Conformal-Style Quantile Analyses for Stochastic Bandits

Stochastic bandit algorithms are usually analyzed under a mean-reward criterion, yet many problems favor arms with strong upper-tail performance, which we study herein. For a fixed miscoverage level $\alpha$, the natural upper-tail target of arm $j$ is the upper endpoint $F_j^{-1}(1-\alpha/2)$ of a central prediction interval. This target can rank arms differently from their means, creating a central mismatch with the classical bandit objective. To this end, we propose ACP-UCB1, a conformal-style policy that combines an adaptive conformal estimate of the upper endpoint with a UCB-type optimism bonus. The technical challenge is that the conformity scores used by ACP-UCB1 are recomputed from evolving empirical quantile estimates and evaluated at an adaptive level. We control this endpoint through reward-quantile concentration, a perturbation argument for recomputed score quantiles, and deterministic localization of the adaptive level. ACP-UCB1 achieves logarithmic upper-quantile regret with per-arm contribution $O(\nicefrac{\log n}{\Delta_j^{\mathrm{ACP}}})$. We also provide metric-specific regret decompositions comparing ACP-UCB1 with UCB1 and use numerical experiments to validate performance and improvement.