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GaussianPSL: Soft partitioning for complex PSL problem

Phuong Mai Dinh, Van-Nam Huynh

Abstract

Many practical applications of multi-objective optimization (MOO), including engineering design, autonomous systems, and machine learning, often yield complex Pareto frontiers (e.g., discontinuous, degenerate, or non-convex), which pose challenges for traditional scalarization and Pareto Set Learning (PSL) methods that struggle to approximate them accurately. In this paper, we propose GaussianPSL, a novel framework that uses soft partitions of the Pareto decision/objective space to address the challenges posed by complex Pareto frontiers. Our method dynamically partitions the space, enabling simple MLP networks to learn localized features within each region and then aggregate this information for the final prediction. This partition-aware strategy enhances both exploration and convergence, reduces sensitivity to initialization, and improves robustness against local optima. Experimental results demonstrate that the proposed approach consistently outperforms standard PSL models in learning complex Pareto fronts while maintaining model simplicity. Overall, GaussianPSL offers a new direction for effective, scalable MOO in challenging frontier geometries.

GaussianPSL: Soft partitioning for complex PSL problem

Abstract

Many practical applications of multi-objective optimization (MOO), including engineering design, autonomous systems, and machine learning, often yield complex Pareto frontiers (e.g., discontinuous, degenerate, or non-convex), which pose challenges for traditional scalarization and Pareto Set Learning (PSL) methods that struggle to approximate them accurately. In this paper, we propose GaussianPSL, a novel framework that uses soft partitions of the Pareto decision/objective space to address the challenges posed by complex Pareto frontiers. Our method dynamically partitions the space, enabling simple MLP networks to learn localized features within each region and then aggregate this information for the final prediction. This partition-aware strategy enhances both exploration and convergence, reduces sensitivity to initialization, and improves robustness against local optima. Experimental results demonstrate that the proposed approach consistently outperforms standard PSL models in learning complex Pareto fronts while maintaining model simplicity. Overall, GaussianPSL offers a new direction for effective, scalable MOO in challenging frontier geometries.

Paper Structure

This paper contains 27 sections, 32 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multi-Objective Optimization
  4. Pareto Optimality
  5. Strong Pareto Optimality
  6. Weak Pareto Optimality.
  7. Pareto Set Learning
  8. Motivation
  9. Limitations of PSL on the weak Pareto frontier
  10. Example: Disconnected Pareto Front in 2D
  11. Proposed Method: GaussianPSL
  12. Soft Partitioning on Decision Space
  13. Adaptive Density Control
  14. Experiments
  15. Baseline Methods
  16. ...and 12 more sections

Figures (4)

  • Figure 1: Illustration of strong and weak Pareto optimality for a two-objective minimization problem. Source: zheng2025weakparetoboundaryachilles
  • Figure 2: The overall architecture of the framework using the soft partitioning on the decision space
  • Figure 3: Top row (under-reconstruction): When small-scale geometry (black outline) is insufficiently covered, clone the respective elipsoid. Bottom row (over-reconstruction): If small-scale geometry is represented by one large splat, split it into two. Source: kerbl3Dgaussians
  • Figure 4: Illustrative example demonstrating the relation between Pareto front, preference rays, and solutions. Source: navon2021learning

Theorems & Definitions (2)

  • Definition 2.1: Pareto Dominance
  • Definition 2.2: Pareto Frontier and Optimal Solution Space