In this study, we propose a novel multi-objective Bayesian optimization (MOBO) method to efficiently identify the Pareto front (PF) defined by risk measures for black-box functions under the presence of input uncertainty (IU). Existing BO methods for Pareto optimization in the presence of IU are risk-specific or without theoretical guarantees, whereas our proposed method addresses general risk measures and has theoretical guarantees. The basic idea of the proposed method is to assume a Gaussian process (GP) model for the black-box function and to construct high-probability bounding boxes for the risk measures using the GP model. Furthermore, in order to reduce the uncertainty of non-dominated bounding boxes, we propose a method of selecting the next evaluation point using a maximin distance defined by the maximum value of a quasi distance based on bounding boxes. As theoretical analysis, we prove that the algorithm can return an arbitrary-accurate solution in a finite number of iterations with high probability, for various risk measures such as Bayes risk, worst-case risk, and value-at-risk. We also give a theoretical analysis that takes into account approximation errors because there exist non-negligible approximation errors (e.g., finite approximation of PFs and sampling-based approximation of bounding boxes) in practice. We confirm that the proposed method outperforms compared with existing methods not only in the setting with IU but also in the setting of ordinary MOBO through numerical experiments.

翻译：本文提出一种新颖的多目标贝叶斯优化（MOBO）方法，用于在输入不确定性（IU）条件下高效识别黑箱函数风险度量定义的帕累托前沿（PF）。现有面向IU下帕累托优化的贝叶斯优化方法要么针对特定风险度量设计，要么缺乏理论保证，而本文方法适用于一般风险度量并具有理论保障。该方法的核心思想是：对黑箱函数建立高斯过程（GP）模型，并基于该模型构建风险度量的高概率边界框。为进一步降低非支配边界框的不确定性，我们提出通过最大化基于边界框的拟距离极大极小值来选择下一个评估点。理论分析表明，对于贝叶斯风险、最坏情形风险、风险价值等多种风险度量，该算法可在有限迭代次数内以高概率返回任意精度的解。考虑到实际应用中存在不可忽略的近似误差（如帕累托前沿的有限近似及边界框的采样近似），我们还给出了包含近似误差的理论分析。数值实验证实，不仅在IU场景下，而且在普通MOBO场景中，本文方法均优于现有方法。