Beyond Pessimism: Offline Learning in KL-regularized Games

We study offline learning in KL-regularized two-player zero-sum games, where policies are optimized under a KL constraint to a fixed reference policy. Prior work relies on pessimistic value estimation to handle distribution shift, yielding only $\widetilde{\mathcal{O}}(1/\sqrt n)$ statistical rates. We develop a new pessimism-free algorithm and analytical framework for KL-regularized games, built on the smoothness of KL-regularized best responses and a stability property of the Nash equilibrium induced by skew symmetry. This yields the first $\widetilde{\mathcal{O}}(1/n)$ sample complexity bound for offline learning in KL-regularized zero-sum games, achieved entirely without pessimism. We further propose an efficient self-play policy optimization algorithm and prove that, with a number of iterations linear in the sample size, it achieves the same fast $\widetilde{\mathcal{O}}(1/n)$ statistical rate as the minimax estimator.