Neural Evolution Strategy for Black-box Pareto Set Learning

Multi-objective optimization problems (MOPs) are prevalent in numerous realworld applications. Recently, Pareto Set Learning (PSL) has emerged as a powerful paradigm for solving MOPs. PSL can produce a neural network for modeling the set of all Pareto optimal solutions. However, applying PSL to black-box objectives, particularly those exhibiting non-separability, high dimensionality, and/or other complex properties, remains very challenging. To address this issue, we propose leveraging evolution strategies (ESs), a class of specialized blackbox optimization algorithms, within the PSL paradigm. Traditional ESs capture the complex dimensional dependencies less efficiently, which can significantly hinder their performance in PSL. To tackle this issue, we suggest encapsulating the dependencies within a neural network, which is then trained using a novel gradient estimation method. The proposed method, termed Neural-ES, is evaluated using a bespoke benchmark suite for black-box PSL. Experimental comparisons with other methods demonstrate the efficiency of Neural-ES, underscoring its ability to learn the Pareto sets of challenging black-box MOPs.

Recently, a novel paradigm called Pareto Set Learning (PSL) has attracted increasing attention [9][10][11][12][13][14][15]. PSL formulates a multi-objective optimization problem (MOP) [4] as a set learning task by decomposing the problem into an infinite number of single-objective subproblems and resolving the optimal solutions for all of them. In this process, a set model, typically a neural network, learns the mapping from an input preference to its corresponding Pareto optimal solution. Consequently, the Pareto set is recovered in both the objective and solution spaces. Unlike traditional multi-objective methods, which only accommodate a limited number of predefined user preferences [16,5], a trained PSL model can provide a tailored optimal solution for any valid decision-maker preference trade-off without requiring re-optimization from scratch.

Despite the rapid development of PSL, it remains largely unclear how to apply this paradigm to a black-box MOP where both the analytic expression and derivatives of the objectives are unknown. On