Bayesian optimization devolves the global optimization of a costly objective function to the global optimization of a sequence of acquisition functions. This inner-loop optimization can be catastrophically difficult if it involves posterior sample paths, especially in higher dimensions. We introduce an efficient global optimization strategy for posterior samples based on global rootfinding. It provides gradient-based optimizers with two sets of judiciously selected starting points, designed to combine exploration and exploitation. The number of starting points can be kept small without sacrificing optimization quality. Remarkably, even with just one point from each set, the global optimum is discovered most of the time. The algorithm scales practically linearly to high dimensions, breaking the curse of dimensionality. For Gaussian process Thompson sampling (GP-TS), we demonstrate remarkable improvement in both inner- and outer-loop optimization, surprisingly outperforming alternatives like EI and GP-UCB in most cases. Our approach also improves the performance of other posterior sample-based acquisition functions, such as variants of entropy search. Furthermore, we propose a sample-average formulation of GP-TS, which has a parameter to explicitly control exploitation and can be computed at the cost of one posterior sample. Our implementation is available at https://github.com/UQUH/TSRoots .

翻译：贝叶斯优化将代价高昂目标函数的全局优化问题转化为一系列采集函数的全局优化问题。若该内层优化涉及后验样本路径，则可能面临灾难性困难，在高维情形下尤为显著。本文提出一种基于全局根寻找的后验样本高效全局优化策略。该策略为基于梯度的优化器提供两组经审慎选择的起始点，旨在平衡探索与利用。即使保持较少的起始点数量，亦不会牺牲优化质量。值得注意的是，仅使用每组中的单个起始点时，大多数情况下仍能发现全局最优解。该算法在实际应用中可实现近似线性的高维扩展，从而突破维度灾难。针对高斯过程汤普森采样（GP-TS），我们在内外层优化中均展示了显著改进，在多数情况下意外超越了如EI和GP-UCB等替代方法。本方法同样提升了其他基于后验样本的采集函数（如各类熵搜索变体）的性能。此外，我们提出了GP-TS的样本平均形式，该形式通过显式参数控制利用程度，且计算成本仅相当于单次后验采样。算法实现已发布于https://github.com/UQUH/TSRoots。