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.

翻译：不完美信息博弈（IIG）是指每个玩家仅能部分观察当前博弈状态的游戏。我们研究如何通过带有轨迹反馈的自我博弈来学习零和IIG中的$\epsilon$-最优策略。我们给出了与问题无关的下界$\widetilde{\mathcal{O}}(H(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$，即高概率下学习这些策略所需的最小实现次数，其中$H$为博弈长度，$A_{\mathcal{X}}$和$B_{\mathcal{Y}}$分别为两个玩家的总动作数。我们还针对该场景提出了两种基于跟随正则化领导者（FTRL）的算法：均衡FTRL，该算法匹配此下界，但需要预先知道信息集结构以定义正则化；以及自适应FTRL，该算法无需此先验知识，通过逐步调整正则化适应观测，仅需$\widetilde{\mathcal{O}}(H^2(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$次实现。