Multi-objective optimization via equivariant deep hypervolume approximation

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 an increasing number of objectives and data points, which restricts its use in those common multiobjective optimization frameworks. To overcome these restrictions, previous work has focused on approximating the hypervolume using deep learning. In this work, we propose a novel deep learning architecture to approximate the hypervolume function, 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. We show through an ablation study that including these symmetries leads to significantly improved model accuracy. 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-theart multi-objective BO methods and EAs on a range of synthetic and real-world benchmark test cases. The results show that our methods are promising for such multi-objective optimization tasks. Imagine, while listening to a lecture you also quickly want to check out the latest news on your phone, so you can appear informed during lunch. As an experienced listener, who knows what lecture material is important, and an excellent reader, who knows how to scan over the headlines, you are confident in your abilities in each of those tasks.