The recently proposed distribution correction estimation (DICE) family of estimators has advanced the state of the art in off-policy evaluation from behavior-agnostic data. While these estimators all perform some form of stationary distribution correction, they arise from different derivations and objective functions. In this paper, we unify these estimators as regularized Lagrangians of the same linear program. The unification allows us to expand the space of DICE estimators to new alternatives that demonstrate improved performance. More importantly, by analyzing the expanded space of estimators both mathematically and empirically we find that dual solutions offer greater flexibility in navigating the tradeoff between optimization stability and estimation bias, and generally provide superior estimates in practice.
Princeton University（普林斯顿大学）；Carnegie Mellon University（卡内基梅隆大学）；University of California, Berkeley（加州大学伯克利分校）；University of Pennsylvania（宾夕法尼亚大学）
Policy optimization, which finds the desired policy by maximizing value functions via optimization techniques, lies at the heart of reinforcement learning (RL). In addition to value maximization, other practical considerations arise as well, including the need of encouraging exploration, and that of ensuring certain structural properties of the learned policy due to safety, resource and operational constraints. These can often be accounted for via regularized RL, which augments the target value function with a structure-promoting regularizer. Focusing on discounted infinite-horizon Markov deci…
In this paper, we present a probability one convergence proof, under suitable conditions, of a certain class of actor-critic algorithms for finding approximate solutions to entropy-regularized MDPs using the machinery of stochastic approximation. To obtain this overall result, we prove the convergence of policy evaluation with general regularizers when using linear approximation architectures and show convergence of entropy-regularized policy improvement.
We present an off-policy actor-critic algorithm for Reinforcement Learning (RL) that combines ideas from gradient-free optimization via stochastic search with learned action-value function. The result is a simple procedure consisting of three steps: i) policy evaluation by estimating a parametric action-value function; ii) policy improvement via the estimation of a local non-parametric policy; and iii) generalization by fitting a parametric policy. Each step can be implemented in different ways, giving rise to several algorithm variants. Our algorithm draws on connections to existing literatur…
University of California, Berkeley（加州大学伯克利分校）；Netflix（奈飞）
We study the problem of off-policy evaluation (OPE) in Reinforcement Learning (RL), where the aim is to estimate the performance of a new policy given historical data that may have been generated by a different policy, or policies. In particular, we introduce a novel doubly-robust estimator for the OPE problem in RL, based on the Targeted Maximum Likelihood Estimation principle from the statistical causal inference literature. We also introduce several variance reduction techniques that lead to impressive performance gains in off-policy evaluation. We show empirically that our estimator unifor…
Natural policy gradient (NPG) methods are among the most widely used policy optimization algorithms in contemporary reinforcement learning. This class of methods is often applied in conjunction with entropy regularization -- an algorithmic scheme that encourages exploration -- and is closely related to soft policy iteration and trust region policy optimization. Despite the empirical success, the theoretical underpinnings for NPG methods remain limited even for the tabular setting. This paper develops $\textit{non-asymptotic}$ convergence guarantees for entropy-regularized NPG methods under sof…
Entropy regularized algorithms such as Soft Q-learning and Soft Actor-Critic, recently showed state-of-the-art performance on a number of challenging reinforcement learning (RL) tasks. The regularized formulation modifies the standard RL objective and thus generally converges to a policy different from the optimal greedy policy of the original RL problem. Practically, it is important to control the sub-optimality of the regularized optimal policy. In this paper, we establish sufficient conditions for convergence of a large class of regularized dynamic programming algorithms, unified under regu…
We propose and study a general framework for regularized Markov decision processes (MDPs) where the goal is to find an optimal policy that maximizes the expected discounted total reward plus a policy regularization term. The extant entropy-regularized MDPs can be cast into our framework. Moreover, under our framework, many regularization terms can bring multi-modality and sparsity, which are potentially useful in reinforcement learning. In particular, we present sufficient and necessary conditions that induce a sparse optimal policy. We also conduct a full mathematical analysis of the proposed…
The University of Tokyo（东京大学）；Tokyo Institute of Technology（东京工业大学）
We analyze the convergence behaviour of a recently proposed algorithm for regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is based on a new interpretation of DAL as a proximal minimization algorithm. We theoretically show under some conditions that DAL converges super-linearly in a non-asymptotic and global sense. Due to a special modelling of sparse estimation problems in the context of machine learning, the assumptions we make are milder and more natural than those made in conventional analysis of augmented Lagrangian algorithms. In addition, the new interpretatio…
Technion - Israel Institute of Technology（以色列理工学院）
Trust region policy optimization (TRPO) is a popular and empirically successful policy search algorithm in Reinforcement Learning (RL) in which a surrogate problem, that restricts consecutive policies to be 'close' to one another, is iteratively solved. Nevertheless, TRPO has been considered a heuristic algorithm inspired by Conservative Policy Iteration (CPI). We show that the adaptive scaling mechanism used in TRPO is in fact the natural "RL version" of traditional trust-region methods from convex analysis. We first analyze TRPO in the planning setting, in which we have access to the model a…
We study policy optimization for Markov decision processes (MDPs) with multiple reward value functions, which are to be jointly optimized according to given criteria such as proportional fairness (smooth concave scalarization), hard constraints (constrained MDP), and max-min trade-off. We propose an Anchor-changing Regularized Natural Policy Gradient (ARNPG) framework, which can systematically incorporate ideas from well-performing first-order methods into the design of policy optimization algorithms for multi-objective MDP problems. Theoretically, the designed algorithms based on the ARNPG fr…
University of Massachusetts（马萨诸塞大学）；University of Wisconsin（威斯康星大学）
We present a novel $l_1$ regularized off-policy convergent TD-learning method (termed RO-TD), which is able to learn sparse representations of value functions with low computational complexity. The algorithmic framework underlying RO-TD integrates two key ideas: off-policy convergent gradient TD methods, such as TDC, and a convex-concave saddle-point formulation of non-smooth convex optimization, which enables first-order solvers and feature selection using online convex regularization. A detailed theoretical and experimental analysis of RO-TD is presented. A variety of experiments are present…
The combination of Monte-Carlo tree search (MCTS) with deep reinforcement learning has led to significant advances in artificial intelligence. However, AlphaZero, the current state-of-the-art MCTS algorithm, still relies on handcrafted heuristics that are only partially understood. In this paper, we show that AlphaZero's search heuristics, along with other common ones such as UCT, are an approximation to the solution of a specific regularized policy optimization problem. With this insight, we propose a variant of AlphaZero which uses the exact solution to this policy optimization problem, and …
Conventional Reinforcement Learning (RL) algorithms usually have one single agent learning to solve the task independently. As a result, the agent can only explore a limited part of the state-action space while the learned behavior is highly correlated to the agent's previous experience, making the training prone to a local minimum. In this work, we empower RL with the capability of teamwork and propose a novel non-local policy optimization framework called Diversity-regularized Collaborative Exploration (DiCE). DiCE utilizes a group of heterogeneous agents to explore the environment simultane…
Graduate School of Science and Technology, Nara Institute of Science and Technology（奈良科学技术大学大学院科学技术研究科）
This paper aims to establish an entropy-regularized value-based reinforcement learning method that can ensure the monotonic improvement of policies at each policy update. Unlike previously proposed lower-bounds on policy improvement in general infinite-horizon MDPs, we derive an entropy-regularization aware lower bound. Since our bound only requires the expected policy advantage function to be estimated, it is scalable to large-scale (continuous) state-space problems. We propose a novel reinforcement learning algorithm that exploits this lower-bound as a criterion for adjusting the degree of a…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Salesforce Research（赛富时研究院）
Offline reinforcement learning enables learning from a fixed dataset, without further interactions with the environment. The lack of environmental interactions makes the policy training vulnerable to state-action pairs far from the training dataset and prone to missing rewarding actions. For training more effective agents, we propose a framework that supports learning a flexible yet well-regularized fully-implicit policy. We further propose a simple modification to the classical policy-matching methods for regularizing with respect to the dual form of the Jensen--Shannon divergence and the int…
University of Edinburgh（爱丁堡大学）；Imperial College London（帝国理工学院）；Vega Protocol（维加协议）；The Alan Turing Institute（艾伦·图灵研究所）
We study the global convergence of policy gradient for infinite-horizon, continuous state and action space, and entropy-regularized Markov decision processes (MDPs). We consider a softmax policy with (one-hidden layer) neural network approximation in a mean-field regime. Additional entropic regularization in the associated mean-field probability measure is added, and the corresponding gradient flow is studied in the 2-Wasserstein metric. We show that the objective function is increasing along the gradient flow. Further, we prove that if the regularization in terms of the mean-field measure is …
Robust Markov decision processes (MDPs) aim to handle changing or partially known system dynamics. To solve them, one typically resorts to robust optimization methods. However, this significantly increases computational complexity and limits scalability in both learning and planning. On the other hand, regularized MDPs show more stability in policy learning without impairing time complexity. Yet, they generally do not encompass uncertainty in the model dynamics. In this work, we aim to learn robust MDPs using regularization. We first show that regularized MDPs are a particular instance of robu…
Wasserstein Policy Gradient for Entropy-Regularized Linear-Quadratic Control
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
Google Research（谷歌研究院）；Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）
Many recent successful (deep) reinforcement learning algorithms make use of regularization, generally based on entropy or Kullback-Leibler divergence. We propose a general theory of regularized Markov Decision Processes that generalizes these approaches in two directions: we consider a larger class of regularizers, and we consider the general modified policy iteration approach, encompassing both policy iteration and value iteration. The core building blocks of this theory are a notion of regularized Bellman operator and the Legendre-Fenchel transform, a classical tool of convex optimization. T…
Academy for Advanced Interdisciplinary Studies, Peking University（北京大学前沿交叉学科研究院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
Regularized MDPs serve as a smooth version of original MDPs. However, biased optimal policy always exists for regularized MDPs. Instead of making the coefficientλof regularized term sufficiently small, we propose an adaptive reduction scheme for λ to approximate optimal policy of the original MDP. It is shown that the iteration complexity for obtaining anε-optimal policy could be reduced in comparison with setting sufficiently smallλ. In addition, there exists strong duality connection between the reduction method and solving the original MDP directly, from which we can derive more adaptive re…
In this work, we study algorithms for learning in infinite-horizon undiscounted Markov decision processes (MDPs) with function approximation. We first show that the regret analysis of the Politex algorithm (a version of regularized policy iteration) can be sharpened from $O(T^{3/4})$ to $O(\sqrt{T})$ under nearly identical assumptions, and instantiate the bound with linear function approximation. Our result provides the first high-probability $O(\sqrt{T})$ regret bound for a computationally efficient algorithm in this setting. The exact implementation of Politex with neural network function ap…
The Australian National University（澳大利亚国立大学）；Data61（Data61研究所）；Microsoft Research Cambridge（微软剑桥研究院）
Entropic regularization of policies in Reinforcement Learning (RL) is a commonly used heuristic to ensure that the learned policy explores the state-space sufficiently before overfitting to a local optimal policy. The primary motivation for using entropy is for exploration and disambiguating optimal policies; however, the theoretical effects are not entirely understood. In this work, we study the more general regularized RL objective and using Fenchel duality; we derive the dual problem which takes the form of an adversarial reward problem. In particular, we find that the optimal policy found …
In this short note we derive a relationship between the Bregman divergence from the current policy to the optimal policy and the suboptimality of the current value function in a regularized Markov decision process. This result has implications for multi-task reinforcement learning, offline reinforcement learning, and regret analysis under function approximation, among others.
Mastercard Corporation（万事达卡公司）；Georgia Institute of Technology（佐治亚理工学院）
We explore the use of policy approximations to reduce the computational cost of learning Nash equilibria in zero-sum stochastic games. We propose a new Q-learning type algorithm that uses a sequence of entropy-regularized soft policies to approximate the Nash policy during the Q-function updates. We prove that under certain conditions, by updating the regularized Q-function, the algorithm converges to a Nash equilibrium. We also demonstrate the proposed algorithm's ability to transfer previous training experiences, enabling the agents to adapt quickly to new environments. We provide a dynamic …
We consider the stochastic contextual bandit problem with additional regularization. The motivation comes from problems where the policy of the agent must be close to some baseline policy which is known to perform well on the task. To tackle this problem we use a nonparametric model and propose an algorithm splitting the context space into bins, and solving simultaneously - and independently - regularized multi-armed bandit instances on each bin. We derive slow and fast rates of convergence, depending on the unknown complexity of the problem. We also consider a new relevant margin condition to…
Many real world tasks exhibit rich structure that is repeated across different parts of the state space or in time. In this work we study the possibility of leveraging such repeated structure to speed up and regularize learning. We start from the KL regularized expected reward objective which introduces an additional component, a default policy. Instead of relying on a fixed default policy, we learn it from data. But crucially, we restrict the amount of information the default policy receives, forcing it to learn reusable behaviors that help the policy learn faster. We formalize this strategy …
Applied and Theoretical Aspects of Robot Intelligence (ATARI) Lab（应用与理论机器人智能实验室）；Munich Institute of Robotics and Machine Intelligence (MIRMI)（慕尼黑机器人与机器智能研究所）；Technical University of Munich（慕尼黑工业大学）
We study the sparse entropy-regularized reinforcement learning (ERL) problem in which the entropy term is a special form of the Tsallis entropy. The optimal policy of this formulation is sparse, i.e.,~at each state, it has non-zero probability for only a small number of actions. This addresses the main drawback of the standard Shannon entropy-regularized RL (soft ERL) formulation, in which the optimal policy is softmax, and thus, may assign a non-negligible probability mass to non-optimal actions. This problem is aggravated as the number of actions is increased. In this paper, we follow the wo…
Tsinghua University（清华大学）；Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）
Model-free deep reinforcement learning (RL) algorithms have been widely used for a range of complex control tasks. However, slow convergence and sample inefficiency remain challenging problems in RL, especially when handling continuous and high-dimensional state spaces. To tackle this problem, we propose a general acceleration method for model-free, off-policy deep RL algorithms by drawing the idea underlying regularized Anderson acceleration (RAA), which is an effective approach to accelerating the solving of fixed point problems with perturbations. Specifically, we first explain how policy i…
We propose a general framework for entropy-regularized average-reward reinforcement learning in Markov decision processes (MDPs). Our approach is based on extending the linear-programming formulation of policy optimization in MDPs to accommodate convex regularization functions. Our key result is showing that using the conditional entropy of the joint state-action distributions as regularization yields a dual optimization problem closely resembling the Bellman optimality equations. This result enables us to formalize a number of state-of-the-art entropy-regularized reinforcement learning algori…
Recent advancement in combining trajectory optimization with function approximation (especially neural networks) shows promise in learning complex control policies for diverse tasks in robot systems. Despite their great flexibility, the large neural networks for parameterizing control policies impose significant challenges. The learned neural control policies are often overcomplex and non-smooth, which can easily cause unexpected or diverging robot motions. Therefore, they often yield poor generalization performance in practice. To address this issue, we propose adVErsarially Regularized pOlic…
Relative Entropy Regularized Reinforcement Learning for Efficient Encrypted Policy Synthesis
School of Aeronautics and Astronautics, Purdue University（航空宇航工程学院，普渡大学）；ASRI, the Department of Electrical and Computer Engineering, Seoul National University（ASRI，电气电子工程系，首尔国立大学）；Japan Society for the Promotion of Science（日本学术振兴会）
Convergence of a L2 regularized Policy Gradient Algorithm for the Multi Armed Bandit
``Octav Mayer'' Institute of Mathematics of the Romanian Academy Bd. Carol I 8, Ia s i 700505, Romania CEREMADE, Universit\'e Paris Dauphine - PSL, CNRS, Paris, France；anonymous institute
Entropy regularization is a popular method in reinforcement learning (RL). Although it has many advantages, it alters the RL objective of the original Markov Decision Process (MDP). Though divergence regularization has been proposed to settle this problem, it cannot be trivially applied to cooperative multi-agent reinforcement learning (MARL). In this paper, we investigate divergence regularization in cooperative MARL and propose a novel off-policy cooperative MARL framework, divergence-regularized multi-agent actor-critic (DMAC). Theoretically, we derive the update rule of DMAC which is natur…
Online interactions with the environment to collect data samples for training a Reinforcement Learning (RL) agent is not always feasible due to economic and safety concerns. The goal of Offline Reinforcement Learning is to address this problem by learning effective policies using previously collected datasets. Standard off-policy RL algorithms are prone to overestimations of the values of out-of-distribution (less explored) actions and are hence unsuitable for Offline RL. Behavior regularization, which constraints the learned policy within the support set of the dataset, has been proposed to t…
School of Engineering and Computer Science（工程与计算机科学学院）；Victoria University of Wellington（惠灵顿维多利亚大学）；National Institute of Water and Atmospheric Research（国家水与大气研究所）
Forward KL Regularized Preference Optimization for Aligning Diffusion Policies
Institute of Artificial Intelligence (TeleAI), China Telecom（中国电信人工智能研究院（TeleAI））；Tsinghua University（清华大学）；Northwestern Polytechnical University Xi’an（西北工业大学（西安））；Hong Kong University of Science and Technology（香港科技大学）；ShenZhen Research Institute of Northwestern Polytechnical University（西北工业大学深圳研究院）
Policy gradient is an efficient technique for improving a policy in a reinforcement learning setting. However, vanilla online variants are on-policy only and not able to take advantage of off-policy data. In this paper we describe a new technique that combines policy gradient with off-policy Q-learning, drawing experience from a replay buffer. This is motivated by making a connection between the fixed points of the regularized policy gradient algorithm and the Q-values. This connection allows us to estimate the Q-values from the action preferences of the policy, to which we apply Q-learning up…
