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.
Categories: stat.ML cs.LG