We study a nonparametric contextual bandit problem where the expected reward functions belong to a H\"older class with smoothness parameter $\beta$. We show how this interpolates between two extremes that were previously studied in isolation: non-differentiable bandits ($\beta\leq1$), where rate-optimal regret is achieved by running separate non-contextual bandits in different context regions, and parametric-response bandits ($\beta=\infty$), where rate-optimal regret can be achieved with minimal or no exploration due to infinite extrapolatability. We develop a novel algorithm that carefully adjusts to all smoothness settings and we prove its regret is rate-optimal by establishing matching upper and lower bounds, recovering the existing results at the two extremes. In this sense, our work bridges the gap between the existing literature on parametric and non-differentiable contextual bandit problems and between bandit algorithms that exclusively use global or local information, shedding light on the crucial interplay of complexity and regret in contextual bandits.

该研究讨论了非参数上下文赌博问题，研究了函数的Hölder类和光滑度参数β之间的插值关系，提出了一种新算法，能够调整到各种光滑度设置，并通过确立匹配的上下限证明其遗憾是速率最优的，从而弥合了现有文献关于参数和非可区分性情境赌徒问题，以及仅使用全局或本地信息的赌徒算法之间的差距，同时也揭示了上下文赌徒问题中的复杂性和遗憾之间关键性的相互影响。

平滑上下文强化学习：连接参数化和不可微性遗憾模式