  We study how to learn $\epsilon$-optimal strategies in zero-sum imperfect
information games (IIG) with trajectory feedback. In this setting, players
update their policies sequentially based on their observations over a fixed
number of episodes, denoted by $T$. Existing procedures suffer from high
variance due to the use of importance sampling over sequences of actions
(Steinberger et al., 2020; McAleer et al., 2022). To reduce this variance, we
consider a fixed sampling approach, where players still update their policies
over time, but with observations obtained through a given fixed sampling
policy. Our approach is based on an adaptive Online Mirror Descent (OMD)
algorithm that applies OMD locally to each information set, using individually
decreasing learning rates and a regularized loss. We show that this approach
guarantees a convergence rate of $\tilde{\mathcal{O}}(T^{-1/2})$ with high
probability and has a near-optimal dependence on the game parameters when
applied with the best theoretical choices of learning rates and sampling
policies. To achieve these results, we generalize the notion of OMD
stabilization, allowing for time-varying regularization with convex increments.
