Controllable Expensive Multi-objective Learning with Warm-starting Bayesian Optimization

TL;DR

This work tackles instability and inefficiency in derivative-free Pareto Set Learning for expensive multi-objective optimization. It introduces Co-PSL, a two-stage framework that first warm-starts Bayesian optimization to produce quality GP priors, and then trains a controllable PSL mapping via a hypernetwork to map preference vectors to Pareto solutions, enabling real-time trade-off control. Empirical results across six synthesis and real-world problems show that Co-PSL delivers more stable and accurate Pareto fronts (lower mean Euclidean distance to truth and improved hypervolume attainment) than baselines in most cases, while reducing costly evaluations. The approach meaningfully enhances robustness and practicality of MOBO in high-cost settings, with avenues for extending Pareto-set evaluation and high-dimensional scalability.

Abstract

Pareto Set Learning (PSL) is a promising approach for approximating the entire Pareto front in multi-objective optimization (MOO) problems. However, existing derivative-free PSL methods are often unstable and inefficient, especially for expensive black-box MOO problems where objective function evaluations are costly. In this work, we propose to address the instability and inefficiency of existing PSL methods with a novel controllable PSL method, called Co-PSL. Particularly, Co-PSL consists of two stages: (1) warm-starting Bayesian optimization to obtain quality Gaussian Processes priors and (2) controllable Pareto set learning to accurately acquire a parametric mapping from preferences to the corresponding Pareto solutions. The former is to help stabilize the PSL process and reduce the number of expensive function evaluations. The latter is to support real-time trade-off control between conflicting objectives. Performances across synthesis and real-world MOO problems showcase the effectiveness of our Co-PSL for expensive multi-objective optimization tasks.

Figures (8)

• Figure 1: Performance of traditional Pareto Set Learning (PSL-MOBO) and our Co-PSL on DTLZ2 problem. We use the Mean Euclidean Distance between truth and estimated Pareto solutions under the same set of reference vectors to determine the performance of the Pareto Set Model, and thus, the quality of the learned Pareto Front. While the main baseline PSL-MOBO exhibits a highly unstable and uncertain performance in obtaining the Pareto front, Co-PSL demonstrates a smoother and more consistent front across optimization iterations.
• Figure 2: 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, with synthesis problems on the top row and real-world problems on the bottom row. The solid line is the mean value averaged among 5 independent runs, and the shaded region is the standard deviation around the mean value.
• Figure 3: Mean Euclidean Distance between the truth Pareto solutions and the predicted Pareto solutions under the same set of preference vectors with respect to the number of training iterations between PSL-MOBO (baseline) and Co-PSL (ours) among the three synthesis problems.
• Figure 4: Ablation studies for Co-PSL on DTLZ2 with Log Hypervolume Difference (left column) and Mean Euclidean Distance (right column), including (a) trade-off between warm-starting and Controllable Pareto Front Learning and (b) trade-off $\beta$ between previous parameter and random initialization for Pareto Set Model parameter initialization.
• Figure 5: Performance of Co-PSL (proposed) and PSL-MOBO (baseline) on DTLZ2 with individual components on Log Hypervolume Difference and Mean Euclidean Distance.

Theorems & Definitions (5)

• Definition 1: Dominance
• Definition 2: Pareto Optimality
• Definition 3: Pareto Set/Front
• Definition 4: Hypervolume
• Definition 5: Hypervolume Improvement