Pessimism for Offline Linear Contextual Bandits using $\ell_p$ Confidence Sets

Abstract

We present a family $\{\hatπ\}_{p\ge 1}$ of pessimistic learning rules for offline learning of linear contextual bandits, relying on confidence sets with respect to different $\ell_p$ norms, where $\hatπ_2$ corresponds to Bellman-consistent pessimism (BCP), while $\hatπ_\infty$ is a novel generalization of lower confidence bound (LCB) to the linear setting. We show that the novel $\hatπ_\infty$ learning rule is, in a sense, adaptively optimal, as it achieves the minimax performance (up to log factors) against all $\ell_q$-constrained problems, and as such it strictly dominates all other predictors in the family, including $\hatπ_2$.