Smooth Contextual Bandits: Bridging the Parametric and Nondifferentiable Regret Regimes

We study a nonparametric contextual bandit problem in which the expected reward functions belong to a Hölder class with smoothness parameter β. We show how this interpolates between two extremes that were previously studied in isolation: nondifferentiable bandits (β at most 1), with which rate-optimal regret is achieved by running separate noncontextual bandits in different context regions, and parametric-response bandits (infinite $\beta$), with which rate-optimal regret can be achieved with minimal or no exploration because of 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 nondifferentiable 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.

Funding: N. Kallus acknowledges support from the National Science Foundation [Grant 1846210]. X. Mao acknowledges support from the National Science Foundation of China [Grant 72201150, Grant 72293561].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2021.2237.