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Smooth Tchebycheff Scalarization for Multi-Objective Optimization
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Proposes smooth Tchebycheff (STCH) scalarization, a log-sum-exp smoothing of the Tchebycheff scalarization enabling fast gradient-based multi-objective optimization that can reach all Pareto solutions.
Problem
Gradient-based multi-objective optimization: linear scalarization misses non-convex Pareto fronts, Tchebycheff scalarization is nonsmooth and converges slowly, and adaptive gradient methods have high per-iteration cost.
Approach
Replace the nonsmooth max in Tchebycheff scalarization with a log-sum-exp (infimal convolution) smoothing with parameter mu, giving a differentiable scalarization that is a uniform approximation of TCH, convex when objectives are convex, finds weakly Pareto optimal points, and can recover all Pareto solutions for small mu.
Key results
- On NYUv2, STCH achieves Delta_p +8.54, second best overall after DB-MTL (+8.91), and far above TCH (-5.67).
- For Pareto set learning, STCH attains lowest Delta HV on 10 of 11 problems (e.g., RocketInjector 1.08e-02 vs TCH 2.73e-02).
- On Office-31, STCH achieves Delta_p +1.48, the best overall and the only method dominating STL on all tasks.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace classical Tchebycheff scalarization with smooth Tchebycheff scalarization | Multi-task learning on NYUv2 · NYUv2 | Δp: -5.67 -> +8.54 | Helped |
| use STCH instead of DB-MTL | Multi-task learning on NYUv2 · NYUv2 | Δp: +8.91 -> +8.54 | Hurt |
| use STCH instead of DB-MTL | Multi-task learning on Office-31 · Office-31 | Δp: +1.05±0.20 -> +1.48±0.31 | Helped |
| replace DB-MTL with STCH | Multi-task learning · QM9 | Δp: -58.10±3.89 -> -58.14±4.18 | No clear effect |
| replace DB-MTL with STCH | Multi-task learning · QM9 | T: 4.56 -> 1.00 | Helped |
| replace Tchebycheff scalarization with STCH scalarization | Pareto set learning · F1-F6 and five engineering design problems | ΔHV | Helped |
| set smoothing parameter μ to 0.01 (ablation) | STCH Pareto set learning · Engineering design problems | ΔHV | Mixed |
| set smoothing parameter μ to 0.1 (ablation) | STCH Pareto set learning · Engineering design problems | ΔHV | Mixed |
| set smoothing parameter μ to 0.5 (ablation) | STCH Pareto set learning · Engineering design problems | ΔHV | Mixed |
| set smoothing parameter μ to 1 (ablation) | STCH Pareto set learning · Engineering design problems | ΔHV | Mixed |
| use a homotopy schedule that gradually decreases μ instead of fixed μ (ablation) | STCH optimization · Experiments | performance | No clear effect |
Surprising findings
- The classic Tchebycheff scalarization performs worse than the single-task baseline on NYUv2 and Office-31, while its smoothed version outperforms it substantially.
Limitations
- Only unconstrained and deterministic multi-objective optimization is handled; constraints need penalty-style extensions not evaluated.
- Smoothing parameter mu is a hyperparameter with no single best value across problems; homotopy schedules did not clearly help.
- Global optimality guarantees are limited to convex objectives or the Pareto-front conditions; non-convex cases give only Pareto stationarity.
Experimental setup
Datasets: NYUv2 Office-31 QM9 six synthetic benchmark problems F1-F6 five engineering design problems (BarTruss, HatchCover, DiskBrake, GearTrain, RocketInjector)
Benchmarks: NYUv2 Office-31 QM9 synthetic benchmarks F1-F6 engineering design problems
Metrics: Delta_p (relative average performance improvement) hypervolume difference (Delta HV) run time mean/median gap
Baselines: Linear scalarization COSMOS EPO TCH MGDA GradNorm PCGrad DB-MTL Nash-MTL IMTL Aligned-MTL MoCo
Builds on: Tchebycheff scalarization infimal convolution smoothing MGDA LibMTL
Closest papers by meaning
- Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization (ICLR 2025 · similarity 0.94)
- SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front (NeurIPS 2026 · similarity 0.85)
- Runtime Analysis of the SMS-EMOA for Many-Objective Optimization (AAAI 2024 · similarity 0.85)
- How to Use the Metropolis Algorithm for Multi-Objective Optimization? (AAAI 2024 · similarity 0.84)
- Learning Configurations for Data-Driven Multi-Objective Optimization (ICML 2025 · similarity 0.84)
- Pareto-frontier Entropy Search with Variational Lower Bound Maximization (ICML 2025 · similarity 0.84)
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