Learning to Approximate: Auto Direction Vector Set Generation for Hypervolume Contribution Approximation

TL;DR

This paper introduces LtA, a machine learning-based method to automatically generate direction vector sets that enhance the approximation accuracy of hypervolume contribution in multi-objective optimization.

Contribution

It proposes a novel learning-based approach to generate direction vector sets for the R2 HVC indicator, improving hypervolume contribution approximation quality.

Findings

LtA outperforms existing vector set generation methods.

The learned vectors lead to more accurate hypervolume contribution estimates.

Experimental results validate the effectiveness of the proposed method.

Abstract

Hypervolume contribution is an important concept in evolutionary multi-objective optimization (EMO). It involves in hypervolume-based EMO algorithms and hypervolume subset selection algorithms. Its main drawback is that it is computationally expensive in high-dimensional spaces, which limits its applicability to many-objective optimization. Recently, an R2 indicator variant (i.e., $R_2^{\text{HVC}}$ indicator) is proposed to approximate the hypervolume contribution. The $R_2^{\text{HVC}}$ indicator uses line segments along a number of direction vectors for hypervolume contribution approximation. It has been shown that different direction vector sets lead to different approximation quality. In this paper, we propose \textit{Learning to Approximate (LtA)}, a direction vector set generation method for the $R_2^{\text{HVC}}$ indicator. The direction vector set is automatically learned from training data. The learned direction vector set can then be used in the $R_2^{\text{HVC}}$ indicator to improve its approximation quality. The usefulness of the proposed LtA method is examined by comparing it with other commonly-used direction vector set generation methods for the $R_2^{\text{HVC}}$ indicator. Experimental results suggest the superiority of LtA over the other methods for generating high quality direction vector sets.