Approximation of Box Decomposition Algorithm for Fast Hypervolume-Based Multi-Objective Optimization

TL;DR

The paper tackles the computational bottleneck of HV-based Bayesian optimization by formalizing and analyzing a deterministic approximation to the Hypervolume Box-Decomposition Algorithm (HBDA). It proves that exact HBDA has super-polynomial memory growth and introduces a pruning-based approximation controlled by a threshold $\alpha$, guaranteeing $K \le 2/\alpha$ hyperrectangles and reducing preprocessing and HV-improvement costs. The authors provide precise time complexities, show how $\alpha=0$ recovers the exact method, and discuss practical defaults used in tools like BoTorch. They also acknowledge limitations, such as the potential for vanishing gradient in approximations, and call for further work to improve robustness for many-objective optimization scenarios. Overall, the work equips HV-based BO with a rigorous, scalable algorithmic description essential for practical deployment beyond low-dimensional objective spaces.

Abstract

Hypervolume (HV)-based Bayesian optimization (BO) is one of the standard approaches for multi-objective decision-making. However, the computational cost of optimizing the acquisition function remains a significant bottleneck, primarily due to the expense of HV improvement calculations. While HV box-decomposition offers an efficient way to cope with the frequent exact improvement calculations, it suffers from super-polynomial memory complexity $O(MN^{\lfloor \frac{M + 1}{2} \rfloor})$ in the worst case as proposed by Lacour et al. (2017). To tackle this problem, Couckuyt et al. (2012) employed an approximation algorithm. However, a rigorous algorithmic description is currently absent from the literature. This paper bridges this gap by providing comprehensive mathematical and algorithmic details of this approximation algorithm.

Figures (2)

• Figure 1: Conceptual visualizations for the notations used in this paper. Left: Visualization of box decomposition. The dominated space $\mathcal{D}$ is white, and the non-dominated space $\mathcal{N}$ is divided by hyperrectangles $B_k$. Box decomposition algorithms identify the lower and upper bounds of each hyperrectangle. Right: Visualization of HV improvement calculation. The HV improvement calculation is the summation of the HV for each intersection of hyperrectangle $B_k$ and the new entry $\boldsymbol{y}^{\mathrm{new}}$ once HBDA is performed. The intersections with $B_1,B_3,B_5,B_7$ exhibit non-zero HVs in this example.
• Figure 2: Step-by-step visualization of Algorithm \ref{['alg:approx-hbda']}. $\mathcal{P} = \{[2, 8], [6, 4], [8, 2]\}$, $\boldsymbol{r} = [10, 10]$, and $\alpha = 0.1$ are used for this example. Red represents the rectangle to split in the current iteration, while blue means these rectangles are in the stack. Green is accepted in Line \ref{['line:acceptance']} owing to the dominance $\boldsymbol{u} \nprec \mathcal{P}$, and gray is discarded. The black rectangles in the final result are the missed rectangles compared to the exact HBDAs. The summation of these rectangles' HVs is the worst-case error in the HV improvement approximation.

Theorems & Definitions (6)

• Proposition 1
• Proof 1
• Proposition 2
• Proof 2
• Proposition 3
• Proof 3