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

TL;DR

This paper introduces a new multi-objective Bayesian optimization method that efficiently finds Pareto fronts for risk measures under input uncertainty, with theoretical guarantees and improved performance over existing methods.

Contribution

The proposed method handles general risk measures with theoretical guarantees and introduces a bounding box approach for input uncertainty in multi-objective Bayesian optimization.

Findings

Outperforms existing methods in input uncertainty settings

Provides finite-iteration guarantees for accurate solutions

Effective for various risk measures like Bayes risk and value-at-risk

Abstract

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.