Alternating Mirror Descent for Constrained Min-Max Games

TL;DR

This paper introduces an alternating mirror descent algorithm for constrained min-max games, demonstrating improved convergence properties over simultaneous methods by analyzing its regret bounds and behavior in constrained and unconstrained settings.

Contribution

It proposes and analyzes an alternating mirror descent algorithm for constrained zero-sum games, showing better convergence than simultaneous methods and connecting to existing gradient descent results.

Findings

Achieves an $O(K^{-2/3})$ regret bound for the proposed algorithm.

Demonstrates divergence of simultaneous mirror descent in constrained settings.

Recovers known results for unconstrained zero-sum games with alternating gradient descent.

Abstract

In this paper we study two-player bilinear zero-sum games with constrained strategy spaces. An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint. We propose and analyze the alternating mirror descent algorithm, in which each player takes turns to take action following the mirror descent algorithm for constrained optimization. We interpret alternating mirror descent as an alternating discretization of a skew-gradient flow in the dual space, and use tools from convex optimization and modified energy function to establish an $O(K^{-2/3})$ bound on its average regret after $K$ iterations. This quantitatively verifies the algorithm's better behavior than the simultaneous version of mirror descent algorithm, which is known to diverge and yields an $O(K^{-1/2})$ average regret bound. In the special case of an unconstrained setting, our results recover the behavior of alternating gradient descent algorithm for zero-sum games which was studied in (Bailey et al., COLT 2020).