Smooth Contextual Bandits: Bridging the Parametric and Non-differentiable Regret Regimes
This work addresses the gap in contextual bandit algorithms for researchers, providing a unified framework that interpolates between extreme cases, though it is incremental in extending existing results.
The paper tackles the problem of nonparametric contextual bandits with expected reward functions in a Hölder class, bridging parametric and non-differentiable regimes by developing a novel algorithm that achieves rate-optimal regret across all smoothness settings, with matching upper and lower bounds.
We study a nonparametric contextual bandit problem where 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: non-differentiable bandits ($β\leq1$), where rate-optimal regret is achieved by running separate non-contextual bandits in different context regions, and parametric-response bandits (satisfying $β=\infty$), 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.