Improving Pareto Set Learning for Expensive Multi-objective Optimization via Stein Variational Hypernetworks

TL;DR

This work targets expensive multi-objective optimization by addressing PSL instability due to surrogate fragmentation and pseudo-local optima. It introduces SVH-PSL, which couples Stein Variational Gradient Descent with Hypernetworks and a per-dimension local kernel to jointly move a diverse set of particles toward the true Pareto front. The approach formulates a controllable PSL with a Chebyshev scalarization, leverages SVGD gradients to update a parametric Pareto-set model, and employs a local kernel to better capture objective-specific geometry. Extensive experiments on synthetic and real-world MOBO benchmarks demonstrate faster convergence and improved Pareto front quality and diversity compared to state-of-the-art baselines. The method offers a robust, scalable path for solving EMOPs where expensive evaluations hinder traditional surrogate-based optimization.

Abstract

Expensive multi-objective optimization problems (EMOPs) are common in real-world scenarios where evaluating objective functions is costly and involves extensive computations or physical experiments. Current Pareto set learning methods for such problems often rely on surrogate models like Gaussian processes to approximate the objective functions. These surrogate models can become fragmented, resulting in numerous small uncertain regions between explored solutions. When using acquisition functions such as the Lower Confidence Bound (LCB), these uncertain regions can turn into pseudo-local optima, complicating the search for globally optimal solutions. To address these challenges, we propose a novel approach called SVH-PSL, which integrates Stein Variational Gradient Descent (SVGD) with Hypernetworks for efficient Pareto set learning. Our method addresses the issues of fragmented surrogate models and pseudo-local optima by collectively moving particles in a manner that smooths out the solution space. The particles interact with each other through a kernel function, which helps maintain diversity and encourages the exploration of underexplored regions. This kernel-based interaction prevents particles from clustering around pseudo-local optima and promotes convergence towards globally optimal solutions. Our approach aims to establish robust relationships between trade-off reference vectors and their corresponding true Pareto solutions, overcoming the limitations of existing methods. Through extensive experiments across both synthetic and real-world MOO benchmarks, we demonstrate that SVH-PSL significantly improves the quality of the learned Pareto set, offering a promising solution for expensive multi-objective optimization problems.

Figures (6)

• Figure 1: Approximate Pareto front comparison after the first 6 function evaluations using PSL-MOBO and our method, SVH-PSL, on the ZDT1 problem (2 objectives, 20 dimensions).
• Figure 2: Comparison of the impact of neighboring particles on a particle with and without the local kernel. Here we are considering $K=10$
• Figure 3: Mean Log Hypervolume Differences between the truth Pareto Front and the learned Pareto Front with respect to the number of expensive evaluations on all MOBO algorithms.
• Figure 4: Illustration of SVH-PSL performance with and without local kernel integration in real-world problems with a complex PF.
• Figure 5: Impact of $\alpha$ on the trade-off between exploration and exploitation in ZDT1 (LHD & Pareto front).

Theorems & Definitions (4)

• Definition 1: Dominance
• Definition 2: Pareto Optimality
• Definition 3: Pareto Set/Front
• Definition 4: Hypervolume