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Noise-Resilient Symbolic Regression with Dynamic Gating Reinforcement Learning
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Proposes NRSR, an RL-based symbolic regression method with an L0 noise-resilient gating module that masks irrelevant input variables and a mixed path entropy bonus, outperforming DSR, HESL, GP-Meld, DGSR and Eureqa on noisy and clean Nguyen benchmarks.
Problem
Symbolic regression struggles to recover correct expressions from data containing many irrelevant (noisy) input variables, which inflates the search space; most SR benchmarks are clean.
Approach
Train an L0-regularized gating layer to select relevant input variables and fold the gates into the RL policy's action mask, combined with a mixed path entropy bonus (single-step hierarchical entropy plus whole-sequence path entropy) and PPO with a quantile-filtered baseline.
Key results
- On Nguyen with 5 noisy inputs, NRSR achieves 89.1% RR and 425K EEN versus 72.5% RR for the best baseline (DGSR).
- With 10 noisy inputs, NRSR keeps 89.1% RR while DSR, HESL, GP-Meld, DGSR and Eureqa drop to 23.2%, 26.4%, 35.1%, 68.5% and 34.3%.
- On clean Nguyen data, NRSR reports 89.7% RR and 408K EEN, above HESL (88.3%) and DSR (83.2%), though Eureqa has lower NMSE (1.97e-3).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add NGM (ablation) | NRSR · Nguyen benchmarks, 10 noisy inputs | RR: 32.9% -> 89.1% | Helped |
| add NGM (ablation) | NRSR · Nguyen benchmarks, 10 noisy inputs | EEN: 1.45M -> 423K | Helped |
| add NGM (ablation) | NRSR · Nguyen benchmarks, 10 noisy inputs | NMSE: 0.093 -> 8.52e-3 | Helped |
| add MPE (ablation) | NRSR · Nguyen benchmarks, clean data | RR: 85.3% -> 89.7% | Helped |
| add MPE (ablation) | NRSR · Nguyen benchmarks, clean data | EEN: 431K -> 408K | Helped |
| add MPE (ablation) | NRSR · Nguyen benchmarks, clean data | NMSE: 5.98e-3 -> 7.66e-3 | Mixed |
| add PPO (ablation) | NRSR · Nguyen benchmarks, clean data | RR: 86.7% -> 89.7% | Helped |
| add PPO (ablation) | NRSR · Nguyen benchmarks, clean data | EEN: 462K -> 408K | Helped |
| add PPO (ablation) | NRSR · Nguyen benchmarks, clean data | NMSE: 8.92e-3 -> 7.66e-3 | Helped |
| use averaged gating layer with Otsu threshold instead of final-epoch gates (ablation) | NGM · NGM test expressions | Accuracy: 60.21% -> 98.92% | Helped |
| scale Otsu threshold to 1.05 (ablation) | NGM · NGM test expressions, 10 noisy inputs | Accuracy: 95.70% -> 98.92% | Helped |
| scale Otsu threshold from 1.05 to 1.1 (ablation) | NGM · NGM test expressions, 10 noisy inputs | Accuracy: 98.92% -> 94.62% | Hurt |
Surprising findings
- Eureqa attains the lowest NMSE on clean data despite the lowest recovery rate (67.4%), which the authors attribute to its algorithms.
- The gating layer taken from the final training epoch gives only 60.21% filter accuracy, much worse than the epoch-averaged gates with Otsu thresholding.
Limitations
- Evaluated only on the 12 Nguyen benchmarks, which use a small operator library and few variables.
- Noisy inputs are synthetic irrelevant variables; real-world noise (e.g., measurement noise) is not studied.
- NRSR fails entirely on Nguyen-12 (0% RR), as do all baselines.
Experimental setup
Datasets: Nguyen SR benchmark suite (12 expressions)
Benchmarks: Nguyen-1 to Nguyen-12 with 5 noise variables Nguyen-1 to Nguyen-12 with 10 noise variables Nguyen-1 to Nguyen-12 clean
Metrics: recovery rate (RR) explored expression number (EEN) normalized mean-square error (NMSE) effective exploration ratio (EER) gate filter accuracy
Baselines: DSR HESL GP-Meld DGSR Eureqa
Builds on: DSR (deep symbolic regression) PPO hierarchical entropy regularization (HESL) GP-Meld Gumbel-softmax / binary concrete L0 relaxation
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