Practical Efficient Global Optimization is No-regret

TL;DR

This paper establishes the first cumulative regret bounds for practical Gaussian process-based Bayesian optimization with a nugget, showing it is a no-regret algorithm for common kernels, supported by theoretical analysis and experiments.

Contribution

It provides the first theoretical regret bounds for practical EGO with a nugget, demonstrating its no-regret property for standard kernels.

Findings

Practical EGO has sublinear cumulative regret bounds.

Practical EGO is a no-regret algorithm for common kernels.

Numerical experiments validate the theoretical results.

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 Mat\'{e}rn kernels ($\nu>\frac{1}{2}$). 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.