[Paper Review] Enhancing Generalization and Scalability for Multi-Objective Optimization with Population Pre-Training

Haokai Hong, Min Jiang+2 more|arXiv (Cornell University)|Dec 11, 2023

Advanced Multi-Objective Optimization Algorithms6 citations

TL;DR

This paper introduces a Pre-Evolved Model (PEM) for multi-objective optimization, leveraging a pre-trained Transformer (PET) to generate high-quality populations by learning evolutionary patterns from diverse MOPs. The PET is pre-evolved on benchmark problems and fine-tuned on new MOPs using iterative evaluation feedback, significantly outperforming state-of-the-art MOEAs on large-scale, many-objective, constrained, and computationally expensive problems.

ABSTRACT

Multi-objective optimization problems (MOPs) require the simultaneous optimization of conflicting objectives. Real-world MOPs often exhibit complex characteristics, including high-dimensional decision spaces, many objectives, or computationally expensive evaluations. While population-based evolutionary computation has shown promise in addressing diverse MOPs through problem-specific adaptations, existing approaches frequently lack generalizability across distinct problem classes. Inspired by pre-training paradigms in machine learning, we propose a Population Pre-trained Model (PPM) that leverages historical optimization knowledge to solve complex MOPs within a unified framework efficiently. PPM models evolutionary patterns via population modeling, addressing two key challenges: (1) handling diverse decision spaces across problems and (2) capturing the interdependency between objective and decision spaces during evolution. To this end, we develop a population transformer architecture that embeds decision spaces of varying scales into a common latent space, enabling knowledge transfer across diverse problems. Furthermore, our architecture integrates objective-space features through objective fusion to enhance population prediction accuracy for complex MOPs. Our approach achieves robust generalization to downstream optimization tasks with up to 5,000 dimensions--five times the training scale and 200 times greater than prior work. Extensive evaluations on standardized benchmarks and out-of-training real-world applications demonstrate the consistent superiority of our method over state-of-the-art algorithms tailored to specific problem classes, improving the performance and generalization of evolutionary computation in solving MOPs.

Motivation & Objective

• Address the challenge of poor generalization and scalability in existing multi-objective evolutionary algorithms (MOEAs) when solving diverse complex MOPs.
• Overcome the inefficiency of MOEAs in handling large-scale decision variables, many objectives, expensive evaluations, and constraints.
• Introduce a novel paradigm—pre-evolving and fine-evolving—inspired by large-scale pre-training in NLP and computer vision, to transfer learned evolutionary knowledge across problems.
• Develop a general-purpose optimization framework that replaces traditional population reproduction operators with a learnable, adaptive model.
• Demonstrate the effectiveness of the PET on both synthetic benchmarks and real-world applications, including constrained, high-dimensional problems with limited evaluations.

Proposed method

• Design a Pre-Evolved Model (PEM) as a general-purpose replacement for population reproduction in MOEAs, enabling transfer learning across MOPs.
• Implement a Pre-Evolved Transformer (PET) based on the classical transformer architecture, with dimension embedding and objective encoding to process solution populations.
• Pre-train the PET on a large corpus of existing MOPs to learn effective evolutionary patterns across diverse problem types.
• Fine-tune the PET on new MOPs by iteratively generating offspring, evaluating them, and updating the model with feedback to improve subsequent generations.
• Integrate the PET into existing MOEAs (e.g., NSGA-II, WOF) by replacing standard reproduction operators like SBX, DE, or PM.
• Use masked training during pre-evolving (where target solutions are hidden) and autoregressive generation during fine-evolving (where generated solutions are evaluated and fed back).

Experimental results

Research questions

• RQ1 Can a pre-trained model learn generalizable evolutionary patterns across diverse MOPs to improve population quality in multi-objective optimization?
• RQ2 Does pre-evolving a transformer-based model lead to better generalization and scalability compared to traditional MOEAs on large-scale, many-objective, and computationally expensive problems?
• RQ3 Can the fine-tuning process of the PET adapt to new MOPs with limited evaluations and constraints, achieving superior performance?
• RQ4 To what extent does pre-training enhance the performance of MOEAs on real-world applications with complex constraints and high dimensionality?
• RQ5 Is the proposed pre-evolving and fine-evolving paradigm effective even when applied to problems not seen during pre-training?

Key findings

• The PET achieved a 91.62 improvement over the suboptimal algorithm on ZDT6 with 3,000 and 5,000 decision variables, demonstrating strong performance on large-scale problems.
• On LSMOP 7–9, the PET significantly outperformed state-of-the-art MOEAs in many-objective (up to 10 objectives), large-scale (2,500–5,000 dimensions), and computationally expensive (1,000 evaluations) settings.
• On the real-world TREE problem with 3,000 dimensions and constraints, the PET achieved the best results on all five instances (TREE 1–5), while some baselines failed to find feasible solutions.
• The ablation study confirmed that an un-pre-evolved PET performs no better than random reproduction operators, proving that pre-training is essential for effectiveness.
• The PET showed emergent capabilities on non-pre-evolved problems, suggesting it can generalize beyond its pre-training distribution.
• Statistical tests confirmed the PET's superiority with p < 0.05 on most benchmarks, with significant improvements marked by '+' in tables.