Approximate Equilibrium Computation for Discrete-Time Linear-Quadratic Mean-Field Games

While the topic of mean-field games (MFGs) has a relatively long history, heretofore there has been limited work concerning algorithms for the computation of equilibrium control policies. In this paper, we develop a computable policy iteration algorithm for approximating the mean-field equilibrium in linear-quadratic MFGs with discounted cost. Given the mean-field, each agent faces a linear-quadratic tracking problem, the solution of which involves a dynamical system evolving in retrograde time. This makes the development of forward-in-time algorithm updates challenging. By identifying a structural property of the mean-field update operator, namely that it preserves sequences of a particular form, we develop a forward-in-time equilibrium computation algorithm. Bounds that quantify the accuracy of the computed mean-field equilibrium as a function of the algorithm's stopping condition are provided. The optimality of the computed equilibrium is validated numerically. In contrast to the most recent/concurrent results, our algorithm appears to be the first to study infinite-horizon MFGs with non-stationary mean-field equilibria, though with focus on the linear quadratic setting.Comment: This paper has been accepted in ACC 202

Online Planning for Decentralized Stochastic Control with Partial History Sharing

In decentralized stochastic control, standard approaches for sequential decision-making, e.g. dynamic programming, quickly become intractable due to the need to maintain a complex information state. Computational challenges are further compounded if agents do not possess complete model knowledge. In this paper, we take advantage of the fact that in many problems agents share some common information, or history, termed partial history sharing. Under this information structure the policy search space is greatly reduced. We propose a provably convergent, online tree-search based algorithm that does not require a closed-form model or explicit communication among agents. Interestingly, our algorithm can be viewed as a generalization of several existing heuristic solvers for decentralized partially observable Markov decision processes. To demonstrate the applicability of the model, we propose a novel collaborative intrusion response model, where multiple agents (defenders) possessing asymmetric information aim to collaboratively defend a computer network. Numerical results demonstrate the performance of our algorithm.Comment: Accepted to American Control Conference (ACC) 201

Reinforcement Learning in Non-Stationary Discrete-Time Linear-Quadratic Mean-Field Games

In this paper, we study large population multi-agent reinforcement learning (RL) in the context of discrete-time linear-quadratic mean-field games (LQ-MFGs). Our setting differs from most existing work on RL for MFGs, in that we consider a non-stationary MFG over an infinite horizon. We propose an actor-critic algorithm to iteratively compute the mean-field equilibrium (MFE) of the LQ-MFG. There are two primary challenges: i) the non-stationarity of the MFG induces a linear-quadratic tracking problem, which requires solving a backwards-in-time (non-causal) equation that cannot be solved by standard (causal) RL algorithms; ii) Many RL algorithms assume that the states are sampled from the stationary distribution of a Markov chain (MC), that is, the chain is already mixed, an assumption that is not satisfied for real data sources. We first identify that the mean-field trajectory follows linear dynamics, allowing the problem to be reformulated as a linear quadratic Gaussian problem. Under this reformulation, we propose an actor-critic algorithm that allows samples to be drawn from an unmixed MC. Finite-sample convergence guarantees for the algorithm are then provided. To characterize the performance of our algorithm in multi-agent RL, we have developed an error bound with respect to the Nash equilibrium of the finite-population game.Comment: To appear in CDC 202

Convergent Policy Optimization for Safe Reinforcement Learning

We study the safe reinforcement learning problem with nonlinear function approximation, where policy optimization is formulated as a constrained optimization problem with both the objective and the constraint being nonconvex functions. For such a problem, we construct a sequence of surrogate convex constrained optimization problems by replacing the nonconvex functions locally with convex quadratic functions obtained from policy gradient estimators. We prove that the solutions to these surrogate problems converge to a stationary point of the original nonconvex problem. Furthermore, to extend our theoretical results, we apply our algorithm to examples of optimal control and multi-agent reinforcement learning with safety constraints

Optimization for Reinforcement Learning: From Single Agent to Cooperative Agents

This article reviews recent advances in multi-agent reinforcement learning algorithms for large-scale control systems and communication networks, which learn to communicate and cooperate. We provide an overview of this emerging field, with an emphasis on the decentralized setting under different coordination protocols. We highlight the evolution of reinforcement learning algorithms from single-agent to multi-agent systems, from a distributed optimization perspective, and conclude with future directions and challenges, in the hope to catalyze the growing synergy among distributed optimization, signal processing, and reinforcement learning communities

Feature-Based Q-Learning for Two-Player Stochastic Games

Consider a two-player zero-sum stochastic game where the transition function can be embedded in a given feature space. We propose a two-player Q-learning algorithm for approximating the Nash equilibrium strategy via sampling. The algorithm is shown to find an $\epsilon$-optimal strategy using sample size linear to the number of features. To further improve its sample efficiency, we develop an accelerated algorithm by adopting techniques such as variance reduction, monotonicity preservation and two-sided strategy approximation. We prove that the algorithm is guaranteed to find an $\epsilon$-optimal strategy using no more than $\tilde{\mathcal{O}}(K/(\epsilon^{2}(1-\gamma)^{4}))$ samples with high probability, where $K$ is the number of features and $\gamma$ is a discount factor. The sample, time and space complexities of the algorithm are independent of original dimensions of the game.Comment: 23 page

Multi-Agent Reinforcement Learning: A Selective Overview of Theories and Algorithms

Recent years have witnessed significant advances in reinforcement learning (RL), which has registered great success in solving various sequential decision-making problems in machine learning. Most of the successful RL applications, e.g., the games of Go and Poker, robotics, and autonomous driving, involve the participation of more than one single agent, which naturally fall into the realm of multi-agent RL (MARL), a domain with a relatively long history, and has recently re-emerged due to advances in single-agent RL techniques. Though empirically successful, theoretical foundations for MARL are relatively lacking in the literature. In this chapter, we provide a selective overview of MARL, with focus on algorithms backed by theoretical analysis. More specifically, we review the theoretical results of MARL algorithms mainly within two representative frameworks, Markov/stochastic games and extensive-form games, in accordance with the types of tasks they address, i.e., fully cooperative, fully competitive, and a mix of the two. We also introduce several significant but challenging applications of these algorithms. Orthogonal to the existing reviews on MARL, we highlight several new angles and taxonomies of MARL theory, including learning in extensive-form games, decentralized MARL with networked agents, MARL in the mean-field regime, (non-)convergence of policy-based methods for learning in games, etc. Some of the new angles extrapolate from our own research endeavors and interests. Our overall goal with this chapter is, beyond providing an assessment of the current state of the field on the mark, to identify fruitful future research directions on theoretical studies of MARL. We expect this chapter to serve as continuing stimulus for researchers interested in working on this exciting while challenging topic.Comment: Invited Chapter in Handbook on RL and Control (Springer Studies in Systems, Decision and Control