Efficient Approximation of Expected Hypervolume Improvement using Gauss-Hermite Quadrature

TL;DR

This paper proposes a new, efficient method using Gauss-Hermite quadrature to approximate the expected hypervolume improvement in multi-objective optimization, offering a cheaper alternative to Monte Carlo methods especially for correlated objectives.

Contribution

It introduces a novel application of Gauss-Hermite quadrature for EHVI approximation, effective for both independent and correlated predictive densities, reducing computational cost.

Findings

Gauss-Hermite quadrature provides accurate EHVI approximations.

Method outperforms Monte Carlo in computational efficiency.

Statistically significant correlation with Monte Carlo results across test problems.

Abstract

Many methods for performing multi-objective optimisation of computationally expensive problems have been proposed recently. Typically, a probabilistic surrogate for each objective is constructed from an initial dataset. The surrogates can then be used to produce predictive densities in the objective space for any solution. Using the predictive densities, we can compute the expected hypervolume improvement (EHVI) due to a solution. Maximising the EHVI, we can locate the most promising solution that may be expensively evaluated next. There are closed-form expressions for computing the EHVI, integrating over the multivariate predictive densities. However, they require partitioning the objective space, which can be prohibitively expensive for more than three objectives. Furthermore, there are no closed-form expressions for a problem where the predictive densities are dependent, capturing the correlations between objectives. Monte Carlo approximation is used instead in such cases, which is not cheap. Hence, the need to develop new accurate but cheaper approximation methods remains. Here we investigate an alternative approach toward approximating the EHVI using Gauss-Hermite quadrature. We show that it can be an accurate alternative to Monte Carlo for both independent and correlated predictive densities with statistically significant rank correlations for a range of popular test problems.