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.

翻译：超体积改进（HVI）因其Pareto兼容性，常被用于多目标贝叶斯优化算法中定义采集函数。本研究不同于聚焦HVI的特定统计矩，旨在给出双目标问题中HVI概率分布的精确表达式。针对高斯过程（GP）建模产生的二元高斯随机变量，我们通过基于单元划分的方法推导了其超体积改进的概率分布。与HVI分布的蒙特卡洛近似相比，我们的精确表达式在数值精度和计算效率上更具优势。基于该分布，我们提出了一种新型采集函数——超体积改进的$\varepsilon$概率（$\varepsilon$-PoHVI）。实验表明，在众多广泛应用的二元测试问题上，当GP模型预测不确定性较大时，$\varepsilon$-PoHVI显著优于其他相关采集函数（如$\varepsilon$-PoI和期望超体积改进）。