arXiv subjects

Zhuoran Yang

Publications and source records attributed to Zhuoran Yang.

At least 145 records · Page 8Linked to original sources

arXiv2021

Natural Actor-Critic Converges Globally for Hierarchical Linear Quadratic Regulator

Multi-agent reinforcement learning has been successfully applied to a number of challenging problems. Despite these empirical successes, theoretical understanding of different algorithms is lacking, primarily due to the curse of dimensionality caused by the exponential growth of the state-action space with the number of agents. We study a fundamental problem of multi-agent linear quadratic regulator (LQR) in a setting where the agents are partially exchangeable. In this setting, we develop a hierarchical actor-critic algorithm, whose computational complexity is independent of the total number of agents, and prove its global linear convergence to the optimal policy. As LQRs are often used to approximate general dynamic systems, this paper provides an important step towards a better understanding of general hierarchical mean-field multi-agent reinforcement learning.

cs.LG↗

arXiv2021

Understanding Implicit Regularization in Over-Parameterized Single Index Model

In this paper, we leverage over-parameterization to design regularization-free algorithms for the high-dimensional single index model and provide theoretical guarantees for the induced implicit regularization phenomenon. Specifically, we study both vector and matrix single index models where the link function is nonlinear and unknown, the signal parameter is either a sparse vector or a low-rank symmetric matrix, and the response variable can be heavy-tailed. To gain a better understanding of the role played by implicit regularization without excess technicality, we assume that the distribution of the covariates is known a priori. For both the vector and matrix settings, we construct an over-parameterized least-squares loss function by employing the score function transform and a robust truncation step designed specifically for heavy-tailed data. We propose to estimate the true parameter by applying regularization-free gradient descent to the loss function. When the initialization is close to the origin and the stepsize is sufficiently small, we prove that the obtained solution achieves minimax optimal statistical rates of convergence in both the vector and matrix cases. In addition, our experimental results support our theoretical findings and also demonstrate that our methods empirically outperform classical methods with explicit regularization in terms of both $\ell_2$-statistical rate and variable selection consistency.

stat.ML↗

arXiv2021

Exponential Bellman Equation and Improved Regret Bounds for Risk-Sensitive Reinforcement Learning

We study risk-sensitive reinforcement learning (RL) based on the entropic risk measure. Although existing works have established non-asymptotic regret guarantees for this problem, they leave open an exponential gap between the upper and lower bounds. We identify the deficiencies in existing algorithms and their analysis that result in such a gap. To remedy these deficiencies, we investigate a simple transformation of the risk-sensitive Bellman equations, which we call the exponential Bellman equation. The exponential Bellman equation inspires us to develop a novel analysis of Bellman backup procedures in risk-sensitive RL algorithms, and further motivates the design of a novel exploration mechanism. We show that these analytic and algorithmic innovations together lead to improved regret upper bounds over existing ones.

cs.LG↗

arXiv2021

Fast Multi-Agent Temporal-Difference Learning via Homotopy Stochastic Primal-Dual Optimization

We study the policy evaluation problem in multi-agent reinforcement learning where a group of agents, with jointly observed states and private local actions and rewards, collaborate to learn the value function of a given policy via local computation and communication over a connected undirected network. This problem arises in various large-scale multi-agent systems, including power grids, intelligent transportation systems, wireless sensor networks, and multi-agent robotics. When the dimension of state-action space is large, the temporal-difference learning with linear function approximation is widely used. In this paper, we develop a new distributed temporal-difference learning algorithm and quantify its finite-time performance. Our algorithm combines a distributed stochastic primal-dual method with a homotopy-based approach to adaptively adjust the learning rate in order to minimize the mean-square projected Bellman error by taking fresh online samples from a causal on-policy trajectory. We explicitly take into account the Markovian nature of sampling and improve the best-known finite-time error bound from $O(1/\sqrt{T})$ to~$O(1/T)$, where $T$ is the total number of iterations.

math.OC↗

arXiv2021

Towards General Function Approximation in Zero-Sum Markov Games

This paper considers two-player zero-sum finite-horizon Markov games with simultaneous moves. The study focuses on the challenging settings where the value function or the model is parameterized by general function classes. Provably efficient algorithms for both decoupled and {coordinated} settings are developed. In the {decoupled} setting where the agent controls a single player and plays against an arbitrary opponent, we propose a new model-free algorithm. The sample complexity is governed by the Minimax Eluder dimension -- a new dimension of the function class in Markov games. As a special case, this method improves the state-of-the-art algorithm by a $\sqrt{d}$ factor in the regret when the reward function and transition kernel are parameterized with $d$-dimensional linear features. In the {coordinated} setting where both players are controlled by the agent, we propose a model-based algorithm and a model-free algorithm. In the model-based algorithm, we prove that sample complexity can be bounded by a generalization of Witness rank to Markov games. The model-free algorithm enjoys a $\sqrt{K}$-regret upper bound where $K$ is the number of episodes.

cs.GT↗

arXiv2021

Randomized Exploration for Reinforcement Learning with General Value Function Approximation

We propose a model-free reinforcement learning algorithm inspired by the popular randomized least squares value iteration (RLSVI) algorithm as well as the optimism principle. Unlike existing upper-confidence-bound (UCB) based approaches, which are often computationally intractable, our algorithm drives exploration by simply perturbing the training data with judiciously chosen i.i.d. scalar noises. To attain optimistic value function estimation without resorting to a UCB-style bonus, we introduce an optimistic reward sampling procedure. When the value functions can be represented by a function class $\mathcal{F}$, our algorithm achieves a worst-case regret bound of $\widetilde{O}(\mathrm{poly}(d_EH)\sqrt{T})$ where $T$ is the time elapsed, $H$ is the planning horizon and $d_E$ is the $\textit{eluder dimension}$ of $\mathcal{F}$. In the linear setting, our algorithm reduces to LSVI-PHE, a variant of RLSVI, that enjoys an $\widetilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret. We complement the theory with an empirical evaluation across known difficult exploration tasks.

cs.LG↗

arXiv2021

Upper Confidence Primal-Dual Reinforcement Learning for CMDP with Adversarial Loss

We consider online learning for episodic stochastically constrained Markov decision processes (CMDPs), which plays a central role in ensuring the safety of reinforcement learning. Here the loss function can vary arbitrarily across the episodes, and both the loss received and the budget consumption are revealed at the end of each episode. Previous works solve this problem under the restrictive assumption that the transition model of the Markov decision processes (MDPs) is known a priori and establish regret bounds that depend polynomially on the cardinalities of the state space $\mathcal{S}$ and the action space $\mathcal{A}$. In this work, we propose a new \emph{upper confidence primal-dual} algorithm, which only requires the trajectories sampled from the transition model. In particular, we prove that the proposed algorithm achieves $\widetilde{\mathcal{O}}(L|\mathcal{S}|\sqrt{|\mathcal{A}|T})$ upper bounds of both the regret and the constraint violation, where $L$ is the length of each episode. Our analysis incorporates a new high-probability drift analysis of Lagrange multiplier processes into the celebrated regret analysis of upper confidence reinforcement learning, which demonstrates the power of "optimism in the face of uncertainty" in constrained online learning.

cs.LG↗

arXiv2021

Provably Efficient Generative Adversarial Imitation Learning for Online and Offline Setting with Linear Function Approximation

In generative adversarial imitation learning (GAIL), the agent aims to learn a policy from an expert demonstration so that its performance cannot be discriminated from the expert policy on a certain predefined reward set. In this paper, we study GAIL in both online and offline settings with linear function approximation, where both the transition and reward function are linear in the feature maps. Besides the expert demonstration, in the online setting the agent can interact with the environment, while in the offline setting the agent only accesses an additional dataset collected by a prior. For online GAIL, we propose an optimistic generative adversarial policy optimization algorithm (OGAP) and prove that OGAP achieves $\widetilde{\mathcal{O}}(H^2 d^{3/2}K^{1/2}+KH^{3/2}dN_1^{-1/2})$ regret. Here $N_1$ represents the number of trajectories of the expert demonstration, $d$ is the feature dimension, and $K$ is the number of episodes. For offline GAIL, we propose a pessimistic generative adversarial policy optimization algorithm (PGAP). For an arbitrary additional dataset, we obtain the optimality gap of PGAP, achieving the minimax lower bound in the utilization of the additional dataset. Assuming sufficient coverage on the additional dataset, we show that PGAP achieves $\widetilde{\mathcal{O}}(H^{2}dK^{-1/2} +H^2d^{3/2}N_2^{-1/2}+H^{3/2}dN_1^{-1/2} \ )$ optimality gap. Here $N_2$ represents the number of trajectories of the additional dataset with sufficient coverage.

cs.LG↗

arXiv2021

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

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 $ε$-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 a single time-scale off-policy AC algorithm.

cs.LG↗

arXiv2021

A Unified Off-Policy Evaluation Approach for General Value Function

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.

cs.LG↗

arXiv2021

Gap-Dependent Bounds for Two-Player Markov Games

As one of the most popular methods in the field of reinforcement learning, Q-learning has received increasing attention. Recently, there have been more theoretical works on the regret bound of algorithms that belong to the Q-learning class in different settings. In this paper, we analyze the cumulative regret when conducting Nash Q-learning algorithm on 2-player turn-based stochastic Markov games (2-TBSG), and propose the very first gap dependent logarithmic upper bounds in the episodic tabular setting. This bound matches the theoretical lower bound only up to a logarithmic term. Furthermore, we extend the conclusion to the discounted game setting with infinite horizon and propose a similar gap dependent logarithmic regret bound. Also, under the linear MDP assumption, we obtain another logarithmic regret for 2-TBSG, in both centralized and independent settings.

cs.LG↗

arXiv2021

A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum

This paper proposes a new algorithm -- the \underline{S}ingle-timescale Do\underline{u}ble-momentum \underline{St}ochastic \underline{A}pprox\underline{i}matio\underline{n} (SUSTAIN) -- for tackling stochastic unconstrained bilevel optimization problems. We focus on bilevel problems where the lower level subproblem is strongly-convex and the upper level objective function is smooth. Unlike prior works which rely on \emph{two-timescale} or \emph{double loop} techniques, we design a stochastic momentum-assisted gradient estimator for both the upper and lower level updates. The latter allows us to control the error in the stochastic gradient updates due to inaccurate solution to both subproblems. If the upper objective function is smooth but possibly non-convex, we show that {\aname}~requires $\mathcal{O}(ε^{-3/2})$ iterations (each using ${\cal O}(1)$ samples) to find an $ε$-stationary solution. The $ε$-stationary solution is defined as the point whose squared norm of the gradient of the outer function is less than or equal to $ε$. The total number of stochastic gradient samples required for the upper and lower level objective functions matches the best-known complexity for single-level stochastic gradient algorithms. We also analyze the case when the upper level objective function is strongly-convex.

math.OC↗

arXiv2021

Single-Timescale Actor-Critic Provably Finds Globally Optimal Policy

We study the global convergence and global optimality of actor-critic, one of the most popular families of reinforcement learning algorithms. While most existing works on actor-critic employ bi-level or two-timescale updates, we focus on the more practical single-timescale setting, where the actor and critic are updated simultaneously. Specifically, in each iteration, the critic update is obtained by applying the Bellman evaluation operator only once while the actor is updated in the policy gradient direction computed using the critic. Moreover, we consider two function approximation settings where both the actor and critic are represented by linear or deep neural networks. For both cases, we prove that the actor sequence converges to a globally optimal policy at a sublinear $O(K^{-1/2})$ rate, where $K$ is the number of iterations. To the best of our knowledge, we establish the rate of convergence and global optimality of single-timescale actor-critic with linear function approximation for the first time. Moreover, under the broader scope of policy optimization with nonlinear function approximation, we prove that actor-critic with deep neural network finds the globally optimal policy at a sublinear rate for the first time.

cs.LG↗

arXiv2021

Multi-Agent Reinforcement Learning: A Selective Overview of Theories and Algorithms

Recent years have witnessed significant advances in reinforcement learning (RL), which has registered great success in solving various sequential decision-making problems in machine learning. Most of the successful RL applications, e.g., the games of Go and Poker, robotics, and autonomous driving, involve the participation of more than one single agent, which naturally fall into the realm of multi-agent RL (MARL), a domain with a relatively long history, and has recently re-emerged due to advances in single-agent RL techniques. Though empirically successful, theoretical foundations for MARL are relatively lacking in the literature. In this chapter, we provide a selective overview of MARL, with focus on algorithms backed by theoretical analysis. More specifically, we review the theoretical results of MARL algorithms mainly within two representative frameworks, Markov/stochastic games and extensive-form games, in accordance with the types of tasks they address, i.e., fully cooperative, fully competitive, and a mix of the two. We also introduce several significant but challenging applications of these algorithms. Orthogonal to the existing reviews on MARL, we highlight several new angles and taxonomies of MARL theory, including learning in extensive-form games, decentralized MARL with networked agents, MARL in the mean-field regime, (non-)convergence of policy-based methods for learning in games, etc. Some of the new angles extrapolate from our own research endeavors and interests. Our overall goal with this chapter is, beyond providing an assessment of the current state of the field on the mark, to identify fruitful future research directions on theoretical studies of MARL. We expect this chapter to serve as continuing stimulus for researchers interested in working on this exciting while challenging topic.

cs.LG↗

arXiv2021

Sample Elicitation

It is important to collect credible training samples $(x,y)$ for building data-intensive learning systems (e.g., a deep learning system). Asking people to report complex distribution $p(x)$, though theoretically viable, is challenging in practice. This is primarily due to the cognitive loads required for human agents to form the report of this highly complicated information. While classical elicitation mechanisms apply to eliciting a complex and generative (and continuous) distribution $p(x)$, we are interested in eliciting samples $x_i \sim p(x)$ from agents directly. We coin the above problem "sample elicitation". This paper introduces a deep learning aided method to incentivize credible sample contributions from self-interested and rational agents. We show that with an accurate estimation of a certain $f$-divergence function we can achieve approximate incentive compatibility in eliciting truthful samples. We then present an efficient estimator with theoretical guarantees via studying the variational forms of the $f$-divergence function. We also show a connection between this sample elicitation problem and $f$-GAN, and how this connection can help reconstruct an estimator of the distribution based on collected samples. Experiments on synthetic data, MNIST, and CIFAR-10 datasets demonstrate that our mechanism elicits truthful samples. Our implementation is available at https://github.com/weijiaheng/Credible-sample-elicitation.git.

cs.LG↗

arXiv2021

Policy Optimization Provably Converges to Nash Equilibria in Zero-Sum Linear Quadratic Games

We study the global convergence of policy optimization for finding the Nash equilibria (NE) in zero-sum linear quadratic (LQ) games. To this end, we first investigate the landscape of LQ games, viewing it as a nonconvex-nonconcave saddle-point problem in the policy space. Specifically, we show that despite its nonconvexity and nonconcavity, zero-sum LQ games have the property that the stationary point of the objective function with respect to the linear feedback control policies constitutes the NE of the game. Building upon this, we develop three projected nested-gradient methods that are guaranteed to converge to the NE of the game. Moreover, we show that all of these algorithms enjoy both globally sublinear and locally linear convergence rates. Simulation results are also provided to illustrate the satisfactory convergence properties of the algorithms. To the best of our knowledge, this work appears to be the first one to investigate the optimization landscape of LQ games, and provably show the convergence of policy optimization methods to the Nash equilibria. Our work serves as an initial step toward understanding the theoretical aspects of policy-based reinforcement learning algorithms for zero-sum Markov games in general.

cs.LG↗

arXiv2021

Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework

This paper develops a Pontryagin Differentiable Programming (PDP) methodology, which establishes a unified framework to solve a broad class of learning and control tasks. The PDP distinguishes from existing methods by two novel techniques: first, we differentiate through Pontryagin's Maximum Principle, and this allows to obtain the analytical derivative of a trajectory with respect to tunable parameters within an optimal control system, enabling end-to-end learning of dynamics, policies, or/and control objective functions; and second, we propose an auxiliary control system in the backward pass of the PDP framework, and the output of this auxiliary control system is the analytical derivative of the original system's trajectory with respect to the parameters, which can be iteratively solved using standard control tools. We investigate three learning modes of the PDP: inverse reinforcement learning, system identification, and control/planning. We demonstrate the capability of the PDP in each learning mode on different high-dimensional systems, including multi-link robot arm, 6-DoF maneuvering quadrotor, and 6-DoF rocket powered landing.

cs.LG↗

arXiv2020

On Function Approximation in Reinforcement Learning: Optimism in the Face of Large State Spaces

The classical theory of reinforcement learning (RL) has focused on tabular and linear representations of value functions. Further progress hinges on combining RL with modern function approximators such as kernel functions and deep neural networks, and indeed there have been many empirical successes that have exploited such combinations in large-scale applications. There are profound challenges, however, in developing a theory to support this enterprise, most notably the need to take into consideration the exploration-exploitation tradeoff at the core of RL in conjunction with the computational and statistical tradeoffs that arise in modern function-approximation-based learning systems. We approach these challenges by studying an optimistic modification of the least-squares value iteration algorithm, in the context of the action-value function represented by a kernel function or an overparameterized neural network. We establish both polynomial runtime complexity and polynomial sample complexity for this algorithm, without additional assumptions on the data-generating model. In particular, we prove that the algorithm incurs an $\tilde{\mathcal{O}}(δ_{\mathcal{F}} H^2 \sqrt{T})$ regret, where $δ_{\mathcal{F}}$ characterizes the intrinsic complexity of the function class $\mathcal{F}$, $H$ is the length of each episode, and $T$ is the total number of episodes. Our regret bounds are independent of the number of states, a result which exhibits clearly the benefit of function approximation in RL.

cs.LG↗