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 \epsilon-optimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound \widetilde{O}(H(A_{X}+B_{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_{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 \widetilde{O}(H^2(A_{X}+B_{Y})/\epsilon^2) realizations without this requirement by progressively adapting the regularization to the observations.