Regret-Based $(\epsilon,\delta)$-optimal Stopping Criteria for Bayesian Optimization

TL;DR

This paper introduces a theoretically grounded stopping criterion for Bayesian optimization using Gaussian processes, ensuring near-optimal solutions efficiently with high probability.

Contribution

It provides tighter regret bounds for GP-UCB and develops a new stopping rule that guarantees $ ext{ extepsilon}$-optimality with high confidence.

Findings

The proposed stopping criterion achieves high probability $ ext{ extepsilon}$-optimal solutions.

Numerical experiments validate the effectiveness and efficiency of the new stopping rule.

Abstract

Bayesian optimization (BO) is a widely used iterative black-box optimization method that utilizes Gaussian process (GP) surrogate models. In practice, BO is typically terminated after a fixed evaluation budget is exhausted, which can incur unnecessary cost and provides no optimality guarantee on solution quality. Recent research in developing a practical stopping criterion has made empirical progress, yet a theoretically sound stopping criterion remains a work in progress. In this work, we present provably tighter instantaneous regret bounds for GP upper confidence bound (GP-UCB) at any given iteration. Then, we propose stopping criteria for GP-UCB based on this tighter bound that ensures an $\epsilon$-optimal solution with high probability $1-\delta$ upon termination. Numerical experiments are performed to validate and demonstrate the effectiveness and efficiency of our stopping criteria.