Multi-objective optimization via equivariant deep hypervolume approximation

TL;DR

This paper introduces DeepHV, a neural network-based approach to efficiently approximate the hypervolume indicator in multi-objective optimization, leveraging equivariance properties for improved accuracy and generalization.

Contribution

The paper proposes DeepHV, a deep neural network that approximates hypervolume using equivariance to scale and permutation, enhancing efficiency and scalability in multi-objective optimization.

Findings

DeepHV achieves comparable accuracy to exact methods.

DeepHV significantly reduces computation time.

DeepHV generalizes well across different objectives and data sets.

Abstract

Optimizing multiple competing objectives is a common problem across science and industry. The inherent inextricable trade-off between those objectives leads one to the task of exploring their Pareto front. A meaningful quantity for the purpose of the latter is the hypervolume indicator, which is used in Bayesian Optimization (BO) and Evolutionary Algorithms (EAs). However, the computational complexity for the calculation of the hypervolume scales unfavorably with increasing number of objectives and data points, which restricts its use in those common multi-objective optimization frameworks. To overcome these restrictions we propose to approximate the hypervolume function with a deep neural network, which we call DeepHV. For better sample efficiency and generalization, we exploit the fact that the hypervolume is scale-equivariant in each of the objectives as well as permutation invariant w.r.t. both the objectives and the samples, by using a deep neural network that is equivariant w.r.t. the combined group of scalings and permutations. We evaluate our method against exact, and approximate hypervolume methods in terms of accuracy, computation time, and generalization. We also apply and compare our methods to state-of-the-art multi-objective BO methods and EAs on a range of synthetic benchmark test cases. The results show that our methods are promising for such multi-objective optimization tasks.