On Improved Regret Bounds In Bayesian Optimization with Gaussian Noise

TL;DR

This paper introduces new pointwise bounds on Gaussian process prediction errors under Gaussian noise, leading to improved convergence rates for Bayesian optimization algorithms like GP-UCB and GP-TS.

Contribution

It establishes novel bounds on GP prediction errors in the frequentist setting, enhancing regret bounds for common Bayesian optimization methods.

Findings

Improved convergence rates for GP-UCB and GP-TS.

New bounds applicable to general BO algorithms with noise.

Enhanced understanding of prediction errors under Gaussian noise.

Abstract

Bayesian optimization (BO) with Gaussian process (GP) surrogate models is a powerful black-box optimization method. Acquisition functions are a critical part of a BO algorithm as they determine how the new samples are selected. Some of the most widely used acquisition functions include upper confidence bound (UCB) and Thompson sampling (TS). The convergence analysis of BO algorithms has focused on the cumulative regret under both the Bayesian and frequentist settings for the objective. In this paper, we establish new pointwise bounds on the prediction error of GP under the frequentist setting with Gaussian noise. Consequently, we prove improved convergence rates of cumulative regret bound for both GP-UCB and GP-TS. Of note, the new prediction error bound under Gaussian noise can be applied to general BO algorithms and convergence analysis, e.g., the asymptotic convergence of expected improvement (EI) with noise.