Smoothness-Adaptive Contextual Bandits

TL;DR

This paper investigates adaptive algorithms for non-parametric contextual bandits that do not require prior knowledge of payoff function smoothness, proposing a policy that achieves near-optimal regret rates under certain conditions.

Contribution

It introduces a framework for smoothness-adaptive algorithms in contextual bandits, overcoming the challenge of unknown payoff smoothness under a self-similarity condition.

Findings

Adaptive policy matches known-smoothness regret rates up to a logarithmic factor.

Proves impossibility of adaptation without self-similarity.

Establishes conditions under which adaptation is feasible.

Abstract

We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. In practice, however, the smoothness of payoff functions is typically not known in advance, and misspecification of smoothness may severely deteriorate the performance of existing methods. In this work, we consider a framework where the smoothness of payoff functions is not known, and study when and how algorithms may adapt to unknown smoothness. First, we establish that designing algorithms that adapt to unknown smoothness of payoff functions is, in general, impossible. However, under a self-similarity condition (which does not reduce the minimax complexity of the dynamic optimization problem at hand), we establish that adapting to unknown smoothness is possible, and further devise a general policy for achieving smoothness-adaptive performance. Our policy infers the smoothness of payoffs throughout the decision-making process, while leveraging the structure of off-the-shelf non-adaptive policies. We establish that for problem settings with either differentiable or non-differentiable payoff functions, this policy matches (up to a logarithmic scale) the regret rate that is achievable when the smoothness of payoffs is known a priori.