Asymptotic Theory for IV-Based Reinforcement Learning with Potential Endogeneity
The University of Hong Kong（香港大学）；Shenzhen Loop Area Institute（深圳河套学院）；Zhejiang University（浙江大学）；The Hong Kong University of Science and Technology（香港科技大学）
In the standard data analysis framework, data is collected (once and for all), and then data analysis is carried out. However, with the advancement of digital technology, decision-makers constantly analyze past data and generate new data through their decisions. We model this as a Markov decision process and show that the dynamic interaction between data generation and data analysis leads to a new type of bias -- reinforcement bias -- that exacerbates the endogeneity problem in standard data analysis. We propose a class of instrument variable (IV)-based reinforcement learning (RL) algorithms to correct for the bias and establish their theoretical properties by incorporating them into a stochastic approximation (SA) framework. Our analysis accommodates iterate-dependent Markovian structures and, therefore, can be used to study RL algorithms with policy improvement. We also provide formulas for inference on optimal policies of the IV-RL algorithms. These formulas highlight how intertemporal dependency of the Markovian environment affects the inference.
National University of Singapore（新加坡国立大学）；Massachusetts Institute of Technology（麻省理工学院）
We consider un-discounted reinforcement learning (RL) in Markov decision processes (MDPs) under temporal drifts, ie, both the reward and state transition distributions are allowed to evolve over time, as long as their respective total variations, quantified by suitable metrics, do not exceed certain variation budgets. This setting captures the endogeneity, exogeneity, uncertainty, and partial feedback in sequential decision-making scenarios, and finds applications in vehicle remarketing and real-time bidding. We first develop the Sliding Window Upper-Confidence bound for Reinforcement Learning…
The Ohio State University（俄亥俄州立大学）；Southern University of Science and Technology（南方科技大学）
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…
McGill University（麦吉尔大学）；University of Liege（列日大学）
This paper provides an analysis of the tradeoff between asymptotic bias (suboptimality with unlimited data) and overfitting (additional suboptimality due to limited data) in the context of reinforcement learning with partial observability. Our theoretical analysis formally characterizes that while potentially increasing the asymptotic bias, a smaller state representation decreases the risk of overfitting. This analysis relies on expressing the quality of a state representation by bounding L1 error terms of the associated belief states. Theoretical results are empirically illustrated when the s…
Northwestern University（西北大学）；Columbia Business School（哥伦比亚商学院）；Columbia University（哥伦比亚大学）
We investigate statistical uncertainty quantification for reinforcement learning (RL) and its implications in exploration policy. Despite ever-growing literature on RL applications, fundamental questions about inference and error quantification, such as large-sample behaviors, appear to remain quite open. In this paper, we fill in the literature gap by studying the central limit theorem behaviors of estimated Q-values and value functions under various RL settings. In particular, we explicitly identify closed-form expressions of the asymptotic variances, which allow us to efficiently construct …
University of Oxford（牛津大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Boston University（波士顿大学）
We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution which is the solution of the Bellman equation, thus giving the optimal control for the problem. In addition, we study the convergence of the limit differential equation to the stationary solution. As a by-product of our analysis, we obtain the limiting behavior of single-layer neural n…
University of Science and Technology of China（中国科学技术大学）
Model-based reinforcement learning algorithms tend to achieve higher sample efficiency than model-free methods. However, due to the inevitable errors of learned models, model-based methods struggle to achieve the same asymptotic performance as model-free methods. In this paper, We propose a Policy Optimization method with Model-Based Uncertainty (POMBU)---a novel model-based approach---that can effectively improve the asymptotic performance using the uncertainty in Q-values. We derive an upper bound of the uncertainty, based on which we can approximate the uncertainty accurately and efficientl…
We consider the problem of off-policy evaluation for reinforcement learning, where the goal is to estimate the expected reward of a target policy $π$ using offline data collected by running a logging policy $μ$. Standard importance-sampling based approaches for this problem suffer from a variance that scales exponentially with time horizon $H$, which motivates a splurge of recent interest in alternatives that break the "Curse of Horizon" (Liu et al. 2018, Xie et al. 2019). In particular, it was shown that a marginalized importance sampling (MIS) approach can be used to achieve an estimation er…
École Polytechnique（巴黎综合理工学院）；University of Warsaw（华沙大学）；The Chinese University of Hong Kong（香港中文大学）
Stochastic approximation (SA) is a key method used in statistical learning. Recently, its non-asymptotic convergence analysis has been considered in many papers. However, most of the prior analyses are made under restrictive assumptions such as unbiased gradient estimates and convex objective function, which significantly limit their applications to sophisticated tasks such as online and reinforcement learning. These restrictions are all essentially relaxed in this work. In particular, we analyze a general SA scheme to minimize a non-convex, smooth objective function. We consider update proced…
Load serving entities with storage units reach sizes and performances that can significantly impact clearing prices in electricity markets. Nevertheless, price endogeneity is rarely considered in storage bidding strategies and modeling the electricity market is a challenging task. Meanwhile, model-free reinforcement learning such as the Actor-Critic are becoming increasingly popular for designing energy system controllers. Yet implementation frequently requires lengthy, data-intense, and unsafe trial-and-error training. To fill these gaps, we implement an online Supervised Actor-Critic (SAC) a…
In this paper, we study gap-dependent regret guarantees for risk-sensitive reinforcement learning based on the entropic risk measure. We propose a novel definition of sub-optimality gaps, which we call cascaded gaps, and we discuss their key components that adapt to the underlying structures of the problem. Based on the cascaded gaps, we derive non-asymptotic and logarithmic regret bounds for two model-free algorithms under episodic Markov decision processes. We show that, in appropriate settings, these bounds feature exponential improvement over existing ones that are independent of gaps. We …
University of California Santa Barbara（加州大学圣巴巴拉分校）；Princeton University（普林斯顿大学）
We present a Reinforcement Learning (RL) algorithm to solve infinite horizon asymptotic Mean Field Game (MFG) and Mean Field Control (MFC) problems. Our approach can be described as a unified two-timescale Mean Field Q-learning: The \emph{same} algorithm can learn either the MFG or the MFC solution by simply tuning the ratio of two learning parameters. The algorithm is in discrete time and space where the agent not only provides an action to the environment but also a distribution of the state in order to take into account the mean field feature of the problem. Importantly, we assume that the …
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Shenzhen Institute of Artificial Intelligence and Robotics for Society（深圳市人工智能与机器人研究院）
In a sequential decision-making problem, off-policy evaluation estimates the expected cumulative reward of a target policy using logged trajectory data generated from a different behavior policy, without execution of the target policy. Reinforcement learning in high-stake environments, such as healthcare and education, is often limited to off-policy settings due to safety or ethical concerns, or inability of exploration. Hence it is imperative to quantify the uncertainty of the off-policy estimate before deployment of the target policy. In this paper, we propose a novel framework that provides…
Stanford University（斯坦福大学）；New York University（纽约大学）
We establish that an optimistic variant of Q-learning applied to a fixed-horizon episodic Markov decision process with an aggregated state representation incurs regret $\tilde{\mathcal{O}}(\sqrt{H^5 M K} + εHK)$, where $H$ is the horizon, $M$ is the number of aggregate states, $K$ is the number of episodes, and $ε$ is the largest difference between any pair of optimal state-action values associated with a common aggregate state. Notably, this regret bound does not depend on the number of states or actions and indicates that asymptotic per-period regret is no greater than $ε$, independent of ho…
Reinforcement Learning agents are expected to eventually perform well. Typically, this takes the form of a guarantee about the asymptotic behavior of an algorithm given some assumptions about the environment. We present an algorithm for a policy whose value approaches the optimal value with probability 1 in all computable probabilistic environments, provided the agent has a bounded horizon. This is known as strong asymptotic optimality, and it was previously unknown whether it was possible for a policy to be strongly asymptotically optimal in the class of all computable probabilistic environme…
Northwestern University（西北大学）；Princeton University（普林斯顿大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
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 pr…
This manuscript surveys reinforcement learning from the perspective of optimization and control with a focus on continuous control applications. It surveys the general formulation, terminology, and typical experimental implementations of reinforcement learning and reviews competing solution paradigms. In order to compare the relative merits of various techniques, this survey presents a case study of the Linear Quadratic Regulator (LQR) with unknown dynamics, perhaps the simplest and best-studied problem in optimal control. The manuscript describes how merging techniques from learning theory an…
When designing algorithms for finite-time-horizon episodic reinforcement learning problems, a common approach is to introduce a fictitious discount factor and use stationary policies for approximations. Empirically, it has been shown that the fictitious discount factor helps reduce variance, and stationary policies serve to save the per-iteration computational cost. Theoretically, however, there is no existing work on convergence analysis for algorithms with this fictitious discount recipe. This paper takes the first step towards analyzing these algorithms. It focuses on two vanilla policy gra…
Oxford University（牛津大学）；Australian National University（澳大利亚国立大学）；Google DeepMind（谷歌DeepMind）
Reinforcement learners are agents that learn to pick actions that lead to high reward. Ideally, the value of a reinforcement learner's policy approaches optimality--where the optimal informed policy is the one which maximizes reward. Unfortunately, we show that if an agent is guaranteed to be "asymptotically optimal" in any (stochastically computable) environment, then subject to an assumption about the true environment, this agent will be either "destroyed" or "incapacitated" with probability 1. Much work in reinforcement learning uses an ergodicity assumption to avoid this problem. Often, do…
This paper develops the first policy gradient method with global optimality guarantee and complexity analysis for robust reinforcement learning under model mismatch. Robust reinforcement learning is to learn a policy robust to model mismatch between simulator and real environment. We first develop the robust policy (sub-)gradient, which is applicable for any differentiable parametric policy class. We show that the proposed robust policy gradient method converges to the global optimum asymptotically under direct policy parameterization. We further develop a smoothed robust policy gradient metho…
Reinforcement learning algorithms typically assume rewards to be sampled from light-tailed distributions, such as Gaussian or bounded. However, a wide variety of real-world systems generate rewards that follow heavy-tailed distributions. We consider such scenarios in the setting of undiscounted reinforcement learning. By constructing a lower bound, we show that the difficulty of learning heavy-tailed rewards asymptotically dominates the difficulty of learning transition probabilities. Leveraging techniques from robust mean estimation, we propose Heavy-UCRL2 and Heavy-Q-Learning, and show that …
Department of Mechanical Engineering and Materials Science, Duke University（杜克大学机械工程与材料科学系）；Duke University（杜克大学）
In this paper, we propose a distributed off-policy actor critic method to solve multi-agent reinforcement learning problems. Specifically, we assume that all agents keep local estimates of the global optimal policy parameter and update their local value function estimates independently. Then, we introduce an additional consensus step to let all the agents asymptotically achieve agreement on the global optimal policy function. The convergence analysis of the proposed algorithm is provided and the effectiveness of the proposed algorithm is validated using a distributed resource allocation exampl…
Department of Statistics, Harvard University（哈佛大学统计系）
Recent progress in reinforcement learning has led to remarkable performance in a range of applications, but its deployment in high-stakes settings remains quite rare. One reason is a limited understanding of the behavior of reinforcement algorithms, both in terms of their regret and their ability to learn the underlying system dynamics---existing work is focused almost exclusively on characterizing rates, with little attention paid to the constants multiplying those rates that can be critically important in practice. To start to address this challenge, we study perhaps the simplest non-bandit …
Berkeley Artificial Intelligence Research（伯克利人工智能研究院）；University of California, Berkeley（加州大学伯克利分校）
Model-based reinforcement learning (RL) algorithms can attain excellent sample efficiency, but often lag behind the best model-free algorithms in terms of asymptotic performance. This is especially true with high-capacity parametric function approximators, such as deep networks. In this paper, we study how to bridge this gap, by employing uncertainty-aware dynamics models. We propose a new algorithm called probabilistic ensembles with trajectory sampling (PETS) that combines uncertainty-aware deep network dynamics models with sampling-based uncertainty propagation. Our comparison to state-of-t…
École Polytechnique（巴黎综合理工学院）；Capital Fund Management（资本基金管理公司）；Imperial College（帝国理工学院）
This paper investigates to what extent one can improve reinforcement learning algorithms. Our study is split in three parts. First, our analysis shows that the classical asymptotic convergence rate $O(1/\sqrt{N})$ is pessimistic and can be replaced by $O((\log(N)/N)^β)$ with $\frac{1}{2}\leq β\leq 1$ and $N$ the number of iterations. Second, we propose a dynamic optimal policy for the choice of the learning rate $(γ_k)_{k\geq 0}$ used in stochastic approximation (SA). We decompose our policy into two interacting levels: the inner and the outer level. In the inner level, we present the \nameref…
Indian Institute of Technology Bombay（印度理工学院孟买分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）
To overcome the curses of dimensionality and modeling of Dynamic Programming (DP) methods to solve Markov Decision Process (MDP) problems, Reinforcement Learning (RL) methods are adopted in practice. Contrary to traditional RL algorithms which do not consider the structural properties of the optimal policy, we propose a structure-aware learning algorithm to exploit the ordered multi-threshold structure of the optimal policy, if any. We prove the asymptotic convergence of the proposed algorithm to the optimal policy. Due to the reduction in the policy space, the proposed algorithm provides rema…
Multi-agent reinforcement learning (MARL) achieves significant empirical successes. However, MARL suffers from the curse of many agents. In this paper, we exploit the symmetry of agents in MARL. In the most generic form, we study a mean-field MARL problem. Such a mean-field MARL is defined on mean-field states, which are distributions that are supported on continuous space. Based on the mean embedding of the distributions, we propose MF-FQI algorithm that solves the mean-field MARL and establishes a non-asymptotic analysis for MF-FQI algorithm. We highlight that MF-FQI algorithm enjoys a "bles…
Balancing exploration and exploitation remains a key challenge in reinforcement learning (RL). State-of-the-art RL algorithms suffer from high sample complexity, particularly in the sparse reward case, where they can do no better than to explore in all directions until the first positive rewards are found. To mitigate this, we propose Rapidly Randomly-exploring Reinforcement Learning (R3L). We formulate exploration as a search problem and leverage widely-used planning algorithms such as Rapidly-exploring Random Tree (RRT) to find initial solutions. These solutions are used as demonstrations to…
Online Estimation and Inference for Robust Policy Evaluation in Reinforcement Learning
School of Mathematics, Shanghai University of Finance and Economics（上海财经大学数学学院）；Stern School of Business, New York University（纽约大学斯特恩商学院）；Daniels School of Business, Purdue University（普渡大学丹尼尔斯商学院）
University at Buffalo（布法罗大学）；University of Utah（犹他大学）
Temporal-difference learning with gradient correction (TDC) is a two time-scale algorithm for policy evaluation in reinforcement learning. This algorithm was initially proposed with linear function approximation, and was later extended to the one with general smooth function approximation. The asymptotic convergence for the on-policy setting with general smooth function approximation was established in [bhatnagar2009convergent], however, the finite-sample analysis remains unsolved due to challenges in the non-linear and two-time-scale update structure, non-convex objective function and the tim…
Stanford University（斯坦福大学）；New York University（纽约大学）
We design a simple reinforcement learning (RL) agent that implements an optimistic version of $Q$-learning and establish through regret analysis that this agent can operate with some level of competence in any environment. While we leverage concepts from the literature on provably efficient RL, we consider a general agent-environment interface and provide a novel agent design and analysis. This level of generality positions our results to inform the design of future agents for operation in complex real environments. We establish that, as time progresses, our agent performs competitively relati…
Model-Free Robust Average-Reward Reinforcement Learning with Sample Complexity Analysis
Department of Electrical and Computer Engineering, University of Central Florida, Orlando, Florida, USA（电气与计算机工程系，中央佛罗里达大学，奥兰多，佛罗里达州，美国）；Department of Computer Science, University of Central Florida, Orlando, Florida, USA（计算机科学系，中央佛罗里达大学，奥兰多，佛罗里达州，美国）
Model-free deep reinforcement learning (RL) agents can learn an effective policy directly from repeated interactions with a black-box environment. However in practice, the algorithms often require large amounts of training experience to learn and generalize well. In addition, classic model-free learning ignores the domain information contained in the state transition tuples. Model-based RL, on the other hand, attempts to learn a model of the environment from experience and is substantially more sample efficient, but suffers from significantly large asymptotic bias owing to the imperfect dynami…
