On the Optimal Sample Complexity of Offline Multi-Armed Bandits with KL Regularization

Kullback-Leibler (KL) regularization is widely used in offline decision-making and offers several benefits, motivating recent work on the sample complexity of offline learning with respect to \emph{KL-regularized performance metrics}. Nevertheless, the exact sample complexity of KL-regularized offline learning remains largely from fully characterized. In this paper, we study this question in the setting of multi-armed bandits (MABs). We provide a sharp analysis of KL-PCB~(Zhao et al., 2026), showing that it achieves a sample complexity of $\tilde{O}(\eta SAC^{\pi^\*}/\epsilon)$ under large regularization $\eta = \tilde{O}(\epsilon^{-1})$, and a sample complexity of $\tilde{\Omega}(SAC^{\pi^\*}/\epsilon^2)$ under small regularization $\eta = \tilde{\Omega}(\epsilon^{-1})$, where $\eta$ is the regularization parameter, $S$ is the number of contexts, $A$ is the number of arms, $C^{\pi^\*}$ is the policy coverage coefficient at the optimal policy $\pi^*$, $\epsilon$ is the desired sub-optimality, and $\tilde{O}$ and $\tilde{\Omega}$ hide all poly-logarithmic factors. We further provide a pair of sharper sample complexity lower bounds, which matches the upper bounds over the entire range of regularization strengths. Overall, our results provide a nearly complete characterization of offline multi-armed bandits with KL regularization.