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

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 with respect to the 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 global sublinear and local linear convergence rates. Simulation results are then provided to validate the proposed algorithms. To the best of our knowledge, this work appears the first 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 of understanding the theoretical aspects of policy-based reinforcement learning algorithms for zero-sum Markov games in general.