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ParetoFlow: Guided Flows in Multi-Objective Optimization
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Proposes ParetoFlow, a flow-matching generative method for offline multi-objective optimization that guides sampling toward the Pareto front using weight-vector predictor guidance, local filtering, and neighboring evolution.
Problem
Offline multi-objective optimization: finding designs that approximate the Pareto front using only an offline dataset of designs and labels, where existing methods rely on evolutionary algorithms, Bayesian optimization, or less advanced generative models.
Approach
Applies flow matching with a multi-objective predictor guidance that assigns each sample a uniform weight vector over predicted objectives, a hypercone-based local filtering for non-convex fronts, and a neighboring evolution step where neighboring weight distributions generate offspring and the best one is kept, combined with a Pareto-set archive.
Key results
- ParetoFlow achieves the best average rank of 3.12 across all Off-MOO-Bench tasks, ahead of MM+IOM (6.73) and MM (6.71) as the best prior methods.
- ParetoFlow ranks first on MORL (1.00) and Sci-Design (2.75), and attains the top hypervolume on ZDT2 (6.79 ± 0.16) among the reported methods.
- Removing the local filtering reduces ZDT2 diversity from 5.144 to 2.080, supporting its role in diversity.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ParetoFlow instead of the compared methods | Off-MOO-Bench · All Tasks | average rank: 6.71 ± 4.31 -> 3.12 ± 3.77 | Helped |
| use uniform weights across samples rather than equal weights for every sample (ablation) | ParetoFlow · ZDT2, C-10/MOP1, MO-Hopper, Zinc, RE23 | hypervolume: Equal: 6.15 ± 0.23, 4.64 ± 0.03, 5.58 ± 0.38, 4.14 ± 0.14, 4.75 ± 0.00; ParetoFlow: 6.79 ± 0.16, 4.77 ± 0.00, 5.69 ± 0.0 | Helped |
| use uniform weights across samples rather than weight only the first objective (ablation) | ParetoFlow · ZDT2, C-10/MOP1, MO-Hopper, Zinc, RE23 | hypervolume: First: 5.58 ± 0.38, 4.59 ± 0.02, 5.25 ± 0.23, 4.00 ± 0.14, 4.89 ± 0.01; ParetoFlow: 6.79 ± 0.16, 4.77 ± 0.00, 5.69 ± 0.0 | Helped |
| add local filtering (ablation) | ParetoFlow · ZDT2, C-10/MOP1, MO-Hopper, Zinc, RE23 | hypervolume: w/o local: 5.78 ± 0.15, 4.65 ± 0.03, 5.44 ± 0.21, 4.36 ± 0.04, 5.13 ± 0.41; ParetoFlow: 6.79 ± 0.16, 4.77 ± 0.00, 5.69 ± | Helped |
| add neighboring evolution (ablation) | ParetoFlow · ZDT2, C-10/MOP1, MO-Hopper, Zinc, RE23 | hypervolume: w/o neighbor: 6.43 ± 0.01, 4.64 ± 0.00, 5.62 ± 0.16, 4.40 ± 0.05, 6.08 ± 0.20; ParetoFlow: 6.79 ± 0.16, 4.77 ± 0.00, 5.6 | Mixed |
| add Pareto set update (ablation) | ParetoFlow · ZDT2, C-10/MOP1, MO-Hopper, Zinc, RE23 | hypervolume: w/o PS: 6.45 ± 0.52, 4.49 ± 0.00, 5.00 ± 0.02, 4.40 ± 0.00, 5.28 ± 0.21; ParetoFlow: 6.79 ± 0.16, 4.77 ± 0.00, 5.69 ± 0. | Mixed |
| increase the number of neighbors K (ablation) | ParetoFlow · MO-Hopper and C-10/MOP1 | normalized hypervolume | No clear effect |
| vary the number of offspring O (ablation) | ParetoFlow · MO-Hopper and C-10/MOP1 | normalized hypervolume | No clear effect |
| increase the number of sampling steps (ablation) | ParetoFlow · MO-Hopper and C-10/MOP1 | normalized hypervolume | Helped |
| increase predictor-guidance scaling gamma from 0 to 2 (ablation) | ParetoFlow · MO-Hopper and C-10/MOP1 | normalized hypervolume | Helped |
| vary noise factor g (ablation) | ParetoFlow · MO-Hopper and C-10/MOP1 | normalized hypervolume | No clear effect |
| use IOM predictors instead of the default multiple-model setup (ablation) | ParetoFlow · ZDT2, C-10/MOP1, MO-Hopper, Zinc, RE23 | hypervolume: ParetoFlow: 6.79 ± 0.16, 4.77 ± 0.00, 5.69 ± 0.03, 4.49 ± 0.06, 6.32 ± 0.46; ParetoFlow + IOM: 6.32 ± 0.22, 4.75 ± 0.01, | Mixed |
Surprising findings
- Generative modeling methods show reduced effectiveness on MO-NAS relative to Sci-Design, which the authors attribute to difficulty modeling high-dimensional discrete data.
- GP-based Bayesian optimization methods often underperform the best offline dataset design D(best) in this offline setting.
Limitations
- Generative performance drops on high-dimensional discrete tasks such as MO-NAS, which the paper attributes to difficulty modeling discrete data.
- Predictor-based guidance is unreliable at small time steps t, so guidance is applied only after a threshold (0.8), which is a tuned heuristic.
- Combinatorial tasks (MO-TSP, MO-CVRP, MO-KP) were excluded because the decoding strategy is complex.
Experimental setup
Datasets: Off-MOO-Bench synthetic functions MO-NAS (NAS-Bench-201 based C-10/MOP tasks) MO-Swimmer MO-Hopper Molecule (GSK3b/JNK3) Regex ZINC RFP four-bar truss / RE tasks MO-Portfolio
Benchmarks: Off-MOO-Bench synthetic functions MO-NAS MORL Sci-Design Real-world engineering (RE) MO-Portfolio
Metrics: hypervolume (100th and 50th percentile) average rank
Baselines: E2E MH MM MOEA/D + MM MOBO MOBO-qParEGO MOBO-JES PROUD LaMBO-2 CorrVAE MOGFN NSGD-III
Builds on: flow matching classifier guidance MOEA/D Off-MOO-Bench Das-Dennis weight generation
Closest papers by meaning
- Preference-Guided Diffusion for Multi-Objective Offline Optimization (NeurIPS 2025 · similarity 0.91)
- Offline Multi-Objective Optimization (ICML 2024 · similarity 0.88)
- Pareto Front-Diverse Batch Multi-Objective Bayesian Optimization (AAAI 2024 · similarity 0.88)
- Interactive Hyperparameter Optimization in Multi-Objective Problems via Preference Learning (AAAI 2024 · similarity 0.86)
- SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front (NeurIPS 2026 · similarity 0.86)
- Are You Concerned about Limited Function Evaluations: Data-Augmented Pareto Set Learning for Expensive Multi-Objective Optimization (AAAI 2024 · similarity 0.86)
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