Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability

Qingyue Zhao, Kaixuan Ji, Heyang Zhao, Quanquan Gu

Keywords: Contextual Bandits, forward Kullback-Leibler divergence, Regularized Objectives

Abstract: *Kullback-Leibler* (KL) regularization is ubiquitous in reinforcement learning algorithms in the form of *reverse* or *forward* KL. Recent studies have demonstrated $\epsilon^{-1}$-type fast rates for decision making under reverse KL regularization, in contrast to the standard $\epsilon^{-2}$-type sample complexity. However, for forward-KL-regularized objectives, existing statistical analyses are either not applicable or result in $\tilde{O}(\epsilon^{-2})$ slow rates. We take the first step towards addressing this problem via a streamlined analysis of forward-KL-regularized offline CBs. We give the first $\tilde{O}(\epsilon^{-1})$ upper bounds in tabular and general function approximation settings, both under notions of *single-policy concentrability*. In particular, our convex-analytical pipeline unifies these settings by exploiting the pessimism principle in a novel way and completely bypasses the proof routines in previous works based on the mean value theorem, which might be of independent interest. Moreover, we provide rate-optimal lower bounds, manifesting the tightness of our upper bounds in terms of statistical rates. Our lower bounds also demonstrate that the forward-KL-regularized sample complexity recovers the unregularized slow rate in the low-regularization regime, similarly to the reverse-KL regularization.

Submission Number: 95