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

Bounding Box-based Multi-objective Bayesian Optimization of Risk Measures under Input Uncertainty