Stopping Bayesian Optimization with Probabilistic Regret Bounds
This work addresses the stopping rule problem in Bayesian optimization for black-box search, offering a more principled alternative to fixed budgets, though it is incremental as it builds on existing Bayesian optimization frameworks.
The paper tackles the problem of when to stop Bayesian optimization by introducing probabilistic regret bounds, specifically an $(ε, δ)$-criterion, and shows that Bayesian optimization with Gaussian process priors satisfies this criterion under mild assumptions, with empirical results demonstrating the approach's effectiveness.
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 $(ε, δ)$-criterion: stop when a solution has been found whose value is within $ε> 0$ of the optimum with probability at least $1 - δ$ 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.