Sharp analysis of linear ensemble sampling

2026-02-08

AI Summary

This work addresses the long-standing gap in theoretical understanding of Ensemble Sampling (ES) for linear stochastic bandits, where tight high-probability regret bounds have remained elusive compared to Thompson Sampling. By modeling the exploration mechanism of discrete-time ES as a time-uniform boundary-crossing problem involving multiple independent Brownian motions, we leverage tools from stochastic process theory, high-dimensional probability, and time-uniform concentration inequalities to establish a sharp regret upper bound. We prove that with an ensemble size of Θ(d log n), ES achieves a high-probability regret bound of Õ(d^{3/2}√n) while maintaining computational complexity comparable to Thompson Sampling, thereby closing the theoretical gap between the two methods. Our analysis further highlights the naturalness and necessity of a continuous-time perspective in deriving such results.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• linear bandits
• ensemble sampling
• regret analysis
• randomized exploration
• stochastic optimization

Innovation

• linear ensemble sampling
• stochastic linear bandits
• Brownian motion
• time-uniform concentration
• randomized exploration