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 softmax parameterization, focusing on discounted Markov decision processes (MDPs). Assuming access to exact policy evaluation, we demonstrate that the algorithm converges linearly -- or even quadratically once it enters a local region around the optimal policy -- when computing optimal value functions of the regularized MDP. Moreover, the algorithm is provably stable vis-à-vis inexactness of policy evaluation. Our convergence results accommodate a wide range of learning rates, and shed light upon the role of entropy regularization in enabling fast convergence.
University of California, Berkeley（加州大学伯克利分校）；Amazon Research（亚马逊研究院）；University of Southern California（南加州大学）；University of Oxford（牛津大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Oxford-Man Institute, University of Oxford（牛津大学牛津曼研究所）
Entropy regularization has been extensively adopted to improve the efficiency, the stability, and the convergence of algorithms in reinforcement learning. This paper analyzes both quantitatively and qualitatively the impact of entropy regularization for Mean Field Game (MFG) with learning in a finite time horizon. Our study provides a theoretical justification that entropy regularization yields time-dependent policies and, furthermore, helps stabilizing and accelerating convergence to the game equilibrium. In addition, this study leads to a policy-gradient algorithm for exploration in MFG. Und…
The outcome of Jacobian singular values regularization was studied for supervised learning problems. It also was shown that Jacobian conditioning regularization can help to avoid the ``mode-collapse'' problem in Generative Adversarial Networks. In this paper, we try to answer the following question: Can information about policy conditioning help to shape a more stable and general policy of reinforcement learning agents? To answer this question, we conduct a study of Jacobian conditioning behavior during policy optimization. To the best of our knowledge, this is the first work that research con…
Korea Advanced Institute of Science and Technology（韩国科学技术院）
In this paper, sample-aware policy entropy regularization is proposed to enhance the conventional policy entropy regularization for better exploration. Exploiting the sample distribution obtainable from the replay buffer, the proposed sample-aware entropy regularization maximizes the entropy of the weighted sum of the policy action distribution and the sample action distribution from the replay buffer for sample-efficient exploration. A practical algorithm named diversity actor-critic (DAC) is developed by applying policy iteration to the objective function with the proposed sample-aware entro…
Entropy regularization is an important idea in reinforcement learning, with great success in recent algorithms like Soft Q Network (SQN) and Soft Actor-Critic (SAC1). In this work, we extend this idea into the on-policy realm. We propose the soft policy gradient theorem (SPGT) for on-policy maximum entropy reinforcement learning. With SPGT, a series of new policy optimization algorithms are derived, such as SPG, SA2C, SA3C, SDDPG, STRPO, SPPO, SIMPALA and so on. We find that SDDPG is equivalent to SAC1. For policy gradient, the policy network is often represented as a Gaussian distribution wit…
Carnegie Mellon University（卡内基梅隆大学）；University of Pennsylvania（宾夕法尼亚大学）
This paper investigates the problem of computing the equilibrium of competitive games, which is often modeled as a constrained saddle-point optimization problem with probability simplex constraints. Despite recent efforts in understanding the last-iterate convergence of extragradient methods in the unconstrained setting, the theoretical underpinnings of these methods in the constrained settings, especially those using multiplicative updates, remain highly inadequate, even when the objective function is bilinear. Motivated by the algorithmic role of entropy regularization in single-agent reinfo…
Entropy regularization is used to get improved optimization performance in reinforcement learning tasks. A common form of regularization is to maximize policy entropy to avoid premature convergence and lead to more stochastic policies for exploration through action space. However, this does not ensure exploration in the state space. In this work, we instead consider the distribution of discounted weighting of states, and propose to maximize the entropy of a lower bound approximation to the weighting of a state, based on latent space state representation. We propose entropy regularization based…
Trust Region Policy Optimization (TRPO) is a popular and empirically successful policy search algorithm in reinforcement learning (RL). It iteratively solved the surrogate problem which restricts consecutive policies to be close to each other. TRPO is an on-policy algorithm. On-policy methods bring many benefits, like the ability to gauge each resulting policy. However, they typically discard all the knowledge about the policies which existed before. In this work, we use a replay buffer to borrow from the off-policy learning setting to TRPO. Entropy regularization is usually used to improve po…
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…
University of California, Berkeley（加州大学伯克利分校）；National University of Singapore（新加坡国立大学）
Deep Reinforcement Learning (Deep RL) has been receiving increasingly more attention thanks to its encouraging performance on a variety of control tasks. Yet, conventional regularization techniques in training neural networks (e.g., $L_2$ regularization, dropout) have been largely ignored in RL methods, possibly because agents are typically trained and evaluated in the same environment, and because the deep RL community focuses more on high-level algorithm designs. In this work, we present the first comprehensive study of regularization techniques with multiple policy optimization algorithms o…
Nara Institute of Science and Technology（奈良先端科学技术大学院大学）
In this paper, we propose cautious policy programming (CPP), a novel value-based reinforcement learning (RL) algorithm that can ensure monotonic policy improvement during learning. Based on the nature of entropy-regularized RL, we derive a new entropy regularization-aware lower bound of policy improvement that only requires estimating the expected policy advantage function. CPP leverages this lower bound as a criterion for adjusting the degree of a policy update for alleviating policy oscillation. Different from similar algorithms that are mostly theory-oriented, we also propose a novel interp…
Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）；Facebook AI Research（脸书人工智能研究院）
Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO) are among the most successful policy gradient approaches in deep reinforcement learning (RL). While these methods achieve state-of-the-art performance across a wide range of challenging tasks, there is room for improvement in the stabilization of the policy learning and how the off-policy data are used. In this paper we revisit the theoretical foundations of these algorithms and propose a new algorithm which stabilizes the policy improvement through a proximity term that constrains the discounted state-action visita…
Fast Policy Learning for Linear Quadratic Control with Entropy Regularization
Department of Industrial Engineering & Operations Research, University of California, Berkeley（工业工程与运营管理系，加州大学伯克利分校）；Department of Mathematics, University of Oxford（数学系，牛津大学）；Department of Management Science & Engineering, Stanford University（管理科学与工程系，斯坦福大学）
Policy-based reinforcement learning methods suffer from the policy collapse problem. We find valued-based reinforcement learning methods with ε-greedy mechanism are capable of enjoying three characteristics, Closed-form Diversity, Objective-invariant Exploration and Adaptive Trade-off, which help value-based methods avoid the policy collapse problem. However, there does not exist a parallel mechanism for policy-based methods that achieves all three characteristics. In this paper, we propose an entropy regularization free mechanism that is designed for policy-based methods, which achieves Close…
JD Intelligent Cities Research, JD Technology（京东科技京东智能城市研究院）；Institute for AI Industry Research (AIR), Tsinghua University（清华大学人工智能产业研究院）
Most prior approaches to offline reinforcement learning (RL) utilize \textit{behavior regularization}, typically augmenting existing off-policy actor critic algorithms with a penalty measuring divergence between the policy and the offline data. However, these approaches lack guaranteed performance improvement over the behavior policy. In this work, we start from the performance difference between the learned policy and the behavior policy, we derive a new policy learning objective that can be used in the offline setting, which corresponds to the advantage function value of the behavior policy,…
Language Technologies Institute（语言技术研究所）；Carnegie Mellon University（卡内基梅隆大学）
In this work, we study the credit assignment problem in reward augmented maximum likelihood (RAML) learning, and establish a theoretical equivalence between the token-level counterpart of RAML and the entropy regularized reinforcement learning. Inspired by the connection, we propose two sequence prediction algorithms, one extending RAML with fine-grained credit assignment and the other improving Actor-Critic with a systematic entropy regularization. On two benchmark datasets, we show the proposed algorithms outperform RAML and Actor-Critic respectively, providing new alternatives to sequence p…
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…
Carnegie Mellon University（卡内基梅隆大学）；Stanford University（斯坦福大学）；University of Chicago（芝加哥大学）
Entropy regularization is known to improve exploration in sequential decision-making problems. We show that this same mechanism can also lead to nearly unbiased and lower-variance estimates of the mean reward in the optimize-and-estimate structured bandit setting. Mean reward estimation (i.e., population estimation) tasks have recently been shown to be essential for public policy settings where legal constraints often require precise estimates of population metrics. We show that leveraging entropy and KL divergence can yield a better trade-off between reward and estimator variance than existin…
In this paper, a sparse Markov decision process (MDP) with novel causal sparse Tsallis entropy regularization is proposed.The proposed policy regularization induces a sparse and multi-modal optimal policy distribution of a sparse MDP. The full mathematical analysis of the proposed sparse MDP is provided.We first analyze the optimality condition of a sparse MDP. Then, we propose a sparse value iteration method which solves a sparse MDP and then prove the convergence and optimality of sparse value iteration using the Banach fixed point theorem. The proposed sparse MDP is compared to soft MDPs wh…
Policy evaluation or value function or Q-function approximation is a key procedure in reinforcement learning (RL). It is a necessary component of policy iteration and can be used for variance reduction in policy gradient methods. Therefore its quality has a significant impact on most RL algorithms. Motivated by manifold regularized learning, we propose a novel kernelized policy evaluation method that takes advantage of the intrinsic geometry of the state space learned from data, in order to achieve better sample efficiency and higher accuracy in Q-function approximation. Applying the proposed …
DeepStock: Reinforcement Learning with Policy Regularizations for Inventory Management
Booth School of Business, University of Chicago（芝加哥大学商学院）；Taobao & Tmall Group（淘宝与天猫集团）；School of Economics, Sichuan University（四川大学经济学院）；Graduate School of Business, Columbia University（哥伦比亚大学商学院）；School of Operations Research and Information Engineering, Cornell University（康奈尔大学运筹学与信息工程学院）
In a reward-free environment, what is a suitable intrinsic objective for an agent to pursue so that it can learn an optimal task-agnostic exploration policy? In this paper, we argue that the entropy of the state distribution induced by finite-horizon trajectories is a sensible target. Especially, we present a novel and practical policy-search algorithm, Maximum Entropy POLicy optimization (MEPOL), to learn a policy that maximizes a non-parametric, $k$-nearest neighbors estimate of the state distribution entropy. In contrast to known methods, MEPOL is completely model-free as it requires neithe…
Revisiting Entropy Regularization: Adaptive Coefficient Unlocks Its Potential for LLM Reinforcement Learning
State Key Lab of Processors, Institute of Computing Technology, CAS（处理器国家重点实验室，计算技术研究所，中国科学院）；University of Science and Technology of China（中国科学技术大学）；University of Chinese Academy of Sciences（中国科学院大学）；StepFun Inc（StepFun公司）
Recently deep reinforcement learning (DRL) has achieved outstanding success on solving many difficult and large-scale RL problems. However the high sample cost required for effective learning often makes DRL unaffordable in resource-limited applications. With the aim of improving sample efficiency and learning performance, we will develop a new DRL algorithm in this paper that seamless integrates entropy-induced and bootstrap-induced techniques for efficient and deep exploration of the learning environment. Specifically, a general form of Tsallis entropy regularizer will be utilized to drive e…
The policy gradient theorem is defined based on an objective with respect to the initial distribution over states. In the discounted case, this results in policies that are optimal for one distribution over initial states, but may not be uniformly optimal for others, no matter where the agent starts from. Furthermore, to obtain unbiased gradient estimates, the starting point of the policy gradient estimator requires sampling states from a normalized discounted weighting of states. However, the difficulty of estimating the normalized discounted weighting of states, or the stationary state distr…
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…
Although deep Reinforcement Learning (RL) has proven successful in a wide range of tasks, one challenge it faces is interpretability when applied to real-world problems. Saliency maps are frequently used to provide interpretability for deep neural networks. However, in the RL domain, existing saliency map approaches are either computationally expensive and thus cannot satisfy the real-time requirement of real-world scenarios or cannot produce interpretable saliency maps for RL policies. In this work, we propose an approach of Distillation with selective Input Gradient Regularization (DIGR) whi…
Mila（米拉研究所）；McGill University（麦吉尔大学）；Google Research（谷歌研究院）；University of Alberta（阿尔伯塔大学）
Entropy regularization is commonly used to improve policy optimization in reinforcement learning. It is believed to help with \emph{exploration} by encouraging the selection of more stochastic policies. In this work, we analyze this claim using new visualizations of the optimization landscape based on randomly perturbing the loss function. We first show that even with access to the exact gradient, policy optimization is difficult due to the geometry of the objective function. Then, we qualitatively show that in some environments, a policy with higher entropy can make the optimization landscape…
Politecnico di Milano（米兰理工大学）；University of Padova（帕多瓦大学）
Learning in a lifelong setting, where the dynamics continually evolve, is a hard challenge for current reinforcement learning algorithms. Yet this would be a much needed feature for practical applications. In this paper, we propose an approach which learns a hyper-policy, whose input is time, that outputs the parameters of the policy to be queried at that time. This hyper-policy is trained to maximize the estimated future performance, efficiently reusing past data by means of importance sampling, at the cost of introducing a controlled bias. We combine the future performance estimate with the …
Adaptive Advantage-Guided Policy Regularization for Offline Reinforcement Learning
National University of Defense Technology（国防科技大学）；Shanghai Institute of Microsystem and Information Technology（上海微系统与信息技术研究所）；Chinese Academy of Sciences（中国科学院）；Delft University of Technology（代尔夫特理工大学）
Nara Institute of Science and Technology（奈良先端科学技术大学院大学）；Osaka University（大阪大学）；Advanced Telecommunication Research（国际电气通信基础技术研究所）
Maximum Tsallis entropy (MTE) framework in reinforcement learning has gained popularity recently by virtue of its flexible modeling choices including the widely used Shannon entropy and sparse entropy. However, non-Shannon entropies suffer from approximation error and subsequent underperformance either due to its sensitivity or the lack of closed-form policy expression. To improve the tradeoff between flexibility and empirical performance, we propose to strengthen their error-robustness by enforcing implicit Kullback-Leibler (KL) regularization in MTE motivated by Munchausen DQN (MDQN). We do …
