# Preference-Shaped Expected Hypervolume and R2 Improvement: Exact Computation and Monotonicity

arXiv: https://arxiv.org/abs/2605.28746 · HTML: https://mlsanity.com/paper/2605.28746/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 20 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 2 of 2)

- [R2-based Hypervolume Contribution Approximation](https://mlsanity.com/paper/1805.06773/) (2018)
- [The Magnitude of Dominated Sets: A Pareto Compliant Indicator Grounded in Metric Geometry](https://mlsanity.com/paper/2604.18147/) (2026)

## Abstract (arXiv)

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.