Probability Distribution of Hypervolume Improvement in Bi-objective Bayesian Optimization

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 - ε-probability of hypervolume improvement (ε-PoHVI). Experimentally, we show that on many widely-applied bi-objective test problems, ε-PoHVI significantly outperforms other related acquisition functions, e.g., ε-PoI, and expected hypervolume improvement, when the GP model exhibits a large the prediction uncertainty.