Full error analysis of policy gradient learning algorithms for exploratory linear quadratic mean-field control problem in continuous time with common noise
We consider reinforcement learning (RL) methods for finding optimal policies in linear quadratic (LQ) mean field control (MFC) problems over an infinite horizon in continuous time, with common noise and entropy regularization. We study policy gradient (PG) learning and first demonstrate convergence in a model-based setting by establishing a suitable gradient domination condition.Next, our main contribution is a comprehensive error analysis, where we prove the global linear convergence and sample complexity of the PG algorithm with two-point gradient estimates in a model-free setting with unknown parameters. In this setting, the parameterized optimal policies are learned from samples of the states and population distribution.Finally, we provide numerical evidence supporting the convergence of our implemented algorithms.
The paper addresses the question of lives versus livelihood in an SIRD model augmented with a macroeconomic structure. The constraints on the availability of health facilities - both infrastructure and health workers determine the probability of receiving treatment which is found to be higher for the patients with severe infection than the patients with mild infection for the specific parametric configuration of the paper. Distinguishing between two types of direct intervention policy - hard lockdown and soft lockdown, the study derives alternative policy options available to the government. T…
Reinforcement learning is a powerful tool to learn the optimal policy of possibly multiple agents by interacting with the environment. As the number of agents grow to be very large, the system can be approximated by a mean-field problem. Therefore, it has motivated new research directions for mean-field control (MFC) and mean-field game (MFG). In this paper, we study the policy gradient method for the linear-quadratic mean-field control and game, where we assume each agent has identical linear state transitions and quadratic cost functions. While most of the recent works on policy gradient for…
We study the convergence of deterministic policy gradient algorithms in continuous state and action space for the prototypical Linear Quadratic Regulator (LQR) problem when the search space is not limited to the family of linear policies. We first provide a counterexample showing that extending the policy class to piecewise linear functions results in local minima of the policy gradient algorithm. To solve this problem, we develop a new approach that involves sequentially increasing a discount factor between iterations of the original policy gradient algorithm. We finally prove that this homot…
This paper studies a continuous-time stochastic linear-quadratic (SLQ) optimal control problem on infinite-horizon. A data-driven policy iteration algorithm is proposed to solve the SLQ problem. Without knowing three system coefficient matrices, this algorithm uses the collected data to iteratively approximate a solution of the corresponding stochastic algebraic Riccati equation (SARE). A simulation example is provided to illustrate the effectiveness and applicability of the algorithm.
We study the sample complexity of approximate policy iteration (PI) for the Linear Quadratic Regulator (LQR), building on a recent line of work using LQR as a testbed to understand the limits of reinforcement learning (RL) algorithms on continuous control tasks. Our analysis quantifies the tension between policy improvement and policy evaluation, and suggests that policy evaluation is the dominant factor in terms of sample complexity. Specifically, we show that to obtain a controller that is within $\varepsilon$ of the optimal LQR controller, each step of policy evaluation requires at most $(n…
We consider the static output feedback control for Linear Quadratic Regulator problems with structured constraints under the assumption that system parameters are unknown. To solve the problem in the model free setting, we propose the policy gradient algorithm based on the gradient projection method and show its global convergence to $\varepsilon$-stationary points. In addition, we introduce a variance reduction technique and show both theoretically and numerically that it significantly reduces the variance in the gradient estimation. We also show in the numerical experiments that the model fr…
We consider a general-sum N-player linear-quadratic game with stochastic dynamics over a finite horizon and prove the global convergence of the natural policy gradient method to the Nash equilibrium. In order to prove the convergence of the method, we require a certain amount of noise in the system. We give a condition, essentially a lower bound on the covariance of the noise in terms of the model parameters, in order to guarantee convergence. We illustrate our results with numerical experiments to show that even in situations where the policy gradient method may not converge in the determinis…
We study derivative-free methods for policy optimization over the class of linear policies. We focus on characterizing the convergence rate of these methods when applied to linear-quadratic systems, and study various settings of driving noise and reward feedback. We show that these methods provably converge to within any pre-specified tolerance of the optimal policy with a number of zero-order evaluations that is an explicit polynomial of the error tolerance, dimension, and curvature properties of the problem. Our analysis reveals some interesting differences between the settings of additive d…
Department of Information Technology（信息技术系）；Uppsala University（乌普萨拉大学）；Sweden（瑞典（注：此处瑞典为国家，按要求不应提取，原数据可能存在错误或混淆，正确机构应为前两者））
Learning to make decisions from observed data in dynamic environments remains a problem of fundamental importance in a number of fields, from artificial intelligence and robotics, to medicine and finance. This paper concerns the problem of learning control policies for unknown linear dynamical systems so as to maximize a quadratic reward function. We present a method to optimize the expected value of the reward over the posterior distribution of the unknown system parameters, given data. The algorithm involves sequential convex programing, and enjoys reliable local convergence and robust stabi…
We propose projection-free sequential algorithms for linear-quadratic dynamics games. These policy gradient based algorithms are akin to Stackelberg leadership model and can be extended to model-free settings. We show that if the leader performs natural gradient descent/ascent, then the proposed algorithm has a global sublinear convergence to the Nash equilibrium. Moreover, if the leader adopts a quasi-Newton policy, the algorithm enjoys a global quadratic convergence. Along the way, we examine and clarify the intricacies of adopting sequential policy updates for LQ games, namely, issues perta…
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…
University of Texas at Dallas（得克萨斯大学达拉斯分校）；University of Melbourne（墨尔本大学）
We present a midpoint policy iteration algorithm to solve linear quadratic optimal control problems in both model-based and model-free settings. The algorithm is a variation of Newton's method, and we show that in the model-based setting it achieves cubic convergence, which is superior to standard policy iteration and policy gradient algorithms that achieve quadratic and linear convergence, respectively. We also demonstrate that the algorithm can be approximately implemented without knowledge of the dynamics model by using least-squares estimates of the state-action value function from traject…
First order policy optimization has been widely used in reinforcement learning. It guarantees to find the optimal policy for the state-feedback linear quadratic regulator (LQR). However, the performance of policy optimization remains unclear for the linear quadratic Gaussian (LQG) control where the LQG cost has spurious suboptimal stationary points. In this paper, we introduce a novel perturbed policy gradient (PGD) method to escape a large class of bad stationary points (including high-order saddles). In particular, based on the specific structure of LQG, we introduce a novel reparameterizati…
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…
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）
This paper studies the robustness of policy iteration in the context of continuous-time infinite-horizon linear quadratic regulation (LQR) problem. It is shown that Kleinman's policy iteration algorithm is inherently robust to small disturbances and enjoys local input-to-state stability in the sense of Sontag. More precisely, whenever the disturbance-induced input term in each iteration is bounded and small, the solutions of the policy iteration algorithm are also bounded and enter a small neighborhood of the optimal solution of the LQR problem. Based on this result, an off-policy data-driven …
Linear quadratic regulator (LQR) is one of the most popular frameworks to tackle continuous Markov decision process tasks. With its fundamental theory and tractable optimal policy, LQR has been revisited and analyzed in recent years, in terms of reinforcement learning scenarios such as the model-free or model-based setting. In this paper, we introduce the \textit{Structured Policy Iteration} (S-PI) for LQR, a method capable of deriving a structured linear policy. Such a structured policy with (block) sparsity or low-rank can have significant advantages over the standard LQR policy: more interp…
This work theoretically studies a ubiquitous reinforcement learning policy for controlling the canonical model of continuous-time stochastic linear-quadratic systems. We show that randomized certainty equivalent policy addresses the exploration-exploitation dilemma in linear control systems that evolve according to unknown stochastic differential equations and their operating cost is quadratic. More precisely, we establish square-root of time regret bounds, indicating that randomized certainty equivalent policy learns optimal control actions fast from a single state trajectory. Further, linear…
Wasserstein Policy Gradient for Entropy-Regularized Linear-Quadratic Control
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
This technical report is concerned with the convergence properties of what we call the split optimal policy iteration for coupled LQR problems; see section 3.1 in the manuscript. Interestingly, the iteration shows different convergence behavior for continuous and discrete time systems: while global convergence holds for both cases, we have local quadratic convergence for the continuous time case, but only linear convergence for the discrete time case - even though quadratic convergence is retained in the limit as the coupling between the subsystems vanishes.
Linear-Quadratic Mean-Field Reinforcement Learning: Convergence of Policy Gradient Methods
Program in Applied and Computational Mathematics, Princeton（普林斯顿大学应用与计算数学项目）；Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning（上海人工智能与深度学习前沿科学中心）；NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学-华东师范大学纽约大学上海数学科学研究所）；NYU Shanghai（上海纽约大学）；Walmart Global Tech（沃尔玛全球科技）
We propose to solve large scale Markowitz mean-variance (MV) portfolio allocation problem using reinforcement learning (RL). By adopting the recently developed continuous-time exploratory control framework, we formulate the exploratory MV problem in high dimensions. We further show the optimality of a multivariate Gaussian feedback policy, with time-decaying variance, in trading off exploration and exploitation. Based on a provable policy improvement theorem, we devise a scalable and data-efficient RL algorithm and conduct large scale empirical tests using data from the S&P 500 stocks. We foun…
We study model-based and model-free policy optimization in a class of nonzero-sum stochastic dynamic games called linear quadratic (LQ) deep structured games. In such games, players interact with each other through a set of weighted averages (linear regressions) of the states and actions. In this paper, we focus our attention to homogeneous weights; however, for the special case of infinite population, the obtained results extend to asymptotically vanishing weights wherein the players learn the sequential weighted mean-field equilibrium. Despite the non-convexity of the optimization in policy …
An information collection problem in a wireless network with random events is considered. Wireless devices report on each event using one of multiple reporting formats. Each format has a different quality and uses different data lengths. Delivering all data in the highest quality format can overload system resources. The goal is to make intelligent format selection and routing decisions to maximize time-averaged information quality subject to network stability. Lyapunov optimization theory can be used to solve such a problem by repeatedly minimizing the linear terms of a quadratic drift-plus-p…
In this paper, we propose a learning approach to analyze dynamic systems with asymmetric information structure. Instead of adopting a game theoretic setting, we investigate an online quadratic optimization problem driven by system noises with unknown statistics. Due to information asymmetry, it is infeasible to use classic Kalman filter nor optimal control strategies for such systems. It is necessary and beneficial to develop a robust approach that learns the probability statistics as time goes forward. Motivated by online convex optimization (OCO) theory, we introduce the notion of regret, wh…
In this paper, we study the global convergence of model-based and model-free policy gradient descent and natural policy gradient descent algorithms for linear quadratic deep structured teams. In such systems, agents are partitioned into a few sub-populations wherein the agents in each sub-population are coupled in the dynamics and cost function through a set of linear regressions of the states and actions of all agents. Every agent observes its local state and the linear regressions of states, called deep states. For a sufficiently small risk factor and/or sufficiently large population, we pro…
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）
This paper considers a distributed reinforcement learning problem for decentralized linear quadratic control with partial state observations and local costs. We propose a Zero-Order Distributed Policy Optimization algorithm (ZODPO) that learns linear local controllers in a distributed fashion, leveraging the ideas of policy gradient, zero-order optimization and consensus algorithms. In ZODPO, each agent estimates the global cost by consensus, and then conducts local policy gradient in parallel based on zero-order gradient estimation. ZODPO only requires limited communication and storage even i…
Many reinforcement learning methods achieve great success in practice but lack theoretical foundation. In this paper, we study the convergence analysis on the problem of the Linear Quadratic Regulator (LQR). The global linear convergence properties and sample complexities are established for several popular algorithms such as the policy gradient algorithm, TD-learning and the actor-critic (AC) algorithm. Our results show that the actor-critic algorithm can reduce the sample complexity compared with the policy gradient algorithm. Although our analysis is still preliminary, it explains the benef…
In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic utility are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The game is analyzed and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy opt…
