AI summaryⓘ
The authors study two methods used in multiobjective Bayesian optimization to decide which solutions to try next: the hypervolume indicator and the R2 indicator. They explore how different mathematical transformations affect the accuracy and behavior of these methods, showing that some preserve key properties while others change the geometry they measure. For the hypervolume side, they clarify relationships between known approaches and point out cases where some properties like variance monotonicity fail. On the R2 side, they demonstrate that the expected improvement measure corresponds to volumes in a transformed scalarization space rather than simple objective space, and propose new algorithms based on this insight.
Bayesian optimizationmultiobjective optimizationhypervolume indicatorR2 indicatorexpected improvementPareto frontscalarizationTchebycheff scalarizationGaussian processintegral representations
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.