The K-Nearest Neighbour UCB algorithm for multi-armed bandits with covariates

TL;DR

This paper introduces the k-Nearest Neighbour UCB algorithm for multi-armed bandits with covariates, achieving near-optimal regret bounds by exploiting low intrinsic data dimensionality and noise conditions, with empirical validation.

Contribution

It proposes a simple, implementable algorithm that adapts to unknown intrinsic dimensions and noise levels, providing theoretical regret guarantees and empirical evidence.

Findings

Regret bounds are minimax optimal up to logarithmic factors.

Algorithm effectively leverages low intrinsic dimensionality and noise conditions.

Empirical results confirm the algorithm's ability to exploit data on unknown sub-manifolds.

Abstract

In this paper we propose and explore the k-Nearest Neighbour UCB algorithm for multi-armed bandits with covariates. We focus on a setting where the covariates are supported on a metric space of low intrinsic dimension, such as a manifold embedded within a high dimensional ambient feature space. The algorithm is conceptually simple and straightforward to implement. The k-Nearest Neighbour UCB algorithm does not require prior knowledge of the either the intrinsic dimension of the marginal distribution or the time horizon. We prove a regret bound for the k-Nearest Neighbour UCB algorithm which is minimax optimal up to logarithmic factors. In particular, the algorithm automatically takes advantage of both low intrinsic dimensionality of the marginal distribution over the covariates and low noise in the data, expressed as a margin condition. In addition, focusing on the case of bounded rewards, we give corresponding regret bounds for the k-Nearest Neighbour KL-UCB algorithm, which is an analogue of the KL-UCB algorithm adapted to the setting of multi-armed bandits with covariates. Finally, we present empirical results which demonstrate the ability of both the k-Nearest Neighbour UCB and k-Nearest Neighbour KL-UCB to take advantage of situations where the data is supported on an unknown sub-manifold of a high-dimensional feature space.