Probability Distribution of Hypervolume Improvement in Bi-objective Bayesian Optimization

TL;DR

This paper addresses the challenge of uncertainty in hypervolume improvement (HVI) within bi-objective Bayesian optimization by deriving the exact distribution of HVI under Gaussian process posteriors. It introduces a generalized HVI and a cell-partitioning approach to obtain conditional PDFs/CDFs, enabling exact computation and improved numerical efficiency over Monte Carlo methods. Building on this, the authors propose the $\varepsilon$-Probability of Hypervolume Improvement ($\varepsilon$-PoHVI), a quantile-based acquisition function that remains robust when HVI exhibits high dispersion, and two practical schemes to adapt $\varepsilon$ over iterations. Empirical results across classical (ZDT), synthetic (WOSGZ), and real-world (RE) bi-objective problems show that $\varepsilon$-PoHVI substantially accelerates convergence compared to $\varepsilon$-PoI and EHVI, particularly when predictive uncertainty is large, highlighting its practical impact for faster, more reliable Pareto-front discovery.

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.

Figures (9)

• Figure 1: For a two-dimensional objective space, we picture the augmented Pareto approximation set $\widetilde{\mathcal{P}}$ by the black dots $\mathbf{y} \textbf{y} ^{(0)},\ldots, \mathbf{y} \textbf{y} ^{(4)}$ and the attainment boundary by the red curve. The posterior distribution of $\mathbf{y} \textbf{y}$ at a point $\mathbf{x}\in\mathcal{X}$ is illustrated by the light gray ellipsoids. The generalized hypervolume improvement of two realizations $\mathbf{a} \textbf{a}$ and $\mathbf{b} \textbf{b}$ are depicted in the shaded area. The objective space $[- \boldsymbol{\infty} \textbf{\infty} , \mathbf{r} \textbf{r} ]$ is partitioned into cells (e.g., $C(1,0)$). When restricting the random point $\mathbf{y} \textbf{y}$ to a cell, its hypervolume can always be expressed in four terms: $\Delta\xspace^+( \mathbf{a} \textbf{a} )|_{C(0,1)}=\lambda($$A_1$$)+\lambda($$A_2$$)+\lambda($$A_3$$)+ \lambda($$A_4$) ($\lambda$ is the Lebesgue measure on $\mathbb{R}\xspace^2$). When a point is dominated by $\widetilde{\mathcal{P}}$, its negative hypervolume improvement can be written similarly: $\Delta\xspace^-( \mathbf{b} \textbf{b} )|_{C(3,3)}=-\lambda($$A_1'$$)-\lambda($$A_2'$$)-\lambda($$A_3'$$)-\lambda($$A_4'$).
• Figure 2: Left: For the Pareto front in Fig. \ref{['fig:2dhv']}, we show the CDF of $\mathbf{y} \textbf{y}$ computed from the exact and MC methods (using 100, 500, and 2 500 samples). Right: The CPU time for the exact and the MC method (with $10^4$ sample points) w.r.t. an increasing number of points of $\mathcal{P}$.
• Figure 3: The log difference between the hypervolume of the best-so-far approximation set $\mathcal{P}$ and the target hypervolume over function evaluations. The target hypervolume is obtained with 1 000 points evenly sampled on the Pareto front of each problem. We show the mean and standard error of the log differences measured from $15$ independent runs of each acquisition function on each problem.
• Figure 4: On each test problem, we show some descriptive statistics: min, max, mean, median, standard deviation (std), and 25%- and 75%-quantiles of the hypervolume (HV) value observed at the last iteration of the BO algorithm. The entries are color-coded relative to the corresponding ones (e.g., we take all the mean values in a column) in the same column/problem, where a more greenish color indicates better performance and vice versa.
• Figure 5: The critical difference (CD) chart obtained with the Nemenyi posthoc testing procedure to a Friedman test. The performance of two acquisition functions significantly differs on a problem set if their average ranks of HV values differ by at least the critical difference shown as the interval atop each chart. The thick horizontal line indicates a clique of acquisition functions with no significant difference.

Theorems & Definitions (5)

• Lemma 1
• proof
• Remark
• Theorem 1
• proof