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Expensive Multi-Objective Bayesian Optimization Based on Diffusion Models

Bingdong Li, Zixiang Di, Yongfan Lu, Hong Qian, Feng Wang, Peng Yang, Ke Tang, Aimin Zhou

TL;DR

This work tackles EMOPs by integrating a composite diffusion model into Pareto Set Learning (PSL) to stabilize solution distributions under limited evaluations. CDM-PSL combines conditional and unconditional diffusion models with an entropy-based gradient weighting and guided denoising to simultaneously improve convergence to the Pareto front and maintain diversity. Extensive experiments on synthetic benchmarks and real-world problems show superior performance over state-of-the-art MOBO methods, with ablations confirming the value of each component. While the approach incurs higher computational cost, it offers a principled path to scalable, high-quality Pareto sets in expensive optimization scenarios, with future work aiming to further scale to very high-dimensional problems using Monte Carlo tree methods.

Abstract

Multi-objective Bayesian optimization (MOBO) has shown promising performance on various expensive multi-objective optimization problems (EMOPs). However, effectively modeling complex distributions of the Pareto optimal solutions is difficult with limited function evaluations. Existing Pareto set learning algorithms may exhibit considerable instability in such expensive scenarios, leading to significant deviations between the obtained solution set and the Pareto set (PS). In this paper, we propose a novel Composite Diffusion Model based Pareto Set Learning algorithm, namely CDM-PSL, for expensive MOBO. CDM-PSL includes both unconditional and conditional diffusion model for generating high-quality samples. Besides, we introduce an information entropy based weighting method to balance different objectives of EMOPs. This method is integrated with the guiding strategy, ensuring that all the objectives are appropriately balanced and given due consideration during the optimization process; Extensive experimental results on both synthetic benchmarks and real-world problems demonstrates that our proposed algorithm attains superior performance compared with various state-of-the-art MOBO algorithms.

Expensive Multi-Objective Bayesian Optimization Based on Diffusion Models

TL;DR

This work tackles EMOPs by integrating a composite diffusion model into Pareto Set Learning (PSL) to stabilize solution distributions under limited evaluations. CDM-PSL combines conditional and unconditional diffusion models with an entropy-based gradient weighting and guided denoising to simultaneously improve convergence to the Pareto front and maintain diversity. Extensive experiments on synthetic benchmarks and real-world problems show superior performance over state-of-the-art MOBO methods, with ablations confirming the value of each component. While the approach incurs higher computational cost, it offers a principled path to scalable, high-quality Pareto sets in expensive optimization scenarios, with future work aiming to further scale to very high-dimensional problems using Monte Carlo tree methods.

Abstract

Multi-objective Bayesian optimization (MOBO) has shown promising performance on various expensive multi-objective optimization problems (EMOPs). However, effectively modeling complex distributions of the Pareto optimal solutions is difficult with limited function evaluations. Existing Pareto set learning algorithms may exhibit considerable instability in such expensive scenarios, leading to significant deviations between the obtained solution set and the Pareto set (PS). In this paper, we propose a novel Composite Diffusion Model based Pareto Set Learning algorithm, namely CDM-PSL, for expensive MOBO. CDM-PSL includes both unconditional and conditional diffusion model for generating high-quality samples. Besides, we introduce an information entropy based weighting method to balance different objectives of EMOPs. This method is integrated with the guiding strategy, ensuring that all the objectives are appropriately balanced and given due consideration during the optimization process; Extensive experimental results on both synthetic benchmarks and real-world problems demonstrates that our proposed algorithm attains superior performance compared with various state-of-the-art MOBO algorithms.
Paper Structure (67 sections, 17 equations, 16 figures, 5 tables, 3 algorithms)

This paper contains 67 sections, 17 equations, 16 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Expensive Multi-objective Optimization
  4. Bayesian Optimization
  5. Diffusion Models
  6. Forward Process
  7. Reverse Process
  8. Conditional Generative Models
  9. Diffusion Model Based Optimization Algorithms
  10. Our Method
  11. Overview
  12. Data Extraction
  13. Diffusion Model Training
  14. Diffusion Process
  15. Noise Prediction
  16. ...and 52 more sections

Figures (16)

  • Figure 1: (left) The framework of CDM-PSL. (right) Diffusion Model Training (DMT), Conditional Generation (CG), and Unconditional Generation (UG). DMT involves learning from the selected samples through multiple steps; CG is designed to create high-quality samples with an optimized distribution; UG is used to generate diverse samples with high efficiency.
  • Figure 2: The HV results of 10 algorithms, evaluated on synthetic test functions and real-world problems ($d=20$). The horizontal axis denotes the FEs after the initialization phase, similarly hereinafter.
  • Figure 3: Ablation results of CDM-PSL on 6 instances.
  • Figure 4: Approximate Pareto fronts obtained by CDM-PSL and MOBO w/o CDM-PSL.
  • Figure 5: The HV values relative to the number of FEs for CDM-PSL with different number of steps $t$.
  • ...and 11 more figures

Theorems & Definitions (3)

  • Definition 2.1: Pareto dominance
  • Definition 2.2: Pareto optimal
  • Definition 2.3: Pareto Set and Pareto Front