Bayesian optimization is a class of global optimization techniques. In Bayesian optimization, the underlying objective function is modeled as a realization of a Gaussian process. Although the Gaussian process assumption implies a random distribution of the Bayesian optimization outputs, quantification of this uncertainty is rarely studied in the literature. In this work, we propose a novel approach to assess the output uncertainty of Bayesian optimization algorithms, which proceeds by constructing confidence regions of the maximum point (or value) of the objective function. These regions can be computed efficiently, and their confidence levels are guaranteed by the uniform error bounds for sequential Gaussian process regression newly developed in the present work. Our theory provides a unified uncertainty quantification framework for all existing sequential sampling policies and stopping criteria.

翻译：贝叶斯优化是一类全局优化技术。在贝叶斯优化中，目标函数被建模为高斯过程的一个实现。尽管高斯过程假设意味着贝叶斯优化输出具有随机分布，但文献中鲜少研究这种不确定性的量化。本文提出一种评估贝叶斯优化算法输出不确定性的新方法，该方法通过构建目标函数最大值点（或最大值）的置信区域来实施。这些区域可高效计算，其置信水平由本文新开发的序列高斯过程回归一致误差界保证。我们的理论为所有现有序列采样策略与停止准则提供了统一的不确定性量化框架。