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Yingbin Liang

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77

ICLR Conference 2025 Conference Paper

A Theoretical Analysis of Self-Supervised Learning for Vision Transformers

  • Yu Huang 0023
  • Zixin Wen
  • Yuejie Chi
  • Yingbin Liang

Self-supervised learning has become a cornerstone in computer vision, primarily divided into reconstruction-based methods like masked autoencoders (MAE) and discriminative methods such as contrastive learning (CL). Recent empirical observations reveal that MAE and CL capture different types of representations: CL tends to focus on global patterns, while MAE adeptly captures **both global and subtle local** information simultaneously. Despite a flurry of recent empirical investigations to shed light on this difference, theoretical understanding remains limited, especially on the dominant architecture **vision transformers** (ViTs). In this paper, to provide rigorous insights, we model the visual data distribution by considering two types of spatial features: dominant global features and comparatively minuscule local features, and study the impact of imbalance among these features. We analyze the training dynamics of one-layer softmax-based ViTs on both MAE and CL objectives using gradient descent. Our analysis shows that as the degree of feature imbalance varies, ViTs trained with the MAE objective effectively learn both global and local features to achieve near-optimal reconstruction, while the CL-trained ViTs favor predominantly global features, even under mild imbalance. These results provide a theoretical explanation for distinct behaviors of MAE and CL observed in empirical studies.

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NeurIPS Conference 2025 Conference Paper

Absorb and Converge: Provable Convergence Guarantee for Absorbing Discrete Diffusion Models

  • Yuchen Liang
  • Renxiang Huang
  • Lifeng Lai
  • Ness Shroff
  • Yingbin Liang

Discrete state space diffusion models have shown significant advantages in applications involving discrete data, such as text and image generation. It has also been observed that their performance is highly sensitive to the choice of rate matrices, particularly between uniform and absorbing rate matrices. While empirical results suggest that absorbing rate matrices often yield better generation quality compared to uniform rate matrices, existing theoretical works have largely focused on the uniform rate matrices case. Notably, convergence guarantees and error analyses for absorbing diffusion models are still missing. In this work, we provide the first finite-time error bounds and convergence rate analysis for discrete diffusion models using absorbing rate matrices. We begin by deriving an upper bound on the KL divergence of the forward process, introducing a surrogate initialization distribution to address the challenge posed by the absorbing stationary distribution, which is a singleton and causes the KL divergence to be ill-defined. We then establish the first convergence guarantees for both the $\tau$-leaping and uniformization samplers under absorbing rate matrices, demonstrating improved rates over their counterparts using uniform rate matrices. Furthermore, under suitable assumptions, we provide convergence guarantees without early stopping. Our analysis introduces several new technical tools to address challenges unique to absorbing rate matrices. These include a Jensen-type argument for bounding forward process convergence, novel techniques for bounding absorbing score functions, and a non-divergent upper bound on the score near initialization that removes the need of early-stopping.

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ICLR Conference 2025 Conference Paper

Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach

  • Yuchen Liang
  • Peizhong Ju
  • Yingbin Liang
  • Ness B. Shroff

Accelerated diffusion models hold the potential to significantly enhance the efficiency of standard diffusion processes. Theoretically, these models have been shown to achieve faster convergence rates than the standard $\mathcal O(1/\epsilon^2)$ rate of vanilla diffusion models, where $\epsilon$ denotes the target accuracy. However, current theoretical studies have established the acceleration advantage only for restrictive target distribution classes, such as those with smoothness conditions imposed along the entire sampling path or with bounded support. In this work, we significantly broaden the target distribution classes with a new accelerated stochastic DDPM sampler. In particular, we show that it achieves accelerated performance for three broad distribution classes not considered before. Our first class relies on the smoothness condition posed only to the target density $q_0$, which is far more relaxed than the existing smoothness conditions posed to all $q_t$ along the entire sampling path. Our second class requires only a finite second moment condition, allowing for a much wider class of target distributions than the existing finite-support condition. Our third class is Gaussian mixture, for which our result establishes the first acceleration guarantee. Moreover, among accelerated DDPM type samplers, our results specialized for bounded-support distributions show an improved dependency on the data dimension $d$. Our analysis introduces a novel technique for establishing performance guarantees via constructing a tilting factor representation of the convergence error and utilizing Tweedie's formula to handle Taylor expansion terms. This new analytical framework may be of independent interest.

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NeurIPS Conference 2025 Conference Paper

Discrete Diffusion Models: Novel Analysis and New Sampler Guarantees

  • Yuchen Liang
  • Yingbin Liang
  • Lifeng Lai
  • Ness Shroff

Discrete diffusion models have recently gained significant prominence in applications involving natural language and graph data. A key factor influencing their effectiveness is the efficiency of discretized samplers. Among these, $\tau$-leaping samplers have become particularly popular due to their theoretical and empirical success. However, existing theoretical analyses of $\tau$-leaping often rely on somewhat restrictive and difficult-to-verify regularity assumptions, and their convergence bounds contain quadratic dependence on the vocabulary size. In this work, we introduce a new analytical approach for discrete diffusion models that removes the need for such assumptions. For the standard $\tau$-leaping method, we establish convergence guarantees in KL divergence that scale linearly with vocabulary size, improving upon prior results with quadratic dependence. Our approach is also more broadly applicable: it provides the first convergence guarantees for other widely used samplers, including the Euler method and Tweedie $\tau$-leaping. Central to our approach is a novel technique based on differential inequalities, offering a more flexible alternative to the traditional Girsanov change-of-measure methods. This technique may also be of independent interest for the analysis of other stochastic processes.

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ICLR Conference 2025 Conference Paper

DUET: Decentralized Bilevel Optimization without Lower-Level Strong Convexity

  • Zhen Qin
  • Zhuqing Liu
  • Songtao Lu
  • Yingbin Liang
  • Jia Liu 0002

Decentralized bilevel optimization (DBO) provides a powerful framework for multi-agent systems to solve local bilevel tasks in a decentralized fashion without the need for a central server. However, most existing DBO methods rely on lower-level strong convexity (LLSC) to guarantee unique solutions and a well-defined hypergradient for stationarity measure, hindering their applicability in many practical scenarios not satisfying LLSC. To overcome this limitation, we introduce a new single-loop DBO algorithm called diminishing quadratically-regularized bilevel decentralized optimization (DUET), which eliminates the need for LLSC by introducing a diminishing quadratic regularization to the lower-level (LL) objective. We show that DUET achieves an iteration complexity of $O(1/T^{1-5p-\frac{11}{4}\tau})$ for approximate KKT-stationary point convergence under relaxed assumptions, where $p$ and $\tau $ are control parameters for LL learning rate and averaging, respectively. In addition, our DUET algorithm incorporates gradient tracking to address data heterogeneity, a key challenge in DBO settings. To the best of our knowledge, this is the first work to tackle DBO without LLSC under decentralized settings with data heterogeneity. Numerical experiments validate the theoretical findings and demonstrate the practical effectiveness of our proposed algorithms.

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ICLR Conference 2025 Conference Paper

Dynamic Loss-Based Sample Reweighting for Improved Large Language Model Pretraining

  • Daouda Sow
  • Herbert Woisetschläger
  • Saikiran Bulusu
  • Shiqiang Wang 0001
  • Hans-Arno Jacobsen
  • Yingbin Liang

Pretraining large language models (LLMs) on vast and heterogeneous datasets is crucial for achieving state-of-the-art performance across diverse downstream tasks. However, current training paradigms treat all samples equally, overlooking the importance or relevance of individual samples throughout the training process. Existing reweighting strategies, which primarily focus on group-level data importance, fail to leverage fine-grained instance-level information and do not adapt dynamically to individual sample importance as training progresses. In this paper, we introduce novel algorithms for dynamic, instance-level data reweighting aimed at improving both the efficiency and effectiveness of LLM pretraining. Our methods adjust the weight of each training sample based on its loss value in an online fashion, allowing the model to dynamically focus on more informative or important samples at the current training stage. In particular, our framework allows us to systematically devise reweighting strategies deprioritizing redundant or uninformative data, which we find tend to work best. Furthermore, we develop a new theoretical framework for analyzing the impact of loss-based reweighting on the convergence of gradient-based optimization, providing the first formal characterization of how these strategies affect convergence bounds. We empirically validate our approach across a spectrum of tasks, from pretraining 7B and 1.4B parameter LLMs to smaller-scale language models and linear regression problems, demonstrating that our loss-based reweighting approach can lead to faster convergence and significantly improved performance.

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ICML Conference 2025 Conference Paper

How Transformers Learn Regular Language Recognition: A Theoretical Study on Training Dynamics and Implicit Bias

  • Ruiquan Huang
  • Yingbin Liang
  • Jing Yang 0002

Language recognition tasks are fundamental in natural language processing (NLP) and have been widely used to benchmark the performance of large language models (LLMs). These tasks also play a crucial role in explaining the working mechanisms of transformers. In this work, we focus on two representative tasks in the category of regular language recognition, known as ’even pairs’ and ’parity check’, the aim of which is to determine whether the occurrences of certain subsequences in a given sequence are even. Our goal is to explore how a one-layer transformer, consisting of an attention layer followed by a linear layer, learns to solve these tasks by theoretically analyzing its training dynamics under gradient descent. While even pairs can be solved directly by a one-layer transformer, parity check need to be solved by integrating Chain-of-Thought (CoT), either into the inference stage of a transformer well-trained for the even pairs task, or into the training of a one-layer transformer. For both problems, our analysis shows that the joint training of attention and linear layers exhibits two distinct phases. In the first phase, the attention layer grows rapidly, mapping data sequences into separable vectors. In the second phase, the attention layer becomes stable, while the linear layer grows logarithmically and approaches in direction to a max-margin hyperplane that correctly separates the attention layer outputs into positive and negative samples, and the loss decreases at a rate of $O(1/t)$. Our experiments validate those theoretical results.

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NeurIPS Conference 2025 Conference Paper

Multi-head Transformers Provably Learn Symbolic Multi-step Reasoning via Gradient Descent

  • Tong Yang
  • Yu Huang
  • Yingbin Liang
  • Yuejie Chi

Transformers have demonstrated remarkable capabilities in multi-step reasoning tasks. However, understandings of the underlying mechanisms by which they acquire these abilities through training remain limited, particularly from a theoretical standpoint. This work investigates how transformers learn to solve symbolic multi-step reasoning problems through chain-of-thought processes, focusing on path-finding in trees. We analyze two intertwined tasks: a backward reasoning task, where the model outputs a path from a goal node to the root, and a more complex forward reasoning task, where the model implements two-stage reasoning by first identifying the goal-to-root path and then reversing it to produce the root-to-goal path. Our theoretical analysis, grounded in the dynamics of gradient descent, shows that trained one-layer transformers can provably solve both tasks with generalization guarantees to unseen trees. In particular, our multi-phase training dynamics for forward reasoning elucidate how different attention heads learn to specialize and coordinate autonomously to solve the two subtasks in a single autoregressive path. These results provide a mechanistic explanation of how trained transformers can implement sequential algorithmic procedures. Moreover, they offer insights into the emergence of reasoning abilities, suggesting that when tasks are structured to take intermediate chain-of-thought steps, even shallow multi-head transformers can effectively solve problems that would otherwise require deeper architectures.

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JMLR Journal 2025 Journal Article

Random Pruning Over-parameterized Neural Networks Can Improve Generalization: A Training Dynamics Analysis

  • Hongru Yang
  • Yingbin Liang
  • Xiaojie Guo
  • Lingfei Wu
  • Zhangyang Wang

It has been observed that applying pruning-at-initialization methods and training the sparse networks can sometimes yield slightly better test performance than training the original dense network. Such experimental observations are yet to be understood theoretically. This work makes the first attempt to study this phenomenon. Specifically, we identify a theoretical minimal setting and study a classification task with a one-hidden-layer neural network, which is randomly pruned according to different rates at the initialization. We show that as long as the pruning rate is below a certain threshold, the network provably exhibits good generalization performance after training.More surprisingly, the generalization bound gets better as the pruning rate mildly gets larger. To complement this positive result, we also show a negative result: there exists a large pruning rate such that while gradient descent is still able to drive the training loss toward zero, the generalization performance is no better than random guessing. This further suggests that pruning can change the feature learning process, which leads to the performance drop of the pruned neural network. To our knowledge, this is the first theory work studying how different pruning rates affect neural networks' performance, suggesting that an appropriate pruning rate might improve the neural network's generalization. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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ICLR Conference 2025 Conference Paper

Theory on Mixture-of-Experts in Continual Learning

  • Hongbo Li 0008
  • Sen Lin 0001
  • Lingjie Duan
  • Yingbin Liang
  • Ness B. Shroff

Continual learning (CL) has garnered significant attention because of its ability to adapt to new tasks that arrive over time. Catastrophic forgetting (of old tasks) has been identified as a major issue in CL, as the model adapts to new tasks. The Mixture-of-Experts (MoE) model has recently been shown to effectively mitigate catastrophic forgetting in CL, by employing a gating network to sparsify and distribute diverse tasks among multiple experts. However, there is a lack of theoretical analysis of MoE and its impact on the learning performance in CL. This paper provides the first theoretical results to characterize the impact of MoE in CL via the lens of overparameterized linear regression tasks. We establish the benefit of MoE over a single expert by proving that the MoE model can diversify its experts to specialize in different tasks, while its router learns to select the right expert for each task and balance the loads across all experts. Our study further suggests an intriguing fact that the MoE in CL needs to terminate the update of the gating network after sufficient training rounds to attain system convergence, which is not needed in the existing MoE studies that do not consider the continual task arrival. Furthermore, we provide explicit expressions for the expected forgetting and overall generalization error to characterize the benefit of MoE in the learning performance in CL. Interestingly, adding more experts requires additional rounds before convergence, which may not enhance the learning performance. Finally, we conduct experiments on both synthetic and real datasets to extend these insights from linear models to deep neural networks (DNNs), which also shed light on the practical algorithm design for MoE in CL.

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ICLR Conference 2025 Conference Paper

Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers

  • Yuchen Liang
  • Peizhong Ju
  • Yingbin Liang
  • Ness B. Shroff

The denoising diffusion model has recently emerged as a powerful generative technique, capable of transforming noise into meaningful data. While theoretical convergence guarantees for diffusion models are well established when the target distribution aligns with the training distribution, practical scenarios often present mismatches. One common case is in the zero-shot conditional diffusion sampling, where the target conditional distribution is different from the (unconditional) training distribution. These score-mismatched diffusion models remain largely unexplored from a theoretical perspective. In this paper, we present the first performance guarantee with explicit dimensional dependencies for general score-mismatched diffusion samplers, focusing on target distributions with finite second moments. We show that score mismatches result in an asymptotic distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions. This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise. Interestingly, the derived convergence upper bound offers useful guidance for designing a novel bias-optimal zero-shot sampler in linear conditional models that minimizes the asymptotic bias. For such bias-optimal samplers, we further establish convergence guarantees with explicit dependencies on dimension and conditioning, applied to several interesting target distributions, including those with bounded support and Gaussian mixtures. Our findings are supported by numerical studies.

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ICLR Conference 2025 Conference Paper

Transformers Provably Learn Two-Mixture of Linear Classification via Gradient Flow

  • Hongru Yang
  • Zhangyang Wang
  • Jason D. Lee
  • Yingbin Liang

Understanding how transformers learn and utilize hidden connections between tokens is crucial to understand the behavior of large language models. To understand this mechanism, we consider the task of two-mixture of linear classification which possesses a hidden correspondence structure among tokens, and study the training dynamics of a symmetric two-headed transformer with ReLU neurons. Motivated by the stage-wise learning phenomenon in our experiments, we design and theoretically analyze a three-stage training algorithm, which can effectively characterize the actual gradient descent dynamics when we simultaneously train the neuron weights and the softmax attention. The first stage is a neuron learning stage, where the neurons align with the underlying signals. The second stage is a attention feature learning stage, where we analyze the feature learning process of how the attention learns to utilize the relationship between the tokens to solve certain hard samples. In the meantime, the attention features evolve from a nearly non-separable state (at the initialization) to a well-separated state. The third stage is a convergence stage, where the population loss is driven towards zero. The key technique in our analysis of softmax attention is to identify a critical sub-system inside a large dynamical system and bound the growth of the non-linear sub-system by a linear system. Finally, we discuss the setting with more than two mixtures. We empirically show the difficulty of generalizing our analysis of the gradient flow dynamics to the case even when the number of mixtures equals three, although the transformer can still successfully learn such distribution. On the other hand, we show by construction that there exists a transformer that can solve mixture of linear classification given any arbitrary number of mixtures.

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ICML Conference 2025 Conference Paper

Unlocking the Power of Rehearsal in Continual Learning: A Theoretical Perspective

  • Junze Deng
  • Qinhang Wu
  • Peizhong Ju
  • Sen Lin 0001
  • Yingbin Liang
  • Ness B. Shroff

Rehearsal-based methods have shown superior performance in addressing catastrophic forgetting in continual learning (CL) by storing and training on a subset of past data alongside new data in current task. While such a concurrent rehearsal strategy is widely used, it remains unclear if this approach is always optimal. Inspired by human learning, where sequentially revisiting tasks helps mitigate forgetting, we explore whether sequential rehearsal can offer greater benefits for CL compared to standard concurrent rehearsal. To address this question, we conduct a theoretical analysis of rehearsal-based CL in overparameterized linear models, comparing two strategies: 1) Concurrent Rehearsal, where past and new data are trained together, and 2) Sequential Rehearsal, where new data is trained first, followed by revisiting past data sequentially. By explicitly characterizing forgetting and generalization error, we show that sequential rehearsal performs better when tasks are less similar. These insights further motivate a novel Hybrid Rehearsal method, which trains similar tasks concurrently and revisits dissimilar tasks sequentially. We characterize its forgetting and generalization performance, and our experiments with deep neural networks further confirm that the hybrid approach outperforms standard concurrent rehearsal. This work provides the first comprehensive theoretical analysis of rehearsal-based CL.

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ICLR Conference 2024 Conference Paper

Doubly Robust Instance-Reweighted Adversarial Training

  • Daouda Sow
  • Sen Lin 0001
  • Zhangyang Wang
  • Yingbin Liang

Assigning importance weights to adversarial data has achieved great success in training adversarially robust networks under limited model capacity. However, existing instance-reweighted adversarial training (AT) methods heavily depend on heuristics and/or geometric interpretations to determine those importance weights, making these algorithms lack rigorous theoretical justification/guarantee. Moreover, recent research has shown that adversarial training suffers from a severe non-uniform robust performance across the training distribution, e.g., data points belonging to some classes can be much more vulnerable to adversarial attacks than others. To address both issues, in this paper, we propose a novel doubly-robust instance reweighted AT framework, which allows to obtain the importance weights via exploring distributionally robust optimization (DRO) techniques, and at the same time boosts the robustness on the most vulnerable examples. In particular, our importance weights are obtained by optimizing the KL-divergence regularized loss function, which allows us to devise new algorithms with a theoretical convergence guarantee. Experiments on standard classification datasets demonstrate that our proposed approach outperforms related state-of-the-art baseline methods in terms of average robust performance, and at the same time improves the robustness against attacks on the weakest data points. Codes can be found in the Supplement.

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ICML Conference 2024 Conference Paper

Improving Sample Efficiency of Model-Free Algorithms for Zero-Sum Markov Games

  • Songtao Feng
  • Ming Yin 0003
  • Yu-Xiang Wang 0003
  • Jing Yang 0002
  • Yingbin Liang

The problem of two-player zero-sum Markov games has recently attracted increasing interests in theoretical studies of multi-agent reinforcement learning (RL). In particular, for finite-horizon episodic Markov decision processes (MDPs), it has been shown that model-based algorithms can find an $\epsilon$-optimal Nash Equilibrium (NE) with the sample complexity of $O(H^3SAB/\epsilon^2)$, which is optimal in the dependence of the horizon $H$ and the number of states $S$ (where $A$ and $B$ denote the number of actions of the two players, respectively). However, none of the existing model-free algorithms can achieve such an optimality. In this work, we propose a model-free stage-based algorithm and show that it achieves the same sample complexity as the best model-based algorithm, and hence for the first time demonstrate that model-free algorithms can enjoy the same optimality in the $H$ dependence as model-based algorithms. The main improvement of the dependency on $H$ arises by leveraging the popular variance reduction technique based on the reference-advantage decomposition previously used only for single-agent RL. However, such a technique relies on a critical monotonicity property of the value function, which does not hold in Markov games due to the update of the policy via the coarse correlated equilibrium (CCE) oracle. Thus, to extend such a technique to Markov games, our algorithm features a key novel design of updating the reference value functions as the pair of optimistic and pessimistic value functions whose value difference is the smallest in the history in order to achieve the desired improvement in the sample efficiency.

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ICML Conference 2024 Conference Paper

In-context Convergence of Transformers

  • Yu Huang 0023
  • Yuan Cheng
  • Yingbin Liang

Transformers have recently revolutionized many domains in modern machine learning and one salient discovery is their remarkable in-context learning capability, where models can solve an unseen task by utilizing task-specific prompts without further parameters fine-tuning. This also inspired recent theoretical studies aiming to understand the in-context learning mechanism of transformers, which however focused only on linear transformers. In this work, we take the first step toward studying the learning dynamics of a one-layer transformer with softmax attention trained via gradient descent in order to in-context learn linear function classes. We consider a structured data model, where each token is randomly sampled from a set of feature vectors in either balanced or imbalanced fashion. For data with balanced features, we establish the finite-time convergence guarantee with near-zero prediction error by navigating our analysis over two phases of the training dynamics of the attention map. More notably, for data with imbalanced features, we show that the learning dynamics take a stage-wise convergence process, where the transformer first converges to a near-zero prediction error for the query tokens of dominant features, and then converges later to a near-zero error for query tokens of under-represented features, via one and four training phases. Our proof features new techniques for analyzing the competing strengths of two types of attention weights, the change of which determines different training phases.

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NeurIPS Conference 2024 Conference Paper

In-Context Learning with Representations: Contextual Generalization of Trained Transformers

  • Tong Yang
  • Yu Huang
  • Yingbin Liang
  • Yuejie Chi

In-context learning (ICL) refers to a remarkable capability of pretrained large language models, which can learn a new task given a few examples during inference. However, theoretical understanding of ICL is largely under-explored, particularly whether transformers can be trained to generalize to unseen examples in a prompt, which will require the model to acquire contextual knowledge of the prompt for generalization. This paper investigates the training dynamics of transformers by gradient descent through the lens of non-linear regression tasks. The contextual generalization here can be attained via learning the template function for each task in-context, where all template functions lie in a linear space with $m$ basis functions. We analyze the training dynamics of one-layer multi-head transformers to {in-contextly} predict unlabeled inputs given partially labeled prompts, where the labels contain Gaussian noise and the number of examples in each prompt are not sufficient to determine the template. Under mild assumptions, we show that the training loss for a one-layer multi-head transformer converges linearly to a global minimum. Moreover, the transformer effectively learns to perform ridge regression over the basis functions. To our knowledge, this study is the first provable demonstration that transformers can learn contextual (i. e. , template) information to generalize to both unseen examples and tasks when prompts contain only a small number of query-answer pairs.

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JMLR Journal 2024 Journal Article

Neural Networks with Sparse Activation Induced by Large Bias: Tighter Analysis with Bias-Generalized NTK

  • Hongru Yang
  • Ziyu Jiang
  • Ruizhe Zhang
  • Yingbin Liang
  • Zhangyang Wang

We study training one-hidden-layer ReLU networks in the neural tangent kernel (NTK) regime, where the networks' biases are initialized to some constant rather than zero. We prove that under such initialization, the neural network will have sparse activation throughout the entire training process, which enables fast training procedures via some sophisticated computational methods. With such initialization, we show that the neural networks possess a different limiting kernel which we call bias-generalized NTK, and we study various properties of the neural networks with this new kernel. We first characterize the gradient descent dynamics. In particular, we show that the network in this case can achieve as fast convergence as the dense network, as opposed to the previous work suggesting that the sparse networks converge slower. In addition, our result improves the previous required width to ensure convergence. Secondly, we study the networks' generalization: we show a width-sparsity dependence, which yields a sparsity-dependent Rademacher complexity and generalization bound. To our knowledge, this is the first sparsity-dependent generalization result via Rademacher complexity. Lastly, we study the smallest eigenvalue of this new kernel. We identify a data-dependent region where we can derive a much sharper lower bound on the NTK's smallest eigenvalue than the worst-case bound previously known. This can lead to improvement in the generalization bound. [abs] [ pdf ][ bib ] &copy JMLR 2024. ( edit, beta )

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NeurIPS Conference 2024 Conference Paper

Non-asymptotic Convergence of Training Transformers for Next-token Prediction

  • Ruiquan Huang
  • Yingbin Liang
  • Jing Yang

Transformers have achieved extraordinary success in modern machine learning due to their excellent ability to handle sequential data, especially in next-token prediction (NTP) tasks. However, the theoretical understanding of their performance in NTP is limited, with existing studies focusing mainly on asymptotic performance. This paper provides a fine-grained non-asymptotic analysis of the training dynamics of a one-layer transformer consisting of a self-attention module followed by a feed-forward layer. We first characterize the essential structural properties of training datasets for NTP using a mathematical framework based on partial orders. Then, we design a two-stage training algorithm, where the pre-processing stage for training the feed-forward layer and the main stage for training the attention layer exhibit fast convergence performance. Specifically, both layers converge sub-linearly to the direction of their corresponding max-margin solutions. We also show that the cross-entropy loss enjoys a linear convergence rate. Furthermore, we show that the trained transformer presents non-trivial prediction ability with dataset shift, which sheds light on the remarkable generalization performance of transformers. Our analysis technique involves the development of novel properties on the attention gradient and further in-depth analysis of how these properties contribute to the convergence of the training process. Our experiments further validate our theoretical findings.

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ICLR Conference 2024 Conference Paper

On the Hardness of Online Nonconvex Optimization with Single Oracle Feedback

  • Ziwei Guan
  • Yi Zhou 0017
  • Yingbin Liang

Online nonconvex optimization has been an active area of research recently. Previous studies either considered the global regret with full information about the objective functions, or studied the local regret with window-smoothed objective functions, which required access to unlimited number of gradient oracles per time step. In this paper, we focus on the more challenging and practical setting, where access to only a single oracle is allowed per time step, and take the local regret of the original (i.e., unsmoothed) objective functions as the performance metric. Specifically, for both settings respectively with a single exact and stochastic gradient oracle feedback, we derive lower bounds on the local regret and show that the classical online (stochastic) gradient descent algorithms are optimal. Moreover, for the more challenging setting with a single function value oracle feedback, we develop an online algorithm based on a one-point running difference gradient estimator, and show that such an algorithm achieves a local regret that a generic stochastic gradient oracle can best achieve.

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ICLR Conference 2024 Conference Paper

Provable Benefits of Multi-task RL under Non-Markovian Decision Making Processes

  • Ruiquan Huang
  • Yuan Cheng
  • Jing Yang 0002
  • Vincent Tan
  • Yingbin Liang

In multi-task reinforcement learning (RL) under Markov decision processes (MDPs), the presence of shared latent structures among multiple MDPs has been shown to yield significant benefits to the sample efficiency compared to single-task RL. In this paper, we investigate whether such a benefit can extend to more general sequential decision making problems such as predictive state representations (PSRs). The main challenge here is that the large and complex model space makes it hard to identify what types of common latent structure of multi-task PSRs can reduce the model complexity and improve sample efficiency. To this end, we posit a joint model class for tasks and use the notion of $\eta$-bracketing number to quantify its complexity; this number also serves as a general metric to capture the similarity of tasks and thus determines the benefit of multi-task over single-task RL. We first study upstream multi-task learning over PSRs, in which all tasks share the same observation and action spaces. We propose a provably efficient algorithm UMT-PSR for finding near-optimal policies for all PSRs, and demonstrate that the advantage of multi-task learning manifests if the joint model class of PSRs has a smaller $\eta$-bracketing number compared to that of individual single-task learning. We further investigate downstream learning, in which the agent needs to learn a new target task that shares some commonalities with the upstream tasks via a similarity constraint. By exploiting the learned PSRs from the upstream, we develop a sample-efficient algorithm that provably finds a near-optimal policy. Upon specialization to some examples with small $\eta$-bracketing numbers, our results further highlight the benefit compared to directly learning a single-task PSR.

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ICLR Conference 2024 Conference Paper

Provably Efficient UCB-type Algorithms For Learning Predictive State Representations

  • Ruiquan Huang
  • Yingbin Liang
  • Jing Yang 0002

The general sequential decision-making problem, which includes Markov decision processes (MDPs) and partially observable MDPs (POMDPs) as special cases, aims at maximizing a cumulative reward by making a sequence of decisions based on a history of observations and actions over time. Recent studies have shown that the sequential decision-making problem is statistically learnable if it admits a low-rank structure modeled by predictive state representations (PSRs). Despite these advancements, existing approaches typically involve oracles or steps that are computationally intractable. On the other hand, the upper confidence bound (UCB) based approaches, which have served successfully as computationally efficient methods in bandits and MDPs, have not been investigated for more general PSRs, due to the difficulty of optimistic bonus design in these more challenging settings. This paper proposes the first known UCB-type approach for PSRs, featuring a novel bonus term that upper bounds the total variation distance between the estimated and true models. We further characterize the sample complexity bounds for our designed UCB-type algorithms for both online and offline PSRs. In contrast to existing approaches for PSRs, our UCB-type algorithms enjoy computational tractability, last-iterate guaranteed near-optimal policy, and guaranteed model accuracy.

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NeurIPS Conference 2024 Conference Paper

Training Dynamics of Transformers to Recognize Word Co-occurrence via Gradient Flow Analysis

  • Hongru Yang
  • Bhavya Kailkhura
  • Zhangyang Wang
  • Yingbin Liang

Understanding the training dynamics of transformers is important to explain the impressive capabilities behind large language models. In this work, we study the dynamics of training a shallow transformer on a task of recognizing co-occurrence of two designated words. In the literature of studying training dynamics of transformers, several simplifications are commonly adopted such as weight reparameterization, attention linearization, special initialization, and lazy regime. In contrast, we analyze the gradient flow dynamics of simultaneously training three attention matrices and a linear MLP layer from random initialization, and provide a framework of analyzing such dynamics via a coupled dynamical system. We establish near minimum loss and characterize the attention model after training. We discover that gradient flow serves as an inherent mechanism that naturally divide the training process into two phases. In Phase 1, the linear MLP quickly aligns with the two target signals for correct classification, whereas the softmax attention remains almost unchanged. In Phase 2, the attention matrices and the MLP evolve jointly to enlarge the classification margin and reduce the loss to a near minimum value. Technically, we prove a novel property of the gradient flow, termed \textit{automatic balancing of gradients}, which enables the loss values of different samples to decrease almost at the same rate and further facilitates the proof of near minimum training loss. We also conduct experiments to verify our theoretical results.

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ICML Conference 2023 Conference Paper

A Near-Optimal Algorithm for Safe Reinforcement Learning Under Instantaneous Hard Constraints

  • Ming Shi 0003
  • Yingbin Liang
  • Ness B. Shroff

In many applications of Reinforcement Learning (RL), it is critically important that the algorithm performs safely, such that instantaneous hard constraints are satisfied at each step, and unsafe states and actions are avoided. However, existing algorithms for “safe” RL are often designed under constraints that either require expected cumulative costs to be bounded or assume all states are safe. Thus, such algorithms could violate instantaneous hard constraints and traverse unsafe states (and actions) in practice. Hence, in this paper, we develop the first near-optimal safe RL algorithm for episodic Markov Decision Processes with unsafe states and actions under instantaneous hard constraints and the linear mixture model. It achieves a regret $\tilde{O}(\frac{d H^3 \sqrt{d K}}{\Delta_c})$ that nearly matches the state-of-the-art regret in the setting with only unsafe actions and that in the unconstrained setting, and is safe at each step, where $d$ is the feature-mapping dimension, $K$ is the number of episodes, $H$ is the episode length, and $\Delta_c$ is a safety-related parameter. We also provide a lower bound $\tilde{\Omega}(\max\{d H \sqrt{K}, \frac{H}{\Delta_c^2}\})$, which indicates that the dependency on $\Delta_c$ is necessary. Further, both our algorithm design and regret analysis involve several novel ideas, which may be of independent interest.

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ICML Conference 2023 Conference Paper

Generalized-Smooth Nonconvex Optimization is As Efficient As Smooth Nonconvex Optimization

  • Ziyi Chen 0002
  • Yi Zhou 0017
  • Yingbin Liang
  • Zhaosong Lu

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are sub-optimal. Instead, these problems have been shown to satisfy certain generalized-smooth conditions, which have not been well understood in the existing literature. In this paper, we propose a notion of $\alpha$-symmetric generalized-smoothness that substantially extends the existing notions and covers many important functions such as high-order polynomials and exponential functions. We study the fundamental properties and establish descent lemmas for the functions in this class. Then, to solve such a large class of nonconvex problems, we design a special deterministic normalized gradient descent algorithm that achieves the optimal iteration complexity $\mathcal{O}(\epsilon^{-2})$, and also prove that the popular SPIDER variance reduction algorithm achieves the optimal sample complexity $\mathcal{O}(\epsilon^{-3})$. Our results show that solving generalized-smooth nonconvex problems is as efficient as solving smooth nonconvex problems.

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AAAI Conference 2023 Conference Paper

Global Convergence of Two-Timescale Actor-Critic for Solving Linear Quadratic Regulator

  • Xuyang Chen
  • Jingliang Duan
  • Yingbin Liang
  • Lin Zhao

The actor-critic (AC) reinforcement learning algorithms have been the powerhouse behind many challenging applications. Nevertheless, its convergence is fragile in general. To study its instability, existing works mostly consider the uncommon double-loop variant or basic models with finite state and action space. We investigate the more practical single-sample two-timescale AC for solving the canonical linear quadratic regulator (LQR) problem, where the actor and the critic update only once with a single sample in each iteration on an unbounded continuous state and action space. Existing analysis cannot conclude the convergence for such a challenging case. We develop a new analysis framework that allows establishing the global convergence to an epsilon-optimal solution with at most an order of epsilon to -2.5 sample complexity. To our knowledge, this is the first finite-time convergence analysis for the single sample two-timescale AC for solving LQR with global optimality. The sample complexity improves those of other variants by orders, which sheds light on the practical wisdom of single sample algorithms. We also further validate our theoretical findings via comprehensive simulation comparisons.

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ICLR Conference 2023 Conference Paper

Improved Sample Complexity for Reward-free Reinforcement Learning under Low-rank MDPs

  • Yuan Cheng
  • Ruiquan Huang
  • Yingbin Liang
  • Jing Yang 0002

In reward-free reinforcement learning (RL), an agent explores the environment first without any reward information, in order to achieve certain learning goals afterwards for any given reward. In this paper we focus on reward-free RL under low-rank MDP models, in which both the representation and linear weight vectors are unknown. Although various algorithms have been proposed for reward-free low-rank MDPs, the corresponding sample complexity is still far from being satisfactory. In this work, we first provide the first known sample complexity lower bound that holds for any algorithm under low-rank MDPs. This lower bound implies it is strictly harder to find a near-optimal policy under low-rank MDPs than under linear MDPs. We then propose a novel model-based algorithm, coined RAFFLE, and show it can both find an $\epsilon$-optimal policy and achieve an $\epsilon$-accurate system identification via reward-free exploration, with a sample complexity significantly improving the previous results. Such a sample complexity matches our lower bound in the dependence on $\epsilon$, as well as on $K$ {in the large $d$ regime}, where $d$ and $K$ respectively denote the representation dimension and action space cardinality. Finally, we provide a planning algorithm (without further interaction with true environment) for RAFFLE to learn a near-accurate representation, which is the first known representation learning guarantee under the same setting.

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JMLR Journal 2023 Journal Article

Lower Bounds and Accelerated Algorithms for Bilevel Optimization

  • Kaiyi Ji
  • Yingbin Liang

Bilevel optimization has recently attracted growing interests due to its wide applications in modern machine learning problems. Although recent studies have characterized the convergence rate for several such popular algorithms, it is still unclear how much further these convergence rates can be improved. In this paper, we address this fundamental question from two perspectives. First, we provide the first-known lower complexity bounds of $\widetilde \Omega\bigg(\sqrt{\frac{L_y\widetilde L_{xy}^2}{\mu_x\mu_y^2}}\bigg)$ and $\widetilde \Omega\big(\frac{1}{\sqrt{\epsilon}}\min\{\kappa_y,\frac{1}{\sqrt{\epsilon^{3}}}\}\big)$ respectively for strongly-convex-strongly-convex and convex-strongly-convex bilevel optimizations. Second, we propose an accelerated bilevel optimizer named AccBiO, for which we provide the first-known complexity bounds without the gradient boundedness assumption (which was made in existing analyses) under the two aforementioned geometries. We also provide significantly tighter upper bounds than the existing complexity when the bounded gradient assumption does hold. We show that AccBiO achieves the optimal results (i.e., the upper and lower bounds match up to logarithmic factors) when the inner-level problem takes a quadratic form with a constant-level condition number. Interestingly, our lower bounds under both geometries are larger than the corresponding optimal complexities of minimax optimization, establishing that bilevel optimization is provably more challenging than minimax optimization. Our theoretical results are validated by numerical experiments. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

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ICLR Conference 2023 Conference Paper

M-L2O: Towards Generalizable Learning-to-Optimize by Test-Time Fast Self-Adaptation

  • Junjie Yang
  • Xuxi Chen
  • Tianlong Chen 0001
  • Zhangyang Wang
  • Yingbin Liang

Learning to Optimize (L2O) has drawn increasing attention as it often remarkably accelerates the optimization procedure of complex tasks by "overfitting" specific task type, leading to enhanced performance compared to analytical optimizers. Generally, L2O develops a parameterized optimization method (i.e., "optimizer") by learning from solving sample problems. This data-driven procedure yields L2O that can efficiently solve problems similar to those seen in training, that is, drawn from the same "task distribution". However, such learned optimizers often struggle when new test problems come with a substantially deviation from the training task distribution. This paper investigates a potential solution to this open challenge, by meta-training an L2O optimizer that can perform fast test-time self-adaptation to a out-of-distribution task, in only a few steps. We theoretically characterize the generalization of L2O, and further show that our proposed framework (termed as M-L2O) provably facilitates rapid task adaptation by locating well-adapted initial points for the optimizer weight. Empirical observations on several classic tasks like LASSO and Quadratic, demonstrate that M-L2O converges significantly faster than vanilla L2O with only $5$ steps of adaptation, echoing our theoretical results. Codes are available in https://github.com/VITA-Group/M-L2O.

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ICLR Conference 2023 Conference Paper

Near-Optimal Adversarial Reinforcement Learning with Switching Costs

  • Ming Shi 0003
  • Yingbin Liang
  • Ness B. Shroff

Switching costs, which capture the costs for changing policies, are regarded as a critical metric in reinforcement learning (RL), in addition to the standard metric of losses (or rewards). However, existing studies on switching costs (with a coefficient that is strictly positive and is independent of the time horizon) have mainly focused on static RL, where the loss distribution is assumed to be fixed during the learning process, and thus practical scenarios where the loss distribution could be non-stationary or even adversarial are not considered. While adversarial RL better models this type of practical scenarios, an open problem remains: how to develop a provably efficient algorithm for adversarial RL with switching costs? This paper makes the first effort towards solving this problem. First, we provide a regret lower-bound that shows that the regret of any algorithm must be larger than $\tilde{\Omega}( ( H S A )^{1/3} T^{2/3} )$, where $T$, $S$, $A$ and $H$ are the number of episodes, states, actions and layers in each episode, respectively. Our lower bound indicates that, due to the fundamental challenge of switching costs in adversarial RL, the best achieved regret (whose dependency on $T$ is $\tilde{O}(\sqrt{T})$) in static RL with switching costs (as well as adversarial RL without switching costs) is no longer achievable. Moreover, we propose two novel switching-reduced algorithms with regrets that match our lower bound when the transition function is known, and match our lower bound within a small factor of $\tilde{O}( H^{1/3} )$ when the transition function is unknown. Our regret analysis demonstrates the near-optimal performance of them.

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NeurIPS Conference 2023 Conference Paper

Non-Convex Bilevel Optimization with Time-Varying Objective Functions

  • Sen Lin
  • Daouda Sow
  • Kaiyi Ji
  • Yingbin Liang
  • Ness Shroff

Bilevel optimization has become a powerful tool in a wide variety of machine learning problems. However, the current nonconvex bilevel optimization considers an offline dataset and static functions, which may not work well in emerging online applications with streaming data and time-varying functions. In this work, we study online bilevel optimization (OBO) where the functions can be time-varying and the agent continuously updates the decisions with online streaming data. To deal with the function variations and the unavailability of the true hypergradients in OBO, we propose a single-loop online bilevel optimizer with window averaging (SOBOW), which updates the outer-level decision based on a window average of the most recent hypergradient estimations stored in the memory. Compared to existing algorithms, SOBOW is computationally efficient and does not need to know previous functions. To handle the unique technical difficulties rooted in single-loop update and function variations for OBO, we develop a novel analytical technique that disentangles the complex couplings between decision variables, and carefully controls the hypergradient estimation error. We show that SOBOW can achieve a sublinear bilevel local regret under mild conditions. Extensive experiments across multiple domains corroborate the effectiveness of SOBOW.

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ICML Conference 2023 Conference Paper

Non-stationary Reinforcement Learning under General Function Approximation

  • Songtao Feng
  • Ming Yin 0003
  • Ruiquan Huang
  • Yu-Xiang Wang 0003
  • Jing Yang 0002
  • Yingbin Liang

General function approximation is a powerful tool to handle large state and action spaces in a broad range of reinforcement learning (RL) scenarios. However, theoretical understanding of non-stationary MDPs with general function approximation is still limited. In this paper, we make the first such an attempt. We first propose a new complexity metric called dynamic Bellman Eluder (DBE) dimension for non-stationary MDPs, which subsumes majority of existing tractable RL problems in static MDPs as well as non-stationary MDPs. Based on the proposed complexity metric, we propose a novel confidence-set based model-free algorithm called SW-OPEA, which features a sliding window mechanism and a new confidence set design for non-stationary MDPs. We then establish an upper bound on the dynamic regret for the proposed algorithm, and show that SW-OPEA is provably efficient as long as the variation budget is not significantly large. We further demonstrate via examples of non-stationary linear and tabular MDPs that our algorithm performs better in small variation budget scenario than the existing UCB-type algorithms. To the best of our knowledge, this is the first dynamic regret analysis in non-stationary MDPs with general function approximation.

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TMLR Journal 2023 Journal Article

Online Min-max Problems with Non-convexity and Non-stationarity

  • Yu Huang
  • Yuan Cheng
  • Yingbin Liang
  • Longbo Huang

Online min-max optimization has recently gained considerable interest due to its rich applications to game theory, multi-agent reinforcement learning, online robust learning, etc. Theoretical understanding in this field has been mainly focused on convex-concave settings. Online min-max optimization with nonconvex geometries, which captures various online deep learning problems, has yet been studied so far. In this paper, we make the first effort and investigate online nonconvex-strongly-concave min-max optimization in the nonstationary environment. We first introduce a natural notion of local Nash equilibrium (NE)-regret, and then propose a novel algorithm coined TSODA to achieve the optimal regret. We further generalize our study to the setting with stochastic first-order feedback, and show that a variation of TSODA can also achieve the same optimal regret in expectation. Our theoretical results and the superior performance of the proposed method are further validated by empirical experiments. To our best knowledge, this is the first exploration of efficient online nonconvex min-max optimization.

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NeurIPS Conference 2023 Conference Paper

Provably Efficient Algorithm for Nonstationary Low-Rank MDPs

  • Yuan Cheng
  • Jing Yang
  • Yingbin Liang

Reinforcement learning (RL) under changing environment models many real-world applications via nonstationary Markov Decision Processes (MDPs), and hence gains considerable interest. However, theoretical studies on nonstationary MDPs in the literature have mainly focused on tabular and linear (mixture) MDPs, which do not capture the nature of unknown representation in deep RL. In this paper, we make the first effort to investigate nonstationary RL under episodic low-rank MDPs, where both transition kernels and rewards may vary over time, and the low-rank model contains unknown representation in addition to the linear state embedding function. We first propose a parameter-dependent policy optimization algorithm called PORTAL, and further improve PORTAL to its parameter-free version of Ada-PORTAL, which is able to tune its hyper-parameters adaptively without any prior knowledge of nonstationarity. For both algorithms, we provide upper bounds on the average dynamic suboptimality gap, which show that as long as the nonstationarity is not significantly large, PORTAL and Ada-PORTAL are sample-efficient and can achieve arbitrarily small average dynamic suboptimality gap with polynomial sample complexity.

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ICLR Conference 2023 Conference Paper

Safe Exploration Incurs Nearly No Additional Sample Complexity for Reward-Free RL

  • Ruiquan Huang
  • Jing Yang 0002
  • Yingbin Liang

Reward-free reinforcement learning (RF-RL), a recently introduced RL paradigm, relies on random action-taking to explore the unknown environment without any reward feedback information. While the primary goal of the exploration phase in RF-RL is to reduce the uncertainty in the estimated model with minimum number of trajectories, in practice, the agent often needs to abide by certain safety constraint at the same time. It remains unclear how such safe exploration requirement would affect the corresponding sample complexity in order to achieve the desired optimality of the obtained policy in planning. In this work, we make a first attempt to answer this question. In particular, we consider the scenario where a safe baseline policy is known beforehand, and propose a unified Safe reWard-frEe ExploraTion (SWEET) framework. We then particularize the SWEET framework to the tabular and the low-rank MDP settings, and develop algorithms coined Tabular-SWEET and Low-rank-SWEET, respectively. Both algorithms leverage the concavity and continuity of the newly introduced truncated value functions, and are guaranteed to achieve zero constraint violation during exploration with high probability. Furthermore, both algorithms can provably find a near-optimal policy subject to any constraint in the planning phase. Remarkably, the sample complexities under both algorithms match or even outperform the state of the art in their constraint-free counterparts up to some constant factors, proving that safety constraint hardly increases the sample complexity for RF-RL.

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ICLR Conference 2023 Conference Paper

Theoretical Characterization of the Generalization Performance of Overfitted Meta-Learning

  • Peizhong Ju
  • Yingbin Liang
  • Ness B. Shroff

Meta-learning has arisen as a successful method for improving training performance by training over many similar tasks, especially with deep neural networks (DNNs). However, the theoretical understanding of when and why overparameterized models such as DNNs can generalize well in meta-learning is still limited. As an initial step towards addressing this challenge, this paper studies the generalization performance of overfitted meta-learning under a linear regression model with Gaussian features. In contrast to a few recent studies along the same line, our framework allows the number of model parameters to be arbitrarily larger than the number of features in the ground truth signal, and hence naturally captures the overparameterized regime in practical deep meta-learning. We show that the overfitted min $\ell_2$-norm solution of model-agnostic meta-learning (MAML) can be beneficial, which is similar to the recent remarkable findings on "benign overfitting" and "double descent" phenomenon in the classical (single-task) linear regression. However, due to the uniqueness of meta-learning such as task-specific gradient descent inner training and the diversity/fluctuation of the ground-truth signals among training tasks, we find new and interesting properties that do not exist in single-task linear regression. We first provide a high-probability upper bound (under reasonable tightness) on the generalization error, where certain terms decrease when the number of features increases. Our analysis suggests that benign overfitting is more significant and easier to observe when the noise and the diversity/fluctuation of the ground truth of each training task are large. Under this circumstance, we show that the overfitted min $\ell_2$-norm solution can achieve an even lower generalization error than the underparameterized solution.

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ICML Conference 2023 Conference Paper

Theory on Forgetting and Generalization of Continual Learning

  • Sen Lin 0001
  • Peizhong Ju
  • Yingbin Liang
  • Ness B. Shroff

Continual learning (CL), which aims to learn a sequence of tasks, has attracted significant recent attention. However, most work has focused on the experimental performance of CL, and theoretical studies of CL are still limited. In particular, there is a lack of understanding on what factors are important and how they affect "catastrophic forgetting" and generalization performance. To fill this gap, our theoretical analysis, under overparameterized linear models, provides the first-known explicit form of the expected forgetting and generalization error for a general CL setup with an arbitrary number of tasks. Further analysis of such a key result yields a number of theoretical explanations about how overparameterization, task similarity, and task ordering affect both forgetting and generalization error of CL. More interestingly, by conducting experiments on real datasets using deep neural networks (DNNs), we show that some of these insights even go beyond the linear models and can be carried over to practical setups. In particular, we use concrete examples to show that our results not only explain some interesting empirical observations in recent studies, but also motivate better practical algorithm designs of CL.

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NeurIPS Conference 2022 Conference Paper

A Unifying Framework of Off-Policy General Value Function Evaluation

  • Tengyu Xu
  • Zhuoran Yang
  • Zhaoran Wang
  • Yingbin Liang

General Value Function (GVF) is a powerful tool to represent both the {\em predictive} and {\em retrospective} knowledge in reinforcement learning (RL). In practice, often multiple interrelated GVFs need to be evaluated jointly with pre-collected off-policy samples. In the literature, the gradient temporal difference (GTD) learning method has been adopted to evaluate GVFs in the off-policy setting, but such an approach may suffer from a large estimation error even if the function approximation class is sufficiently expressive. Moreover, none of the previous work have formally established the convergence guarantee to the ground truth GVFs under the function approximation settings. In this paper, we address both issues through the lens of a class of GVFs with causal filtering, which cover a wide range of RL applications such as reward variance, value gradient, cost in anomaly detection, stationary distribution gradient, etc. We propose a new algorithm called GenTD for off-policy GVFs evaluation and show that GenTD learns multiple interrelated multi-dimensional GVFs as efficiently as a single canonical scalar value function. We further show that unlike GTD, the learned GVFs by GenTD are guaranteed to converge to the ground truth GVFs as long as the function approximation power is sufficiently large. To our best knowledge, GenTD is the first off-policy GVF evaluation algorithm that has global optimality guarantee.

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UAI Conference 2022 Conference Paper

Data sampling affects the complexity of online SGD over dependent data

  • Shaocong Ma
  • Ziyi Chen 0002
  • Yi Zhou 0017
  • Kaiyi Ji
  • Yingbin Liang

Conventional machine learning applications typically assume that data samples are independently and identically distributed (i. i. d.). However, practical scenarios often involve a data-generating process that produces highly dependent data samples, which are known to heavily bias the stochastic optimization process and slow down the convergence of learning. In this paper, we conduct a fundamental study on how different stochastic data sampling schemes affect the sample complexity of online stochastic gradient descent (SGD) over highly dependent data. Specifically, with a $\phi$-mixing process of data, we show that online SGD with proper periodic data-subsampling achieves an improved sample complexity over the standard online SGD in the full spectrum of the data dependence level. Interestingly, even subsampling a subset of data samples can accelerate the convergence of online SGD over highly dependent data. Moreover, we show that online SGD with mini-batch sampling can further substantially improve the sample complexity over online SGD with periodic data-subsampling over highly dependent data. Numerical experiments validate our theoretical results.

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UAI Conference 2022 Conference Paper

Deterministic policy gradient: Convergence analysis

  • Huaqing Xiong
  • Tengyu Xu
  • Lin Zhao 0009
  • Yingbin Liang
  • Wei Zhang 0013

The deterministic policy gradient (DPG) method proposed in Silver et al. [2014] has been demonstrated to exhibit superior performance particularly for applications with multi-dimensional and continuous action spaces. However, it remains unclear whether DPG converges, and if so, how fast it converges and whether it converges as efficiently as other PG methods. In this paper, we provide a theoretical analysis of DPG to answer those questions. We study the single timescale DPG (often the case in practice) in both on-policy and off-policy settings, and show that both algorithms attain an $\epsilon$-accurate stationary policy with a sample complexity of $\mathcal{O}(\epsilon^{-2})$. Moreover, we establish the convergence rate for DPG under Gaussian noise exploration, which is widely adopted in practice to improve the performance of DPG. To our best knowledge, this is the first non-asymptotic convergence characterization for DPG methods.

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ICLR Conference 2022 Conference Paper

Model-Based Offline Meta-Reinforcement Learning with Regularization

  • Sen Lin 0001
  • Jialin Wan
  • Tengyu Xu
  • Yingbin Liang
  • Junshan Zhang

Existing offline reinforcement learning (RL) methods face a few major challenges, particularly the distributional shift between the learned policy and the behavior policy. Offline Meta-RL is emerging as a promising approach to address these challenges, aiming to learn an informative meta-policy from a collection of tasks. Nevertheless, as shown in our empirical studies, offline Meta-RL could be outperformed by offline single-task RL methods on tasks with good quality of datasets, indicating that a right balance has to be delicately calibrated between "exploring" the out-of-distribution state-actions by following the meta-policy and "exploiting" the offline dataset by staying close to the behavior policy. Motivated by such empirical analysis, we propose model-based offline $\text{\bf Me}$ta-RL with $\text{\bf r}$egularized $\text{\bf P}$olicy $\text{\bf O}$ptimization (MerPO), which learns a meta-model for efficient task structure inference and an informative meta-policy for safe exploration of out-of-distribution state-actions. In particular, we devise a new meta-Regularized model-based Actor-Critic (RAC) method for within-task policy optimization, as a key building block of MerPO, using both conservative policy evaluation and regularized policy improvement; and the intrinsic tradeoff therein is achieved via striking the right balance between two regularizers, one based on the behavior policy and the other on the meta-policy. We theoretically show that the learnt policy offers guaranteed improvement over both the behavior policy and the meta-policy, thus ensuring the performance improvement on new tasks via offline Meta-RL. Our experiments corroborate the superior performance of MerPO over existing offline Meta-RL methods.

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NeurIPS Conference 2022 Conference Paper

On the Convergence Theory for Hessian-Free Bilevel Algorithms

  • Daouda Sow
  • Kaiyi Ji
  • Yingbin Liang

Bilevel optimization has arisen as a powerful tool in modern machine learning. However, due to the nested structure of bilevel optimization, even gradient-based methods require second-order derivative approximations via Jacobian- or/and Hessian-vector computations, which can be costly and unscalable in practice. Recently, Hessian-free bilevel schemes have been proposed to resolve this issue, where the general idea is to use zeroth- or first-order methods to approximate the full hypergradient of the bilevel problem. However, we empirically observe that such approximation can lead to large variance and unstable training, but estimating only the response Jacobian matrix as a partial component of the hypergradient turns out to be extremely effective. To this end, we propose a new Hessian-free method, which adopts the zeroth-order-like method to approximate the response Jacobian matrix via taking difference between two optimization paths. Theoretically, we provide the convergence rate analysis for the proposed algorithms, where our key challenge is to characterize the approximation and smoothness properties of the trajectory-dependent estimator, which can be of independent interest. This is the first known convergence rate result for this type of Hessian-free bilevel algorithms. Experimentally, we demonstrate that the proposed algorithms outperform baseline bilevel optimizers on various bilevel problems. Particularly, in our experiment on few-shot meta-learning with ResNet-12 network over the miniImageNet dataset, we show that our algorithm outperforms baseline meta-learning algorithms, while other baseline bilevel optimizers do not solve such meta-learning problems within a comparable time frame.

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ICLR Conference 2022 Conference Paper

PER-ETD: A Polynomially Efficient Emphatic Temporal Difference Learning Method

  • Ziwei Guan
  • Tengyu Xu
  • Yingbin Liang

Emphatic temporal difference (ETD) learning (Sutton et al., 2016) is a successful method to conduct the off-policy value function evaluation with function approximation. Although ETD has been shown to converge asymptotically to a desirable value function, it is well-known that ETD often encounters a large variance so that its sample complexity can increase exponentially fast with the number of iterations. In this work, we propose a new ETD method, called PER-ETD (i.e., PEriodically Restarted-ETD), which restarts and updates the follow-on trace only for a finite period for each iteration of the evaluation parameter. Further, PER-ETD features a design of the logarithmical increase of the restart period with the number of iterations, which guarantees the best trade-off between the variance and bias and keeps both vanishing sublinearly. We show that PER-ETD converges to the same desirable fixed point as ETD, but improves the exponential sample complexity of ETD to be polynomials. Our experiments validate the superior performance of PER-ETD and its advantage over ETD.

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NeurIPS Conference 2022 Conference Paper

Provable Benefit of Multitask Representation Learning in Reinforcement Learning

  • Yuan Cheng
  • Songtao Feng
  • Jing Yang
  • Hong Zhang
  • Yingbin Liang

As representation learning becomes a powerful technique to reduce sample complexity in reinforcement learning (RL) in practice, theoretical understanding of its advantage is still limited. In this paper, we theoretically characterize the benefit of representation learning under the low-rank Markov decision process (MDP) model. We first study multitask low-rank RL (as upstream training), where all tasks share a common representation, and propose a new multitask reward-free algorithm called REFUEL. REFUEL learns both the transition kernel and the near-optimal policy for each task, and outputs a well-learned representation for downstream tasks. Our result demonstrates that multitask representation learning is provably more sample-efficient than learning each task individually, as long as the total number of tasks is above a certain threshold. We then study the downstream RL in both online and offline settings, where the agent is assigned with a new task sharing the same representation as the upstream tasks. For both online and offline settings, we develop a sample-efficient algorithm, and show that it finds a near-optimal policy with the suboptimality gap bounded by the sum of the estimation error of the learned representation in upstream and a vanishing term as the number of downstream samples becomes large. Our downstream results of online and offline RL further capture the benefit of employing the learned representation from upstream as opposed to learning the representation of the low-rank model directly. To the best of our knowledge, this is the first theoretical study that characterizes the benefit of representation learning in exploration-based reward-free multitask RL for both upstream and downstream tasks.

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NeurIPS Conference 2022 Conference Paper

Provable Generalization of Overparameterized Meta-learning Trained with SGD

  • Yu Huang
  • Yingbin Liang
  • Longbo Huang

Despite the empirical success of deep meta-learning, theoretical understanding of overparameterized meta-learning is still limited. This paper studies the generalization of a widely used meta-learning approach, Model-Agnostic Meta-Learning (MAML), which aims to find a good initialization for fast adaptation to new tasks. Under a mixed linear regression model, we analyze the generalization properties of MAML trained with SGD in the overparameterized regime. We provide both upper and lower bounds for the excess risk of MAML, which captures how SGD dynamics affect these generalization bounds. With such sharp characterizations, we further explore how various learning parameters impact the generalization capability of overparameterized MAML, including explicitly identifying typical data and task distributions that can achieve diminishing generalization error with overparameterization, and characterizing the impact of adaptation learning rate on both excess risk and the early stopping time. Our theoretical findings are further validated by experiments.

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JMLR Journal 2022 Journal Article

Theoretical Convergence of Multi-Step Model-Agnostic Meta-Learning

  • Kaiyi Ji
  • Junjie Yang
  • Yingbin Liang

As a popular meta-learning approach, the model-agnostic meta-learning (MAML) algorithm has been widely used due to its simplicity and effectiveness. However, the convergence of the general multi-step MAML still remains unexplored. In this paper, we develop a new theoretical framework to provide such convergence guarantee for two types of objective functions that are of interest in practice: (a) resampling case (e.g., reinforcement learning), where loss functions take the form in expectation and new data are sampled as the algorithm runs; and (b) finite-sum case (e.g., supervised learning), where loss functions take the finite-sum form with given samples. For both cases, we characterize the convergence rate and the computational complexity to attain an $\epsilon$-accurate solution for multi-step MAML in the general nonconvex setting. In particular, our results suggest that an inner-stage stepsize needs to be chosen inversely proportional to the number $N$ of inner-stage steps in order for $N$-step MAML to have guaranteed convergence. From the technical perspective, we develop novel techniques to deal with the nested structure of the meta gradient for multi-step MAML, which can be of independent interest. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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NeurIPS Conference 2022 Conference Paper

Will Bilevel Optimizers Benefit from Loops

  • Kaiyi Ji
  • Mingrui Liu
  • Yingbin Liang
  • Lei Ying

Bilevel optimization has arisen as a powerful tool for solving a variety of machine learning problems. Two current popular bilevel optimizers AID-BiO and ITD-BiO naturally involve solving one or two sub-problems, and consequently, whether we solve these problems with loops (that take many iterations) or without loops (that take only a few iterations) can significantly affect the overall computational efficiency. Existing studies in the literature cover only some of those implementation choices, and the complexity bounds available are not refined enough to enable rigorous comparison among different implementations. In this paper, we first establish unified convergence analysis for both AID-BiO and ITD-BiO that are applicable to all implementation choices of loops. We then specialize our results to characterize the computational complexity for all implementations, which enable an explicit comparison among them. Our result indicates that for AID-BiO, the loop for estimating the optimal point of the inner function is beneficial for overall efficiency, although it causes higher complexity for each update step, and the loop for approximating the outer-level Hessian-inverse-vector product reduces the gradient complexity. For ITD-BiO, the two loops always coexist, and our convergence upper and lower bounds show that such loops are necessary to guarantee a vanishing convergence error, whereas the no-loop scheme suffers from an unavoidable non-vanishing convergence error. Our numerical experiments further corroborate our theoretical results.

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ICML Conference 2021 Conference Paper

Bilevel Optimization: Convergence Analysis and Enhanced Design

  • Kaiyi Ji
  • Junjie Yang
  • Yingbin Liang

Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensive convergence rate analysis for two popular algorithms respectively based on approximate implicit differentiation (AID) and iterative differentiation (ITD). For the AID-based method, we orderwisely improve the previous convergence rate analysis due to a more practical parameter selection as well as a warm start strategy, and for the ITD-based method we establish the first theoretical convergence rate. Our analysis also provides a quantitative comparison between ITD and AID based approaches. For stochastic bilevel optimization, we propose a novel algorithm named stocBiO, which features a sample-efficient hypergradient estimator using efficient Jacobian- and Hessian-vector product computations. We provide the convergence rate guarantee for stocBiO, and show that stocBiO outperforms the best known computational complexities orderwisely with respect to the condition number $\kappa$ and the target accuracy $\epsilon$. We further validate our theoretical results and demonstrate the efficiency of bilevel optimization algorithms by the experiments on meta-learning and hyperparameter optimization.

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ICML Conference 2021 Conference Paper

CRPO: A New Approach for Safe Reinforcement Learning with Convergence Guarantee

  • Tengyu Xu
  • Yingbin Liang
  • Guanghui Lan

In safe reinforcement learning (SRL) problems, an agent explores the environment to maximize an expected total reward and meanwhile avoids violation of certain constraints on a number of expected total costs. In general, such SRL problems have nonconvex objective functions subject to multiple nonconvex constraints, and hence are very challenging to solve, particularly to provide a globally optimal policy. Many popular SRL algorithms adopt a primal-dual structure which utilizes the updating of dual variables for satisfying the constraints. In contrast, we propose a primal approach, called constraint-rectified policy optimization (CRPO), which updates the policy alternatingly between objective improvement and constraint satisfaction. CRPO provides a primal-type algorithmic framework to solve SRL problems, where each policy update can take any variant of policy optimization step. To demonstrate the theoretical performance of CRPO, we adopt natural policy gradient (NPG) for each policy update step and show that CRPO achieves an $\mathcal{O}(1/\sqrt{T})$ convergence rate to the global optimal policy in the constrained policy set and an $\mathcal{O}(1/\sqrt{T})$ error bound on constraint satisfaction. This is the first finite-time analysis of primal SRL algorithms with global optimality guarantee. Our empirical results demonstrate that CRPO can outperform the existing primal-dual baseline algorithms significantly.

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ICML Conference 2021 Conference Paper

Doubly Robust Off-Policy Actor-Critic: Convergence and Optimality

  • Tengyu Xu
  • Zhuoran Yang
  • Zhaoran Wang 0001
  • Yingbin Liang

Designing off-policy reinforcement learning algorithms is typically a very challenging task, because a desirable iteration update often involves an expectation over an on-policy distribution. Prior off-policy actor-critic (AC) algorithms have introduced a new critic that uses the density ratio for adjusting the distribution mismatch in order to stabilize the convergence, but at the cost of potentially introducing high biases due to the estimation errors of both the density ratio and value function. In this paper, we develop a doubly robust off-policy AC (DR-Off-PAC) for discounted MDP, which can take advantage of learned nuisance functions to reduce estimation errors. Moreover, DR-Off-PAC adopts a single timescale structure, in which both actor and critics are updated simultaneously with constant stepsize, and is thus more sample efficient than prior algorithms that adopt either two timescale or nested-loop structure. We study the finite-time convergence rate and characterize the sample complexity for DR-Off-PAC to attain an $\epsilon$-accurate optimal policy. We also show that the overall convergence of DR-Off-PAC is doubly robust to the approximation errors that depend only on the expressive power of approximation functions. To the best of our knowledge, our study establishes the first overall sample complexity analysis for single time-scale off-policy AC algorithm.

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NeurIPS Conference 2021 Conference Paper

Faster Non-asymptotic Convergence for Double Q-learning

  • Lin Zhao
  • Huaqing Xiong
  • Yingbin Liang

Double Q-learning (Hasselt, 2010) has gained significant success in practice due to its effectiveness in overcoming the overestimation issue of Q-learning. However, the theoretical understanding of double Q-learning is rather limited. The only existing finite-time analysis was recently established in (Xiong et al. 2020), where the polynomial learning rate adopted in the analysis typically yields a slower convergence rate. This paper tackles the more challenging case of a constant learning rate, and develops new analytical tools that improve the existing convergence rate by orders of magnitude. Specifically, we show that synchronous double Q-learning attains an $\epsilon$-accurate global optimum with a time complexity of $\tilde{\Omega}\left(\frac{\ln D}{(1-\gamma)^7\epsilon^2} \right)$, and the asynchronous algorithm achieves a time complexity of $\tilde{\Omega}\left(\frac{L}{(1-\gamma)^7\epsilon^2} \right)$, where $D$ is the cardinality of the state-action space, $\gamma$ is the discount factor, and $L$ is a parameter related to the sampling strategy for asynchronous double Q-learning. These results improve the existing convergence rate by the order of magnitude in terms of its dependence on all major parameters $(\epsilon, 1-\gamma, D, L)$. This paper presents a substantial step toward the full understanding of the fast convergence of double-Q learning.

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UAI Conference 2021 Conference Paper

Finite-time theory for momentum Q-learning

  • Bowen Weng
  • Huaqing Xiong
  • Lin Zhao 0009
  • Yingbin Liang
  • Wei Zhang 0013

Existing studies indicate that momentum ideas in conventional optimization can be used to improve the performance of Q-learning algorithms. However, the finite-time analysis for momentum-based Q-learning algorithms is only available for the tabular case without function approximation. This paper analyzes a class of momentum-based Q-learning algorithms with finite-time convergence guarantee. Specifically, we propose the MomentumQ algorithm, which integrates the Nesterov’s and Polyak’s momentum schemes, and generalizes the existing momentum-based Q-learning algorithms. For the infinite state-action space case, we establish the convergence guarantee for MomentumQ with linear function approximation under Markovian sampling. In particular, we characterize a finite-time convergence rate which is provably faster than the vanilla Q-learning. This is the first finite-time analysis for momentum-based Q-learning algorithms with function approximation. For the tabular case under synchronous sampling, we also obtain a finite-time convergence rate that is slightly better than the SpeedyQ (Azar et al. , NIPS 2011). Finally, we demonstrate through various experiments that the proposed MomentumQ outperforms other momentum-based Q-learning algorithms.

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AAAI Conference 2021 Conference Paper

Non-asymptotic Convergence of Adam-type Reinforcement Learning Algorithms under Markovian Sampling

  • Huaqing Xiong
  • Tengyu Xu
  • Yingbin Liang
  • Wei Zhang

Despite the wide applications of Adam in reinforcement learning (RL), the theoretical convergence of Adam-type RL algorithms has not been established. This paper provides the first such convergence analysis for two fundamental RL algorithms of policy gradient (PG) and temporal difference (TD) learning that incorporate AMSGrad updates (a standard alternative of Adam in theoretical analysis), referred to as PG- AMSGrad and TD-AMSGrad, respectively. Moreover, our analysis focuses on Markovian sampling for both algorithms. We show that under general nonlinear function approximation, PG-AMSGrad with a constant stepsize converges to a neighborhood of a stationary point at the rate of O(1/T) (where T denotes the number of iterations), and with a diminishing stepsize converges exactly to a stationary point at the rate of O(log2 T/ √ T). Furthermore, under linear function approximation, TD-AMSGrad with a constant stepsize converges to a neighborhood of the global optimum at the rate of O(1/T), and with a diminishing stepsize converges exactly to the global optimum at the rate of O(log T/ √ T). Our study develops new techniques for analyzing the Adamtype RL algorithms under Markovian sampling.

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NeurIPS Conference 2021 Conference Paper

Provably Faster Algorithms for Bilevel Optimization

  • Junjie Yang
  • Kaiyi Ji
  • Yingbin Liang

Bilevel optimization has been widely applied in many important machine learning applications such as hyperparameter optimization and meta-learning. Recently, several momentum-based algorithms have been proposed to solve bilevel optimization problems faster. However, those momentum-based algorithms do not achieve provably better computational complexity than $\mathcal{\widetilde O}(\epsilon^{-2})$ of the SGD-based algorithm. In this paper, we propose two new algorithms for bilevel optimization, where the first algorithm adopts momentum-based recursive iterations, and the second algorithm adopts recursive gradient estimations in nested loops to decrease the variance. We show that both algorithms achieve the complexity of $\mathcal{\widetilde O}(\epsilon^{-1. 5})$, which outperforms all existing algorithms by the order of magnitude. Our experiments validate our theoretical results and demonstrate the superior empirical performance of our algorithms in hyperparameter applications.

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ICLR Conference 2021 Conference Paper

Proximal Gradient Descent-Ascent: Variable Convergence under KŁ Geometry

  • Ziyi Chen 0002
  • Yi Zhou 0017
  • Tengyu Xu
  • Yingbin Liang

The gradient descent-ascent (GDA) algorithm has been widely applied to solve minimax optimization problems. In order to achieve convergent policy parameters for minimax optimization, it is important that GDA generates convergent variable sequences rather than convergent sequences of function value or gradient norm. However, the variable convergence of GDA has been proved only under convexity geometries, and it is lack of understanding in general nonconvex minimax optimization. This paper fills such a gap by studying the convergence of a more general proximal-GDA for regularized nonconvex-strongly-concave minimax optimization. Specifically, we show that proximal-GDA admits a novel Lyapunov function, which monotonically decreases in the minimax optimization process and drives the variable sequences to a critical point. By leveraging this Lyapunov function and the KL geometry that parameterizes the local geometries of general nonconvex functions, we formally establish the variable convergence of proximal-GDA to a certain critical point $x^*$, i.e., $x_t\to x^*, y_t\to y^*(x^*)$. Furthermore, over the full spectrum of the KL-parameterized geometry, we show that proximal-GDA achieves different types of convergence rates ranging from sublinear convergence up to finite-step convergence, depending on the geometry associated with the KL parameter. This is the first theoretical result on the variable convergence for nonconvex minimax optimization.

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IJCAI Conference 2020 Conference Paper

Analysis of Q-learning with Adaptation and Momentum Restart for Gradient Descent

  • Bowen Weng
  • Huaqing Xiong
  • Yingbin Liang
  • Wei Zhang

Existing convergence analyses of Q-learning mostly focus on the vanilla stochastic gradient descent (SGD) type of updates. Despite the Adaptive Moment Estimation (Adam) has been commonly used for practical Q-learning algorithms, there has not been any convergence guarantee provided for Q-learning with such type of updates. In this paper, we first characterize the convergence rate for Q-AMSGrad, which is the Q-learning algorithm with AMSGrad update (a commonly adopted alternative of Adam for theoretical analysis). To further improve the performance, we propose to incorporate the momentum restart scheme to Q-AMSGrad, resulting in the so-called Q-AMSGradR algorithm. The convergence rate of Q-AMSGradR is also established. Our experiments on a linear quadratic regulator problem demonstrate that the two proposed Q-learning algorithms outperform the vanilla Q-learning with SGD updates. The two algorithms also exhibit significantly better performance than the DQN learning method over a batch of Atari 2600 games.

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NeurIPS Conference 2020 Conference Paper

Convergence of Meta-Learning with Task-Specific Adaptation over Partial Parameters

  • Kaiyi Ji
  • Jason D. Lee
  • Yingbin Liang
  • H. Vincent Poor

Although model-agnostic meta-learning (MAML) is a very successful algorithm in meta-learning practice, it can have high computational cost because it updates all model parameters over both the inner loop of task-specific adaptation and the outer-loop of meta initialization training. A more efficient algorithm ANIL (which refers to almost no inner loop) was proposed recently by Raghu et al. 2019, which adapts only a small subset of parameters in the inner loop and thus has substantially less computational cost than MAML as demonstrated by extensive experiments. However, the theoretical convergence of ANIL has not been studied yet. In this paper, we characterize the convergence rate and the computational complexity for ANIL under two representative inner-loop loss geometries, i. e. , strongly-convexity and nonconvexity. Our results show that such a geometric property can significantly affect the overall convergence performance of ANIL. For example, ANIL achieves a faster convergence rate for a strongly-convex inner-loop loss as the number $N$ of inner-loop gradient descent steps increases, but a slower convergence rate for a nonconvex inner-loop loss as $N$ increases. Moreover, our complexity analysis provides a theoretical quantification on the improved efficiency of ANIL over MAML. The experiments on standard few-shot meta-learning benchmarks validate our theoretical findings.

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NeurIPS Conference 2020 Conference Paper

Finite-Time Analysis for Double Q-learning

  • Huaqing Xiong
  • Lin Zhao
  • Yingbin Liang
  • Wei Zhang

Although Q-learning is one of the most successful algorithms for finding the best action-value function (and thus the optimal policy) in reinforcement learning, its implementation often suffers from large overestimation of Q-function values incurred by random sampling. The double Q-learning algorithm proposed in~\citet{hasselt2010double} overcomes such an overestimation issue by randomly switching the update between two Q-estimators, and has thus gained significant popularity in practice. However, the theoretical understanding of double Q-learning is rather limited. So far only the asymptotic convergence has been established, which does not characterize how fast the algorithm converges. In this paper, we provide the first non-asymptotic (i. e. , finite-time) analysis for double Q-learning. We show that both synchronous and asynchronous double Q-learning are guaranteed to converge to an $\epsilon$-accurate neighborhood of the global optimum by taking $\tilde{\Omega}\left(\left( \frac{1}{(1-\gamma)^6\epsilon^2}\right)^{\frac{1}{\omega}} +\left(\frac{1}{1-\gamma}\right)^{\frac{1}{1-\omega}}\right)$ iterations, where $\omega\in(0, 1)$ is the decay parameter of the learning rate, and $\gamma$ is the discount factor. Our analysis develops novel techniques to derive finite-time bounds on the difference between two inter-connected stochastic processes, which is new to the literature of stochastic approximation.

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ICML Conference 2020 Conference Paper

History-Gradient Aided Batch Size Adaptation for Variance Reduced Algorithms

  • Kaiyi Ji
  • Zhe Wang 0021
  • Bowen Weng
  • Yi Zhou 0017
  • Wei Zhang 0013
  • Yingbin Liang

Variance-reduced algorithms, although achieve great theoretical performance, can run slowly in practice due to the periodic gradient estimation with a large batch of data. Batch-size adaptation thus arises as a promising approach to accelerate such algorithms. However, existing schemes either apply prescribed batch-size adaption rule or exploit the information along optimization path via additional backtracking and condition verification steps. In this paper, we propose a novel scheme, which eliminates backtracking line search but still exploits the information along optimization path by adapting the batch size via history stochastic gradients. We further theoretically show that such a scheme substantially reduces the overall complexity for popular variance-reduced algorithms SVRG and SARAH/SPIDER for both conventional nonconvex optimization and reinforcement learning problems. To this end, we develop a new convergence analysis framework to handle the dependence of the batch size on history stochastic gradients. Extensive experiments validate the effectiveness of the proposed batch-size adaptation scheme.

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NeurIPS Conference 2020 Conference Paper

Improving Sample Complexity Bounds for (Natural) Actor-Critic Algorithms

  • Tengyu Xu
  • Zhe Wang
  • Yingbin Liang

The actor-critic (AC) algorithm is a popular method to find an optimal policy in reinforcement learning. In the infinite horizon scenario, the finite-sample convergence rate for the AC and natural actor-critic (NAC) algorithms has been established recently, but under independent and identically distributed (i. i. d. ) sampling and single-sample update at each iteration. In contrast, this paper characterizes the convergence rate and sample complexity of AC and NAC under Markovian sampling, with mini-batch data for each iteration, and with actor having general policy class approximation. We show that the overall sample complexity for a mini-batch AC to attain an $\epsilon$-accurate stationary point improves the best known sample complexity of AC by an order of $\mathcal{O}(\epsilon^{-1}\log(1/\epsilon))$, and the overall sample complexity for a mini-batch NAC to attain an $\epsilon$-accurate globally optimal point improves the existing sample complexity of NAC by an order of $\mathcal{O}(\epsilon^{-2}/\log(1/\epsilon))$. Moreover, the sample complexity of AC and NAC characterized in this work outperforms that of policy gradient (PG) and natural policy gradient (NPG) by a factor of $\mathcal{O}((1-\gamma)^{-3})$ and $\mathcal{O}((1-\gamma)^{-4}\epsilon^{-2}/\log(1/\epsilon))$, respectively. This is the first theoretical study establishing that AC and NAC attain orderwise performance improvement over PG and NPG under infinite horizon due to the incorporation of critic.

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IJCAI Conference 2020 Conference Paper

Proximal Gradient Algorithm with Momentum and Flexible Parameter Restart for Nonconvex Optimization

  • Yi Zhou
  • Zhe Wang
  • Kaiyi Ji
  • Yingbin Liang
  • Vahid Tarokh

Various types of parameter restart schemes have been proposed for proximal gradient algorithm with momentum to facilitate their convergence in convex optimization. However, under parameter restart, the convergence of proximal gradient algorithm with momentum remains obscure in nonconvex optimization. In this paper, we propose a novel proximal gradient algorithm with momentum and parameter restart for solving nonconvex and nonsmooth problems. Our algorithm is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global sub-linear convergence rate in nonconvex and nonsmooth optimization; and 3) have guaranteed convergence to a critical point and have various types of asymptotic convergence rates depending on the parameterization of local geometry in nonconvex and nonsmooth optimization. Numerical experiments demonstrate the convergence and effectiveness of our proposed algorithm.

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ICLR Conference 2020 Conference Paper

Reanalysis of Variance Reduced Temporal Difference Learning

  • Tengyu Xu
  • Zhe Wang 0021
  • Yi Zhou 0017
  • Yingbin Liang

Temporal difference (TD) learning is a popular algorithm for policy evaluation in reinforcement learning, but the vanilla TD can substantially suffer from the inherent optimization variance. A variance reduced TD (VRTD) algorithm was proposed by \cite{korda2015td}, which applies the variance reduction technique directly to the online TD learning with Markovian samples. In this work, we first point out the technical errors in the analysis of VRTD in \cite{korda2015td}, and then provide a mathematically solid analysis of the non-asymptotic convergence of VRTD and its variance reduction performance. We show that VRTD is guaranteed to converge to a neighborhood of the fixed-point solution of TD at a linear convergence rate. Furthermore, the variance error (for both i.i.d.\ and Markovian sampling) and the bias error (for Markovian sampling) of VRTD are significantly reduced by the batch size of variance reduction in comparison to those of vanilla TD. As a result, the overall computational complexity of VRTD to attain a given accurate solution outperforms that of TD under Markov sampling and outperforms that of TD under i.i.d.\ sampling for a sufficiently small conditional number.

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AAAI Conference 2020 Conference Paper

Robust Stochastic Bandit Algorithms under Probabilistic Unbounded Adversarial Attack

  • Ziwei Guan
  • Kaiyi Ji
  • Donald J. Bucci Jr.
  • Timothy Y. Hu
  • Joseph Palombo
  • Michael Liston
  • Yingbin Liang

The multi-armed bandit formalism has been extensively studied under various attack models, in which an adversary can modify the reward revealed to the player. Previous studies focused on scenarios where the attack value either is bounded at each round or has a vanishing probability of occurrence. These models do not capture powerful adversaries that can catastrophically perturb the revealed reward. This paper investigates the attack model where an adversary attacks with a certain probability at each round, and its attack value can be arbitrary and unbounded if it attacks. Furthermore, the attack value does not necessarily follow a statistical distribution. We propose a novel sample median-based and explorationaided UCB algorithm (called med-E-UCB) and a medianbased -greedy algorithm (called med- -greedy). Both of these algorithms are provably robust to the aforementioned attack model. More specifically we show that both algorithms achieve O(log T) pseudo-regret (i. e. , the optimal regret without attacks). We also provide a high probability guarantee of O(log T) regret with respect to random rewards and random occurrence of attacks. These bounds are achieved under arbitrary and unbounded reward perturbation as long as the attack probability does not exceed a certain constant threshold. We provide multiple synthetic simulations of the proposed algorithms to verify these claims and showcase the inability of existing techniques to achieve sublinear regret. We also provide experimental results of the algorithm operating in a cognitive radio setting using multiple software-defined radios.

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JMLR Journal 2020 Journal Article

Spectral Algorithms for Community Detection in Directed Networks

  • Zhe Wang
  • Yingbin Liang
  • Pengsheng Ji

Community detection in large social networks is affected by degree heterogeneity of nodes. The D-SCORE algorithm for directed networks was introduced to reduce this effect by taking the element-wise ratios of the singular vectors of the adjacency matrix before clustering. Meaningful results were obtained for the statistician citation network, but rigorous analysis on its performance was missing. First, this paper establishes theoretical guarantee for this algorithm and its variants for the directed degree-corrected block model (Directed-DCBM). Second, this paper provides significant improvements for the original D-SCORE algorithms by attaching the nodes outside of the community cores using the information of the original network instead of the singular vectors. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

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UAI Conference 2019 Conference Paper

Cubic Regularization with Momentum for Nonconvex Optimization

  • Zhe Wang 0021
  • Yi Zhou 0017
  • Yingbin Liang
  • Guanghui Lan

Momentum is a popular technique to accelerate the convergence in practical training, and its impact on convergence guarantee has been well-studied for first-order algorithms. However, such a successful acceleration technique has not yet been proposed for second-order algorithms in nonconvex optimization. In this paper, we apply the momentum scheme to cubic regularized (CR) Newton’s method and explore the potential for acceleration. Our numerical experiments on various nonconvex optimization problems demonstrate that the momentum scheme can substantially facilitate the convergence of cubic regularization, and perform even better than the Nesterov’s acceleration scheme for CR. Theoretically, we prove that CR under momentum achieves the best possible convergence rate to a second-order stationary point for nonconvex optimization. Moreover, we study the proposed algorithm for solving problems satisfying an error bound condition and establish a local quadratic convergence rate. Then, particularly for finite-sum problems, we show that the proposed algorithm can allow computational inexactness that reduces the overall sample complexity without degrading the convergence rate.

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NeurIPS Conference 2019 Conference Paper

Finite-Sample Analysis for SARSA with Linear Function Approximation

  • Shaofeng Zou
  • Tengyu Xu
  • Yingbin Liang

SARSA is an on-policy algorithm to learn a Markov decision process policy in reinforcement learning. We investigate the SARSA algorithm with linear function approximation under the non-i. i. d. \ setting, where a single sample trajectory is available. With a Lipschitz continuous policy improvement operator that is smooth enough, SARSA has been shown to converge asymptotically. However, its non-asymptotic analysis is challenging and remains unsolved due to the non-i. i. d. samples, and the fact that the behavior policy changes dynamically with time. In this paper, we develop a novel technique to explicitly characterize the stochastic bias of a type of stochastic approximation procedures with time-varying Markov transition kernels. Our approach enables non-asymptotic convergence analyses of this type of stochastic approximation algorithms, which may be of independent interest. Using our bias characterization technique and a gradient descent type of analysis, we further provide the finite-sample analysis on the mean square error of the SARSA algorithm. In the end, we present a fitted SARSA algorithm, which includes the original SARSA algorithm and its variant as special cases. This fitted SARSA algorithm provides a framework for \textit{iterative} on-policy fitted policy iteration, which is more memory and computationally efficient. For this fitted SARSA algorithm, we also present its finite-sample analysis.

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ICML Conference 2019 Conference Paper

Improved Zeroth-Order Variance Reduced Algorithms and Analysis for Nonconvex Optimization

  • Kaiyi Ji
  • Zhe Wang 0021
  • Yi Zhou 0017
  • Yingbin Liang

Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al. 2018b, and show that both ZO-SVRG-Coord-Rand and ZO-SVRG-Coord (under our new analysis) outperform other exiting SVRG-type zeroth-order methods as well as ZO-GD and ZO-SGD. Second, the existing SPIDER-type algorithm SPIDER-SZO (Fang et al. , 2018) has superior theoretical performance, but suffers from the generation of a large number of Gaussian random variables as well as a $\sqrt{\epsilon}$-level stepsize in practice. In this paper, we develop a new algorithm ZO-SPIDER-Coord, which is free from Gaussian variable generation and allows a large constant stepsize while maintaining the same convergence rate and query complexity, and we further show that ZO-SPIDER-Coord automatically achieves a linear convergence rate as the iterate enters into a local PL region without restart and algorithmic modification.

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NeurIPS Conference 2019 Conference Paper

SpiderBoost and Momentum: Faster Variance Reduction Algorithms

  • Zhe Wang
  • Kaiyi Ji
  • Yi Zhou
  • Yingbin Liang
  • Vahid Tarokh

SARAH and SPIDER are two recently developed stochastic variance-reduced algorithms, and SPIDER has been shown to achieve a near-optimal first-order oracle complexity in smooth nonconvex optimization. However, SPIDER uses an accuracy-dependent stepsize that slows down the convergence in practice, and cannot handle objective functions that involve nonsmooth regularizers. In this paper, we propose SpiderBoost as an improved scheme, which allows to use a much larger constant-level stepsize while maintaining the same near-optimal oracle complexity, and can be extended with proximal mapping to handle composite optimization (which is nonsmooth and nonconvex) with provable convergence guarantee. In particular, we show that proximal SpiderBoost achieves an oracle complexity of O(min{n^{1/2}\epsilon^{-2}, \epsilon^{-3}}) in composite nonconvex optimization, improving the state-of-the-art result by a factor of O(min{n^{1/6}, \epsilon^{-1/3}}). We further develop a novel momentum scheme to accelerate SpiderBoost for composite optimization, which achieves the near-optimal oracle complexity in theory and substantial improvement in experiments.

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NeurIPS Conference 2019 Conference Paper

Two Time-scale Off-Policy TD Learning: Non-asymptotic Analysis over Markovian Samples

  • Tengyu Xu
  • Shaofeng Zou
  • Yingbin Liang

Gradient-based temporal difference (GTD) algorithms are widely used in off-policy learning scenarios. Among them, the two time-scale TD with gradient correction (TDC) algorithm has been shown to have superior performance. In contrast to previous studies that characterized the non-asymptotic convergence rate of TDC only under identical and independently distributed (i. i. d. ) data samples, we provide the first non-asymptotic convergence analysis for two time-scale TDC under a non-i. i. d. \ Markovian sample path and linear function approximation. We show that the two time-scale TDC can converge as fast as O(log t/t^(2/3)) under diminishing stepsize, and can converge exponentially fast under constant stepsize, but at the cost of a non-vanishing error. We further propose a TDC algorithm with blockwisely diminishing stepsize, and show that it asymptotically converges with an arbitrarily small error at a blockwisely linear convergence rate. Our experiments demonstrate that such an algorithm converges as fast as TDC under constant stepsize, and still enjoys comparable accuracy as TDC under diminishing stepsize.

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NeurIPS Conference 2018 Conference Paper

Convergence of Cubic Regularization for Nonconvex Optimization under KL Property

  • Yi Zhou
  • Zhe Wang
  • Yingbin Liang

Cubic-regularized Newton's method (CR) is a popular algorithm that guarantees to produce a second-order stationary solution for solving nonconvex optimization problems. However, existing understandings of convergence rate of CR are conditioned on special types of geometrical properties of the objective function. In this paper, we explore the asymptotic convergence rate of CR by exploiting the ubiquitous Kurdyka-Lojasiewicz (KL) property of the nonconvex objective functions. In specific, we characterize the asymptotic convergence rate of various types of optimality measures for CR including function value gap, variable distance gap, gradient norm and least eigenvalue of the Hessian matrix. Our results fully characterize the diverse convergence behaviors of these optimality measures in the full parameter regime of the KL property. Moreover, we show that the obtained asymptotic convergence rates of CR are order-wise faster than those of first-order gradient descent algorithms under the KL property.

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JMLR Journal 2018 Journal Article

Distributed Proximal Gradient Algorithm for Partially Asynchronous Computer Clusters

  • Yi Zhou
  • Yingbin Liang
  • Yaoliang Yu
  • Wei Dai
  • Eric P. Xing

With ever growing data volume and model size, an error-tolerant, communication efficient, yet versatile distributed algorithm has become vital for the success of many large-scale machine learning applications. In this work we propose m-PAPG, an implementation of the flexible proximal gradient algorithm in model parallel systems equipped with the partially asynchronous communication protocol. The worker machines communicate asynchronously with a controlled staleness bound $s$ and operate at different frequencies. We characterize various convergence properties of m-PAPG: 1) Under a general non-smooth and non-convex setting, we prove that every limit point of the sequence generated by m-PAPG is a critical point of the objective function; 2) Under an error bound condition of convex objective functions, we prove that the optimality gap decays linearly for every $s$ steps; 3) Under the Kurdyka-Łojasiewicz inequality and a sufficient decrease assumption, we prove that the sequences generated by m-PAPG converge to the same critical point, provided that a proximal Lipschitz condition is satisfied. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

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NeurIPS Conference 2018 Conference Paper

Minimax Estimation of Neural Net Distance

  • Kaiyi Ji
  • Yingbin Liang

An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper bounds match not only in the order of the sample size but also in terms of the norm of the parameter matrices of neural networks, which justifies the empirical neural net distance as a good approximation of the true neural net distance for training GANs in practice.

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JMLR Journal 2017 Journal Article

A Nonconvex Approach for Phase Retrieval: Reshaped Wirtinger Flow and Incremental Algorithms

  • Huishuai Zhang
  • Yingbin Liang
  • Yuejie Chi

We study the problem of solving a quadratic system of equations, i.e., recovering a vector signal $\boldsymbol{x}\in \mathbb{R}^n$ from its magnitude measurements $y_i=|\langle \boldsymbol{a}_i, \boldsymbol{x}\rangle|, i=1,..., m$. We develop a gradient descent algorithm (referred to as RWF for reshaped Wirtinger flow) by minimizing the quadratic loss of the magnitude measurements. Comparing with Wirtinger flow (WF) (Candes et al., 2015), the loss function of RWF is nonconvex and nonsmooth, but better resembles the least-squares loss when the phase information is also available. We show that for random Gaussian measurements, RWF enjoys linear convergence to the true signal as long as the number of measurements is $\mathcal{O}(n)$. This improves the sample complexity of WF ($\mathcal{O}(n\log n)$), and achieves the same sample complexity as truncated Wirtinger flow (TWF) (Chen and Candes, 2015), but without any sophisticated truncation in the gradient loop. Furthermore, RWF costs less computationally than WF, and runs faster numerically than both WF and TWF. We further develop an incremental (stochastic) version of RWF (IRWF) and connect it with the randomized Kaczmarz method for phase retrieval. We demonstrate that IRWF outperforms existing incremental as well as batch algorithms with experiments. [abs] [ pdf ][ bib ] &copy JMLR 2017. ( edit, beta )

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ICML Conference 2017 Conference Paper

Convergence Analysis of Proximal Gradient with Momentum for Nonconvex Optimization

  • Qunwei Li
  • Yi Zhou 0017
  • Yingbin Liang
  • Pramod K. Varshney

In this work, we investigate the accelerated proximal gradient method for nonconvex programming (APGnc). The method compares between a usual proximal gradient step and a linear extrapolation step, and accepts the one that has a lower function value to achieve a monotonic decrease. In specific, under a general nonsmooth and nonconvex setting, we provide a rigorous argument to show that the limit points of the sequence generated by APGnc are critical points of the objective function. Then, by exploiting the Kurdyka-Lojasiewicz (KL) property for a broad class of functions, we establish the linear and sub-linear convergence rates of the function value sequence generated by APGnc. We further propose a stochastic variance reduced APGnc (SVRG-APGnc), and establish its linear convergence under a special case of the KL property. We also extend the analysis to the inexact version of these methods and develop an adaptive momentum strategy that improves the numerical performance.

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ICML Conference 2016 Conference Paper

Provable Non-convex Phase Retrieval with Outliers: Median TruncatedWirtinger Flow

  • Huishuai Zhang
  • Yuejie Chi
  • Yingbin Liang

Solving systems of quadratic equations is a central problem in machine learning and signal processing. One important example is phase retrieval, which aims to recover a signal from only magnitudes of its linear measurements. This paper focuses on the situation when the measurements are corrupted by arbitrary outliers, for which the recently developed non-convex gradient descent Wirtinger flow (WF) and truncated Wirtinger flow (TWF) algorithms likely fail. We develop a novel median-TWF algorithm that exploits robustness of sample median to resist arbitrary outliers in the initialization and the gradient update in each iteration. We show that such a non-convex algorithm provably recovers the signal from a near-optimal number of measurements composed of i. i. d. Gaussian entries, up to a logarithmic factor, even when a constant portion of the measurements are corrupted by arbitrary outliers. We further show that median-TWF is also robust when measurements are corrupted by both arbitrary outliers and bounded noise. Our analysis of performance guarantee is accomplished by development of non-trivial concentration measures of median-related quantities, which may be of independent interest. We further provide numerical experiments to demonstrate the effectiveness of the approach.

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NeurIPS Conference 2016 Conference Paper

Reshaped Wirtinger Flow for Solving Quadratic System of Equations

  • Huishuai Zhang
  • Yingbin Liang

We study the problem of recovering a vector $\bx\in \bbR^n$ from its magnitude measurements $y_i=|\langle \ba_i, \bx\rangle|, i=1, .. ., m$. Our work is along the line of the Wirtinger flow (WF) approach \citet{candes2015phase}, which solves the problem by minimizing a nonconvex loss function via a gradient algorithm and can be shown to converge to a global optimal point under good initialization. In contrast to the smooth loss function used in WF, we adopt a nonsmooth but lower-order loss function, and design a gradient-like algorithm (referred to as reshaped-WF). We show that for random Gaussian measurements, reshaped-WF enjoys geometric convergence to a global optimal point as long as the number $m$ of measurements is at the order of $\cO(n)$, where $n$ is the dimension of the unknown $\bx$. This improves the sample complexity of WF, and achieves the same sample complexity as truncated-WF \citet{chen2015solving} but without truncation at gradient step. Furthermore, reshaped-WF costs less computationally than WF, and runs faster numerically than both WF and truncated-WF. Bypassing higher-order variables in the loss function and truncations in the gradient loop, analysis of reshaped-WF is simplified.

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NeurIPS Conference 2015 Conference Paper

Analysis of Robust PCA via Local Incoherence

  • Huishuai Zhang
  • Yi Zhou
  • Yingbin Liang

We investigate the robust PCA problem of decomposing an observed matrix into the sum of a low-rank and a sparse error matrices via convex programming Principal Component Pursuit (PCP). In contrast to previous studies that assume the support of the error matrix is generated by uniform Bernoulli sampling, we allow non-uniform sampling, i. e. , entries of the low-rank matrix are corrupted by errors with unequal probabilities. We characterize conditions on error corruption of each individual entry based on the local incoherence of the low-rank matrix, under which correct matrix decomposition by PCP is guaranteed. Such a refined analysis of robust PCA captures how robust each entry of the low rank matrix combats error corruption. In order to deal with non-uniform error corruption, our technical proof introduces a new weighted norm and develops/exploits the concentration properties that such a norm satisfies.

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