]. In terms of posterior learning methods, when P = Q = \u039b1 = \u039b2 = I and M > 1, S2VGD reduces to SVGD; when M = 1, it reduces to learning an MAP solution. 3.3 Structural BNNs Revisited We can leverage the layer decomposition and Householder \ufb02ow above to construct an equivalent BNN by approximating the \u2113th MVG layer in (5) with standard Gaussian weight matrices: p(C|\u03bb) = N (C\u2113; 0, \u03bb) , p(v(p) k\u2113|\u03c6) = N \u0010 v(p) k\u2113; 0, \u03c6I \u0011 , p(v(q) k\u2113|\u03c6) = N \u0010 v(q) k\u2113; 0, \u03c6I \u0011 , p(\u039b(1)|\u03c8) = N \u0010 \u039b(1) \u2113; 0, \u03c8I \u0011 p(\u039b(2)|\u03c8) = N \u0010 \u039b(2) \u2113; 0, \u03c8I \u0011 , \u03bb, \u03c6, \u03c8 \u223cInv-Gamma(\u00b7; a\u2113, b\u2113). (14) The forms of the likelihood for the last layer are de\ufb01ned according to the speci\ufb01c applications. For regression problems on real-valued response y: y|x, WL \u223cN(y; f(x, WL), \u03b3I) \u03b3 \u223cInv-Gamma(\u00b7; aL, bL) , (15) with f(\u00b7) a neural network de\ufb01ned in (4). For classi\ufb01cation problems on discrete labels y: y|x, WL \u223cCategorical(y; Softmax(f(x, WL)). (16) Note that WL can follow the same proposed techniques to reduce parameter size. Therefore, standard SVGD algorithms can be applied to sample from the posterior distribution of each model parameter. Intuitively, in SVGD the kernel function governs the interactions between particles, which employs this information to accelerate convergence and provide better \fRuiyi Zhang1 Chunyuan Li1 Changyou Chen2 Lawrence Carin1 performance. Similarly, the Householder \ufb02ow, encoding structural information, controls the interactions between weights in each particle. 4 Sequential Decision-Making A principal motivation for the proposed S2VGD framework is sequential decision-making, including contextual multi-arm bandits (CMABs) and Markov decision processes (MDPs). A challenge in sequential decision-making in the face of uncertainty is the exploration/exploitation trade-o\ufb00: the trade-o\ufb00between either taking actions that are most rewarding according to the current knowledge, or taking exploratory actions, which may be less immediately rewarding, but may lead to better-informed decisions in the future. In a Bayesian setting, the exploration/exploration tradeo\ufb00is naturally addressed by imposing uncertainty into the parameters of a policy model. 4.1 CMABs and Stein Thompson Sampling CMABs model stochastic, discrete-time and \ufb01nite action-state space control problems. A CMAB is formally de\ufb01ned as a tuple C = \u27e8S, A, Ps, Pr, r\u27e9, where S is the state/context space, A the action/arm space, r \u2208R is the reward, Ps and Pr are the unknown environment distributions to draw the context and reward, respectively. At each time step t, the agent (i) \ufb01rst observes a context st \u2208S, drawn i.i.d. over time from Ps; then (ii) chooses an action at at \u2208A and observes a stochastic reward rt(at, st), which is drawn i.i.d. over time from Pr(\u00b7|at, st), conditioned on the current context and action. The agent makes decisions via a policy \u03c0(a|s) that maps each context to a distribution over actions, yielding the probability of choosing action a in state s. The goal in CMABs is to learn a policy to maximize the expected total reward in T interactions: J(\u03c0) = EPs,\u03c0,Pr PT t=1 rt. We represent the policy using a \u03b8-parameterized neural network \u03c0\u03b8(a|s), where MVG priors p(\u03b8) are employed on the weights. At each time t, given the past observations Dt \u225c{d}t j=1, where dj = (sj, aj, rj), the posterior distribution of \u03b8t is updated as p(\u03b8t|Dt) \u221d Qt j=1 p(dj|\u03b8)p(\u03b8). Thompson sampling [Thompson, 1933] is a popular method to solve CMABs [Li et al., 2011]. It approximates the posterior p(\u03b8|Dt) in an online manner. At each step, Thompson sampling (i) \ufb01rst draws a set of parameter samples, then (ii) picks the action by maximizing the expected reward over current step, i.e., at = arg maxa Er\u223cPr(\u00b7|a,st;\u03b8t)rt, (iii) collects data samples after observing the reward rt, and (iv) updates posterior of the policy. We apply the proposed S2VGD for the updates in the \ufb01nal step, and call the new procedure Stein Thompson sampling, summarized in Algorithm 1. Note our Stein Thompson sampling is a general scheme for exploration/exploitation balance in CMABs. The techniques in [Russo et al., 2017, Kawale et al., 2015] can be adapted in this framework; we leave this for future work. Algorithm 1 Stein Thompson Sampling Require: D = \u2205; initialize particles \u03980 = {\u03b8i}M i=1; 1: for t = 0, 1, 2, . . . , T do 2: Receive context st \u223cPs; 3: Draw a particle \u02c6 \u03b8t from \u0398t; 4: Select at = arg maxa Er\u223cPr(\u00b7|a,st; \u02c6 \u03b8t)rt; 5: Observe reward rt \u223cPr, by performing at; 6: Collect observation: Dt+1 = Dt \u222a(st, at, rt); 7: Update \u0398t+1, according to SVGD in (9); 8: end for 4.2 MDPs and Stein Policy Gradient An MDP is a sequential decision-making procedure in a Markovian dynamical system. It can be seen as an extension of the CMAB, by replacing the context with the notion of a system state, that may dynamically change according to the performed actions and previous state. Formally, an MDP de\ufb01nes a tuple M = \u27e8S, A, Ps, Pr, r, \u03b3\u27e9, which is similar to a CMAB C except that (i) the next state st+1 is now conditioned on state st and action at, i.e., st+1 \u223cPs(\u00b7|st, at); and (ii) a discount factor 0 < \u03b3 < 1 for the reward is considered. The goal is to \ufb01nd a policy \u03c0(a|s) to maximize the discounted expected reward: J(\u03c0) = EPs,\u03c0,Pr PT t=1 \u03b3trt. Policy gradient [Sutton and Barto, 1998] is a family of reinforcement learning methods that solves MDPs by iteratively updating the parameters \u03b8 of the policy to maximize J(\u03b8) \u225cJ(\u03c0\u03b8(a|s)). Instead of searching for a single policy parameterized by \u03b8, we consider adopting an MVG prior for p(\u03b8), and learning its variational posterior distribution q(\u03b8) using S2VGD. Following [Liu et al., 2017], the objective function is modi\ufb01ed as: max q {Eq(\u03b8)[J(\u03b8)] \u2212\u03b1KL(q\u2225p)}, (17) where \u03b1 \u2208 [0, +\u221e) is the temperature hyperparameter to balance exploitation and exploration in the policy. The optimal distribution is shown to have a simple closed form [Liu et al., 2017]: q(\u03b8) \u221dexp \u0012 1 \u03b1J(\u03b8) \u0013 p(\u03b8). (18) \fLearning Structural Weight Uncertainty for Sequential Decision-Making We iteratively approximate the target distribution as: \u25b3\u03b8i = \u03f5 M M X j=1 [\u2207\u03b8j \u0012 1 \u03b1J(\u03b8j) + log p(\u03b8j) \u0013 \u03ba(\u03b8i, \u03b8j) + \u2207\u03b8j\u03ba(\u03b8j, \u03b8i)], (19) where J(\u03b8) can be approximated with REINFORCE [Williams, 1992] or advantage actor critic [Schulman et al., 2016]. We note two advantages of S2VGD in sequential decision-making: (i) the structural priors can characterize the \ufb02exible weight uncertainty, thus providing better exploration-exploitation when learning the policies; (ii) the e\ufb03cient approximation scheme provides accurate representation of the true posterior while maintaining similar online-processing speed. 5 Experiments To demonstrate the e\ufb00ectiveness of our S2VGD, we \ufb01rst conduct experiments on the standard regression and classi\ufb01cation tasks, with real datasets (two synthetic experiments on classi\ufb01cation and regression are given in the Supplementary Material). The superiority of S2VGD is further demonstrated in the experiments on contextual bandits and reinforcement learning. We compare S2VGD with related Bayesian learning algorithms, including VMG [Louizos and Welling, 2016], PBP_MV [Sun et al., 2017], and SVGD [Liu and Wang, 2016]. The RMSprop optimizer is employed if there is no speci\ufb01c declaration. For SVGDbased methods, we use a RBF kernel \u03ba(\u03b8, \u03b8\u2032) = exp(\u2212\u2225\u03b8 \u2212\u03b8\u2032\u22252 2/h), with the bandwidth set to h = med2/ log M. [Oates et al., 2016, Gorham and Mackey, 2017] Here med is the median of the pairwise distance between particles. The hyper-parameters a\u2113= 1, b\u2113= 0.1. The experimental codes of this paper are available at: https : //github.com/zhangry868/S2VGD. We \ufb01rst study the role of hyperparameters in S2VGD: the number of Householder transformations K and the number of particles M. This is investigated by a classi\ufb01cation task from [Liu and Wang, 2016] on the Covertype dataset with 581,012 data points and 54 features. Figure 2: Impact of K and M. We perform 5 runs for each setting and report the mean of testing accuracy in Figure 2. As expected, increasing M or K improves the performance, as they lead to a more accurate approximation. Interestingly, when M is small, increasing K gives signi\ufb01cant improvement. Furthermore, when M is large, the change of K yields similar performance. Therefore, we set K = 1 and M = 20 unless otherwise speci\ufb01ed. 5.1 Regression We use a single-layer BNN for regression tasks. Following [Li et al., 2015], 10 UCI public datasets are considered: 100 hidden units for 2 large datasets (Protein and YearPredict), and 50 hidden units for the other 8 small datasets. We repeat the experiments 20 times for all datasets except for Protein and YearPredict, which we repeat 5 times and once, respectively, for computation considerations [Sun et al., 2017]. The batch size for the two large datasets is set to 1000, while it is 100 for the small datasets. The datasets are randomly split into 90% training and 10% testing. We adopt the root mean squared error (RMSE) and test log-likelihood as the evaluation criteria. The experimental results are shown in Table 1, from which we observe that i) weight structure information is useful (SVGD is the only method without structure, and it yields inferior performance); ii) algorithms with non-parametric assumptions, i.e., the Stein-based methods, perform better; and iii) when combined with structure information, our method achieves state-of-the-art results. 5.2 Classi\ufb01cation We perform the classi\ufb01cation tasks on the standard MNIST dataset, which consists of handwritten digits of size 28 \u00d7 28, with 50,000 images for training and 10,000 for testing. A two-layer model 784-X-X-10 with ReLU activation function is used, and X is the number of hidden units for each layer. The training epoch is set to 100. The test errors for network (X-X) sizes 400400 and 800-800 are reported in Table 2. We observe that the Bayesian methods generally perform better than their optimization counterparts. The proposed S2VGD improves SVGD by a signi\ufb01cant margin. Increasing K also improves performance, demonstrating the advantages of incorporating structured weight uncertainty into the model. See [Li et al., 2016a, Louizos and Welling, 2016, Blundell et al., 2015] for details on the other methods with which we compare. We wish to verify that the performance gain of S2VGD is due to the special structural design of the network architecture, rather than the increasing number of model parameters. This is demonstrated by training a NN with 415-415 hidden units using SVGD, which yields test error 1.49%. It has slightly more parameters than our 400-400 network trained by S2VGD (K=10), but worse performance. \fRuiyi Zhang1 Chunyuan Li1 Changyou Chen2 Lawrence Carin1 Table 1: Averaged predictions with standard deviations in terms of RMSE and log-likelihood on test sets. Test RMSE Test Log likelihood Dataset VMG PBP_MV SVGD S2VGD VMG PBP_MV SVGD S2VGD Boston 2.70 \u00b1 0.13 2.76 \u00b1 0.43 2.96\u00b10.10 2.56 \u00b1 0.33 -2.46 \u00b1 0.09 -3.01 \u00b1 0.26 -2.50\u00b10.03 \u22122.43 \u00b1 0.10 Energy 0.54 \u00b1 0.02 0.48 \u00b1 0.04 1.37\u00b10.05 0.38 \u00b1 0.02 -1.06 \u00b1 0.03 -2.37 \u00b1 0.03 -1.77\u00b10.02 \u22120.55 \u00b1 0.04 Concrete 4.89 \u00b1 0.12 4.66 \u00b1 0.44 5.32\u00b10.10 4.25 \u00b1 0.37 -3.01 \u00b1 0.03 -3.22 \u00b1 0.05 -3.08\u00b10.02 \u22122.90 \u00b1 0.07 Kin8nm 0.08 \u00b1 0.00 0.08 \u00b1 0.00 0.09\u00b10.00 0.07 \u00b1 0.00 1.10 \u00b1 0.01 0.78 \u00b1 0.02 0.98\u00b10.01 1.15 \u00b1 0.01 Naval 0.00 \u00b1 0.00 0.00 \u00b1 0.00 0.00 \u00b1 0.00 0.00 \u00b1 0.00 2.46 \u00b1 0.00 4.37 \u00b1 0.17 4.09\u00b10.01 4.79 \u00b1 0.05 CCPP 4.04 \u00b1 0.04 3.91 \u00b1 0.09 4.03\u00b10.03 3.84 \u00b1 0.08 -2.82 \u00b1 0.01 -2.81 \u00b1 0.02 -2.82\u00b10.01 \u22122.77 \u00b1 0.02 Winequality 0.61 \u00b1 0.04 0.61 \u00b1 0.02 0.61\u00b10.01 0.59 \u00b1 0.02 -0.95 \u00b1 0.01 -0.99 \u00b1 0.07 -0.93\u00b10.01 \u22120.90 \u00b1 0.03 Yacht 0.48 \u00b1 0.18 0.53 \u00b1 0.14 0.86\u00b10.05 0.47 \u00b1 0.11 -1.30 \u00b1 0.02 -1.67 \u00b1 0.24 -1.23\u00b10.04 \u22120.81 \u00b1 0.14 Protein 4.13 \u00b1 0.02 4.38 \u00b1 0.01 4.61\u00b10.01 4.15 \u00b1 0.04 \u22122.84 \u00b1 0.00 -2.91 \u00b1 0.03 -2.95\u00b10.00 -2.84 \u00b1 0.01 YearPredict 8.78 \u00b1 NA 8.84 \u00b1 NA 8.68 \u00b1 NA 8.73 \u00b1 NA -3.59 \u00b1 NA -3.58\u00b1NA -3.62 \u00b1 NA \u22123.57 \u00b1 NA (a) Simulation results on Mushroom (b) New Article Recommendation (c) Particle Size on Yahoo!Today Figure 3: Experimental results of Contextual Bandits Table 2: Classi\ufb01cation error of FNN on MNIST. Method Test Error 400-400 800-800 S2VGD (K=10) 1.36% 1.30% S2VGD (K=1) 1.43% 1.39% SVGD 1.53% 1.47% SGLD 1.64% 1.41% RMSprop 1.59% 1.43% RMSspectral 1.65% 1.56% SGD 1.72% 1.47% VMG, variational dropout 1.15% BPB, Gaussian 1.82% 1.99% BPB, scale mixture 1.32% 1.34% SGD, dropout 1.51% 1.33% 5.3 Contextual Bandits Simulation We \ufb01rst simulate a contextual-bandit problem with the UCI mushrooms dataset. Following [Blundell et al., 2015], the provided features of each mushroom are regarded as the context. A reward of 5 is given when an agent eats an edible mushroom. Otherwise, if a mushroom is poisonous and the agent eats it, a reward of -10 or 5 will be received, both with probability 0.5; if the agent decides not to eat the mushroom, it receives a reward of 0. We use a twolayer BNN with ReLU and 50 hidden units to represent the policy. We compared our method with a standard baseline, \u03b5-greedy policy with \u03b5 = 0% (pure greedy), 1%, 3%, respectively [Sutton and Barto, 1998]. We evaluate the performance of di\ufb00erent algorithm by cumulative regret [Sutton and Barto, 1998], a measure of the loss caused by playing suboptimal bandit arms. The results are plotted in Figure 3(a). Thompson sampling with 3 di\ufb00erent strategies to update policy are considered: S2VGD, SVGD and VMG. S2VGD shows lower regret at the beginning of learning than SVGD, and the lowest \ufb01nal cumulative regret among all methods. We hypothesize that our method captures the internal weight correlation, and the structural uncertainty can e\ufb00ectively help the agent learn to make less mistakes in exploration with less observations. News Article Recommendation We consider personalized news article recommendation on Yahoo! [Li et al., 2010], where each time a user visits the portal, a news article from a dynamic pool of candidates is recommended based on the user\u2019s pro\ufb01le (context). The dataset contains 45,811,883 user visits to the Today Module in a 10-day period in May 2009. For each visit, both the user and each of the 20 candidate articles are associated with a feature vector of 6 dimensions [Li et al., 2010]. The goal is to recommend an article to a user based on its behavior, or, formally, maximize the total number of clicks on the recommended articles. The procedure is regraded as a CMAB problem, where articles are treated as arms. The reward is de\ufb01ned to be 1 if the article is clicked on and 0 otherwise. A one-layer NN with ReLU and 50 hidden units is used as the policy \fLearning Structural Weight Uncertainty for Sequential Decision-Making network. The classic LinUCB [Li et al., 2010] is also used as the baseline. The performance is evaluated by an unbiased o\ufb04ine evaluation protocol: the average normalized accumulated click-through-rate (CTR) in every 20000 observations [Li et al., 2010, 2011]. The normalized CTRs are plotted in Figure 3(b). It is clear that S2VGD consistently outperforms other methods. The fact that S2VGD and VMG perform better than SVGD and the baseline LinUCB indicates that structural information helps algorithms to better balance exploration and exploitation. To further verify the in\ufb02uence of the particle size M on the sequential decision-making, we vary the M from 1 to 50 on the-\ufb01rst-day data. All algorithms are repeated 10 times and their mean performances are plotted in Figure 3(c). We observe that the CTR keeps improving when M becomes larger. S2VGD dominates the performance of SVGD with much higher CTRs, the gap becomes larger as M increases. Since larger M typically leads to more accurate posterior estimation of the policy, indicating again that the accurately learned structural uncertainty are bene\ufb01cial for CMABs. 5.4 Reinforcement Learning We apply our S2VGD to policy gradient learning. All experiments are conducted with the OpenAI rllab toolkit [Duan et al., 2016]. Three classical continuous control tasks are considered: Cartpole Swing-Up, Double Pendulum, and Cartpole. Following the settings in [Liu et al., 2017, Houthooft et al., 2016], the policy is parameterized as a two-layer (25-10 hidden units) neural network with tanh as the activation function. The maximal length of horizon is set to 500. SVGD and S2VGD use a sample size of 10000 for policy gradient estimation, and M = 16. For the easy task, Cartpole, all agents are trained for 100 episodes. For the two complex tasks, Cartpole Swing-Up and Double Pendulum, all agents are trained up to 1000 episodes. We consider two di\ufb00erent methods to estimate the gradients: REINFORCE [Williams, 1992] and advantage actor critic (A2C) [Schulman et al., 2016] Figure 4 plots the mean (dark curves) and standard derivation (light areas) of discounted rewards over 5 runs. In all tasks and value-estimation setups, S2VGD converges faster than SVGD and \ufb01nally converges to higher average rewards. The results are even comparable to [Houthooft et al., 2016], in which a subtle reward mechanism is incorporated to encourage exploration. It demonstrates that simply adding structural information on policy networks using S2VGD improves the agent\u2019s exploration ability. We also add a baseline method called SVGD* that applies SVGD to train a network of similar size (25-16 hidden units) with the one reparameterized by S2VGD (K=4). The fact that Figure 4: Learning curves by S2VGD and SVGD with REINFORCE (left) and A2C (right). S2VGD (K=4) converges better than SVGD* demonstrates that our structural uncertainty is key to excellent performance. 6"
+                }
+            ],
+            "Zhiqiang Wang": [
+                {
+                    "url": "http://arxiv.org/abs/2311.02315v1",
+                    "title": "Counting Manatee Aggregations using Deep Neural Networks and Anisotropic Gaussian Kernel",
+                    "abstract": "Manatees are aquatic mammals with voracious appetites. They rely on sea grass\nas the main food source, and often spend up to eight hours a day grazing. They\nmove slow and frequently stay in group (i.e. aggregations) in shallow water to\nsearch for food, making them vulnerable to environment change and other risks.\nAccurate counting manatee aggregations within a region is not only biologically\nmeaningful in observing their habit, but also crucial for designing safety\nrules for human boaters, divers, etc., as well as scheduling nursing,\nintervention, and other plans. In this paper, we propose a deep learning based\ncrowd counting approach to automatically count number of manatees within a\nregion, by using low quality images as input. Because manatees have unique\nshape and they often stay in shallow water in groups, water surface reflection,\nocclusion, camouflage etc. making it difficult to accurately count manatee\nnumbers. To address the challenges, we propose to use Anisotropic Gaussian\nKernel (AGK), with tunable rotation and variances, to ensure that density\nfunctions can maximally capture shapes of individual manatees in different\naggregations. After that, we apply AGK kernel to different types of deep neural\nnetworks primarily designed for crowd counting, including VGG, SANet, Congested\nScene Recognition network (CSRNet), MARUNet etc. to learn manatee densities and\ncalculate number of manatees in the scene. By using generic low quality images\nextracted from surveillance videos, our experiment results and comparison show\nthat AGK kernel based manatee counting achieves minimum Mean Absolute Error\n(MAE) and Root Mean Square Error (RMSE). The proposed method works particularly\nwell for counting manatee aggregations in environments with complex background.",
+                    "authors": "Zhiqiang Wang, Yiran Pang, Cihan Ulus, Xingquan Zhu",
+                    "published": "2023-11-04",
+                    "updated": "2023-11-04",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Recent advancements in Artificial Intelligence (AI) have allowed it to be used in a wide spectrum of fields1. Especially, AI applications in ecology to save species have gained more importance in the last decade with the increasing number of endangered animals. There are many ways to use computational methods to help save endangered species, such as detection the presence of the species2 and counting species to collect information about numbers and density etc. In this paper, we are proposing a deep neural network based method to help save manatees, an endangered species. Human activities impact environment in numerous ways including deforestation, overpopulation, poaching, over-fishing, and climate change. These negative impacts directly affect the physical environment, posing risks and generating opportunities for wildlife. Sometime, this dispersion benefits species, mostly invasive species3. In most cases, however, human-assisted dispersion adversely affects the natural world. As a result, wildlife populations are declining at an alarming rate, and the extinction rates are now up to 100 times higher than the normal extinction rate4. Manatees are one of the wildlife species being affected by human-related threads5. There are four species of manatees in the world: Trichechus inunguis (Amazonian manatee), Trichechus pygmaeus (dwarf manatee), Trichechus senegalensis (West African manatee) and Trichechus manatus (West Indian manatee)6. Amazonian manatees and dwarf manatees populate in freshwater habitats. The dwarf manatees are contentious species only found in the river Aripua\u00f1a in Brasil and closely linked to the Amazonian manatee7,8. The West African manatees are distributed from Angola to Senegal8. The West Indian manatee prefers shallow coastal habitats such as rivers and estuaries. They can be found from Brazil to Florida and all the way around the Caribbean islands. The West Indian Manatee has two subspecies; Florida Manatee and Antillean Manatee, and both of them are considered endangered by IUCN (International Union for Conservation of Nature) 9. While some manatee species, such as the Florida Manatee, have a relatively limited geographical distribution, some manatee types inhabit a wide geographical range. As a result of this wide range, individual manatee migrations occur10. For example, during the 2021 winter, a manatee migration from Florida to the Mexican coast was observed11. Furthermore, some types of manatees, such as Antillean manatees, can live in a diverse range of ecosystems. In some cases, they inhabit clear saltwater in Belize and Mexico, visit rivers, reefs, and freshwater lagoons, whereas they live in salt or fresh water in Chiapas and Tabasco states with insufficient visibility12,13. Such diverse living habits and behaviors make it inherently difficult to track them. 1 arXiv:2311.02315v1  [cs.CV]  4 Nov 2023 \fOver the last decades, manatee populations have been continuously decreasing. Manatees tend to live as a group or individuals. Most of their populations exists as tiny isolates and are also low in density14. Furthermore, they are frequently scattered throughout huge bodies of water and display evasive behavior due to hunting pressure, making detection and counting challenging with existing approaches. Knowing the number of manatees and their gathering pattern in real-time is vital for understanding their population dynamics. The timeliness and accuracy of the count data upon which choices are made frequently determine the efficacy of management decision-making. In other words, improvements in counting techniques may portend better ecological results from management decisions. Meanwhile, manatees rely on sea grass as their primary food source. Because sea grass requires sunlight and shallow water to grow, manatees tend to stay in shallow water to hunt food, making them vulnerable to the environment, e.g. they have very little room/time to move away from (avoid) oncoming boats, resulting in deadly collisions if the boat drivers are not aware that they are approaching manatees. Using aerial survey data, counting estimates for manatees in southern Florida, USA, was developed, and environmental and temporal factors were discovered to impact distributions15,16. However, aerial surveys are time-consuming and costly, and the accuracy depends on factors such as observer bias, weather, and time of the day. Consequently, less time-consuming and less costly counting methods gain more importance in detecting the number of manatees. Furthermore, it is also crucial to have a method to provide a real-time count to allow ecologists to be aware of the threat early and act proactively to protect manatees. Although there are few researches applied to wildlife counting17, most of the existing counting methods18\u201320 are applied to crowds to count the number of people, due to their relevance to important applications such as urban planning and public safety. Fortunately, such advanced techniques in crowd counting can also be generalized to other fields such as wildlife counting, by taking specific characteristics of the objects into consideration. In this paper, we propose to use crowd counting methods to count manatee aggregations. Our goal is to accurately estimate number of manatees in a specific region, using low quality images as input. Due to numerous factors, as we have elaborated above, manatee counting is a challenging task. \u2022 Occlusion: Because manatees tend to live in herd. They frequently block each others when viewing from the surface. As a result, small manatees are likely to be partially or completely blocked from the view. \u2022 Distributions and Distortion: Due to diverse living habits and behaviors, manatees often present in different population density, perspective distortions, and lightning conditions. Without sufficient training data for each type of scene, it is difficult for a model to obtain accurate results for counting. \u2022 Reflections and Camouflage: Furthermore, water reflections tend to make manatees invisible in reflection areas, counting manatees from images captured from surface mounted camera is difficult. \u2022 Background: Finally, the high similarity of appearance between manatees and some elements in the background, such as fishes, rocks, imposes additional challenge to manatee counting. In order to address the above challenges and accurately estimate the density of the manatee, we propose a deep neural network based crowd counting method, which learn to estimate manatee density within an input image. Our method considers distortions caused by the perspective between the water space and the image plane. Furthermore, since the shape of the manatee is closer to an ellipse than a circle, we propose a method that uses an Anisotropic Gaussian kernel (AGK) to best represent the manatee contour, and estimate manatee density in the scene. By formatting manatee counting as a deep neural network density estimation learning task, our approach balances the labeling costs vs. counting efficiency. As a result, our method delivers a simple and high throughput solution for manatee counting requiring very little labeling efforts. Contribution Our research brings the following three unique contributions to enrich our data and algorithm design for domain specific tasks: \u2022 Deep Learning for Counting Manatees: We are among the first to introduce deep learning method to automatically count manatee through low-resolution images captured from surface mounted camera. This pioneering study not only addresses the technical challenges of counting in complex outdoor environments but also offers potential ways to aid endangered species. \u2022 Anisotropic Gaussian Kernel with Line Label Annotation: We introduce Anisotropic Gaussian Kernel combined with line label annotations to generate density map. This novel method can represent the unique shapes of manatees and deliver more precise counting results. \u2022 Manatee Counting Dataset: To validate our method and facilitate further research in this domain, we have developed a comprehensive manatee counting dataset, published through Github for public access (https://github.com/yeyimilk/deeplearning-for-manatee-counting). 2/18 \fFigure 1. Comparison of different types of labels with their difficulties for the labeling work. Related Work Our research is closely related to two research tasks: (1) learning to count number of objects within a scene; and (2) obtaining labels to support counting. Counting Methods Given the above-mentioned specific significance of counting, an increasing number of scholars have attempted to address the counting issue. Existing methods in the field mainly fall into three groups: detection-based21\u201323, regression-based, and density estimation based approaches including Convolutional Neural Network (CNN) based density estimation techniques24. It is worth noting that with the superb performance of deep learning methods, models based on CNN have largely dominated a variety of counting tasks. Previous counting methods mainly focused on counting people, although some methods focused on cars, animals, cells etc. Early works22,23,25 focused on counting people, using detection-based approach. Detection-based methods usually detect a head or person by using a sliding-window-like detector to count the number of people26. Recently many object detection tools (YOLO27, R-CNN28 etc.) are developed for object detection in sparse scenes. Nevertheless, these approaches do not show good performance in congested scenes because they require extraction of low-level characteristics. Crowd counting, especially in real complex environments, has received increasing attention in various domains. A recent survey29 indicates that majority count-related studies predominantly center around yield estimation, phenotyping, livestock monitoring, and insect monitoring, which together constitute approximately 97% of the applications29. A modified version of the Inception-ResNet architecture was employed to count tomatoes and simulated synthetic images were also used to enhance accuracy, although the system struggles to count green fruits30. A novel rice plant counting network, termed RPNet31, has four modules: feature encoder, attention block, initial density map generator, and attention map generator. Results indicate that RPNet outperforms MCNN, CSRNet, SANet, TasselNetV2, and FIDTM on certain high-resolution image datasets. Many detection-based and segmentation-based network have been proposed for counting fruits or plants, including those specific to blueberries, wheat spikes, panicles, pistachios and grapes clusters29. The application of deep learning models, particularly CNN combined with aerial imagery captured through UAV (Unmanned Aerial Vehicle)32 has shown significant promise in detection and counting. Models such as NasNet, Xception, YOLOv4, and YOLOv5 have demonstrated high accuracy, even under challenging conditions. While these models succeed in detecting cattle in various conditions, challenges like occlusion and diverse cattle breeds remain areas for improvement33\u201335. Our method addresses the unique challenges posed by counting submerged animals in outdoor open water environment, especially when dealing with long-shape entities such as manatees, from overhead webcam low quality images. Annotation and Labeling for Counting Labeling is a critical step to support learning. In order to learn to estimate number of objects in a scene, it is necessary to provide supervision (i.e. label information) for the learning algorithms. Such label information varies from labeling the whole image to each single object in the image. Total Count Annotation Using total count as a label in counting tasks can reduce annotation costs. This type of annotation is often referred to as Weakly-Supervised. Earlier counting frameworks36,37 mostly used this approach. Borstel et al.38 proposed a weakly-supervised solution based on the Gaussian process for density estimation. The training samples are divided into multiple subregions in their work, but each region is still annotated with only count labels. Modern counting methods try to dig for more information. Yang et al.39 used a soft-label sorting network to sort multiple images in the presence of only numerical labels. Then they used a regression network with a shared backbone to obtain the final number of people. MATT40 and Sam et al.41 introduced a small number of point-level annotations and trained mainly using total count annotations. Their work shows that only a small 3/18 \fnumber of point annotation samples can improve counting accuracy. JCTNet42 and TransCrowd43 use performant Transformer structures to regress the overall number directly. Further, CrowdMLP44 uses a more concise multilayer perceptron (MLP) as the overall architecture and introduces a self-supervised agent task to impose spatial cues implicitly. Although the total count annotation-based approach has the lowest annotation cost, due to the lack of spatial information, the average counting performance is lower than that of the density map-based approach. Bounding Box Annotation In order to provide accurate information of the object in the scene, bounding box annotations draws a minimum outer rectangle for each object appearing in the image. This allows underlying learning algorithms to precisely learn properties of the objects. Nevertheless, drawing precise rectangles is difficult and labor-intensive, especially for images with a high density of objects. In addition, bounding Box-based annotation methods cannot provide accurate predictions in extreme situations, such as low resolution or severe occlusion. Dot Annotation Dot annotation, one of the most commonly used techniques in recent years, marks each counted object as a dot, where the dot is usually located at the center of the object, e.g. the center of a person\u2019s head. As a result, dot annotation creates a locality point map where the sum of the point map equals to the total number of objects, and the learning can be carried out using information from the dots. Dot annotation tremendously reduces the labeling efforts, compared to others like bounding box annotation, and is also able to handle extreme cases like object occlusions. In reality, it is, however, difficult to train a neural network (NN) using only sparse points. Therefore, point annotations are usually transformed into density maps, using transformation process defined as follows. DI = \u2211 x\u2208L\u2208I K\u03c3(x) (1) Where K\u03c3(\u00b7) is a kernel function (e.g. a Gaussian kernel) and DI is the density map of image I (generated using kernel functions). L is the set of labeled dot/point locations in the input image I, and \u03c3 is the scale parameter of the 2-D Gaussian kernel. When integrated, the density map contains per-pixel density information of the scene, which results in the count of object in the image. Intuitively, the transformation in Eq. (1) blurs each point annotation according to the scale parameter \u03c3, and settings of \u03c3 represent different types of density mapping. The most basic way is to keep \u03c3 as a fixed value, and an existing work45 has studied commonly used \u03c3 parameters. One limitation of using a fixed \u03c3 value is that it prevents the density map label from capturing perspective information of the counted objects. When the scales of the objects varies significantly in the image, a fixed \u03c3 value results in low accuracy. Alternatively, an adaptive kernel19 uses adaptive \u03c3 values by taking average distance obtained from k-nearest neighbor algorithm into consideration. Intuitively, it produces a severely blurred density map in a highly dispersed regions of the target to accommodate the scale variation due to perspective. However, this approach relies on the uniform distribution of the counted objects. When discrete targets are present, the k-nearest algorithm becomes unreliable. Modern approaches try to introduce additional information to produce more reliable adaptive kernels. Liu et al.46, Shi et al.47, Yan et al.48, and Zhang et al.19 use a perspective map to smooth the final density map so that larger counting targets close to the camera have larger and smoother Gaussian regions in an attempt to eliminate the error due to perspective distortion. However, perspective mapping is unavailable in manatee counting because the water is not on the same plane. Some work49 uses depth camera to obtain relative depth of the target to the camera to estimate the size of the target. However, depth acquisition of underwater objects is still an open problem, and is uncommon (due to cost) in general surveillance/tracking systems. Figure 1 shows an example of different labeling/annotation approaches on an image with 21 manatees in the scene. From left to right, the labeling costs and difficulty increase. The left most image is labelled with a number 21, which is the number of manatees in the scene (the label does not provide any additional information about manatees, such as locations, orientation etc.). Dot annotation provides circular points to mark each manatees, and bounding-box annotation uses a rectangle box to outline region of each manatee. For comparisons, our proposed work is based on line-segment annotation where a single line-segment is used to label/mark each manatee. Proposed Method Recently, density maps have been commonly used for presenting crowd counting because they can represent the distribution of the crowd. In our research, we propose to use Anisographic Gaussian Kernel (AGK) based crowd counting approach for manatee counting. In the following, we will first introduce kernel density based counting, and then propose manatee customized crowd counting framework. 4/18 \fBase Network Following recent approaches, we perform counting based on the density estimation framework. The input of the framework is an image I \u2208Rw\u00d7h, represented as a w\u00d7h matrix where w and h denote image width and height respectively. The ground-truth density map DI \u2208Rw\u00d7h is used to train deep neural networks (DNNs) by imposing normalized 2-D Gaussian at manatee locations provided by the annotations. The deep neural network is trained to predict the density map \u02c6 DI \u2208Rw\u00d7h for an image I, such that the network predicted density map output \u02c6 DI is sufficiently close to the ground-truth DI. Kernel Function A kernel, commonly used in machine learning to perform classification and clustering, is a non-linear mapping of two vectors in a feature space, through the dot product of two vectors. Given two d\u2212dimensional vectors xi,xj \u2208Rd, and a transformation function \u03d5(x) defined for each vector, a kernel mapping between xi and xj is a function defined as dot product \u03d5(xi) and \u03d5(x j) as follows: K(xi,xj) = \u03d5(xi)T\u03d5(xj) (2) For example, for 2\u2212dimensional vectors xi,x j \u2208R2, with xi = [xi,1,xi,2] and xj = [xj,1,xj,2] a simple polynomial kernel is defined as K(xi,xj) = (1+xiTxj)2 = (1+xi,1xj,1 +xi,2xj,2)2 = 1+x2 i,1x2 j,1 +x2 i,2x2 j,2 +2xi,1xj,1 +2xi,2xj,2 +2xi,1xi,2xj,1x j,2 = \u03d5(xi)T\u03d5(xj) (3) where \u03d5(xi) = [1,x2 i,1,x2 i,2, \u221a 2xi,1, \u221a 2xi,2, \u221a 2xi,1xi,2], and \u03d5(xj) is defined similarly. The kernel function should satisfy the following three properties: symmetrical, non-negative, and the area under the curve of the function must be equal to 1. There are some well-known examples of kernels satisfying specific properties, such as Gaussian kernel, multivariate Student kernel, and Laplacian kernel. The Gaussian kernel can be expressed as K\u03c3(xi,xj) = ( 1 \u221a 2\u03c0\u03c3 )dexp(\u2212 \r \rxi \u2212x j \r \r2 2\u03c32 ) (4) Similar to polynomial kernel, Gaussian kernel function in Eq. (4) can be transformed as dot product between two vectors, \u03d5(xi) and \u03d5(xj), by using Taylor series to expand the kernel function into an infinite series of products. The original Gaussian kernel has the same spread (\u03c3) for all feature dimensions. As a result, it is difficult to represent a high dimensional space and differentiate features more important to capture the decisions (or classifications)50. Kernel Density Map Generation Given an image I \u2208Rw\u00d7h represented as a w\u00d7h array, and a set of n labelled points p1,\u00b7\u00b7\u00b7 ,pn in the image (e.g. using dot annotation), the kernel density map intends to generate a density map DI \u2208Rw\u00d7h of I, with respect to the labelled points L (so the density map is primarily focused on labelled points). The kernel density estimator (KDE) is a non-parametric estimator used to estimate the univariate or multivariate densities based on kernels as weights51. One direct way to create a kernel density map DI is to compare each pixel in I to each labeled point pi \u2208R2, using 2-D location to represent pixels and points. Denote xi,j \u2208R2 the 2-D location of a pixel of image I located at [i, j], kernel density map DI can be calculated using kernel transformation below DI[i, j] = 1 n n \u2211 k=1 K\u03c3(xi, j,pk); (5) where i = 1,\u00b7\u00b7\u00b7 ,w; j = 1,\u00b7\u00b7\u00b7 ,h are indices of image width and height, and K\u03c3 is a kernel function with bandwidth \u03c3. 5/18 \fDensity Map to Counting After obtaining the density map of an image, the number of manatees can be calculated by using element-wise summation of all points\u2019 density value as follows CI = w \u2211 i=1 h \u2211 j=1 DI[i, j] (6) where CI denotes ground-truth manatee numbers in the image I. For each labeled image, the annotations have ground-truth CI value, so density map needs to be normalized accordingly to ensure that the sum of element-wise density map equals to its CI value. Figure 2 shows an example of an input image I with dot annotations (left panel), and its density map DI based on labeled points (right panel). The ground-truth CI is 21, because there are 21 manatees within the image. After obtaining the density map DI, we will train neural network using input image I, and using density map DI as expected output of the network. A sufficiently trained neural network is therefore capable of learn to detect an input image\u2019s manatee locations as a density map. By calculating element-wise sum of the predicted density map, using Eq. (6), we can calculate number of manatees in the scene (detailed in the latter section). Figure 2. Left panel: An image with dot labels of manatees, and Right panel: Density map of the image generated by applying Gaussian distributions to labeled points Manatee Customized Crowd Counting Line-Label for Manatee Counting Although point label are commonly used for generating density map in crowd counting, a manatee\u2019s body is oval-shaped, meaning that point annotations cannot effectively capture orientation and shape of manatees for accurate counting. An alternative solutions is to use line-segment labels, e.g., a straight line segment, to mark each manatee. A challenge of using a line-segment to annotate each manatee is that a line consists of an infinite number of points, making it difficult to obtain a density map of the input image. In this paper, we propose to use a limited number of points to represent the segment, and then use Gaussian kernel to denote each point for density map generation. More specifically, given two endpoints, xa = [xa,x,xa,y] and xb = [xb,x,xb,y], of a line-segment, we generate \u2308|xb,x \u2212xa,x|+1\u2309 number of Gaussian kernels using position evenly distributed on the line (where \u2308\u00b7\u2309denotes a ceiling function). In addition, to better capture manatee shapes, we generate an oval shaped Gaussian kernel for each dot, using \u03c3 value, which is adjusted according to the position of current point xi = [xi,x,xi,y] comparing to the two endpoints of the line segment, as defined in Eq. (7), where a is a constant value (i.e. a parameter). \u03c3 = \u03c3basic +a\u00b7min(\u2225xi \u2212xa\u2225,\u2225xi \u2212xb\u2225) (7) After obtaining \u03c3 value, we can use point xi\u2019s current location as mean (\u00b5) and generate a Gaussian distribution corresponding to point xi. Repeating this process, one can generate many Gaussian kernels for each line-segment. The total number of Gaussian kernels for each input image (which often contains many line segments) are used to normalize Gaussian kernels such that the density map of the whole image follows a distribution. To illustrate the density map generation using line-segment, Figure 3 shows examples of using single point and a linesegment to generate manatee customized density map. From left to right, Figure 3(I) denotes a single point at (14,14), while the subfigure (II) is the density map generated by given head coordination of the Gaussian kernel with \u03c3 = 4. Figure 3(III) is a line-segment from point (5,5) to point (25,25) which includes 21 points in total, and Figure 3(IV) denotes the density map generated by the given points in Figure 3(III) where the result has been normalized. In this subfigure, the smallest \u03c3, 6/18 \f\u03c3basic = 3, is 3 at the two endpoints, while the largest \u03c3 value is 5 corresponding to the Gaussian kernel at the center point of the line-segment at (15,15). The results in Figure 3 show that using a line-segment, combined with Gaussian kernels, can provide a customized density map resembling a manatee\u2019s oval-shape for counting estimation. Figure 3. (I) A single point at (14,14); (II) Gaussian with \u03c3 = 4; (III) line-segment from point (5,5) to (25,25); (IV) Gaussian with \u03c3basic = 3 and a = 0.2; (V) Anisotropic Gaussian with \u03c3x1 = 8.24,\u03c3x2 = 2.06 for line-segment in (III) Anisotropic Gaussian Kernel for Manatee Counting While the oval-shaped density map in Figure 3(IV) is resembles to manatees\u2019 shape and orientation, it does not effectively capture the width of manatees\u2019 body, where the length of a manatee is much longer than its width. AGK kernel is a modification of the Gaussian kernel (4). Instead of using a single kernel parameter for all features, AGK kernel uses different kernel parameters to differentiate Gaussian distributions along each feature dimension. The AGK kernel function can be defined as K\u03c3(xi,xj) = exp(\u2212 d \u2211 k=1 \r \rxi,k \u2212xj,k \r \r2 2\u03c32 k ) (8) where k = 1,\u00b7\u00b7\u00b7 ,d denotes the feature indices. Because manatee\u2019s density maps are 2-D images, we have d = 2 for manatee counting. To generate a density map that most closely resembles to each individual manatee, parameter \u03c3k is dynamically adjusted using Eq. (9). ( \u03c31 = \u2225xa\u2212xb\u2225 2 \u00d7 FWHM \u03b1 \u03c32 = \u03c31 A R (9) in Eq. (9), \u2225xa \u2212xb\u2225denotes the length of the underlying line-segment. FWHM denotes Full Width at Half Maximum of the Gaussian distribution defined by the line-segment. It is used to adjust the \u03c3 to make the distribution more centralized. Intuitively, given a Gaussian distribution with \u03c3 standard deviation value, FWHM and \u03c3 satisfy following relationship: FWHM = 2 \u221a 2ln2\u03c3 \u22482.355\u03c3. To allow flexibility, we use a parameter \u03b1 to penalize the FWHM, and control Gaussian distribution spread. Overall, \u03c31 is adjusted based on the length of the line-segment and the FWHM. For \u03c32, it is also adjusted based on the \u03c31 value and the empirical Aspect Ratio (A R), i.e. length divided by width, of the underlying object. In our experiments, we empirically set a fixed A Rvalue for all experiments. Figure 4. Left: An image with line labels of manatees, Middle: Density map of the image generated by using generic Gaussian kernels (line-segment labels), and Right: Density map generated by using AGK kernels (line-segment labels) 7/18 \fFigure 3(V) shows an example of the density map generated for a ling between (5,5) and (25,25) by setting the AGK parameters with \u03c31 = 8.24 and \u03c32 = 2.06. Comparing Figure 2(IV) and (V), i.e. density maps generated using simple Gaussian kernel vs. Anisotropic Gaussian kernel (AGK) respectively, the results show that AGK based density map is more resemblance to the manatee shape, and therefore will result in more accurate counting for manatees. In Figure 4, we report a manatee aggregation with line-segment labels (left panel), the density map generated from Gaussian kernels (middle), and the density map generated from AGK kernels, by adding oval shaped Gaussian distributions over the lines. Comparing two density maps, we can find that density map from AGK kernels provide more accurate density estimation with respect to individual manatee\u2019s shape and position. For example, the two manatees to the upper-right scene are represented as two circular-shaped areas, whereas AGK kernel\u2019s density shows clear oval-shapes with accurate orientation and aspect ratios. Figure 5. A conceptual view of the deep neural network based manatee counting workflow. From left to middle: An input image (I), the line-segment labels of the image, the density map DI of the image. From right to middle: An image I is applied to a neural network (NN) to predict its density map \u02c6 DI. The loss function in the middle is used to regulate and train the NN to ensure that \u02c6 DI maximally approximates to the ground-truth DI. CSRNet Learning to Predict Density Map After obtaining density maps for each labeled images, our next step is to use original images to train a deep neural network. Once the network is sufficiently trained, for each input image, the network will output a density map maximally approximating to the ground-truth density map of the input image. More specifically, given a labelled training deadset I with |I | images where each image I is labeled and has a ground-truth density map DI. Denote f\u0398() a deep neural network regulated by a number of tunable parameters \u0398, and for each labeled input image Ii \u2208I and its density map DIi \u2208D, the network will output a predicted density map \u02c6 DIi, the neural network aims to learn optimal parameters \u0398\u2217, such that total loss \u2113() with respect to the predicted density map is minimized. \u0398\u2217= argmin \u0398 \u2211 Ii\u2208I \u2113(f\u0398(I),DIi) (10) To learn to predict density map, we employ Congested Scene Recognition network (CSRNet) as our basic model. CSRNet is designed to understand highly congested scenes and perform accurate count estimation as well as produce high-quality density maps45. The CSRNet consists of two major components, including a CNN (convolutional neural network) as the front-end for 2-D feature extraction and a dilated CNN for the back-end to replace pooling operations. In our design, we use VGG1652, a pretrained CNN network, as the front-end of CSRNet because it has strong transfer learning ability and flexibility to concatenate the back-end for density map generation. The fully-connected layers of VGG-16 is removed in the CSRNet. For the back-end, a 2-D dilated convolution can be defined as y(m,n) = M \u2211 i=1 N \u2211 j=1 x(m+r \u00d7i,n+r \u00d7 j)w(i, j) (11) where y(m,n) is the output, x(m+r \u00d7i,n+r \u00d7 j) is the input, w(i, j) is the filter with M length and N width respectively, and r is the dilation rate. When r = 1, the dilation convolution layer is the same as a normal convolution layer. In the dilated convolution layer, a k \u00d7 k kernel filter will be enlarged to (k + (k \u22121)(r \u22121)) \u00d7 (k + (k \u22121)(r \u22121)) with dilated stride r. Because of the application of the dilated convolution, the CSRNet can maintain the resolution of feature map. Most importantly, the output from dilated convolution contains more detailed information. Overall Framework Algorithm 1 outlines major steps of the proposed framework for manatee counting using DNN and AGK kernels. In addition, Figure 5 also shows conceptual flow of the framework with respect to input images, annotations, density maps, and DNN network training. The general workflow can be splitted into two phrases, annotation and training phrase, and prediction phrase. During annotation and training phrase, all training images are labelled, and each original image and its labels are used to generate AGK 8/18 \fdensity map (detailed in Algorithm 3). After this step, all training images and their density maps are used to train a deep neural network, by using Eq. (13) to calculate the loss between the DNN output and each training image ground-truth density map. The loss is used to update weights of the DNN until the network converges. After that, it enters into prediction phrase that we can use the trained NN to predict the number of manatees in a test image. Density Map Generation One of the main steps in Algorithm 1 is to generate AGK density map (Algorithm 3) for each image I (this step can also be replaced by Algorithm 2 to generate normal Gaussian Kernel Density Map). For Gaussian distributions, the probability density values are continuous infinitely with respect to the input space. In reality, the distributions are restricted to a 2-D window, limited by the image size. From distribution perspective, for a generated Gaussian distribution, the sum all values within the 2-D array should be equal to 1. Therefore, we truncate each single Gaussian distribution to a specific region, and all distributions\u2019 output, and further normalize the values to ensure that they follow a distribution. In the following, we explain steps to generate three types of density map, dot-label, line-label, and line-label Anisotropic Gaussian, respectively. Dot-label Gaussian Density Map To obtain a dot-label Gaussian density map, n Gaussian distributions are generated based on the number of dots (i.e. labels) within the image. Then these n 2D arrays are added to their corresponding point positions in the map, which is an image-sized initialized 2D array without the color dimension. Line-label Gaussian density map To get a line-label Gaussian density map, its general algorithm is shown as Algorithm 2. Firstly, a w by h shape 2D array, O, is initialized with 0 where w and h are the with and height of the Image I, respectively. Then get each l from labels, L, where the label contains two points, start point and end point of the line. A series of points is generated pixel by pixel from start point to end point. For each of the point, a \u03c3 value is created according to the Eq. (7) followed by creating its corresponding window size. After that, a Gaussian distribution array can be generated and added to a temporary density map o at its position. Once Gaussian distributions are generated for all of points of the line-segment, the o is normalized because one line label is only regarded as one count. Finally, the o is added back to the output O. Line-label Anisotropic Gaussian Density Map Algorithm 3 lists steps to generate a line-label anisotropic Gaussian density map. An array of O is initialized with 0 which is the same as it in generating a line-label Gaussian density map. For each of the label, l, in labels, L, first, the length, len, of the line-segment is calculated. With the length of the segment, \u03c31 is calculated according to Eq. (9) followed by \u03c32. Then, generating anisotropic Gaussian distribution based on \u03c31 and \u03c32 within a 2-D array, N (\u00b5,\u03c3), with the shape of len by len. After that, it rotates the generated distribution respecting to the slope of the label. Before adding N \u25e6(\u00b5,\u03c3) back to the output O at the position of the center of the line-segment, N \u25e6(\u00b5,\u03c3) is normalized. Once all the distributions have been created for the labels, anisotropic Gaussian density map is presented for the image I. Limitations The proposed anisotropic Gaussian kernel relies Gaussian distributions to generate a density map to approximate manatee shapes. By doing so, it relies on assumption that manatee shapes are largely visible and are close to the AGK kernel shape. As a result, several potential limitations may hinder this approach from obtaining high accuracy counting results: \u2022 Close-up Views: When manatees are situated too close to the camera, they may appear disproportionately large, with only parts of their bodies being captured. In such cases, line-labels fail to accurately mirror the shape of the manatee. The resultant density graph might not correspond well to the actual manatee shape, and result in poor counting performance. \u2022 Distant Views: On the opposite end, manatees that are extremely further away from the camera appear diminutive, resembling dots. Under these conditions, the differentiation between line-label and dot-label becomes negligible, resulting in comparable accuracy for both annotation methods. These real-world scenarios outline the importance of understanding the inherent constraints of different labeling/annotation approaches for real-world applications. Our experiments and comparisons demonstrate the strength and niche of AGK kernels in counting manatee under different environment conditions. Experiments Benchmark Data In order to validate the performance of proposed framework using low resolution images, manatee surveillance video clips from \u201cSave the manatee Club\u201d 1, which are captured from webcams placed at Blue Spring State Park, are collected to create our benchmark dataset. After using FFmpeg to generate images from the video clips and removing similar images, we obtain 784 images, consisting of different number of manatees, as our testbed. 1https://www.savethemanatee.org/manatees/manatee-webcams/ 9/18 \fAlgorithm 1: DNNs for Manatee Counting Using AGK Data: (1) I : training image set; (2) T : test image set; 1 Define: (1) D: density map set; 2 (2) f\u0398(): a DNN, e.g. a CSRNet or other DNNs; 3 (3) a(I): label an image I and return annotations; 4 === Annotation and training phase === 5 D \u2190{} 6 for I \u2208I do 7 L \u2190a(I) ; // obtaining training image I annotations 8 D \u2190AGKDensity(I,L) ; // obtaining I\u2019s AGK density map 9 D \u2190D \u222aD 10 end 11 f\u0398\u2217() \u2190argmin \u0398 \u2113( f\u0398(I ),D) ; // training DNN network using Eq. (10) 12 13 === Prediction phase === 14 for T \u2208T do 15 \u02c6 DT \u2190f\u0398\u2217(T) ; // predicting density map 16 CT \u2190Calculate manatee numbers in T using Eq. (6) 17 end Algorithm 2: LineDensity() Gaussian Kernel Density Map Generation via Line Labels Data: (1) I: an input image; (2) L: line labels of image I 1 Define: (1) ls and ls: two endpoints of line segment l 2 (2) N (\u00b5,\u03c3): a multidimensional Gaussian distribution with \u00b5 mean and \u03c3 spreads 3 Input: (1) a: expanding factor; 4 (2) \u03c3basic: basic value for generating \u03c3; 5 Output: O: density map of image I 6 [w,h] \u2190Width and height of image I O \u2208Rw\u00d7h \u21900 ; // create an empty density map 7 for l \u2208L do 8 [xa,xb] \u2190[ls,le] ; // two ends of a line-segment 9 P \u2190Evenly select \u2308|xa.x \u2212xb.x|+1\u2309points from line-segment l 10 \u25e6\u2208Rw\u00d7h \u21900 ; // temporary density map 11 for p \u2208P do 12 \u00b5 \u2190p // Gaussian distribution mean 13 \u03c3 \u2190\u03c3basic +a\u00b7min(\u2225p\u2212xa\u2225,\u2225p\u2212xb\u2225) ; // Gaussian distribution spread Eq. (7) 14 N (\u00b5,\u03c3) \u2190Generate 2D Gaussian Distribution 15 \u25e6\u2190\u25e6+N (\u00b5,\u03c3) ; // add current Gaussian distribution to the temporary density map 16 end 17 Normalize temporary density map \u25e6 18 O \u2190O +\u25e6 19 end 20 return O Deduplication Because manatee images are collected from video clips and manatees rarely show rapid movement, some of the images are similar to each other. To avoid data duplication, duplicate images are dropped by using VGG-16 to extract layer-wise features for each image and then a method is employed to calculate the difference between two images, Ia and Ib, with the formula being defined as follows: Disance(Ia,Ib) = \u2211 j\u2208F 1 CjHjWj \f \f|\u03c8j(Ia)\u2212\u03c8j(Ib) \f \f|2 2 (12) F is the outputs from the last activation layer in each group of VGG-16\u2019s 5 groups, namely F = {\u03c8j|j = 1,2,3,4,5} = {layer2,layer4,layer7,layer10,layer13}. Cj,Hj,Wj are the three dimensions of the image, channel, height, and width, respec10/18 \fAlgorithm 3: AGKDensity() AGK Density Map Generation via Line Labels Data: (1) I: an input image; (2) L: line labels of image I 1 Define: (1) ls and ls: two endpoints of line segment l 2 (2) N (\u00b5,\u03c3): a multidimensional Gaussian distribution with \u00b5 mean and \u03c3 spreads 3 Output: O: density map of image I. [w,h] \u2190Width and height of image I 4 O \u2208Rw\u00d7h \u21900 ; // create an empty density map 5 for l \u2208L do 6 [xa,xb] \u2190[ls,le] ; // two ends of a line-segment 7 \u00b5 = [\u00b51,\u00b52] \u2190xa+xb 2 ; // calculate AGK mean 8 \u03c3 = [\u03c31,\u03c32] \u2190Eq. (9) ; // calculate AGK spreads 9 N (\u00b5,\u03c3) \u2190Generate 2-D Gaussian Distribution 10 \u25e6\u2190Calculate slope between xa and xb 11 N \u25e6(\u00b5,\u03c3) \u2190Align 2-D Gaussian Distribution N (\u00b5,\u03c3) based on the line-segment slope 12 Normalize N \u25e6(\u00b5,\u03c3) 13 O \u2190O +N \u25e6(\u00b5,\u03c3) // add current Gaussian distribution to the density map 14 end 15 return O tively. \u03c8j(ix) denotes the jth group ReLu output of the image ix. The smaller value of Distance(Ia,Ib), the more similarity of the two images is considered. When the value of Distance(Ia,Ib) is less than 2, the two images are considered highly similar, then one of the two images is dropped. After comparing similarities of the images, 415 of 784 images are discarded and only 369 images are kept as dataset. Dataset Characteristics All 369 benchmark images have the same size. According to the number of manatees in each image, we separate each image into three density levels: low, medium, and high. We define that, if an image has less than 5 manatees, then it is of low density level. Continue with medium level if it has more or equal to 5 but less than 20 manatees. If an image with more than 19 manatees, it is of high density level. Figure 6 reports the manatee density distributions in the benchmark images at three levels. Images with low density level have the highest quantity (181), which is almost equal to the total amount of medium and high level which have 84 and 104 images, respectively. Figure 6. Distribution of benchmark images w.r.t. different manatee density levels. The x-axis denotes manatee density levels and the y-axis denotes number of images. Experimental Settings Manatee Annotations To validate the performance of using Anisotropic Gaussian Kernel (AGK) for manatee counting, we use dot and line labels to annotate images for performance comparison. In order to reduce duplicate manual label work, we draw a line over each manatee (from tail to head) and make sure the line crosses over the center point of the manatee (the number of labelled line segments within an image is the number of manatees in the image). Consequently, two endpoints of a line for each manatee are obtained and saved into a JSON file for further usage. Assuming xa = [xa,x,xa,y] and xb = [xb,x,xb,y] are the two endpoints of a labelled line segment obtained from previous stage, [ xa,x+xb,x 2 , xa,y+xb,y 2 ] is used as point label for the same manatee. 11/18 \fFigure 7. Examples of algorithm performance with respect to different manatee densities in the scene. The first row shows original images with increasing manatee density from left to right. The second and third rows show ground-truth density map (2nd row) and predicted density map (3rd row) using dot labels. The fourth and fifth rows show ground-truth density map (4th row) and predicted density map (5th row) using line labels and generic Gaussian kernels. The sixth and seventh rows show ground-truth density map (6th row) and predicted density map (7th row) using line labels and anisotropic Gaussian kernels Parameter Settings We validate the performance of multiple baseline methods on our manatee dataset over three labels. Compared to traditional point labeling methods45, our line labeling adds only a small amount of additional time and obtains an overall better performance. In this section, we report the evaluation metrics and perform an ablation study on the three labeled datasets to study their performance. We evaluate the object counting task on an NVIDIA V100 GPU card. The watermark on the surveillance video images at the lower right corner is filled with solid black to avoid additional interference. During training, the initial learning rate is 1e-4, and the Adam optimizer is used. For better training and to prevent overfitting, random flips were used for augmentation. For all networks, the batch size is set to 4. The network directly converts the input image into a density map in training. During the density map generations, some hyperparameters are used in Algorithms 2 and 3. They are the expanding factor, a = 0.2, the base value of \u03c3, \u03c3basic = 15, the empirical Aspect Ratio, A R = 4, full width at half maximum FWHM = 2.355\u22482 \u221a 2ln2, and the FWHM penalizer \u03b1 in Eq. (9) is set as \u03b1 = 4. Intuitively, the FWHM and \u03b1 parameters in Eq. (9) are determined such that the \u03c31 value is roughly 1 4 of the length of the underlying line-segment ||xa \u2212xb||. Same as in previous studies45, we chose the MSE loss to measure pixel difference between ground truth and predicted density maps, with loss value being calculated as follows: \u2113() = 1 |T | |T | \u2211 i=1 ||DTi \u2212\u02c6 DTi||2 2 (13) where T denotes the test set and |T | denotes number of images in the test set. For each test image Ti, DTi and \u02c6 DTi denote 12/18 \fTable 1. Manatee counting performance using different DNNs over different density levels and label types Method Low Medium High Overall MAE RMSE MAE RMSE MAE RMSE MAE RMSE MCNN-dot 3.640 3.812 3.511 4.924 26.982 28.352 11.377 12.363 MCNN-line 3.603 3.753 3.556 4.912 26.998 28.375 11.386 12.346 MCNN-anisotropy 2.471 2.859 4.170 5.562 28.080 29.408 11.573 12.610 SANet-dot 2.048 3.032 2.833 3.909 4.993 7.510 3.291 4.817 SANet-line 1.512 2.645 2.268 3.418 4.197 6.882 2.659 4.315 SANet-anisotropy 1.847 2.619 3.876 5.047 6.042 8.586 3.921 5.417 VGG-dot 3.921 4.721 2.702 3.522 15.601 17.199 7.408 8.481 VGG-line 3.723 4.540 2.678 3.535 16.217 17.748 7.539 8.608 VGG-anisotropy 3.405 4.125 2.550 3.488 15.311 16.888 7.088 8.167 MARUNet-dot 2.312 2.738 4.292 5.685 28.150 29.447 10.045 15.982 MARUNet-line 2.653 3.578 4.347 5.797 27.046 28.498 9.913 15.583 MARUNet-anisotropy 3.325 4.122 3.830 5.030 25.063 26.677 9.567 14.651 CSRNet-dot 1.344 1.925 2.981 3.825 4.978 6.569 3.101 4.106 CSRNet-line 2.256 3.341 3.669 4.67 5.03 6.614 3.652 4.875 CSRNet-anisotropy 1.588 2.19 2.94 3.84 4.506 5.856 3.011 3.962 ground-truth and predicted density maps, respectively. ||\u00b7||2 2 represents the Euclidean distance. Performance Metrics We use mean absolute error (MAE) and root mean square error (RMSE) in the experiments to evaluate algorithm performance. MAE = 1 |T | |T | \u2211 i=1 |CTi \u2212\u02c6 CTi| (14) RMSE = v u u t 1 |T | |T | \u2211 i=1 (CTi \u2212\u02c6 CTi)2 (15) where CTi and \u02c6 CTi are the ground-truth number and predicted number of manatees in image Ti, respectively, calculated using Eq. (6). Baselines MCNN18, SANet53, VGG52, and CSRNet45 are some of the commonly used networks for counting problem. Meanwhile, attention mechanisms were also adopted for counting task recently, such as MARUNet54. Those models are implemented and trained for comparisons to demonstrate the effectiveness of our proposed method. In the following, we use X-dot, X-line, and X-anisotropy to denote that X neural network is trained by using three different types of density maps, which are generated by point label with Gaussian kernel, line label with Gaussian kernel, and line label with anisotropy kernel, respectively. \u2022 MCNN18: MCNN employs a multi-column CNN structure that is adaptive to detect heads of varying sizes, addressing scale variations inherent in crowd images and generating density maps for crowd estimation. \u2022 SANet53: The Scale Aggregation Network employs a feature map encoder to aggregate multi-scale features from original images. Then it uses a density map estimator to fuse these features before generating high-resolution density maps. This architecture ensures robust crowd counting across varied crowd densities. \u2022 VGG52: The VGG network, characterized by its depth and 3x3 convolutions, is known for its robust feature extraction capabilities and has been a foundational architecture in various visual tasks, including crowd counting. \u2022 CSRNet45: CSRNet uses the simplified VGG-16 as front-end structure for feature extraction and a dilated convolution network as back-end, which can handle the scale variations in crowd counting tasks effectively. \u2022 MARUNet54: MARUNet, Multi-level Attention Refined UNet, integrates a density map estimator and a crowd region recognizer, facilitating the network\u2019s focus on crowd regions and providing a strong baseline in crowd density map generation. 13/18 \fExperimental Results Figure 7 reports two examples for each level of the three density levels, low, medium and high with three different density maps. The first row are the original images that each image has 2, 4, 7, 11, 27 and 42 manatees separately. The second row is their Gaussian kernel density maps generated from point labels and the fourth row is their Gaussian kernel density maps generated by line labels while the sixth row is their anisotropic Gaussian kernel density maps. And third, fifth and seventh row are their corresponding estimated results from trained models. The number at the bottom right corner on the each of the density map is the real or estimated amount of manatee in the image. When comparing density maps from the training results with their original counterparts, as shown in Figure 7, it shows that as the number of manatees in the image increases, AGK density maps can represent their original maps more closely compared to the other two density maps. The reason is that in high-density maps, manatees appear relatively smaller and possess similar sizes. As a result, the AGK-generated density provides a more accurate representation of their shapes with the given experimental hyper-parameters. In contrast, the density maps generated from GK-point and GK-line tend to distribute more widely rather than concentrating in specific manatees\u2019 areas. For low and medium density maps, manatees tend to appear larger within the images. In extreme cases, a single manatee might occupy more than one-third of the entire image with only part of its body appearing in the image. Under such conditions, all three types of density maps: AGK, GK-dot, and GK-line, struggle to represent the manatees accurately, resulting in deteriorated performance. Interestingly, in these specific scenarios, GK-dot and GK-line potentially outperform AGK. This is because of their inherent capability to represent larger objects more effectively. For AGK, constrained by preset hyper-parameters, it is more adept at depicting crowded manatees, which are often of smaller sizes. The results in Table 1 show that the anisotropy method has the lowest MAE and RMSE values across the VGG, MARUNet, and CSRNet models for the overall dataset. This shows the efficacy of our proposed line-label anisotropic Gaussian density map. Although this method does not surpass the other two when applied to the MCNN model, the performance metrics are remarkably close among all three methods. For the SANet model, the line Gaussian method yielded the most favorable results. Among all five DNNs, SANet and CSRNet have better performance than others especially in high density maps, where MCNNs and MARUNets have extremely high MAE and RMSE values. The performance of SANet and CSRNet are very similar. In low-density scenarios, CSRNet shows slight advantages, but in medium to high-density scenarios, SANet performs better. Overall, SANet has a lower MAE while CSRNet has better stability. Interestingly, among the four compared networks, SANet is the only one that performed best under the line-label. This may be attributed to its unique scale aggregation architecture. This empirical success proves that our proposed line-label is indeed effective for crowd manatee counting task. CSRNet-dot has the best MAE and RMSE values in low density cases while it is outperformed by CSRNet-anisotropy in high density and overall cases. Specifically, in the low density map, where manatee localizations are more distinct, CSRNet-dot registers the lowest MAE and RMSE values of 1.344 and 1.925, respectively. When the density increases, the proposed CSRNet-anisotropy shows superior MAE values in medium to high density scenarios compared to both CSRNet-dot and CSRNet-line. The superior performance of CSRNet-dot in low densities can be attributed to the clearer object locations enabled by dot annotations, given that CSRNet can effectively separate localization from counting tasks. In terms of overall performance in CSRNet, which tested over the all data, our proposed CSRNet-anisotropy method outperforms the other two methods both in the MAE metric and in the RMSE metric, with a MAE value of 3.011 and a RMSE value of 3.962. The MAE overall performance of our proposed method is 2.99% better than the CSRNet-dot method and 21.29% better than the CSRNet-line method. And the RMSE overall performance of our proposed method is 3.63% better than the CSRNet-dot method and 23.04% better than the CSRNet-line method. In some cases, the proposed anisotropy kernel does not consistently outperform other models, especially when combined with DNN networks like MCNN and SANet. MCNN struggles to identify the optimal method across the three different density levels. In contrast, SANet produces its best performance with the line kernel. Such discrepancies could stem from the intrinsic model structures associated with image scaling in MCNN and SANet. It becomes apparent that our method may not be universally effective for all DNN models. Another noteworthy observation is the subpar performance of our method in low-density map levels across four out of five tested models. This might be attributed to the fact that, in low-density images, manatees often appear larger or may only partially be in the frame. Such scenarios might not be ideally represented by the density maps generated through the anisotropy kernel especially with the given experimental hyperparameters which used for all images. In these instances, Gaussian kernels with dot or line annotations might provide a better fitting representation. In the subsequent section, to further substantiate the efficacy of our method, we apply the proposed method to wheat head counting, where objects have regular consistent shapes. 14/18 \fFigure 8. The original wheat image and its three types of corresponding density maps (left to right): dot based GK, line based GK, and line based AGK Additional Results: Wheat head counting In the previous section, we validated our proposed method in the newly created manatee counting dataset. Given the intricate backgrounds and the variability in manatee shapes, often resulting from varying camera distances, the label can not present the manatee over different density levels as shown and analyzed in prior sections. To further validate the ability and adaptability of our method, we have opted to test it on the \"Global Wheat Head Dataset 2021\"55. Unlike manatees, most wheat heads in these images possess consistent, elongated rectangle-like shapes. We posit that our proposed line annotation with AGK could intuitively represent such forms more effectively. Figure 8 shows an example of original wheat head images and three corresponding density maps that are generated by a dot-based Gaussian Kernel(GK), a line-based GK and a line-based anisotropic GK. It is obvious that line-based AGK density map has a better representation of the wheat heads in terms of position, shape, and direction. Table 2. Wheat head counting performance in CSRNet over different label types Method MAE RMSE Yolo-v5(GWC-2021)56 NA 67.0 CSRNet-dot 30.1 37.6 CSRNet-line 20.3 28.1 CSRNet-anisotropy 15.7 20.2 As illustrated in Table 2, Yolo-v5(GWC-2021)56 has the poorest performance in terms of RMSE. All three models based on CSRNet outperform Yolo-v5, with CSRNet-anisotropy has outstanding MAE of 15.7 and RMSE of 20.2. Yolo-v5 is detection based algorithm that while it has good bounding box accuracy, it misses numerous wheat heads, especially those that are smaller in size. The average size of those missed wheat heads is about 35% smaller than the average size of all wheat heads. For our density estimation-based method, instead of seeking the precise location and boundary of each wheat head directly, it determines the total number of wheat heads by estimating the density of wheat heads within a particular region. This method is particularly good at detecting smaller and densely distributed wheat heads since it emphasizes the overall quantity estimation rather than pinpointing individual targets. When the MAE and RMSE values are close, it often indicates there are not significant outliers. Such consistency can suggest that the model\u2019s predictions are relatively accurate, with minimal discrepancies. As shown in Table 2, the proximity of the MAE and RMSE values reinforces this notion of accuracy in the CSRNet\u2019s predictions, especially CSRNet-anisotropy which proves the effectiveness of our method. Throughout the training process, we noticed that the model effectively capture the macro structure of the wheat. For point label, the model had to work hard to fit the density map precisely to a circular region. To address this challenge, we introduced AGK to generate density map by using line labels, which differ significantly from the traditional circular Gaussian region. This label utilizes an elliptical anisotropic Gaussian distribution to present the wheat\u2019s location and shape. In this distribution, the core line represents the exact location of the wheat head, while the surrounding decay area outlines the potential uncertainty of the position. Compared to the conventional circular Gaussian region, this method endows the model with enhanced adaptability in wheat head counting. 15/18"
+                }
+            ]
+        },
+        "edge_feat": {}
+    }
+}
\ No newline at end of file