# Probability Distribution of Hypervolume Improvement in Bi-objective Bayesian Optimization

arXiv: https://arxiv.org/abs/2205.05505 · HTML: https://mlsanity.com/paper/2205.05505/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 48 references that are in the map.
- First posted on arXiv 11 May 2022.

## Builds on (most cited first, 7 of 7)

- [Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2006.05078/) (2020)
- [Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement](https://mlsanity.com/paper/2105.08195/) (2021)
- [Max-value Entropy Search for Multi-Objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/2009.01721/) (2020)
- [An Easy-to-use Real-world Multi-objective Optimization Problem Suite](https://mlsanity.com/paper/2009.12867/) (2020)
- [Random Hypervolume Scalarizations for Provable Multi-Objective Black Box Optimization](https://mlsanity.com/paper/2006.04655/) (2020)
- [Multi-objective Bayesian Optimization using Pareto-frontier Entropy](https://mlsanity.com/paper/1906.00127/) (2019)
- [Efficient Computation of Expected Hypervolume Improvement Using Box Decomposition Algorithms](https://mlsanity.com/paper/1904.12672/) (2019)

## Built on by (most cited first, 3 of 3)

- [MMD-Newton Method for Multi-objective Optimization](https://mlsanity.com/paper/2505.14610/) (2025)
- [Large Language Models as Amortized Pareto-Front Generators for Constrained Bi-Objective Convex Optimization](https://mlsanity.com/paper/2605.12106/) (2026)
- [Converge Then Diversify: Decoupling Convergence and Diversity in Multi-Objective Bayesian Optimisation](https://mlsanity.com/paper/2609.13396/) (2026)

## Abstract (arXiv)

Hypervolume improvement (HVI) is commonly employed in multi-objective Bayesian optimization algorithms to define acquisition functions due to its Pareto-compliant property. Rather than focusing on specific statistical moments of HVI, this work aims to provide the exact expression of HVI's probability distribution for bi-objective problems. Considering a bi-variate Gaussian random variable resulting from Gaussian process (GP) modeling, we derive the probability distribution of its hypervolume improvement via a cell partition-based method. Our exact expression is superior in numerical accuracy and computation efficiency compared to the Monte Carlo approximation of HVI's distribution. Utilizing this distribution, we propose a novel acquisition function - varepsilon-probability of hypervolume improvement (varepsilon-PoHVI). Experimentally, we show that on many widely-applied bi-objective test problems, varepsilon-PoHVI significantly outperforms other related acquisition functions, e.g., varepsilon-PoI, and expected hypervolume improvement, when the GP model exhibits a large the prediction uncertainty.