Stopping Bayesian Optimization with Probabilistic Regret Bounds

TL;DR

This paper proposes probabilistic stopping criteria for Bayesian optimization, providing theoretical guarantees and practical algorithms, with empirical validation demonstrating improved stopping decisions based on confidence in solutions.

Contribution

It introduces a probabilistic stopping rule based on the $(oldsymbol{ ext{ε, δ}})$-criterion for Bayesian optimization, with theoretical analysis and a practical Monte Carlo evaluation method.

Findings

Bayesian optimization satisfies the $( ext{ε, δ})$-criterion under mild conditions.

The proposed Monte Carlo method for stopping rules is sample-efficient and robust.

Empirical results highlight the advantages and limitations of probabilistic stopping criteria.

Abstract

Bayesian optimization is a popular framework for efficiently tackling black-box search problems. As a rule, these algorithms operate by iteratively choosing what to evaluate next until some predefined budget has been exhausted. We investigate replacing this de facto stopping rule with criteria based on the probability that a point satisfies a given set of conditions. We focus on the prototypical example of an $(\epsilon, \delta)$-criterion: stop when a solution has been found whose value is within $\epsilon > 0$ of the optimum with probability at least $1 - \delta$ under the model. For Gaussian process priors, we show that Bayesian optimization satisfies this criterion under mild technical assumptions. Further, we give a practical algorithm for evaluating Monte Carlo stopping rules in a manner that is both sample efficient and robust to estimation error. These findings are accompanied by empirical results which demonstrate the strengths and weaknesses of the proposed approach.