Contextual Bandits for Unbounded Context Distributions

TL;DR

This paper advances the analysis of nonparametric contextual bandits by addressing unbounded context distributions, proposing two nearest neighbor methods with theoretical regret bounds that are near-optimal.

Contribution

It introduces two novel nearest neighbor algorithms for unbounded contexts and provides regret bounds that match minimax lower bounds up to logarithmic factors.

Findings

First method achieves minimax optimal regret under weak margin and light-tailed contexts.

Second adaptive method attains near-optimal regret with data-driven k selection.

The bounds extend understanding of contextual bandits to unbounded context distributions.

Abstract

Nonparametric contextual bandit is an important model of sequential decision making problems. Under $\alpha$-Tsybakov margin condition, existing research has established a regret bound of $\tilde{O}\left(T^{1-\frac{\alpha+1}{d+2}}\right)$ for bounded supports. However, the optimal regret with unbounded contexts has not been analyzed. The challenge of solving contextual bandit problems with unbounded support is to achieve both exploration-exploitation tradeoff and bias-variance tradeoff simultaneously. In this paper, we solve the nonparametric contextual bandit problem with unbounded contexts. We propose two nearest neighbor methods combined with UCB exploration. The first method uses a fixed $k$. Our analysis shows that this method achieves minimax optimal regret under a weak margin condition and relatively light-tailed context distributions. The second method uses adaptive $k$. By a proper data-driven selection of $k$, this method achieves an expected regret of $\tilde{O}\left(T^{1-\frac{(\alpha+1)\beta}{\alpha+(d+2)\beta}}+T^{1-\beta}\right)$, in which $\beta$ is a parameter describing the tail strength. This bound matches the minimax lower bound up to logarithm factors, indicating that the second method is approximately optimal.