Bayesian optimization is a popular framework for efficiently finding high-quality solutions to difficult problems based on limited prior information. As a rule, these algorithms operate by iteratively choosing what to try next until some predefined budget has been exhausted. We investigate replacing this de facto stopping rule with 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. Given access to the prior distribution of problems, we show how to verify this condition in practice using a limited number of draws from the posterior. For Gaussian process priors, we prove that Bayesian optimization with the proposed criterion stops in finite time and returns a point that satisfies the $(\epsilon, \delta)$-criterion under mild assumptions. These findings are accompanied by extensive empirical results which demonstrate the strengths and weaknesses of this approach.

翻译：贝叶斯优化是一种基于有限先验信息高效求解困难问题优质解的流行框架。通常，这些算法通过迭代选择下一步探索目标直至预设预算耗尽的方式运行。我们研究用 $(\epsilon, \delta)$-准则替代这一默认停止规则：当在模型假设下找到的解以至少 $1 - \delta$ 的概率与最优值的差距在 $\epsilon > 0$ 范围内时停止。给定问题先验分布的前提下，我们展示了如何利用后验分布的有限采样在实际中验证该条件。针对高斯过程先验，我们证明采用所提准则的贝叶斯优化在温和假设下会在有限时间内停止，并返回满足 $(\epsilon, \delta)$-准则的解。这些发现辅以大量实证结果，阐明了该方法的优势与局限。