On Improved Regret Bounds In Bayesian Optimization with Gaussian Noise

TL;DR

This work addresses regret analysis in Bayesian optimization with Gaussian-process surrogates under Gaussian noise in the frequentist setting. It introduces a pointwise prediction error bound for GP posteriors and extends it to general discrete sets, enabling refined cumulative regret bounds for GP-UCB and GP-TS. For squared exponential and Matérn kernels, the authors prove regret rates that match Bayesian benchmarks up to logarithmic factors, improving prior frequentist results and narrowing gaps to lower bounds. The results have practical impact for black-box optimization where noise is Gaussian, offering tighter guarantees for common BO algorithms.

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.

Theorems & Definitions (43)

• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Proof 1
• Remark 3.4
• Lemma 3.6
• Proof 2
• Lemma 3.7
• Lemma 3.8