Projected policy gradient under the simplex parameterization, policy gradient and natural policy gradient under the softmax parameterization, are fundamental algorithms in reinforcement learning. There have been a flurry of recent activities in studying these algorithms from the theoretical aspect. Despite this, their convergence behavior is still not fully understood, even given the access to exact policy evaluations. In this paper, we focus on the discounted MDP setting and conduct a systematic study of the aforementioned policy optimization methods. Several novel results are presented, including 1) global linear convergence of projected policy gradient for any constant step size, 2) sublinear convergence of softmax policy gradient for any constant step size, 3) global linear convergence of softmax natural policy gradient for any constant step size, 4) global linear convergence of entropy regularized softmax policy gradient for a wider range of constant step sizes than existing result, 5) tight local linear convergence rate of entropy regularized natural policy gradient, and 6) a new and concise local quadratic convergence rate of soft policy iteration without the assumption on the stationary distribution under the optimal policy. New and elementary analysis techniques have been developed to establish these results.
Traditional policy gradient methods are fundamentally flawed. Natural gradients converge quicker and better, forming the foundation of contemporary Reinforcement Learning such as Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO). This lecture note aims to clarify the intuition behind natural policy gradients, focusing on the thought process and the key mathematical constructs.
Analysis of On-policy Policy Gradient Methods under the Distribution Mismatch
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University（上海交通大学自动化与智能感知学院）；Key Laboratory of System Control and Information Processing, Ministry of Education of China（教育部系统控制与信息处理重点实验室）
ETH Zurich（苏黎世联邦理工学院）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
Despite their success, policy gradient methods suffer from high variance of the gradient estimate, which can result in unsatisfactory sample complexity. Recently, numerous variance-reduced extensions of policy gradient methods with provably better sample complexity and competitive numerical performance have been proposed. After a compact survey on some of the main variance-reduced REINFORCE-type methods, we propose ProbAbilistic Gradient Estimation for Policy Gradient (PAGE-PG), a novel loopless variance-reduced policy gradient method based on a probabilistic switch between two types of update…
Owing to the growth of interest in Reinforcement Learning in the last few years, gradient based policy control methods have been gaining popularity for Control problems as well. And rightly so, since gradient policy methods have the advantage of optimizing a metric of interest in an end-to-end manner, along with being relatively easy to implement without complete knowledge of the underlying system. In this paper, we study the global convergence of gradient-based policy optimization methods for quadratic control of discrete-time and model-free Markovian jump linear systems (MJLS). We surmount m…
KTH Royal Institute of Technology（KTH皇家理工学院）；ETH Zurich（苏黎世联邦理工学院）；University of Pennsylvania（宾夕法尼亚大学）
We study stochastic policy gradient methods from the perspective of control-theoretic limitations. Our main result is that ill-conditioned linear systems in the sense of Doyle inevitably lead to noisy gradient estimates. We also give an example of a class of stable systems in which policy gradient methods suffer from the curse of dimensionality. Our results apply to both state feedback and partially observed systems.
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…
Policy gradient methods in reinforcement learning update policy parameters by taking steps in the direction of an estimated gradient of policy value. In this paper, we consider the statistically efficient estimation of policy gradients from off-policy data, where the estimation is particularly non-trivial. We derive the asymptotic lower bound on the feasible mean-squared error in both Markov and non-Markov decision processes and show that existing estimators fail to achieve it in general settings. We propose a meta-algorithm that achieves the lower bound without any parametric assumptions and …
In the paper, we design a novel Bregman gradient policy optimization framework for reinforcement learning based on Bregman divergences and momentum techniques. Specifically, we propose a Bregman gradient policy optimization (BGPO) algorithm based on the basic momentum technique and mirror descent iteration. Meanwhile, we further propose an accelerated Bregman gradient policy optimization (VR-BGPO) algorithm based on the variance reduced technique. Moreover, we provide a convergence analysis framework for our Bregman gradient policy optimization under the nonconvex setting. We prove that our BG…
This paper studies a distributed policy gradient in collaborative multi-agent reinforcement learning (MARL), where agents over a communication network aim to find the optimal policy to maximize the average of all agents' local returns. Due to the non-concave performance function of policy gradient, the existing distributed stochastic optimization methods for convex problems cannot be directly used for policy gradient in MARL. This paper proposes a distributed policy gradient with variance reduction and gradient tracking to address the high variances of policy gradient, and utilizes importance …
Columbia University（哥伦比亚大学）；Simons Institute, UC Berkeley（加州大学伯克利分校西蒙斯研究所）
We revisit the finite time analysis of policy gradient methods in the one of the simplest settings: finite state and action MDPs with a policy class consisting of all stochastic policies and with exact gradient evaluations. There has been some recent work viewing this setting as an instance of smooth non-linear optimization problems and showing sub-linear convergence rates with small step-sizes. Here, we take a different perspective based on connections with policy iteration and show that many variants of policy gradient methods succeed with large step-sizes and attain a linear rate of converg…
Folklore suggests that policy gradient can be more robust to misspecification than its relative, approximate policy iteration. This paper studies the case of state-aggregated representations, where the state space is partitioned and either the policy or value function approximation is held constant over partitions. This paper shows a policy gradient method converges to a policy whose regret per-period is bounded by $ε$, the largest difference between two elements of the state-action value function belonging to a common partition. With the same representation, both approximate policy iteration …
Ohio State University（俄亥俄州立大学）；Adobe Research（奥多比研究院）
In this paper, we propose a new algorithm PPG (Proximal Policy Gradient), which is close to both VPG (vanilla policy gradient) and PPO (proximal policy optimization). The PPG objective is a partial variation of the VPG objective and the gradient of the PPG objective is exactly same as the gradient of the VPG objective. To increase the number of policy update iterations, we introduce the advantage-policy plane and design a new clipping strategy. We perform experiments in OpenAI Gym and Bullet robotics environments for ten random seeds. The performance of PPG is comparable to PPO, and the entrop…
We propose a general and systematic strategy to compile arbitrary quantum channels without using ancillary qubits, based on proximal policy optimization -- a powerful deep reinforcement learning algorithm. We rigorously prove that, in sharp contrast to the case of compiling unitary gates, it is impossible to compile an arbitrary channel to arbitrary precision with any given finite elementary channel set, regardless of the length of the decomposition sequence. However, for a fixed accuracy $ε$ one can construct a universal set with constant number of $ε$-dependent elementary channels, such that…
Off-policy stochastic actor-critic methods rely on approximating the stochastic policy gradient in order to derive an optimal policy. One may also derive the optimal policy by approximating the action-value gradient. The use of action-value gradients is desirable as policy improvement occurs along the direction of steepest ascent. This has been studied extensively within the context of natural gradient actor-critic algorithms and more recently within the context of deterministic policy gradients. In this paper we briefly discuss the off-policy stochastic counterpart to deterministic action-val…
Policy gradient methods with actor-critic schemes demonstrate tremendous empirical successes, especially when the actors and critics are parameterized by neural networks. However, it remains less clear whether such "neural" policy gradient methods converge to globally optimal policies and whether they even converge at all. We answer both the questions affirmatively in the overparameterized regime. In detail, we prove that neural natural policy gradient converges to a globally optimal policy at a sublinear rate. Also, we show that neural vanilla policy gradient converges sublinearly to a statio…
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…
The goal of policy gradient approaches is to find a policy in a given class of policies which maximizes the expected return. Given a differentiable model of the policy, we want to apply a gradient-ascent technique to reach a local optimum. We mainly use gradient ascent, because it is theoretically well researched. The main issue is that the policy gradient with respect to the expected return is not available, thus we need to estimate it. As policy gradient algorithms also tend to require on-policy data for the gradient estimate, their biggest weakness is sample efficiency. For this reason, mos…
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…
Offline reinforcement learning, wherein one uses off-policy data logged by a fixed behavior policy to evaluate and learn new policies, is crucial in applications where experimentation is limited such as medicine. We study the estimation of policy value and gradient of a deterministic policy from off-policy data when actions are continuous. Targeting deterministic policies, for which action is a deterministic function of state, is crucial since optimal policies are always deterministic (up to ties). In this setting, standard importance sampling and doubly robust estimators for policy value and …
Policy gradient methods have been successfully applied to many complex reinforcement learning problems. However, policy gradient methods suffer from high variance, slow convergence, and inefficient exploration. In this work, we introduce a maximum entropy policy optimization framework which explicitly encourages parameter exploration, and show that this framework can be reduced to a Bayesian inference problem. We then propose a novel Stein variational policy gradient method (SVPG) which combines existing policy gradient methods and a repulsive functional to generate a set of diverse but well-b…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Improving the sample efficiency in reinforcement learning has been a long-standing research problem. In this work, we aim to reduce the sample complexity of existing policy gradient methods. We propose a novel policy gradient algorithm called SRVR-PG, which only requires $O(1/ε^{3/2})$ episodes to find an $ε$-approximate stationary point of the nonconcave performance function $J(\boldsymbolθ)$ (i.e., $\boldsymbolθ$ such that $\|\nabla J(\boldsymbolθ)\|_2^2\leqε$). This sample complexity improves the existing result $O(1/ε^{5/3})$ for stochastic variance reduced policy gradient algorithms by a …
Policy gradient methods are widely used for control in reinforcement learning, particularly for the continuous action setting. There have been a host of theoretically sound algorithms proposed for the on-policy setting, due to the existence of the policy gradient theorem which provides a simplified form for the gradient. In off-policy learning, however, where the behaviour policy is not necessarily attempting to learn and follow the optimal policy for the given task, the existence of such a theorem has been elusive. In this work, we solve this open problem by providing the first off-policy pol…
Deterministic Policy Gradient (DPG) removes a level of randomness from standard randomized-action Policy Gradient (PG), and demonstrates substantial empirical success for tackling complex dynamic problems involving Markov decision processes. At the same time, though, DPG loses its ability to learn in a model-free (i.e., actor-only) fashion, frequently necessitating the use of critics in order to obtain consistent estimates of the associated policy-reward gradient. In this work, we introduce Zeroth-order Deterministic Policy Gradient (ZDPG), which approximates policy-reward gradients via two-po…
University of Cambridge（剑桥大学）；DeepMind；Max Planck Institute（马克斯·普朗克研究所）；Uber AI Labs（Uber人工智能实验室）；UC Berkeley（加州大学伯克利分校）
Off-policy model-free deep reinforcement learning methods using previously collected data can improve sample efficiency over on-policy policy gradient techniques. On the other hand, on-policy algorithms are often more stable and easier to use. This paper examines, both theoretically and empirically, approaches to merging on- and off-policy updates for deep reinforcement learning. Theoretical results show that off-policy updates with a value function estimator can be interpolated with on-policy policy gradient updates whilst still satisfying performance bounds. Our analysis uses control variate…
Two of the leading approaches for model-free reinforcement learning are policy gradient methods and $Q$-learning methods. $Q$-learning methods can be effective and sample-efficient when they work, however, it is not well-understood why they work, since empirically, the $Q$-values they estimate are very inaccurate. A partial explanation may be that $Q$-learning methods are secretly implementing policy gradient updates: we show that there is a precise equivalence between $Q$-learning and policy gradient methods in the setting of entropy-regularized reinforcement learning, that "soft" (entropy-re…
Policy-gradient methods are widely used for learning control policies. They can be easily distributed to multiple workers and reach state-of-the-art results in many domains. Unfortunately, they exhibit large variance and subsequently suffer from high-sample complexity since they aggregate gradients over entire trajectories. At the other extreme, planning methods, like tree search, optimize the policy using single-step transitions that consider future lookahead. These approaches have been mainly considered for value-based algorithms. Planning-based algorithms require a forward model and are com…
Off-policy learning refers to the problem of learning the value function of a way of behaving, or policy, while following a different policy. Gradient-based off-policy learning algorithms, such as GTD and TDC/GQ, converge even when using function approximation and incremental updates. However, they have been developed for the case of a fixed behavior policy. In control problems, one would like to adapt the behavior policy over time to become more greedy with respect to the existing value function. In this paper, we present the first gradient-based learning algorithms for this problem, which re…
In Reinforcement Learning, the optimal action at a given state is dependent on policy decisions at subsequent states. As a consequence, the learning targets evolve with time and the policy optimization process must be efficient at unlearning what it previously learnt. In this paper, we discover that the policy gradient theorem prescribes policy updates that are slow to unlearn because of their structural symmetry with respect to the value target. To increase the unlearning speed, we study a novel policy update: the gradient of the cross-entropy loss with respect to the action maximizing $q$, b…
We study the estimation of policy gradients for continuous-time systems with known dynamics. By reframing policy learning in continuous-time, we show that it is possible construct a more efficient and accurate gradient estimator. The standard back-propagation through time estimator (BPTT) computes exact gradients for a crude discretization of the continuous-time system. In contrast, we approximate continuous-time gradients in the original system. With the explicit goal of estimating continuous-time gradients, we are able to discretize adaptively and construct a more efficient policy gradient e…
Reinforcement learning methods for robotics are increasingly successful due to the constant development of better policy gradient techniques. A precise (low variance) and accurate (low bias) gradient estimator is crucial to face increasingly complex tasks. Traditional policy gradient algorithms use the likelihood-ratio trick, which is known to produce unbiased but high variance estimates. More modern approaches exploit the reparametrization trick, which gives lower variance gradient estimates but requires differentiable value function approximators. In this work, we study a different type of s…
Columbia University（哥伦比亚大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Due to the high variance of policy gradients, on-policy optimization algorithms are plagued with low sample efficiency. In this work, we propose Augment-Reinforce-Merge (ARM) policy gradient estimator as an unbiased low-variance alternative to previous baseline estimators on tasks with binary action space, inspired by the recent ARM gradient estimator for discrete random variable models. We show that the ARM policy gradient estimator achieves variance reduction with theoretical guarantees, and leads to significantly more stable and faster convergence of policies parameterized by neural network…
The policy gradient approach is a flexible and powerful reinforcement learning method particularly for problems with continuous actions such as robot control. A common challenge in this scenario is how to reduce the variance of policy gradient estimates for reliable policy updates. In this paper, we combine the following three ideas and give a highly effective policy gradient method: (a) the policy gradients with parameter based exploration, which is a recently proposed policy search method with low variance of gradient estimates, (b) an importance sampling technique, which allows us to reuse …
Policy gradient methods are an attractive approach to multi-agent reinforcement learning problems due to their convergence properties and robustness in partially observable scenarios. However, there is a significant performance gap between state-of-the-art policy gradient and value-based methods on the popular StarCraft Multi-Agent Challenge (SMAC) benchmark. In this paper, we introduce semi-on-policy (SOP) training as an effective and computationally efficient way to address the sample inefficiency of on-policy policy gradient methods. We enhance two state-of-the-art policy gradient algorithm…
Peking University（北京大学）；University of Rochester（罗切斯特大学）；Kwai Inc.（快手公司）；Duke University（杜克大学）
In this paper, we propose a novel algorithm named STOchastic Recursive Momentum for Policy Gradient (STORM-PG), which operates a SARAH-type stochastic recursive variance-reduced policy gradient in an exponential moving average fashion. STORM-PG enjoys a provably sharp $O(1/ε^3)$ sample complexity bound for STORM-PG, matching the best-known convergence rate for policy gradient algorithm. In the mean time, STORM-PG avoids the alternations between large batches and small batches which persists in comparable variance-reduced policy gradient methods, allowing considerably simpler parameter tuning. …
Baidu Research（百度研究院）；Dartmouth College（达特茅斯学院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Recent advances in policy gradient methods and deep learning have demonstrated their applicability for complex reinforcement learning problems. However, the variance of the performance gradient estimates obtained from the simulation is often excessive, leading to poor sample efficiency. In this paper, we apply the stochastic variance reduced gradient descent (SVRG) to model-free policy gradient to significantly improve the sample-efficiency. The SVRG estimation is incorporated into a trust-region Newton conjugate gradient framework for the policy optimization. On several Mujoco tasks, our meth…
University of Cambridge（剑桥大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Google Brain（谷歌大脑）；DeepMind；UC Berkeley（加州大学伯克利分校）；Uber AI Labs（优步人工智能实验室）
Model-free deep reinforcement learning (RL) methods have been successful in a wide variety of simulated domains. However, a major obstacle facing deep RL in the real world is their high sample complexity. Batch policy gradient methods offer stable learning, but at the cost of high variance, which often requires large batches. TD-style methods, such as off-policy actor-critic and Q-learning, are more sample-efficient but biased, and often require costly hyperparameter sweeps to stabilize. In this work, we aim to develop methods that combine the stability of policy gradients with the efficiency …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Recently, policy optimization for control purposes has received renewed attention due to the increasing interest in reinforcement learning. In this paper, we investigate the global convergence of gradient-based policy optimization methods for quadratic optimal control of discrete-time Markovian jump linear systems (MJLS). First, we study the optimization landscape of direct policy optimization for MJLS, with static state feedback controllers and quadratic performance costs. Despite the non-convexity of the resultant problem, we are still able to identify several useful properties such as coerc…
Markov Decision Processes are classically solved using Value Iteration and Policy Iteration algorithms. Recent interest in Reinforcement Learning has motivated the study of methods inspired by optimization, such as gradient ascent. Among these, a popular algorithm is the Natural Policy Gradient, which is a mirror descent variant for MDPs. This algorithm forms the basis of several popular Reinforcement Learning algorithms such as Natural actor-critic, TRPO, PPO, etc, and so is being studied with growing interest. It has been shown that Natural Policy Gradient with constant step size converges w…
Built on our previous study on green simulation assisted policy gradient (GS-PG) focusing on trajectory-based reuse, in this paper, we consider infinite-horizon Markov Decision Processes and create a new importance sampling based policy gradient optimization approach to support dynamic decision making. The existing GS-PG method was designed to learn from complete episodes or process trajectories, which limits its applicability to low-data situations and flexible online process control. To overcome this limitation, the proposed approach can selectively reuse the most related partial trajectorie…
Autonomous Learning Laboratory（自主学习实验室）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
Many popular policy gradient methods for reinforcement learning follow a biased approximation of the policy gradient known as the discounted approximation. While it has been shown that the discounted approximation of the policy gradient is not the gradient of any objective function, little else is known about its convergence behavior or properties. In this paper, we show that if the discounted approximation is followed such that the discount factor is increased slowly at a rate related to a decreasing learning rate, the resulting method recovers the standard guarantees of gradient ascent on th…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
In this paper, we revisit and improve the convergence of policy gradient (PG), natural PG (NPG) methods, and their variance-reduced variants, under general smooth policy parametrizations. More specifically, with the Fisher information matrix of the policy being positive definite: i) we show that a state-of-the-art variance-reduced PG method, which has only been shown to converge to stationary points, converges to the globally optimal value up to some inherent function approximation error due to policy parametrization; ii) we show that NPG enjoys a lower sample complexity; iii) we propose SRVR-…
