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SPREAD: Sampling-based Pareto front Refinement via Efficient Adaptive Diffusion

Sedjro Salomon Hotegni, Sebastian Peitz

TL;DR

SPREAD presents a diffusion-based framework for multi-objective optimization that refines candidate solutions via conditional DDPMs guided by adaptive multiple-gradient-descent directions and a diversity-promoting repulsion term. By conditioning diffusion sampling on objective values and integrating MGD-inspired guidance, SPREAD achieves both fast convergence toward Pareto optima and broad front coverage. The approach extends naturally to offline and Bayesian MO models through surrogate models and limited evaluations, with empirical results showing superior hypervolume and diversity across diverse benchmarks. The work demonstrates practical scalability and robustness, offering a versatile tool for large-scale, expensive MOO tasks and suggesting future improvements for constraint handling.

Abstract

Developing efficient multi-objective optimization methods to compute the Pareto set of optimal compromises between conflicting objectives remains a key challenge, especially for large-scale and expensive problems. To bridge this gap, we introduce SPREAD, a generative framework based on Denoising Diffusion Probabilistic Models (DDPMs). SPREAD first learns a conditional diffusion process over points sampled from the decision space and then, at each reverse diffusion step, refines candidates via a sampling scheme that uses an adaptive multiple gradient descent-inspired update for fast convergence alongside a Gaussian RBF-based repulsion term for diversity. Empirical results on multi-objective optimization benchmarks, including offline and Bayesian surrogate-based settings, show that SPREAD matches or exceeds leading baselines in efficiency, scalability, and Pareto front coverage. Code is available at https://github.com/safe-autonomous-systems/moo-spread .

SPREAD: Sampling-based Pareto front Refinement via Efficient Adaptive Diffusion

TL;DR

SPREAD presents a diffusion-based framework for multi-objective optimization that refines candidate solutions via conditional DDPMs guided by adaptive multiple-gradient-descent directions and a diversity-promoting repulsion term. By conditioning diffusion sampling on objective values and integrating MGD-inspired guidance, SPREAD achieves both fast convergence toward Pareto optima and broad front coverage. The approach extends naturally to offline and Bayesian MO models through surrogate models and limited evaluations, with empirical results showing superior hypervolume and diversity across diverse benchmarks. The work demonstrates practical scalability and robustness, offering a versatile tool for large-scale, expensive MOO tasks and suggesting future improvements for constraint handling.

Abstract

Developing efficient multi-objective optimization methods to compute the Pareto set of optimal compromises between conflicting objectives remains a key challenge, especially for large-scale and expensive problems. To bridge this gap, we introduce SPREAD, a generative framework based on Denoising Diffusion Probabilistic Models (DDPMs). SPREAD first learns a conditional diffusion process over points sampled from the decision space and then, at each reverse diffusion step, refines candidates via a sampling scheme that uses an adaptive multiple gradient descent-inspired update for fast convergence alongside a Gaussian RBF-based repulsion term for diversity. Empirical results on multi-objective optimization benchmarks, including offline and Bayesian surrogate-based settings, show that SPREAD matches or exceeds leading baselines in efficiency, scalability, and Pareto front coverage. Code is available at https://github.com/safe-autonomous-systems/moo-spread .

Paper Structure

This paper contains 60 sections, 2 theorems, 38 equations, 16 figures, 17 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Generative Modeling for Multi-Objective Optimization
  4. Gradient-based Methods for Pareto Set Discovery
  5. Preliminaries
  6. Denoising Diffusion Probabilistic Models (DDPMs)
  7. Multi-Objective Optimization (MOO)
  8. Multiple Gradient Descent (MGD)
  9. Method
  10. Training
  11. Sampling
  12. Extension towards surrogate-based optimization
  13. Offline MOO:
  14. Bayesian MOO:
  15. Experiments
  16. ...and 45 more sections

Key Result

Theorem 1

Let $\mathbf{X} \subset \mathcal{X}$ be a training dataset with distribution $P_{\mathbf{X}}$. Let $\Xi\in(0,\infty)^m$, independent of $\mathbf{X}$, and define the training label For a conditioning value $\mathbf{c}$ in the support of $\mathbf{C}$, denote by $P_{\mathbf{X}\mid \mathbf{C}=\mathbf{c}}$ the true conditional data distribution and by $Q_\theta(\cdot\mid \mathbf{c})$ the distribution

Figures (16)

  • Figure 3: Bayesian MOO. Log‑hypervolume difference (LHD) over 20 post‑initialization steps (totaling 100 function evaluations) on nine MOBO benchmarks: Branin and Currin, ZDT1, ZDT2, ZDT3, Penicillin Production, DTLZ2, DTLZ5, DTLZ7, and Car Side Impact (RE41).
  • Figure 4: Empirical validation of the objective improvement guaranteed by Theorem 1. The figure shows the movement of random initial points (blue) toward the corresponding sampled points (red) on ZDT1, ZDT2, ZDT3 and RE21 in the online setting. Guidance is disabled so that only the effect of the diffusion model is visualized.
  • Figure 5: Hypervolume bi-objective example, corresponding to the shaded region defined by the obtained solutions (red) and the reference point (blue).
  • Figure 6: (Bayesian) MGD+RBF baseline and ablation study without guidance. The random seed is fixed to $1000$.
  • Figure 7: Approximate Pareto optimal points for multiple benchmark problems. Solutions from $5$ independent runs are merged, and the non-dominated points are shown.
  • ...and 11 more figures

Theorems & Definitions (7)

  • Definition 1: Pareto Stationarity
  • Definition 2: Dominance
  • Definition 3: Pareto Optimality
  • Theorem 1: Objective Improvement
  • Theorem 2
  • proof
  • proof