In this paper, we make the key delineation on the roles of resolution and statistical uncertainty in hierarchical bandits-based black-box optimization algorithms, guiding a more general analysis and a more efficient algorithm design. We introduce the \textit{optimum-statistical collaboration}, an algorithm framework of managing the interaction between optimization error flux and statistical error flux evolving in the optimization process. We provide a general analysis of this framework without specifying the forms of statistical error and uncertainty quantifier. Our framework and its analysis, due to their generality, can be applied to a large family of functions and partitions that satisfy different local smoothness assumptions and have different numbers of local optimums, which is much richer than the class of functions studied in prior works. Our framework also inspires us to propose a better measure of the statistical uncertainty and consequently a variance-adaptive algorithm \texttt{VHCT}. In theory, we prove the algorithm enjoys rate-optimal regret bounds under different local smoothness assumptions; in experiments, we show the algorithm outperforms prior efforts in different settings.

翻译：本文清晰界定了分层赌博机类黑箱优化算法中分辨率与统计不确定性的角色，从而引导出更通用的分析和更高效的算法设计。我们提出了最优统计协作这一算法框架，用于管理优化过程中优化误差通量与统计误差通量之间的交互作用。我们对该框架进行了通用分析，无需指定统计误差及不确定性量化器的具体形式。由于该框架及其分析具有通用性，因此可应用于满足不同局部光滑性假设且具有不同局部最优值数量的大类函数与划分，其涵盖范围远超先前工作中研究的函数类。该框架还启发我们提出一种更优的统计不确定性度量方法，进而设计出方差自适应算法VHCT。理论上，我们证明了该算法在不同局部光滑性假设下均具有最优的遗憾界；实验表明，该算法在多种设置下均优于先前工作。