A recent development in Bayesian optimization is the use of local optimization strategies, which can deliver strong empirical performance on high-dimensional problems compared to traditional global strategies. The "folk wisdom" in the literature is that the focus on local optimization sidesteps the curse of dimensionality; however, little is known concretely about the expected behavior or convergence of Bayesian local optimization routines. We first study the behavior of the local approach, and find that the statistics of individual local solutions of Gaussian process sample paths are surprisingly good compared to what we would expect to recover from global methods. We then present the first rigorous analysis of such a Bayesian local optimization algorithm recently proposed by M\"uller et al. (2021), and derive convergence rates in both the noisy and noiseless settings.

翻译：贝叶斯优化领域近期的一个发展方向是采用局部优化策略，相较于传统全局策略，该方法在高维问题上展现出强大的实证性能。文献中普遍认为，聚焦局部优化能够规避维度灾难；然而，关于贝叶斯局部优化流程的预期行为或收敛性，目前尚缺乏具体认知。我们首先研究局部方法的行为特征，发现高斯过程样本路径的局部解统计特性出人意料地优于全局方法的预期恢复效果。随后，我们针对Müller等人(2021)近期提出的此类贝叶斯局部优化算法，首次开展严谨分析，并推导出含噪与无噪场景下的收敛速率。