Adapting to game trees in zero-sum imperfect information games

TL;DR

This paper investigates learning near-optimal strategies in zero-sum imperfect information games through self-play, establishing lower bounds and proposing two algorithms that adapt to game structure and observations.

Contribution

It provides a problem-independent lower bound on sample complexity and introduces two FTRL algorithms, one requiring prior game structure knowledge and the other adapting online.

Findings

Lower bound of (H(A_X+B_Y))/(. )^2 on realizations needed

Balanced FTRL matches the lower bound but needs game structure knowledge

Adaptive FTRL achieves near-optimal sample complexity without prior structure knowledge

Abstract

Imperfect information games (IIG) are games in which each player only partially observes the current game state. We study how to learn $\epsilon$-optimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound $\widetilde{\mathcal{O}}(H(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ on the required number of realizations to learn these strategies with high probability, where $H$ is the length of the game, $A_{\mathcal{X}}$ and $B_{\mathcal{Y}}$ are the total number of actions for the two players. We also propose two Follow the Regularized leader (FTRL) algorithms for this setting: Balanced FTRL which matches this lower bound, but requires the knowledge of the information set structure beforehand to define the regularization; and Adaptive FTRL which needs $\widetilde{\mathcal{O}}(H^2(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ realizations without this requirement by progressively adapting the regularization to the observations.