The convergence of policy gradient algorithms in reinforcement learning hinges on the optimization landscape of the underlying optimal control problem. Theoretical insights into these algorithms can often be acquired from analyzing those of linear quadratic control. However, most of the existing literature only considers the optimization landscape for static full-state or output feedback policies (controllers). We investigate the more challenging case of dynamic output-feedback policies for linear quadratic regulation (abbreviated as dLQR), which is prevalent in practice but has a rather complicated optimization landscape. We first show how the dLQR cost varies with the coordinate transformation of the dynamic controller and then derive the optimal transformation for a given observable stabilizing controller. At the core of our results is the uniqueness of the stationary point of dLQR when it is observable, which is in a concise form of an observer-based controller with the optimal similarity transformation. These results shed light on designing efficient algorithms for general decision-making problems with partially observed information.
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.
Decentralized execution is one core demand in cooperative multi-agent reinforcement learning (MARL). Recently, most popular MARL algorithms have adopted decentralized policies to enable decentralized execution and use gradient descent as their optimizer. However, there is hardly any theoretical analysis of these algorithms taking the optimization method into consideration, and we find that various popular MARL algorithms with decentralized policies are suboptimal in toy tasks when gradient descent is chosen as their optimization method. In this paper, we theoretically analyze two common classes of algorithms with decentralized policies -- multi-agent policy gradient methods and value-decomposition methods to prove their suboptimality when gradient descent is used. In addition, we propose the Transformation And Distillation (TAD) framework, which reformulates a multi-agent MDP as a special single-agent MDP with a sequential structure and enables decentralized execution by distilling the learned policy on the derived ``single-agent" MDP. This approach uses a two-stage learning paradigm to address the optimization problem in cooperative MARL, maintaining its performance guarantee. Empirically, we implement TAD-PPO based on PPO, which can theoretically perform optimal policy learning in the finite multi-agent MDPs and shows significant outperformance on a large set of cooperative multi-agent tasks.
In lifelong learning, an agent learns throughout its entire life without resets, in a constantly changing environment, as we humans do. Consequently, lifelong learning comes with a plethora of research problems such as continual domain shifts, which result in non-stationary rewards and environment dynamics. These non-stationarities are difficult to detect and cope with due to their continuous nature. Therefore, exploration strategies and learning methods are required that are capable of tracking the steady domain shifts, and adapting to them. We propose Reactive Exploration to track and react to continual domain shifts in lifelong reinforcement learning, and to update the policy correspondingly. To this end, we conduct experiments in order to investigate different exploration strategies. We empirically show that representatives of the policy-gradient family are better suited for lifelong learning, as they adapt more quickly to distribution shifts than Q-learning. Thereby, policy-gradient methods profit the most from Reactive Exploration and show good results in lifelong learning with continual domain shifts. Our code is available at: https://github.com/ml-jku/reactive-exploration.
We study the performance of policy gradient methods for the subclass of Markov games known as Markov potential games (MPGs), which extends the notion of normal-form potential games to the stateful setting and includes the important special case of the fully cooperative setting where the agents share an identical reward function. Our focus in this paper is to study the convergence of the policy gradient method for solving MPGs under softmax policy parameterization, both tabular and parameterized with general function approximators such as neural networks. We first show the asymptotic convergence of this method to a Nash equilibrium of MPGs for tabular softmax policies. Second, we derive the finite-time performance of the policy gradient in two settings: 1) using the log-barrier regularization, and 2) using the natural policy gradient under the best-response dynamics (NPG-BR). Finally, extending the notion of price of anarchy (POA) and smoothness in normal-form games, we introduce the POA for MPGs and provide a POA bound for NPG-BR. To our knowledge, this is the first POA bound for solving MPGs. To support our theoretical results, we empirically compare the convergence rates and POA of policy gradient variants for both tabular and neural softmax policies.
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.
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, the most popular one is the least-squares criterion of A2C. Despite its practical success, the criterion is not the only one possible. In this paper we for the first time investigate the performance of the one called Empirical Variance(EV). We observe in the experiments that not only EV-criterion performs not worse than A2C but sometimes can be considerably better. Apart from that, we also prove some theoretical guarantees of the actual variance reduction under very general assumptions and show that A2C least-squares goal functional is an upper bound for EV goal. Our experiments indicate that in terms of variance reduction EV-based methods are much better than A2C and allow stronger variance reduction.
Solving a reinforcement learning (RL) problem poses two competing challenges: fitting a potentially discontinuous value function, and generalizing well to new observations. In this paper, we analyze the learning dynamics of temporal difference algorithms to gain novel insight into the tension between these two objectives. We show theoretically that temporal difference learning encourages agents to fit non-smooth components of the value function early in training, and at the same time induces the second-order effect of discouraging generalization. We corroborate these findings in deep RL agents trained on a range of environments, finding that neural networks trained using temporal difference algorithms on dense reward tasks exhibit weaker generalization between states than randomly initialized networks and networks trained with policy gradient methods. Finally, we investigate how post-training policy distillation may avoid this pitfall, and show that this approach improves generalization to novel environments in the ProcGen suite and improves robustness to input perturbations.
We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with $1\le\alpha\le2$ which holds in a wide range of applications in machine learning and signal processing. This condition ensures that any first-order stationary point is a global optimum. We prove that the total sample complexity of SCRN in achieving $\epsilon$-global optimum is $\mathcal{O}(\epsilon^{-7/(2\alpha)+1})$ for $1\le\alpha< 3/2$ and $\mathcal{\tilde{O}}(\epsilon^{-2/(\alpha)})$ for $3/2\le\alpha\le 2$. SCRN improves the best-known sample complexity of stochastic gradient descent. Even under a weak version of gradient dominance property, which is applicable to policy-based reinforcement learning (RL), SCRN achieves the same improvement over stochastic policy gradient methods. Additionally, we show that the average sample complexity of SCRN can be reduced to ${\mathcal{O}}(\epsilon^{-2})$ for $\alpha=1$ using a variance reduction method with time-varying batch sizes. Experimental results in various RL settings showcase the remarkable performance of SCRN compared to first-order methods.
The popular Proximal Policy Optimization (PPO) algorithm approximates the solution in a clipped policy space. Does there exist better policies outside of this space? By using a novel surrogate objective that employs the sigmoid function (which provides an interesting way of exploration), we found that the answer is ``YES'', and the better policies are in fact located very far from the clipped space. We show that PPO is insufficient in ``off-policyness'', according to an off-policy metric called DEON. Our algorithm explores in a much larger policy space than PPO, and it maximizes the Conservative Policy Iteration (CPI) objective better than PPO during training. To the best of our knowledge, all current PPO methods have the clipping operation and optimize in the clipped policy space. Our method is the first of this kind, which advances the understanding of CPI optimization and policy gradient methods. Code is available at https://github.com/raincchio/P3O.
Variance-reduced gradient estimators for policy gradient methods have been one of the main focus of research in the reinforcement learning in recent years as they allow acceleration of the estimation process. We propose a variance-reduced policy-gradient method, called SHARP, which incorporates second-order information into stochastic gradient descent (SGD) using momentum with a time-varying learning rate. SHARP algorithm is parameter-free, achieving $\epsilon$-approximate first-order stationary point with $O(\epsilon^{-3})$ number of trajectories, while using a batch size of $O(1)$ at each iteration. Unlike most previous work, our proposed algorithm does not require importance sampling which can compromise the advantage of variance reduction process. Moreover, the variance of estimation error decays with the fast rate of $O(1/t^{2/3})$ where $t$ is the number of iterations. Our extensive experimental evaluations show the effectiveness of the proposed algorithm on various control tasks and its advantage over the state of the art in practice.
We study average-reward Markov decision processes (AMDPs) and develop novel first-order methods with strong theoretical guarantees for both policy optimization and policy evaluation. Compared with intensive research efforts in finite sample analysis of policy gradient methods for discounted MDPs, existing studies on policy gradient methods for AMDPs mostly focus on regret bounds under restrictive assumptions, and they often lack guarantees on the overall sample complexities. Towards this end, we develop an average-reward stochastic policy mirror descent (SPMD) method for solving AMDPs with and without regularizers and provide convergence guarantees in terms of the long-term average reward. For policy evaluation, existing on-policy methods suffer from sub-optimal convergence rates as well as failure in handling insufficiently random policies due to the lack of exploration in the action space. To remedy these issues, we develop a variance-reduced temporal difference (VRTD) method with linear function approximation for randomized policies along with optimal convergence guarantees, and design an exploratory VRTD method that resolves the exploration issue and provides comparable convergence guarantees. By combining the policy evaluation and policy optimization parts, we establish sample complexity results for solving AMDPs under both generative and Markovian noise models. It is worth noting that when linear function approximation is utilized, our algorithm only needs to update in the low-dimensional parameter space and thus can handle MDPs with large state and action spaces.
We introduce a constrained optimization method for policy gradient reinforcement learning, which uses a virtual trust region to regulate each policy update. In addition to using the proximity of one single old policy as the normal trust region, we propose forming a second trust region through another virtual policy representing a wide range of past policies. We then enforce the new policy to stay closer to the virtual policy, which is beneficial if the old policy performs poorly. More importantly, we propose a mechanism to automatically build the virtual policy from a memory of past policies, providing a new capability for dynamically learning appropriate virtual trust regions during the optimization process. Our proposed method, dubbed Memory-Constrained Policy Optimization (MCPO), is examined in diverse environments, including robotic locomotion control, navigation with sparse rewards and Atari games, consistently demonstrating competitive performance against recent on-policy constrained policy gradient methods.
A major challenge in multi-agent systems is that the system complexity grows dramatically with the number of agents as well as the size of their action spaces, which is typical in real world scenarios such as autonomous vehicles, robotic teams, network routing, etc. It is hence in imminent need to design decentralized or independent algorithms where the update of each agent is only based on their local observations without the need of introducing complex communication/coordination mechanisms. In this work, we study the finite-time convergence of independent entropy-regularized natural policy gradient (NPG) methods for potential games, where the difference in an agent's utility function due to unilateral deviation matches exactly that of a common potential function. The proposed entropy-regularized NPG method enables each agent to deploy symmetric, decentralized, and multiplicative updates according to its own payoff. We show that the proposed method converges to the quantal response equilibrium (QRE) -- the equilibrium to the entropy-regularized game -- at a sublinear rate, which is independent of the size of the action space and grows at most sublinearly with the number of agents. Appealingly, the convergence rate further becomes independent with the number of agents for the important special case of identical-interest games, leading to the first method that converges at a dimension-free rate. Our approach can be used as a smoothing technique to find an approximate Nash equilibrium (NE) of the unregularized problem without assuming that stationary policies are isolated.
We propose a parameterization of a nonlinear dynamic controller based on the recurrent equilibrium network, a generalization of the recurrent neural network. We derive constraints on the parameterization under which the controller guarantees exponential stability of a partially observed dynamical system with sector bounded nonlinearities. Finally, we present a method to synthesize this controller using projected policy gradient methods to maximize a reward function with arbitrary structure. The projection step involves the solution of convex optimization problems. We demonstrate the proposed method with simulated examples of controlling nonlinear plants, including plants modeled with neural networks.
Cooperative multi-agent problems often require coordination between agents, which can be achieved through a centralized policy that considers the global state. Multi-agent policy gradient (MAPG) methods are commonly used to learn such policies, but they are often limited to problems with low-level action spaces. In complex problems with large state and action spaces, it is advantageous to extend MAPG methods to use higher-level actions, also known as options, to improve the policy search efficiency. However, multi-robot option executions are often asynchronous, that is, agents may select and complete their options at different time steps. This makes it difficult for MAPG methods to derive a centralized policy and evaluate its gradient, as centralized policy always select new options at the same time. In this work, we propose a novel, conditional reasoning approach to address this problem and demonstrate its effectiveness on representative option-based multi-agent cooperative tasks through empirical validation. Find code and videos at: \href{https://sites.google.com/view/mahrlsupp/}{https://sites.google.com/view/mahrlsupp/}
In modern robotics, effectively computing optimal control policies under dynamically varying environments poses substantial challenges to the off-the-shelf parametric policy gradient methods, such as the Deep Deterministic Policy Gradient (DDPG) and Twin Delayed Deep Deterministic policy gradient (TD3). In this paper, we propose a systematic methodology to dynamically learn a sequence of optimal control policies non-parametrically, while autonomously adapting with the constantly changing environment dynamics. Specifically, our non-parametric kernel-based methodology embeds a policy distribution as the features in a non-decreasing Euclidean space, therefore allowing its search space to be defined as a very high (possible infinite) dimensional RKHS (Reproducing Kernel Hilbert Space). Moreover, by leveraging the similarity metric computed in RKHS, we augmented our non-parametric learning with the technique of AdaptiveH- adaptively selecting a time-frame window of finishing the optimal part of whole action-sequence sampled on some preceding observed state. To validate our proposed approach, we conducted extensive experiments with multiple classic benchmarks and one simulated robotics benchmark equipped with dynamically changing environments. Overall, our methodology has outperformed the well-established DDPG and TD3 methodology by a sizeable margin in terms of learning performance.
Despite its popularity in the reinforcement learning community, a provably convergent policy gradient method for continuous space-time control problems with nonlinear state dynamics has been elusive. This paper proposes proximal gradient algorithms for feedback controls of finite-time horizon stochastic control problems. The state dynamics are nonlinear diffusions with control-affine drift, and the cost functions are nonconvex in the state and nonsmooth in the control. The system noise can degenerate, which allows for deterministic control problems as special cases. We prove under suitable conditions that the algorithm converges linearly to a stationary point of the control problem, and is stable with respect to policy updates by approximate gradient steps. The convergence result justifies the recent reinforcement learning heuristics that adding entropy regularization or a fictitious discount factor to the optimization objective accelerates the convergence of policy gradient methods. The proof exploits careful regularity estimates of backward stochastic differential equations.
Reinforcement learning constantly deals with hard integrals, for example when computing expectations in policy evaluation and policy iteration. These integrals are rarely analytically solvable and typically estimated with the Monte Carlo method, which induces high variance in policy values and gradients. In this work, we propose to replace Monte Carlo samples with low-discrepancy point sets. We combine policy gradient methods with Randomized Quasi-Monte Carlo, yielding variance-reduced formulations of policy gradient and actor-critic algorithms. These formulations are effective for policy evaluation and policy improvement, as they outperform state-of-the-art algorithms on standardized continuous control benchmarks. Our empirical analyses validate the intuition that replacing Monte Carlo with Quasi-Monte Carlo yields significantly more accurate gradient estimates.
We examine global non-asymptotic convergence properties of policy gradient methods for multi-agent reinforcement learning (RL) problems in Markov potential games (MPG). To learn a Nash equilibrium of an MPG in which the size of state space and/or the number of players can be very large, we propose new independent policy gradient algorithms that are run by all players in tandem. When there is no uncertainty in the gradient evaluation, we show that our algorithm finds an $\epsilon$-Nash equilibrium with $O(1/\epsilon^2)$ iteration complexity which does not explicitly depend on the state space size. When the exact gradient is not available, we establish $O(1/\epsilon^5)$ sample complexity bound in a potentially infinitely large state space for a sample-based algorithm that utilizes function approximation. Moreover, we identify a class of independent policy gradient algorithms that enjoys convergence for both zero-sum Markov games and Markov cooperative games with the players that are oblivious to the types of games being played. Finally, we provide computational experiments to corroborate the merits and the effectiveness of our theoretical developments.