We present a family $\{\widehat{\pi}_p\}_{p\ge 1}$ of pessimistic learning rules for offline learning of linear contextual bandits, relying on confidence sets with respect to different $\ell_p$ norms, where $\widehat{\pi}_2$ corresponds to Bellman-consistent pessimism (BCP), while $\widehat{\pi}_\infty$ is a novel generalization of lower confidence bound (LCB) to the linear setting. We show that the novel $\widehat{\pi}_\infty$ 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 $\widehat{\pi}_2$.

We present a family \{\widehat{\pi}_p\}_{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 \widehat{\pi}_2 corresponds to Bellman-consistent pessimism (BCP), while \widehat{\pi}_\infty is a novel generalization of lower confidence bound (LCB) to the linear setting. We show that the novel \widehat{\pi}_\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 \widehat{\pi}_2 .