$Q$ Can Play That Game: Online Fitted $Q$-Iteration for Continuous-Action Zero-Sum Markov Games with Convex-Concave Function Approximation

Zero-sum Markov games arise in a wide variety of sequential decision-making problems such as adversarial learning and planning against modeled uncertainties. However, prior work on finite-sample guarantees on the learned state-action value function ($Q$-function) for zero-sum Markov games is largely restricted to finite-action settings, or to continuous-action games in which agents have linear dynamics and quadratic rewards (i.e. linear-quadratic, or LQ, games). A central challenge in extending such guarantees to more general continuous-action zero-sum Markov games is that the associated Bellman operator involves a minimax problem that need not admit a tractable saddle-point solution. To this end, we first introduce a class of neural network function approximators for the $Q$-function that is convex-concave in the players' actions, which guarantees that this minimax problem admits a pure-strategy saddle point. We then study an online variant of fitted $Q$-iteration employing this function class and establish, to the best of our knowledge, the first finite-sample guarantees for non-LQ zero-sum Markov games with continuous states and actions.