🤖 AI Summary
This work addresses the weak theoretical convergence guarantees for cumulative regret in Bayesian optimization (BO) under Gaussian noise. First, it establishes the first tight, pointwise prediction error bound for Gaussian process (GP) surrogate models in noisy settings. Leveraging this bound, the paper rigorously analyzes cumulative regret—within a frequentist framework—for acquisition functions including Upper Confidence Bound (UCB) and Thompson Sampling (TS), significantly tightening existing upper bounds for GP-UCB and GP-TS and yielding improved convergence rates. The analysis is further extended to Expected Improvement (EI) and related acquisition strategies, yielding a unified asymptotic convergence theory. The new error bound is broadly applicable, enabling principled theoretical performance evaluation and design of diverse noisy BO algorithms. Overall, this work provides a foundational theoretical tool for robust BO under observation 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.