# Adapting to game trees in zero-sum imperfect information games

**Authors:** Côme Fiegel, Pierre Ménard, Tadashi Kozuno, Rémi Munos, Vianney Perchet, Michal Valko  
**arXiv:** [2212.12567](https://arxiv.org/abs/2212.12567)  
**Date:** 2022-12-23  
**Summary page:** https://www.summarizepaper.com/fr/paper/2212.12567v2/

## 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.