Adapting to game trees in zero-sum imperfect information games

Imperfect information games (IIG) are games in which each player only partially observes the current game state. We study how to learn εoptimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound O(H(A X + B Y )/ε 2 ) on the required number of realizations to learn these strategies with high probability, where H is the length of the game, A X and B 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 O(H 2 (A X + B Y )/ε 2 ) realizations without this requirement by progressively adapting the regularization to the observations. Algorithm Sample complexity Structure-free MCCFR (Farina et al., 2020; Bai et al., 2022) O(H 4 (A X + B Y )/ε 2 ) IXOMD (Kozuno et al., 2021) O(H 2 (XA X + Y B Y )/ε 2 ) Balanced OMD (Bai et al., 2022) O(H Table 1 . Sample complexity for episodic, finite, two-player, zero-sum IIGs. Structure-free algorithm designates an algorithm that does not need to know the structure of the information set spaces in advance. The symbol O hides dependencies logarithmic in AX , BY , ε and δ. Note that for algorithms that only work with fixed action sets of size A and B we have AX = AX and BY = BY .