Towards a Sharp Analysis of Offline Policy Learning for $f$-Divergence-Regularized Contextual Bandits

Abstract

Many offline reinforcement learning algorithms are underpinned by $f$-divergence regularization, but their sample complexity *defined with respect to regularized objectives* still lacks tight analyses, especially in terms of concrete data coverage conditions. In this paper, we study the exact concentrability requirements to achieve the $\tilde{\Theta}(\epsilon^{-1})$ sample complexity for offline $f$-divergence-regularized contextual bandits. For reverse Kullback-Leibler (KL) divergence, arguably the most commonly used one, we achieve an $\tilde{O}(\epsilon^{-1})$ sample complexity under single-policy concentrability for the first time via a novel pessimism-based analysis, surpassing existing $\tilde{O}(\epsilon^{-1})$ bound under all-policy concentrability and $\tilde{O}(\epsilon^{-2})$ bound under single-policy concentrability. We also propose a near-matching lower bound, demonstrating that a multiplicative dependency on single-policy concentrability is necessary to maximally exploit the curvature property of reverse KL. Moreover, for $f$-divergences with strongly convex $f$, to which reverse KL *does not* belong, we show that the sharp sample complexity $\tilde{\Theta}(\epsilon^{-1})$ is achievable even without pessimistic estimation or single-policy concentrability. We further corroborate our theoretical insights with numerical experiments and extend our analysis to contextual dueling bandits. We believe these results take a significant step towards a comprehensive understanding of objectives with $f$-divergence regularization.