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 $\mathcal{O}(H(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ 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 Regularize 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 $\mathcal{O}(H^2(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ plays without this requirement by progressively adapting the regularization to the observations.

翻译：信息不便游戏 (IIG) 是每个玩家只部分观察当前游戏状态的游戏。 我们通过轨迹反馈, 研究如何在零和 IIG 中学习$\ epsilon$- 最佳策略。 我们给出一个自定义的较低约束值$mathcal{O} (H( A\mathcal{B ⁇ mathcal{Y ⁇ )/\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\在\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\在\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\