Sharp analysis of linear ensemble sampling

TL;DR

This paper provides a detailed analysis of linear ensemble sampling in stochastic linear bandits, demonstrating near-optimal regret bounds with computational efficiency, and introduces a novel continuous-time perspective for understanding randomized exploration.

Contribution

It offers the first sharp regret analysis of linear ensemble sampling using a continuous-time Brownian motion approach, bridging the gap to Thompson sampling.

Findings

Achieves $ ilde O(d^{3/2}rac{1}{ oot n})$ regret with ensemble size $m= heta(d ext{log} n)$

Introduces a new perspective by reducing analysis to a time-uniform exceedance problem for Brownian motions

Shows that continuous-time analysis is natural and perhaps necessary for sharp bounds in ensemble sampling

Abstract

We analyse linear ensemble sampling (ES) with standard Gaussian perturbations in stochastic linear bandits. We show that for ensemble size $m=\Theta(d\log n)$, ES attains $\tilde O(d^{3/2}\sqrt n)$ high-probability regret, closing the gap to the Thompson sampling benchmark while keeping computation comparable. The proof brings a new perspective on randomized exploration in linear bandits by reducing the analysis to a time-uniform exceedance problem for $m$ independent Brownian motions. Intriguingly, this continuous-time lens is not forced; it appears natural--and perhaps necessary: the discrete-time problem seems to be asking for a continuous-time solution, and we know of no other way to obtain a sharp ES bound.