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出现的一个进展是局部优化策略的应用，该策略在高维问题中展现出比传统全局策略更优的实证表现。文献中的"经验共识"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程序的预期行为或收敛性，目前尚缺乏具体认知。我们首先研究了局部优化方法的行为特性，发现高斯过程样本路径的单个局部解在统计特性上出人意料地优于我们预期通过全局方法所能获取的结果。继而，我们针对Müller等人(2021)近期提出的贝叶斯局部优化算法进行了首次严谨分析，分别在含噪声和无噪声场景下推导出其收敛速率。