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Gliding over the Pareto Front with Uniform Designs
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Proposes UMOD, which uses fill distance and a max-packing surrogate to generate K uniformly spaced Pareto-optimal objectives via bi-level optimization over preference angles with a neural PF model.
Problem
Generating K uniform, representative Pareto-optimal solutions that cover the entire Pareto front; minimizing fill distance is a nested min-max-min problem that is hard to optimize.
Approach
Replaces fill distance minimization with maximizing the minimum pairwise distance (max-packing) between Pareto objectives, proven to bound fill distance within a factor of 4; optimizes preference angles via bi-level optimization with a neural network surrogate for the PF mapping.
Key results
- On ZDT1, UMOD achieves fill distance 1.04 and IGD 5.19, the best among eight methods (SMS-EMOA attains the highest HV at 1.04 with a rank tie).
- On three-objective DTLZ2, UMOD obtains the best uniformity-related ranks (Uniform 3.17, SUniform 1.18, Fill distance 2.37 vs. 2.59 for the next best).
- On the Adult fairness problem, UMOD achieves delta-Uniformity 0.1049 versus 0.0421 for the next-best method (HVGrad).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use UMOD instead of SMS-EMOA | Pareto-objective selection · ZDT1 | Uniform: 1.85 -> 2.07 | Helped |
| Use UMOD instead of SMS-EMOA | Pareto-objective selection · ZDT1 | Fill Distance: 1.16 -> 1.04 | Helped |
| Use UMOD instead of SMS-EMOA | Pareto-objective selection · RE21 | Uniform: 1.26 -> 1.62 | Helped |
| Use UMOD instead of SMS-EMOA | Pareto-objective selection · RE21 | Fill Distance: 1.09 -> 0.83 | Helped |
| Use UMOD instead of LMPFE | Pareto-objective selection · DTLZ2 | Uniform: 2.45 -> 3.17 | Helped |
| Use UMOD instead of NSGA3 | Pareto-objective selection · DTLZ2 | Fill Distance: 2.59 -> 2.37 | Helped |
| Use UMOD instead of HVGrad | Fairness classifier · Adult | δUnif: 0.0421 -> 0.1049 | Helped |
| Use UMOD instead of HVGrad | Fairness classifier · Adult | HV: 0.6865 -> 0.6800 | Hurt |
| Use UMOD instead of HVGrad | Fairness classifier · Compass | δUnif: 0.2199 -> 0.3801 | Helped |
| Use UMOD instead of HVGrad | Fairness classifier · Compass | HV: 0.7625 -> 0.7262 | Hurt |
Surprising findings
- HV is a coarse measure of uniformity: SMS-EMOA gets the highest HV but not the best uniformity indicators.
- Maximizing minimal pairwise distances also yields much lower IGD, an empirical link not fully explained by theory.
Limitations
- Theoretical guarantees apply only to connected, compact Pareto fronts (DTLZ7 excluded).
- Surrogate network introduces approximation error bounded by 2 epsilon.
- Experiments limited to synthetic and moderately sized real-world problems; large-scale applications left as future work.
Experimental setup
Datasets: Adult Compass ZDT1-6 DTLZ RE21 RE22 RE33 RE37 RE41 RE42
Benchmarks: ZDT1-6 DTLZ2 RE21 RE37 RE41 RE42 Adult fairness Compass fairness
Metrics: Hypervolume IGD Sparsity Spacing Uniformity Smooth Uniformity Fill distance Span
Baselines: MOEA/D MOEA/D-AWA NSGA-III SMS-EMOA LMPFE DEA-GNG Subset selection EPO PMGDA HVGrad
Builds on: MOEA/D PMGDA Das-Dennis preference initialization IGD indicator
Closest papers by meaning
- ParetoFlow: Guided Flows in Multi-Objective Optimization (ICLR 2025 · similarity 0.86)
- Preference-Guided Diffusion for Multi-Objective Offline Optimization (NeurIPS 2025 · similarity 0.84)
- SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front (NeurIPS 2026 · similarity 0.84)
- A Many-Objective Problem Where Crossover Is Provably Indispensable (AAAI 2025 · similarity 0.84)
- Runtime Analysis of the SMS-EMOA for Many-Objective Optimization (AAAI 2024 · similarity 0.84)
- Pareto Front-Diverse Batch Multi-Objective Bayesian Optimization (AAAI 2024 · similarity 0.84)
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