Practical Efficient Global Optimization is No-regret

2026-03-26

AI Summary

This work addresses the widely used yet theoretically underexplored efficient global optimization (EGO) algorithm with a nugget term, establishing for the first time a sublinear cumulative regret bound. By integrating a Gaussian process surrogate model, the expected improvement acquisition function, and nugget-based regularization of the covariance matrix, we rigorously prove that this practical EGO variant is no-regret under both squared exponential and Matérn kernels (with ν > 1/2). Our analysis quantifies the influence of the nugget on the regret bound and is complemented by empirical validation, which not only confirms the sublinear regret upper bound but also offers principled guidance for selecting the nugget parameter in practice.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Online Learning & Bandits

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Efficient global optimization (EGO) is one of the most widely used noise-free Bayesian optimization algorithms.It comprises the Gaussian process (GP) surrogate model and expected improvement (EI) acquisition function. In practice, when EGO is applied, a scalar matrix of a small positive value (also called a nugget or jitter) is usually added to the covariance matrix of the deterministic GP to improve numerical stability. We refer to this EGO with a positive nugget as the practical EGO. Despite its wide adoption and empirical success, to date, cumulative regret bounds for practical EGO have yet to be established. In this paper, we present for the first time the cumulative regret upper bound of practical EGO. In particular, we show that practical EGO has sublinear cumulative regret bounds and thus is a no-regret algorithm for commonly used kernels including the squared exponential (SE) and Matérn kernels ($ν>\frac{1}{2}$). Moreover, we analyze the effect of the nugget on the regret bound and discuss the theoretical implication on its choice. Numerical experiments are conducted to support and validate our findings.

Problem

• Efficient Global Optimization
• Cumulative Regret
• No-regret Algorithm
• Gaussian Process
• Nugget

Innovation

• Bayesian optimization
• expected improvement
• Gaussian process
• cumulative regret
• nugget