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

arXiv: https://arxiv.org/abs/2301.11588 · HTML: https://mlsanity.com/paper/2301.11588/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 27 Jan 2023.

## Builds on (most cited first, 14 of 14)

- [Max-value Entropy Search for Efficient Bayesian Optimization](https://mlsanity.com/paper/1703.01968/) (2017)
- [Information Directed Sampling and Bandits with Heteroscedastic Noise](https://mlsanity.com/paper/1801.09667/) (2018)
- [Distributionally Robust Bayesian Optimization](https://mlsanity.com/paper/2002.09038/) (2020)
- [Multi-objective Bayesian Optimization using Pareto-frontier Entropy](https://mlsanity.com/paper/1906.00127/) (2019)
- [Uncertainty-Aware Search Framework for Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2204.05944/) (2022)
- [Robust Multi-Objective Bayesian Optimization Under Input Noise](https://mlsanity.com/paper/2202.07549/) (2022)
- [Value-at-Risk Optimization with Gaussian Processes](https://mlsanity.com/paper/2105.06126/) (2021)
- [Mean-Variance Analysis in Bayesian Optimization under Uncertainty](https://mlsanity.com/paper/2009.08166/) (2020)
- [Risk-averse Heteroscedastic Bayesian Optimization](https://mlsanity.com/paper/2111.03637/) (2021)
- [Bayesian Optimization for Cascade-type Multi-stage Processes](https://mlsanity.com/paper/2111.08330/) (2021)
- [Active learning for distributionally robust level-set estimation](https://mlsanity.com/paper/2102.04000/) (2021)
- [A Robust Multi-Objective Bayesian Optimization Framework Considering Input Uncertainty](https://mlsanity.com/paper/2202.12848/) (2022)
- [Bayesian Quadrature Optimization for Probability Threshold Robustness Measure](https://mlsanity.com/paper/2006.11986/) (2020)
- [Bayesian Optimization for Distributionally Robust Chance-constrained Problem](https://mlsanity.com/paper/2201.13112/) (2022)

## Built on by (most cited first, 5 of 5)

- [Regret Analysis for Randomized Gaussian Process Upper Confidence Bound](https://mlsanity.com/paper/2409.00979/) (2024)
- [Active Learning for Level Set Estimation Using Randomized Straddle Algorithms](https://mlsanity.com/paper/2408.03144/) (2024)
- [On Regret Bounds of Thompson Sampling for Bayesian Optimization](https://mlsanity.com/paper/2603.09276/) (2026)
- [Randomized Kriging Believer for Parallel Bayesian Optimization with Regret Bounds](https://mlsanity.com/paper/2603.01470/) (2026)
- [Towards Regret Guarantees for One-Step Lookahead Bayesian Optimization](https://mlsanity.com/paper/2606.00956/) (2026)

## Abstract (arXiv)

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.