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：在本研究中，我们提出了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）存在的情况下，高效识别黑箱函数风险度量所定义的帕累托前沿（PF）。现有的针对输入不确定性下帕累托优化的贝叶斯优化方法或局限于特定风险类型，或缺乏理论保证，而本文提出的方法适用于一般风险度量并具有理论保证。该方法的核心思想是：为黑箱函数建立高斯过程（GP）模型，并利用该模型构建风险度量的高概率边界框。此外，为降低非支配边界框的不确定性，我们提出了一种基于边界框最大准距离的最大最小距离选择下一评估点的方法。理论分析证明：对于贝叶斯风险、最坏情况风险、风险价值等多种风险度量，该算法能以高概率在有限迭代次数内返回任意精度的解。我们还给出了考虑近似误差（如帕累托前沿的有限近似和边界框的采样近似）的理论分析，因实际应用中存在不可忽略的近似误差。通过数值实验，我们验证了本文方法不仅在输入不确定性场景下优于现有方法，在普通多目标贝叶斯优化场景中也表现出色。