The Kalai-Smorodinski solution for many-objective Bayesian optimization

TL;DR

This paper introduces a Bayesian optimization method focused on the Kalai-Smorodinsky solution from game theory, tailored for many-objective problems, ensuring fair marginal gains and invariance to objective transformations.

Contribution

It proposes a novel Bayesian optimization algorithm that efficiently finds the Kalai-Smorodinsky solution in high-dimensional objective spaces using copula transformations.

Findings

Effective on problems with up to nine objectives

Ensures equal marginal gains across objectives

Invariance to monotonic transformations of objectives

Abstract

An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends to grow larger with the number of objectives. We thus propose to focus on a specific solution originating from game theory, the Kalai-Smorodinsky solution, which possesses attractive properties. In particular, it ensures equal marginal gains over all objectives. We further make it insensitive to a monotonic transformation of the objectives by considering the objectives in the copula space. A novel tailored algorithm is proposed to search for the solution, in the form of a Bayesian optimization algorithm: sequential sampling decisions are made based on acquisition functions that derive from an instrumental Gaussian process prior. Our approach is tested on four problems with respectively four, six, eight, and nine objectives. The method is available in the Rpackage GPGame available on CRAN at https://cran.r-project.org/package=GPGame.