Optimal-Point Variance Reduction For Bayesian Optimization With Regret Guarantee

2026-05-30

AI Summary

Existing single-step lookahead Bayesian optimization methods rely on complex approximations and lack theoretical regret guarantees. This work proposes the Optimum Variance Reduction (OVR) approach, which requires only posterior sampling and Monte Carlo approximation, augmented with a regularization mechanism to encourage exploration. For the first time, we establish a global uniform error bound for single-step lookahead Bayesian optimization based on Monte Carlo approximation and prove that the regularized OVR algorithm achieves an asymptotically vanishing upper bound on Bayesian expected simple regret. Both theoretical analysis and numerical experiments corroborate the convergence and effectiveness of the proposed method.

Abstract

This paper studies a one-step lookahead Bayesian optimization (BO) method and its theoretical guarantee. Although the empirical effectiveness of one-step lookahead BO methods, such as entropy search, has been studied extensively, they often rely on computationally intractable approximations, and their regret guarantees remain underdeveloped. Thus, this paper proposes a one-step lookahead BO method called optimal-point variance reduction (OVR), which requires only posterior sampling and Monte Carlo approximations. We obtain a uniform error bound over an input domain for the Monte Carlo estimation in OVR. Furthermore, we show that the regularized OVR, with the slight modification to promote exploration, achieves a vanishing Bayesian expected simple regret upper bound. Finally, we demonstrate the effectiveness of OVR through numerical experiments.

Problem

• Bayesian optimization
• one-step lookahead
• regret guarantee
• variance reduction
• posterior sampling

Innovation

• Bayesian optimization
• one-step lookahead
• variance reduction
• regret guarantee
• Monte Carlo approximation