Probability Distribution of Hypervolume Improvement in Bi-objective Bayesian Optimization

TL;DR

This paper derives the exact probability distribution of hypervolume improvement in bi-objective Bayesian optimization, enabling more accurate and efficient acquisition functions, and demonstrates its superiority over existing methods in test problems.

Contribution

It provides the first exact distribution of HVI for bi-objective problems and introduces a novel acquisition function, $ ext{ extepsilon}$-PoHVI, improving optimization performance.

Findings

$ ext{ extepsilon}$-PoHVI outperforms other acquisition functions in large uncertainty scenarios.

Exact HVI distribution improves numerical accuracy and computational efficiency.

Proposed method enhances bi-objective Bayesian optimization effectiveness.

Abstract

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.