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

Yu Inatsu, Shion Takeno, Hiroyuki Hanada, Kazuki Iwata, Ichiro Takeuchi
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:4564-4572, 2024.

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 performs as well or better than existing methods not only in the setting with IU but also in the setting of ordinary MOBO through numerical experiments.

Cite this Paper


BibTeX
@InProceedings{pmlr-v238-inatsu24a, title = {Bounding Box-based Multi-objective {B}ayesian Optimization of Risk Measures under Input Uncertainty}, author = {Inatsu, Yu and Takeno, Shion and Hanada, Hiroyuki and Iwata, Kazuki and Takeuchi, Ichiro}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {4564--4572}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/inatsu24a/inatsu24a.pdf}, url = {https://proceedings.mlr.press/v238/inatsu24a.html}, 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 performs as well or better than existing methods not only in the setting with IU but also in the setting of ordinary MOBO through numerical experiments.} }
Endnote
%0 Conference Paper %T Bounding Box-based Multi-objective Bayesian Optimization of Risk Measures under Input Uncertainty %A Yu Inatsu %A Shion Takeno %A Hiroyuki Hanada %A Kazuki Iwata %A Ichiro Takeuchi %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-inatsu24a %I PMLR %P 4564--4572 %U https://proceedings.mlr.press/v238/inatsu24a.html %V 238 %X 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 performs as well or better than existing methods not only in the setting with IU but also in the setting of ordinary MOBO through numerical experiments.
APA
Inatsu, Y., Takeno, S., Hanada, H., Iwata, K. & Takeuchi, I.. (2024). Bounding Box-based Multi-objective Bayesian Optimization of Risk Measures under Input Uncertainty. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:4564-4572 Available from https://proceedings.mlr.press/v238/inatsu24a.html.

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