Policy Optimization Provably Converges to Nash Equilibria in Zero-Sum Linear Quadratic Games

TL;DR

This paper proves that policy optimization methods for zero-sum linear quadratic games globally converge to Nash equilibria, despite the nonconvex landscape, and introduces algorithms with guaranteed convergence rates.

Contribution

It is the first work to analyze the optimization landscape of LQ games and establish provable convergence of policy optimization to Nash equilibria.

Findings

Stationary points in LQ games are Nash equilibria.

Developed three algorithms with global sublinear and local linear convergence.

Simulation results confirm the algorithms' convergence properties.

Abstract

We study the global convergence of policy optimization for finding the Nash equilibria (NE) in zero-sum linear quadratic (LQ) games. To this end, we first investigate the landscape of LQ games, viewing it as a nonconvex-nonconcave saddle-point problem in the policy space. Specifically, we show that despite its nonconvexity and nonconcavity, zero-sum LQ games have the property that the stationary point of the objective function with respect to the linear feedback control policies constitutes the NE of the game. Building upon this, we develop three projected nested-gradient methods that are guaranteed to converge to the NE of the game. Moreover, we show that all of these algorithms enjoy both globally sublinear and locally linear convergence rates. Simulation results are also provided to illustrate the satisfactory convergence properties of the algorithms. To the best of our knowledge, this work appears to be the first one to investigate the optimization landscape of LQ games, and provably show the convergence of policy optimization methods to the Nash equilibria. Our work serves as an initial step toward understanding the theoretical aspects of policy-based reinforcement learning algorithms for zero-sum Markov games in general.