arXiv subjects

Zhaoran Wang

Publications and source records attributed to Zhaoran Wang.

At least 145 records · Page 8Linked to original sources

arXiv2021

A Primal-Dual Approach to Constrained Markov Decision Processes

In many operations management problems, we need to make decisions sequentially to minimize the cost while satisfying certain constraints. One modeling approach to study such problems is constrained Markov decision process (CMDP). When solving the CMDP to derive good operational policies, there are two key challenges: one is the prohibitively large state space and action space; the other is the hard-to-compute transition kernel. In this work, we develop a sampling-based primal-dual algorithm to solve CMDPs. Our approach alternatively applies regularized policy iteration to improve the policy and subgradient ascent to maintain the constraints. Under mild regularity conditions, we show that the algorithm converges at rate $ O(\log(T)/\sqrt{T})$, where T is the number of iterations. When the CMDP has a weakly coupled structure, our approach can substantially reduce the dimension of the problem through an embedded decomposition. We apply the algorithm to two important applications with weakly coupled structures: multi-product inventory management and multi-class queue scheduling, and show that it generates controls that outperform state-of-art heuristics.

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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.

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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.

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arXiv2020

Provably Efficient Safe Exploration via Primal-Dual Policy Optimization

We study the Safe Reinforcement Learning (SRL) problem using the Constrained Markov Decision Process (CMDP) formulation in which an agent aims to maximize the expected total reward subject to a safety constraint on the expected total value of a utility function. We focus on an episodic setting with the function approximation where the Markov transition kernels have a linear structure but do not impose any additional assumptions on the sampling model. Designing SRL algorithms with provable computational and statistical efficiency is particularly challenging under this setting because of the need to incorporate both the safety constraint and the function approximation into the fundamental exploitation/exploration tradeoff. To this end, we present an \underline{O}ptimistic \underline{P}rimal-\underline{D}ual Proximal Policy \underline{OP}timization (OPDOP) algorithm where the value function is estimated by combining the least-squares policy evaluation and an additional bonus term for safe exploration. We prove that the proposed algorithm achieves an $\tilde{O}(d H^{2.5}\sqrt{T})$ regret and an $\tilde{O}(d H^{2.5}\sqrt{T})$ constraint violation, where $d$ is the dimension of the feature mapping, $H$ is the horizon of each episode, and $T$ is the total number of steps. These bounds hold when the reward/utility functions are fixed but the feedback after each episode is bandit. Our bounds depend on the capacity of the state-action space only through the dimension of the feature mapping and thus our results hold even when the number of states goes to infinity. To the best of our knowledge, we provide the first provably efficient online policy optimization algorithm for CMDP with safe exploration in the function approximation setting.

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arXiv2020

End-to-End Learning and Intervention in Games

In a social system, the self-interest of agents can be detrimental to the collective good, sometimes leading to social dilemmas. To resolve such a conflict, a central designer may intervene by either redesigning the system or incentivizing the agents to change their behaviors. To be effective, the designer must anticipate how the agents react to the intervention, which is dictated by their often unknown payoff functions. Therefore, learning about the agents is a prerequisite for intervention. In this paper, we provide a unified framework for learning and intervention in games. We cast the equilibria of games as individual layers and integrate them into an end-to-end optimization framework. To enable the backward propagation through the equilibria of games, we propose two approaches, respectively based on explicit and implicit differentiation. Specifically, we cast the equilibria as the solutions to variational inequalities (VIs). The explicit approach unrolls the projection method for solving VIs, while the implicit approach exploits the sensitivity of the solutions to VIs. At the core of both approaches is the differentiation through a projection operator. Moreover, we establish the correctness of both approaches and identify the conditions under which one approach is more desirable than the other. The analytical results are validated using several real-world problems.

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arXiv2020

Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach

Structural equation models (SEMs) are widely used in sciences, ranging from economics to psychology, to uncover causal relationships underlying a complex system under consideration and estimate structural parameters of interest. We study estimation in a class of generalized SEMs where the object of interest is defined as the solution to a linear operator equation. We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using the stochastic gradient descent. We consider both 2-layer and multi-layer NNs with ReLU activation functions and prove global convergence in an overparametrized regime, where the number of neurons is diverging. The results are established using techniques from online learning and local linearization of NNs, and improve in several aspects the current state-of-the-art. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.

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arXiv2020

Provable Fictitious Play for General Mean-Field Games

We propose a reinforcement learning algorithm for stationary mean-field games, where the goal is to learn a pair of mean-field state and stationary policy that constitutes the Nash equilibrium. When viewing the mean-field state and the policy as two players, we propose a fictitious play algorithm which alternatively updates the mean-field state and the policy via gradient-descent and proximal policy optimization, respectively. Our algorithm is in stark contrast with previous literature which solves each single-agent reinforcement learning problem induced by the iterates mean-field states to the optimum. Furthermore, we prove that our fictitious play algorithm converges to the Nash equilibrium at a sublinear rate. To the best of our knowledge, this seems the first provably convergent single-loop reinforcement learning algorithm for mean-field games based on iterative updates of both mean-field state and policy.

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arXiv2020

Nearly Dimension-Independent Sparse Linear Bandit over Small Action Spaces via Best Subset Selection

We consider the stochastic contextual bandit problem under the high dimensional linear model. We focus on the case where the action space is finite and random, with each action associated with a randomly generated contextual covariate. This setting finds essential applications such as personalized recommendation, online advertisement, and personalized medicine. However, it is very challenging as we need to balance exploration and exploitation. We propose doubly growing epochs and estimating the parameter using the best subset selection method, which is easy to implement in practice. This approach achieves $ \tilde{\mathcal{O}}(s\sqrt{T})$ regret with high probability, which is nearly independent in the ``ambient'' regression model dimension $d$. We further attain a sharper $\tilde{\mathcal{O}}(\sqrt{sT})$ regret by using the \textsc{SupLinUCB} framework and match the minimax lower bound of low-dimensional linear stochastic bandit problems. Finally, we conduct extensive numerical experiments to demonstrate the applicability and robustness of our algorithms empirically.

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arXiv2020

Single-Timescale Stochastic Nonconvex-Concave Optimization for Smooth Nonlinear TD Learning

Temporal-Difference (TD) learning with nonlinear smooth function approximation for policy evaluation has achieved great success in modern reinforcement learning. It is shown that such a problem can be reformulated as a stochastic nonconvex-strongly-concave optimization problem, which is challenging as naive stochastic gradient descent-ascent algorithm suffers from slow convergence. Existing approaches for this problem are based on two-timescale or double-loop stochastic gradient algorithms, which may also require sampling large-batch data. However, in practice, a single-timescale single-loop stochastic algorithm is preferred due to its simplicity and also because its step-size is easier to tune. In this paper, we propose two single-timescale single-loop algorithms which require only one data point each step. Our first algorithm implements momentum updates on both primal and dual variables achieving an $O(\varepsilon^{-4})$ sample complexity, which shows the important role of momentum in obtaining a single-timescale algorithm. Our second algorithm improves upon the first one by applying variance reduction on top of momentum, which matches the best known $O(\varepsilon^{-3})$ sample complexity in existing works. Furthermore, our variance-reduction algorithm does not require a large-batch checkpoint. Moreover, our theoretical results for both algorithms are expressed in a tighter form of simultaneous primal and dual side convergence.

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arXiv2020

Global Convergence of Policy Gradient for Linear-Quadratic Mean-Field Control/Game in Continuous Time

Reinforcement learning is a powerful tool to learn the optimal policy of possibly multiple agents by interacting with the environment. As the number of agents grow to be very large, the system can be approximated by a mean-field problem. Therefore, it has motivated new research directions for mean-field control (MFC) and mean-field game (MFG). In this paper, we study the policy gradient method for the linear-quadratic mean-field control and game, where we assume each agent has identical linear state transitions and quadratic cost functions. While most of the recent works on policy gradient for MFC and MFG are based on discrete-time models, we focus on the continuous-time models where some analyzing techniques can be interesting to the readers. For both MFC and MFG, we provide policy gradient update and show that it converges to the optimal solution at a linear rate, which is verified by a synthetic simulation. For MFG, we also provide sufficient conditions for the existence and uniqueness of the Nash equilibrium.

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arXiv2020

Deep Reinforcement Learning with Robust and Smooth Policy

Deep reinforcement learning (RL) has achieved great empirical successes in various domains. However, the large search space of neural networks requires a large amount of data, which makes the current RL algorithms not sample efficient. Motivated by the fact that many environments with continuous state space have smooth transitions, we propose to learn a smooth policy that behaves smoothly with respect to states. We develop a new framework -- \textbf{S}mooth \textbf{R}egularized \textbf{R}einforcement \textbf{L}earning ($\textbf{SR}^2\textbf{L}$), where the policy is trained with smoothness-inducing regularization. Such regularization effectively constrains the search space, and enforces smoothness in the learned policy. Moreover, our proposed framework can also improve the robustness of policy against measurement error in the state space, and can be naturally extended to distribubutionally robust setting. We apply the proposed framework to both on-policy (TRPO) and off-policy algorithm (DDPG). Through extensive experiments, we demonstrate that our method achieves improved sample efficiency and robustness.

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arXiv2020

Accelerating Nonconvex Learning via Replica Exchange Langevin Diffusion

Langevin diffusion is a powerful method for nonconvex optimization, which enables the escape from local minima by injecting noise into the gradient. In particular, the temperature parameter controlling the noise level gives rise to a tradeoff between ``global exploration'' and ``local exploitation'', which correspond to high and low temperatures. To attain the advantages of both regimes, we propose to use replica exchange, which swaps between two Langevin diffusions with different temperatures. We theoretically analyze the acceleration effect of replica exchange from two perspectives: (i) the convergence in χ^2-divergence, and (ii) the large deviation principle. Such an acceleration effect allows us to faster approach the global minima. Furthermore, by discretizing the replica exchange Langevin diffusion, we obtain a discrete-time algorithm. For such an algorithm, we quantify its discretization error in theory and demonstrate its acceleration effect in practice.

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arXiv2020

Dynamic Regret of Policy Optimization in Non-stationary Environments

We consider reinforcement learning (RL) in episodic MDPs with adversarial full-information reward feedback and unknown fixed transition kernels. We propose two model-free policy optimization algorithms, POWER and POWER++, and establish guarantees for their dynamic regret. Compared with the classical notion of static regret, dynamic regret is a stronger notion as it explicitly accounts for the non-stationarity of environments. The dynamic regret attained by the proposed algorithms interpolates between different regimes of non-stationarity, and moreover satisfies a notion of adaptive (near-)optimality, in the sense that it matches the (near-)optimal static regret under slow-changing environments. The dynamic regret bound features two components, one arising from exploration, which deals with the uncertainty of transition kernels, and the other arising from adaptation, which deals with non-stationary environments. Specifically, we show that POWER++ improves over POWER on the second component of the dynamic regret by actively adapting to non-stationarity through prediction. To the best of our knowledge, our work is the first dynamic regret analysis of model-free RL algorithms in non-stationary environments.

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arXiv2020

Generative Adversarial Imitation Learning with Neural Networks: Global Optimality and Convergence Rate

Generative adversarial imitation learning (GAIL) demonstrates tremendous success in practice, especially when combined with neural networks. Different from reinforcement learning, GAIL learns both policy and reward function from expert (human) demonstration. Despite its empirical success, it remains unclear whether GAIL with neural networks converges to the globally optimal solution. The major difficulty comes from the nonconvex-nonconcave minimax optimization structure. To bridge the gap between practice and theory, we analyze a gradient-based algorithm with alternating updates and establish its sublinear convergence to the globally optimal solution. To the best of our knowledge, our analysis establishes the global optimality and convergence rate of GAIL with neural networks for the first time.

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arXiv2020

Learning Zero-Sum Simultaneous-Move Markov Games Using Function Approximation and Correlated Equilibrium

We develop provably efficient reinforcement learning algorithms for two-player zero-sum finite-horizon Markov games with simultaneous moves. To incorporate function approximation, we consider a family of Markov games where the reward function and transition kernel possess a linear structure. Both the offline and online settings of the problems are considered. In the offline setting, we control both players and aim to find the Nash Equilibrium by minimizing the duality gap. In the online setting, we control a single player playing against an arbitrary opponent and aim to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm. We show that our algorithm is computationally efficient and provably achieves an $\tilde O(\sqrt{d^3 H^3 T} )$ upper bound on the duality gap and regret, where $d$ is the linear dimension, $H$ the horizon and $T$ the total number of timesteps. Our results do not require additional assumptions on the sampling model. Our setting requires overcoming several new challenges that are absent in Markov decision processes or turn-based Markov games. In particular, to achieve optimism with simultaneous moves, we construct both upper and lower confidence bounds of the value function, and then compute the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash Equilibrium of a general-sum game is computationally hard, our algorithm instead solves for a Coarse Correlated Equilibrium (CCE), which can be obtained efficiently. To our best knowledge, such a CCE-based scheme for optimism has not appeared in the literature and might be of interest in its own right.

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arXiv2020

On the Global Optimality of Model-Agnostic Meta-Learning

Model-agnostic meta-learning (MAML) formulates meta-learning as a bilevel optimization problem, where the inner level solves each subtask based on a shared prior, while the outer level searches for the optimal shared prior by optimizing its aggregated performance over all the subtasks. Despite its empirical success, MAML remains less understood in theory, especially in terms of its global optimality, due to the nonconvexity of the meta-objective (the outer-level objective). To bridge such a gap between theory and practice, we characterize the optimality gap of the stationary points attained by MAML for both reinforcement learning and supervised learning, where the inner-level and outer-level problems are solved via first-order optimization methods. In particular, our characterization connects the optimality gap of such stationary points with (i) the functional geometry of inner-level objectives and (ii) the representation power of function approximators, including linear models and neural networks. To the best of our knowledge, our analysis establishes the global optimality of MAML with nonconvex meta-objectives for the first time.

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arXiv2020

Provably Efficient Causal Reinforcement Learning with Confounded Observational Data

Empowered by expressive function approximators such as neural networks, deep reinforcement learning (DRL) achieves tremendous empirical successes. However, learning expressive function approximators requires collecting a large dataset (interventional data) by interacting with the environment. Such a lack of sample efficiency prohibits the application of DRL to critical scenarios, e.g., autonomous driving and personalized medicine, since trial and error in the online setting is often unsafe and even unethical. In this paper, we study how to incorporate the dataset (observational data) collected offline, which is often abundantly available in practice, to improve the sample efficiency in the online setting. To incorporate the possibly confounded observational data, we propose the deconfounded optimistic value iteration (DOVI) algorithm, which incorporates the confounded observational data in a provably efficient manner. More specifically, DOVI explicitly adjusts for the confounding bias in the observational data, where the confounders are partially observed or unobserved. In both cases, such adjustments allow us to construct the bonus based on a notion of information gain, which takes into account the amount of information acquired from the offline setting. In particular, we prove that the regret of DOVI is smaller than the optimal regret achievable in the pure online setting by a multiplicative factor, which decreases towards zero when the confounded observational data are more informative upon the adjustments. Our algorithm and analysis serve as a step towards causal reinforcement learning.

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arXiv2020

Risk-Sensitive Reinforcement Learning: Near-Optimal Risk-Sample Tradeoff in Regret

We study risk-sensitive reinforcement learning in episodic Markov decision processes with unknown transition kernels, where the goal is to optimize the total reward under the risk measure of exponential utility. We propose two provably efficient model-free algorithms, Risk-Sensitive Value Iteration (RSVI) and Risk-Sensitive Q-learning (RSQ). These algorithms implement a form of risk-sensitive optimism in the face of uncertainty, which adapts to both risk-seeking and risk-averse modes of exploration. We prove that RSVI attains an $\tilde{O}\big(λ(|β| H^2) \cdot \sqrt{H^{3} S^{2}AT} \big)$ regret, while RSQ attains an $\tilde{O}\big(λ(|β| H^2) \cdot \sqrt{H^{4} SAT} \big)$ regret, where $λ(u) = (e^{3u}-1)/u$ for $u>0$. In the above, $β$ is the risk parameter of the exponential utility function, $S$ the number of states, $A$ the number of actions, $T$ the total number of timesteps, and $H$ the episode length. On the flip side, we establish a regret lower bound showing that the exponential dependence on $|β|$ and $H$ is unavoidable for any algorithm with an $\tilde{O}(\sqrt{T})$ regret (even when the risk objective is on the same scale as the original reward), thus certifying the near-optimality of the proposed algorithms. Our results demonstrate that incorporating risk awareness into reinforcement learning necessitates an exponential cost in $|β|$ and $H$, which quantifies the fundamental tradeoff between risk sensitivity (related to aleatoric uncertainty) and sample efficiency (related to epistemic uncertainty). To the best of our knowledge, this is the first regret analysis of risk-sensitive reinforcement learning with the exponential utility.

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