Regret Analysis of Posterior Sampling-Based Expected Improvement for Bayesian Optimization

TL;DR

This paper provides a theoretical regret analysis of a posterior sampling-based expected improvement method in Bayesian optimization, showing it achieves sublinear regret bounds under Gaussian process assumptions and demonstrating its effectiveness through experiments.

Contribution

It introduces a regret analysis for a randomized EI variant using posterior sampling, which was previously lacking in theoretical understanding.

Findings

Achieves sublinear Bayesian cumulative regret bounds.

Demonstrates effectiveness through numerical experiments.

Provides theoretical insights into posterior sampling-based EI.

Abstract

Bayesian optimization is a powerful tool for optimizing an expensive-to-evaluate black-box function. In particular, the effectiveness of expected improvement (EI) has been demonstrated in a wide range of applications. However, theoretical analyses of EI are limited compared with other theoretically established algorithms. This paper analyzes a randomized variant of EI, which evaluates the EI from the maximum of the posterior sample path. We show that this posterior sampling-based random EI achieves the sublinear Bayesian cumulative regret bounds under the assumption that the black-box function follows a Gaussian process. Finally, we demonstrate the effectiveness of the proposed method through numerical experiments.