Policy gradient (PG) gives rise to a rich class of reinforcement learning (RL) methods. Recently, there has been an emerging trend to accelerate the existing PG methods such as REINFORCE by the \emph{variance reduction} techniques. However, all existing variance-reduced PG methods heavily rely on an uncheckable importance weight assumption made for every single iteration of the algorithms. In this paper, a simple gradient truncation mechanism is proposed to address this issue. Moreover, we design a Truncated Stochastic Incremental Variance-Reduced Policy Gradient (TSIVR-PG) method, which is able to maximize not only a cumulative sum of rewards but also a general utility function over a policy's long-term visiting distribution. We show an $\tilde{\mathcal{O}}(ε^{-3})$ sample complexity for TSIVR-PG to find an $ε$-stationary policy. By assuming the overparameterizaiton of policy and exploiting the hidden convexity of the problem, we further show that TSIVR-PG converges to global $ε$-optimal policy with $\tilde{\mathcal{O}}(ε^{-2})$ samples.
Auburn University（奥本大学）；University of Iowa（爱荷华大学）；Facebook AI Research（Facebook 人工智能研究院）；Huawei Inc.（华为公司）
Off-policy policy optimization is a challenging problem in reinforcement learning (RL). The algorithms designed for this problem often suffer from high variance in their estimators, which results in poor sample efficiency, and have issues with convergence. A few variance-reduced on-policy policy gradient algorithms have been recently proposed that use methods from stochastic optimization to reduce the variance of the gradient estimate in the REINFORCE algorithm. However, these algorithms are not designed for the off-policy setting and are memory-inefficient, since they need to collect and stor…
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 Southern California（南加州大学）；T.U. Kaiserslautern（凯泽斯劳滕工业大学）
We study the variance of the REINFORCE policy gradient estimator in environments with continuous state and action spaces, linear dynamics, quadratic cost, and Gaussian noise. These simple environments allow us to derive bounds on the estimator variance in terms of the environment and noise parameters. We compare the predictions of our bounds to the empirical variance in simulation experiments.
Robust Data-Collection Policy Learning for Low-Variance Online Policy Evaluation
California Institute of Technology（加州理工学院）；Massachusetts Institute of Technology（麻省理工学院）；Purdue University（普渡大学）；University of California, Riverside（加州大学河滨分校）；University of Virginia（弗吉尼亚大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
The Technion - Israel Institute of Technology（以色列理工学院）
In this paper we extend temporal difference policy evaluation algorithms to performance criteria that include the variance of the cumulative reward. Such criteria are useful for risk management, and are important in domains such as finance and process control. We propose both TD(0) and LSTD(lambda) variants with linear function approximation, prove their convergence, and demonstrate their utility in a 4-dimensional continuous state space problem.
Politecnico di Milano（米兰理工大学）；Inria（法国国家信息与自动化研究所）
In this paper, we propose a novel reinforcement- learning algorithm consisting in a stochastic variance-reduced version of policy gradient for solving Markov Decision Processes (MDPs). Stochastic variance-reduced gradient (SVRG) methods have proven to be very successful in supervised learning. However, their adaptation to policy gradient is not straightforward and needs to account for I) a non-concave objective func- tion; II) approximations in the full gradient com- putation; and III) a non-stationary sampling pro- cess. The result is SVRPG, a stochastic variance- reduced policy gradient algo…
Deep reinforcement learning (DRL) on Markov decision processes (MDPs) with continuous action spaces is often approached by directly training parametric policies along the direction of estimated policy gradients (PGs). Previous research revealed that the performance of these PG algorithms depends heavily on the bias-variance tradeoffs involved in estimating and using PGs. A notable approach towards balancing this tradeoff is to merge both on-policy and off-policy gradient estimations. However existing PG merging methods can be sample inefficient and are not suitable to train deterministic polic…
Policy gradient methods have demonstrated success in reinforcement learning tasks that have high-dimensional continuous state and action spaces. However, policy gradient methods are also notoriously sample inefficient. This can be attributed, at least in part, to the high variance in estimating the gradient of the task objective with Monte Carlo methods. Previous research has endeavored to contend with this problem by studying control variates (CVs) that can reduce the variance of estimates without introducing bias, including the early use of baselines, state dependent CVs, and the more recent…
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 …
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We show that on-policy policy gradient (PG) and its variance reduction variants can be derived by taking finite difference of function evaluations supplied by estimators from the importance sampling (IS) family for off-policy evaluation (OPE). Starting from the doubly robust (DR) estimator (Jiang & Li, 2016), we provide a simple derivation of a very general and flexible form of PG, which subsumes the state-of-the-art variance reduction technique (Cheng et al., 2019) as its special case and immediately hints at further variance reduction opportunities overlooked by existing literature. We analy…
For reinforcement learning on complex stochastic systems where many factors dynamically impact the output trajectories, it is desirable to effectively leverage the information from historical samples collected in previous iterations to accelerate policy optimization. Classical experience replay allows agents to remember by reusing historical observations. However, the uniform reuse strategy that treats all observations equally overlooks the relative importance of different samples. To overcome this limitation, we propose a general variance reduction based experience replay (VRER) framework tha…
Stanford University（斯坦福大学）；Mila - University of Montreal（米拉研究所 - 蒙特利尔大学）
Off-policy policy estimators that use importance sampling (IS) can suffer from high variance in long-horizon domains, and there has been particular excitement over new IS methods that leverage the structure of Markov decision processes. We analyze the variance of the most popular approaches through the viewpoint of conditional Monte Carlo. Surprisingly, we find that in finite horizon MDPs there is no strict variance reduction of per-decision importance sampling or stationary importance sampling, comparing with vanilla importance sampling. We then provide sufficient conditions under which the p…
Yale University（耶鲁大学）；University of California, Los Angeles（加州大学洛杉矶分校）
We study the off-policy evaluation (OPE) problem in reinforcement learning with linear function approximation, which aims to estimate the value function of a target policy based on the offline data collected by a behavior policy. We propose to incorporate the variance information of the value function to improve the sample efficiency of OPE. More specifically, for time-inhomogeneous episodic linear Markov decision processes (MDPs), we propose an algorithm, VA-OPE, which uses the estimated variance of the value function to reweight the Bellman residual in Fitted Q-Iteration. We show that our al…
Princeton University（普林斯顿大学）；University of Rochester（罗切斯特大学）；Tencent AI Lab（腾讯人工智能实验室）；Kwai AI Lab（快手人工智能实验室）；Northwestern University（西北大学）
Many complex domains, such as robotics control and real-time strategy (RTS) games, require an agent to learn a continuous control. In the former, an agent learns a policy over $\mathbb{R}^d$ and in the latter, over a discrete set of actions each of which is parametrized by a continuous parameter. Such problems are naturally solved using policy based reinforcement learning (RL) methods, but unfortunately these often suffer from high variance leading to instability and slow convergence. Unnecessary variance is introduced whenever policies over bounded action spaces are modeled using distribution…
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-…
University of Oxford（牛津大学）；Auburn University（奥本大学）
We present a mean-variance policy iteration (MVPI) framework for risk-averse control in a discounted infinite horizon MDP optimizing the variance of a per-step reward random variable. MVPI enjoys great flexibility in that any policy evaluation method and risk-neutral control method can be dropped in for risk-averse control off the shelf, in both on- and off-policy settings. This flexibility reduces the gap between risk-neutral control and risk-averse control and is achieved by working on a novel augmented MDP directly. We propose risk-averse TD3 as an example instantiating MVPI, which outperfo…
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…
This paper studies the problem of data collection for policy evaluation in Markov decision processes (MDPs). In policy evaluation, we are given a target policy and asked to estimate the expected cumulative reward it will obtain in an environment formalized as an MDP. We develop theory for optimal data collection within the class of tree-structured MDPs by first deriving an oracle data collection strategy that uses knowledge of the variance of the reward distributions. We then introduce the Reduced Variance Sampling (ReVar) algorithm that approximates the oracle strategy when the reward varianc…
Universiteit van Amsterdam（阿姆斯特丹大学）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
In this paper, we propose World Model Policy Gradient (WMPG), an approach to reduce the variance of policy gradient estimates using learned world models (WM's). In WMPG, a WM is trained online and used to imagine trajectories. The imagined trajectories are used in two ways. Firstly, to calculate a without-replacement estimator of the policy gradient. Secondly, the return of the imagined trajectories is used as an informed baseline. We compare the proposed approach with AC and MAC on a set of environments of increasing complexity (CartPole, LunarLander and Pong) and find that WMPG has better sa…
Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气与计算机工程系）
Off-policy Actor-Critic algorithms have demonstrated phenomenal experimental performance but still require better explanations. To this end, we show its policy evaluation error on the distribution of transitions decomposes into: a Bellman error, a bias from policy mismatch, and a variance term from sampling. By comparing the magnitude of bias and variance, we explain the success of the Emphasizing Recent Experience sampling and 1/age weighted sampling. Both sampling strategies yield smaller bias and variance and are hence preferable to uniform sampling.
Reinforcement learning algorithms are typically geared towards optimizing the expected return of an agent. However, in many practical applications, low variance in the return is desired to ensure the reliability of an algorithm. In this paper, we propose on-policy and off-policy actor-critic algorithms that optimize a performance criterion involving both mean and variance in the return. Previous work uses the second moment of return to estimate the variance indirectly. Instead, we use a much simpler recently proposed direct variance estimator which updates the estimates incrementally using tem…
The Ohio State University（俄亥俄州立大学）；The University of Utah（犹他大学）
Temporal difference (TD) learning is a popular algorithm for policy evaluation in reinforcement learning, but the vanilla TD can substantially suffer from the inherent optimization variance. A variance reduced TD (VRTD) algorithm was proposed by Korda and La (2015), which applies the variance reduction technique directly to the online TD learning with Markovian samples. In this work, we first point out the technical errors in the analysis of VRTD in Korda and La (2015), and then provide a mathematically solid analysis of the non-asymptotic convergence of VRTD and its variance reduction perform…
UC Berkeley（加州大学伯克利分校）；University of Washington（华盛顿大学）；OpenAI（开放人工智能研究中心）
Policy gradient methods have enjoyed great success in deep reinforcement learning but suffer from high variance of gradient estimates. The high variance problem is particularly exasperated in problems with long horizons or high-dimensional action spaces. To mitigate this issue, we derive a bias-free action-dependent baseline for variance reduction which fully exploits the structural form of the stochastic policy itself and does not make any additional assumptions about the MDP. We demonstrate and quantify the benefit of the action-dependent baseline through both theoretical analysis as well as…
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 …
The Technion - Israel Institute of Technology（以色列理工学院）
Managing risk in dynamic decision problems is of cardinal importance in many fields such as finance and process control. The most common approach to defining risk is through various variance related criteria such as the Sharpe Ratio or the standard deviation adjusted reward. It is known that optimizing many of the variance related risk criteria is NP-hard. In this paper we devise a framework for local policy gradient style algorithms for reinforcement learning for variance related criteria. Our starting point is a new formula for the variance of the cost-to-go in episodic tasks. Using this for…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；McGill University（麦吉尔大学）；Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；Princeton University（普林斯顿大学）；Northwestern University（西北大学）；Google Research（谷歌研究院）；Deepmind Montreal（深度思维蒙特利尔分所）
Learning policies from fixed offline datasets is a key challenge to scale up reinforcement learning (RL) algorithms towards practical applications. This is often because off-policy RL algorithms suffer from distributional shift, due to mismatch between dataset and the target policy, leading to high variance and over-estimation of value functions. In this work, we propose variance regularization for offline RL algorithms, using stationary distribution corrections. We show that by using Fenchel duality, we can avoid double sampling issues for computing the gradient of the variance regularizer. T…
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. …
Policy gradient methods are very attractive in reinforcement learning due to their model-free nature and convergence guarantees. These methods, however, suffer from high variance in gradient estimation, resulting in poor sample efficiency. To mitigate this issue, a number of variance-reduction approaches have been proposed. Unfortunately, in the challenging problems with delayed rewards, these approaches either bring a relatively modest improvement or do reduce variance at expense of introducing a bias and undermining convergence. The unbiased methods of gradient estimation, in general, only p…
Risk-Sensitive Deep RL: Variance-Constrained Actor-Critic Provably Finds Globally Optimal Policy
Northwestern University（西北大学）；Peking University（北京大学）；University of Science and Technology of China（中国科学技术大学）；Duke University（杜克大学）；Yale University（耶鲁大学）；Pennsylvania State University（宾夕法尼亚州立大学）
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…
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…
We present and prove properties of a new offline policy evaluator for an exploration learning setting which is superior to previous evaluators. In particular, it simultaneously and correctly incorporates techniques from importance weighting, doubly robust evaluation, and nonstationary policy evaluation approaches. In addition, our approach allows generating longer histories by careful control of a bias-variance tradeoff, and further decreases variance by incorporating information about randomness of the target policy. Empirical evidence from synthetic and realworld exploration learning problem…
Mila, McGill University（米拉研究所，麦吉尔大学）；Mila, University of Montreal（米拉研究所，蒙特利尔大学）；Google Research, Brain Team（谷歌研究院，大脑团队）；Google DeepMind（谷歌DeepMind）
Bandit and reinforcement learning (RL) problems can often be framed as optimization problems where the goal is to maximize average performance while having access only to stochastic estimates of the true gradient. Traditionally, stochastic optimization theory predicts that learning dynamics are governed by the curvature of the loss function and the noise of the gradient estimates. In this paper we demonstrate that this is not the case for bandit and RL problems. To allow our analysis to be interpreted in light of multi-step MDPs, we focus on techniques derived from stochastic optimization prin…
This paper investigates methods for estimating the optimal stochastic control policy for a Markov Decision Process with unknown transition dynamics and an unknown reward function. This form of model-free reinforcement learning comprises many real world systems such as playing video games, simulated control tasks, and real robot locomotion. Existing methods for estimating the optimal stochastic control policy rely on high variance estimates of the policy descent. However, these methods are not guaranteed to find the optimal stochastic policy, and the high variance gradient estimates make conver…
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…
Google Research（谷歌研究院）；Princeton（普林斯顿大学）；Purdue University（普渡大学）；Carnegie Mellon University（卡内基梅隆大学）
We study the sample complexity of reducing reinforcement learning to a sequence of empirical risk minimization problems over the policy space. Such reductions-based algorithms exhibit local convergence in the function space, as opposed to the parameter space for policy gradient algorithms, and thus are unaffected by the possibly non-linear or discontinuous parameterization of the policy class. We propose a variance-reduced variant of Conservative Policy Iteration that improves the sample complexity of producing a $\varepsilon$-functional local optimum from $O(\varepsilon^{-4})$ to $O(\varepsil…
École Polytechnique（巴黎综合理工学院）；HSE University（俄罗斯国立高等经济大学）；University of Duisburg-Essen（杜伊斯堡-埃森大学）
Policy-gradient methods in Reinforcement Learning(RL) are very universal and widely applied in practice but their performance suffers from the high variance of the gradient estimate. Several procedures were proposed to reduce it including actor-critic(AC) and advantage actor-critic(A2C) methods. Recently the approaches have got new perspective due to the introduction of Deep RL: both new control variates(CV) and new sub-sampling procedures became available in the setting of complex models like neural networks. The vital part of CV-based methods is the goal functional for the training of the CV…
This paper introduces a method for constructing an upper bound for exploration policy using either the weighted variance of return sequences or the weighted temporal difference (TD) error. We demonstrate that the variance of the return sequence for a specific state-action pair is an important information source that can be leveraged to guide exploration in reinforcement learning. The intuition is that fluctuation in the return sequence indicates greater uncertainty in the near future returns. This divergence occurs because of the cyclic nature of value-based reinforcement learning; the evolvin…
Low Variance Off-policy Evaluation with State-based Importance Sampling
Agency for Science, Technology and Research（新加坡科技研究局）；Manning College of Information and Computer Sciences（曼宁信息与计算机科学学院）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
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 …
Tencent Inc.（腾讯公司）；University of Manchester（曼彻斯特大学）
Recent success in Deep Reinforcement Learning (DRL) methods has shown that policy optimization with respect to an off-policy distribution via importance sampling is effective for sample reuse. In this paper, we show that the use of importance sampling could introduce high variance in the objective estimate. Specifically, we show in a principled way that the variance of importance sampling estimate grows quadratically with importance ratios and the large ratios could consequently jeopardize the effectiveness of surrogate objective optimization. We then propose a technique called sample dropout …
We revisit the stochastic variance-reduced policy gradient (SVRPG) method proposed by Papini et al. (2018) for reinforcement learning. We provide an improved convergence analysis of SVRPG and show that it can find an $ε$-approximate stationary point of the performance function within $O(1/ε^{5/3})$ trajectories. This sample complexity improves upon the best known result $O(1/ε^2)$ by a factor of $O(1/ε^{1/3})$. At the core of our analysis is (i) a tighter upper bound for the variance of importance sampling weights, where we prove that the variance can be controlled by the parameter distance be…
A central challenge to applying many off-policy reinforcement learning algorithms to real world problems is the variance introduced by importance sampling. In off-policy learning, the agent learns about a different policy than the one being executed. To account for the difference importance sampling ratios are often used, but can increase variance in the algorithms and reduce the rate of learning. Several variations of importance sampling have been proposed to reduce variance, with per-decision importance sampling being the most popular. However, the update rules for most off-policy algorithms…
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…
Dynamic optimization of mean and variance in Markov decision processes (MDPs) is a long-standing challenge caused by the failure of dynamic programming. In this paper, we propose a new approach to find the globally optimal policy for combined metrics of steady-state mean and variance in an infinite-horizon undiscounted MDP. By introducing the concepts of pseudo mean and pseudo variance, we convert the original problem to a bilevel MDP problem, where the inner one is a standard MDP optimizing pseudo mean-variance and the outer one is a single parameter selection problem optimizing pseudo mean. …
