On the convergence rate of noisy Bayesian Optimization with Expected Improvement

TL;DR

This paper advances the theoretical understanding of Bayesian optimization with Expected Improvement (EI), providing new convergence rates and error bounds for noisy observations under Gaussian process assumptions.

Contribution

It introduces the first asymptotic error bounds for GP-EI with noise, extending convergence analysis beyond RKHS to GP priors, and analyzes EI's exploration-exploitation properties.

Findings

Established asymptotic error bounds for GP-EI with noise.

Derived improved convergence rates for noisy and noise-free cases.

Extended theoretical analysis of EI beyond traditional RKHS assumptions.

Abstract

Expected improvement (EI) is one of the most widely used acquisition functions in Bayesian optimization (BO). Despite its proven success in applications for decades, important open questions remain on the theoretical convergence behaviors and rates for EI. In this paper, we contribute to the convergence theory of EI in three novel and critical areas. First, we consider objective functions that fit under the Gaussian process (GP) prior assumption, whereas existing works mostly focus on functions in the reproducing kernel Hilbert space (RKHS). Second, we establish for the first time the asymptotic error bound and its corresponding rate for GP-EI with noisy observations under the GP prior assumption. Third, by investigating the exploration and exploitation properties of the non-convex EI function, we establish improved error bounds of GP-EI for both the noise-free and noisy cases.