The Kalai-Smorodinski solution for many-objective Bayesian optimization

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 R package GPGame available on CRAN at https://cran.r-project.org/ package=GPGame.

Budgeted Multi-Objective Optimization with a Focus on the Central Part of the Pareto Front -- Extended Version

“…In the field of Game Theory, our definition of the center of a Pareto front corresponds to a particular case of the Kalai-Smorodinsky equilibrium [45,11], taking the Nadir as disagreement point d ≡ N. This equilibrium aims at equalizing the ratios of maximal gains of the players, which is the appealing property for the center of a Pareto front as an implicitly preferred point. Recently, it has been used for solving many-objective problems in a Bayesian setting [11]. In general, C is different from the neutral solution [78] and from knee points [14].…”