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Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization
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Proposes Tchebycheff set (TCH-Set) and smooth Tchebycheff set (STCH-Set) scalarizations to find a small set of K solutions that collectively address many (m>100) objectives, with Pareto-optimality and convergence guarantees.
Problem
Many-objective optimization where a dense Pareto set is exponentially large in the number of objectives; the goal is a few solutions (K << m) that each objective is well addressed by at least one solution.
Approach
Extends Tchebycheff scalarization to a set of solutions by taking the min over solutions per objective before the max over objectives, then smooths both max and min with log-sum-exp so the set can be optimized by plain gradient descent with theoretical guarantees.
Key results
- On convex problems with m=128 and K=3, STCH-Set achieves mean worst objective 6.08e-01 versus 1.86e+00 for SoM and 4.64e+00 for linear scalarization.
- On noisy mixed linear regression (sigma=0.1, K=5), STCH-Set reaches worst-objective 2.10 versus 2.50 for SoM and 38.4 for LS.
- STCH-Set achieves lowest worst-objective value in all reported comparisons, and TCH-Set is significantly worse than STCH-Set, showing the importance of smoothness.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use STCH-Set instead of TCH-Set scalarization (ablation) | Set scalarization for convex optimization · Convex optimization, m = 128, K = 3–10 | worst objective value: 1.02e+00 -> 6.08e-01 (K = 3); 1.95e-01 -> 7.99e-02 (K = 10) | Helped |
| Use STCH-Set instead of TCH-Set scalarization (ablation) | Set scalarization for convex optimization · Convex optimization, m = 128, K = 3–10 | average objective value: 3.46e-01 -> 2.12e-01 (K = 3); 5.92e-02 -> 1.34e-02 (K = 10) | Helped |
| Use STCH-Set instead of SoM | Set optimization for convex optimization · Convex optimization, m = 128, K = 3–10 | average objective value: 2.02e-01 -> 2.12e-01 (K = 3); 3.40e-02 -> 1.78e-02 (K = 8) | Mixed |
| Use STCH-Set instead of SoM | Set optimization for noisy mixed linear regression · Noisy mixed linear regression, σ = 0.1, K = 5 | worst objective value: 2.50e+00 -> 2.10e+00 | Helped |
| Use STCH-Set instead of SoM | Set optimization for noisy mixed linear regression · Noisy mixed linear regression, σ = 0.1, K = 5 | average objective value: 1.88e-01 -> 4.72e-01 | Hurt |
| Use STCH-Set instead of SoM | Set optimization for noisy mixed nonlinear regression · Noisy mixed nonlinear regression, σ = 0.1, K = 5 | worst objective value: 1.68e+01 -> 7.43e+00 | Helped |
| Use STCH-Set instead of SoM | Set optimization for noisy mixed nonlinear regression · Noisy mixed nonlinear regression, σ = 0.1, K = 5 | average objective value: 8.54e-01 -> 1.89e+00 | Hurt |
Surprising findings
- The non-smooth TCH-Set optimal solutions need not be weakly Pareto optimal in general; only the worst weighted objective affects the value.
- STCH-Set does not always win on average objective; SoM achieves better mean average value in some settings (e.g., K=3 on m=128 convex).
Limitations
- Only the deterministic setting where all objective values are available is studied.
- Assumptions of no redundant solutions and positive preferences may not hold in extreme cases.
- Experiments are on synthetic convex and regression problems rather than real-world applications.
Experimental setup
Datasets: synthetic convex quadratic objectives (m=128, 1024) synthetic noisy mixed linear regression data (m=1000) synthetic noisy mixed nonlinear regression data
Benchmarks: synthetic convex many-objective quadratic problems (m=128, 1024) noisy mixed linear regression noisy mixed nonlinear regression
Metrics: mean worst objective value mean average objective value Wilcoxon rank-sum test
Baselines: linear scalarization Tchebycheff scalarization smooth Tchebycheff scalarization MosT SoM TCH-Set
Builds on: Tchebycheff scalarization smooth Tchebycheff scalarization MGDA MosT
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