Kernel Single-Index Bandits: Estimation, Inference, and Learning

TL;DR

This paper introduces a kernelized $ ext{ε}$-greedy algorithm for single-index contextual bandits, enabling flexible semiparametric learning, valid inference, and finite-time regret guarantees in adaptive, dependent data settings.

Contribution

It develops a novel kernelized $ ext{ε}$-greedy method with asymptotic inference tools for single-index bandits, addressing adaptive sampling and dependent observations.

Findings

Asymptotic normality for index estimators under adaptive sampling.

Valid confidence intervals for reward functions via a functional CLT.

Finite-time regret bounds of $ ilde{O}( oot T)$ under Lipschitz conditions.

Abstract

We study contextual bandits with finitely many actions in which the reward of each arm follows a single-index model with an arm-specific index parameter and an unknown nonparametric link function. We consider a regime in which arms correspond to stable decision options and covariates evolve adaptively under the bandit policy. This setting creates significant statistical challenges: the sampling distribution depends on the allocation rule, observations are dependent over time, and inverse-propensity weighting induces variance inflation. We propose a kernelized $\varepsilon$-greedy algorithm that combines Stein-based estimation of the index parameters with inverse-propensity-weighted kernel ridge regression for the reward functions. This approach enables flexible semiparametric learning while retaining interpretability. Our analysis develops new tools for inference with adaptively collected data. We establish asymptotic normality for the single-index estimator under adaptive sampling, yielding valid confidence regions, and derive a directional functional central limit theorem for the RKHS estimator, which provides asymptotically valid pointwise confidence intervals. The analysis relies on concentration bounds for inverse-weighted Gram matrices together with martingale central limit theorems. We further obtain finite-time regret guarantees, including $\tilde{O}(\sqrt{T})$ rates under common-link Lipschitz conditions, showing that semiparametric structure can be exploited without sacrificing statistical efficiency. These results provide a unified framework for simultaneous learning and inference in single-index contextual bandits.