🤖 AI Summary
This work addresses a critical gap in preference-guided expected improvement criteria for multi-objective Bayesian optimization: the lack of a systematic understanding of the interplay among geometric structure, monotonicity, and exact computation. By analytically examining the geometric nature of hypervolume and the R2 indicator in both objective and scalarized spaces, the study reveals that the R2-based expected improvement corresponds to a volume in scalarized space rather than a weighted hypervolume in objective space. Building on this insight, the authors unify various hypervolume variants under a common computational perspective and introduce two exact calculation approaches—an ER2I algorithm based on finite summation for discrete settings and an integral-based method leveraging Gaussian surrogate models. Furthermore, they establish an achievement scalarizing optimization framework grounded in scalar Gaussian expected improvement and rigorously analyze its Pareto compliance and monotonicity properties.
📝 Abstract
This paper studies preference-shaped expected improvement criteria for Bayesian multiobjective optimization. We consider two indicator families which are often used for similar algorithmic purposes, but which are geometrically different. The hypervolume indicator is based on a dystopian reference point and measures dominated volume in objective space. The R2 indicator is based on a utopian point and evaluates approximation sets through weighted Tchebycheff scalarization envelopes. The purpose of the paper is to make precise which preference transformations preserve exact computation, Pareto compatibility, and monotonicity properties, and which transformations change the underlying geometry. On the hypervolume side, we revisit canonical EHVI through the Deng representation, formulate product-density weighted EHVI in desirability coordinates, discuss cone-based EHVI as ordinary EHVI after a linear cone transformation, and separate these cases from truncated EHVI, where variance monotonicity may fail. On the R2 side, we prove that exact integral R2 improvement is not, in general, an ordinary objective-space weighted hypervolume. The obstruction is lower-dimensional: Lebesgue-density hypervolume cannot see certain boundary contributions that Tchebycheff scalarizations still detect. We then show that exact integral R2 improvement is exactly a scalarization-space volume, namely the measure of the Tchebycheff shadow between the incumbent scalarization envelope and the reference envelope. This representation yields finite-sum ER2I algorithms for discrete R2, quadrature methods for exact integral R2, and an achievement-space Gaussian surrogate formulation in which ER2I is an integral of scalar Gaussian expected improvements.