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.

翻译：优化多个相互冲突的目标是科学和工业中的常见问题。这些目标之间固有的不可调和的权衡导致需要探索其帕累托前沿。为此目的的一个有意义的量是超体积指标，它被用于贝叶斯优化和进化算法中。然而，计算超体积的计算复杂度随着目标数量和数据点的增加而显著增长，这限制了其在常见多目标优化框架中的应用。为了克服这些限制，我们提出使用深度神经网络来近似超体积函数，我们称之为DeepHV。为了提高样本效率和泛化能力，我们利用超体积在每个目标上具有尺度等变性以及关于目标和样本均具有置换不变性这一事实，采用了一个对缩放和置换的联合群具有等变性的深度神经网络。我们在准确性、计算时间和泛化能力方面，将我们的方法与精确和近似的超体积方法进行了评估。我们还将我们的方法应用于一系列合成基准测试案例，并与最先进的多目标贝叶斯优化方法和进化算法进行了比较。结果表明，我们的方法对于此类多目标优化任务具有前景。