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Racing Control Variable Genetic Programming for Symbolic Regression
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Proposes Racing-CVGP, which runs multiple control-variable experiment schedules in parallel with GP-style selection that prunes poor schedules early, to speed up multi-variable symbolic regression.
Problem
Symbolic regression with many input variables requires massive data and long training time; CVGP's fixed experiment schedule (order of controlled variables) is often suboptimal and delays discovery.
Approach
Maintain a population of expressions each tagged with its own experiment schedule, run GP within control-variable rounds, and use fitness-based racing selection to keep promising schedules and terminate unfavorable ones early, with a freeze operation for stable constants.
Key results
- On Trigonometric (4,4,6) median NMSE below 0.016 for Racing-CVGP versus 0.028 for CVGP and 0.044 for GP; for 8-variable expressions CVGP and GP time out at 48 hours while Racing-CVGP returns NMSE median 0.095.
- On Livermore2 n=6, Racing-CVGP median NMSE 0.001 vs Eureqa 0.026 and GPMeld 0.144.
- Racing-CVGP takes less running time than CVGP and the baselines on trigonometric datasets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Racing-CVGP instead of the strongest listed baseline | Trigonometric datasets, n=4–8 · Trigonometric (4,4,6), (5,5,5), (6,6,10), and (8,8,12) | NMSE (50%): 0.024 -> 0.016; 0.063 -> 0.043; 0.104 -> 0.069; 0.232 -> 0.095 | Helped |
| Use Racing-CVGP instead of the strongest listed baseline | Livermore2 · Livermore2 n=4, n=5, and n=6 | NMSE (50%): 0.029 -> <1e-6; 0.049 -> 0.004; 0.026 -> 0.001 | Helped |
| Use Racing-CVGP instead of the strongest listed baseline | Feynman n=4 · Feynman (n=4) | NMSE (50%): 0.026 -> 0.0154 | Helped |
| Use Racing-CVGP instead of the strongest listed baseline | Feynman n=5 · Feynman (n=5) | NMSE (50%): 0.434 -> 0.577 | Hurt |
| Use Racing-CVGP instead of the strongest listed baseline | Feynman datasets · Feynman n=4 and n=5 | NMSE (75%): 0.708 -> 0.195; 0.943 -> 0.790 | Helped |
| Use a better experiment schedule rather than CVGP's default schedule (ablation) | CVGP · Trigonometric (4,4,6), 10 expressions | NMSE | Helped |
| Use global optimizers rather than local optimizers for fitting open constants (ablation) | Racing-CVGP · Random expressions involving 4 variables | NMSE | Helped |
| Use global optimizers rather than local optimizers for fitting open constants (ablation) | Racing-CVGP · Random expressions involving 4 variables | CPU time | Hurt |
| Use SHGO instead of local optimizers (ablation) | Racing-CVGP · 10 randomly generated expressions involving 4 variables | NMSE | Helped |
| Use SHGO instead of local optimizers (ablation) | Racing-CVGP · 10 randomly generated expressions involving 4 variables | Time | Hurt |
| Race across experiment schedules rather than run CVGP with all schedules (ablation) | Racing-CVGP · Trigonometric datasets, n=4 | Running time | Helped |
| Use a non-default experiment schedule instead of the default schedule (ablation) | CVGP · Trigonometric (4,4,6), 10 expressions | NMSE | Helped |
Surprising findings
- Eureqa achieves the best median NMSE on the Feynman n=5 setting (0.434 vs 0.577 for Racing-CVGP).
Limitations
- Relies on a data oracle that allows querying data with specified controlled variables, which may not be available in practice.
- Baselines were given fixed datasets rather than the same oracle, so comparison is not fully matched.
- Eureqa outperforms it on some Feynman settings.
Experimental setup
Datasets: Trigonometric datasets Livermore2 Feynman
Benchmarks: Trigonometric datasets Livermore2 Feynman
Metrics: NMSE MSE RMSE NRMSE running time
Baselines: GP Eureqa PQT VPG DSR GPMeld CVGP
Builds on: Control Variable Genetic Programming genetic programming BFGS
Closest papers by meaning
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