Alternating Mirror Descent for Constrained Min-Max Games
This provides a theoretical improvement for solving constrained two-player zero-sum games, such as those with probability simplex constraints, though it is incremental relative to prior work on unconstrained cases.
The paper tackles constrained min-max games by proposing alternating mirror descent, achieving an O(K^{-2/3}) average regret bound after K iterations, which improves over the O(K^{-1/2}) bound of simultaneous mirror descent.
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).