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Enumerating Safe Regions in Deep Neural Networks with Provable Probabilistic Guarantees
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Introduces the AllDNN-Verification problem (enumerating all safe regions of a DNN input domain) and proposes ε-ProVe, a sampling-based underestimation of output reachable sets via statistical tolerance limits with interval refinement, giving probabilistic coverage and safety guarantees.
Problem
Enumerating all safe regions of a DNN's input property domain is #P-hard; existing verifiers give only binary safe/unsafe answers or counts, not the location of safe regions.
Approach
Approximate the output reachable set of each region by the min/max of n random samples (underestimation), using statistical prediction of tolerance limits to obtain a probabilistic lower bound on the safe rate, combined with iterative interval refinement to enumerate safe hyperrectangles.
Key results
- On ACAS Xu phi_2 models, ε-ProVe's safe rate underestimates the exact count by a mean of about 1.27% across models, typically within seconds to ~30 s.
- On Model_2_20, ε-ProVe returns 335 safe regions with 78.50% safe rate in 0.4 s versus 79.1% exact count taking 234 min.
- Among 252 safe hyperrectangles for Model_2_68, only 2 had nonzero violation rate and all were below the 0.05% bound implied by R=99.5%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ε-ProVe instead of exact counting | Model_2_20 safe-region estimation · Model_2_20 | computation time: 234min -> 0.4s | Helped |
| Use ε-ProVe instead of exact counting | Model_2_56 safe-region estimation · Model_2_56 | computation time: 196min -> 0.3s | Helped |
| Use ε-ProVe instead of Monte Carlo sampling | Model_MN_1 safe-region estimation · Model_MN_1 | safe rate: 67.59% -> 64.72% | Hurt |
| Use ε-ProVe instead of Monte Carlo sampling | Model_MN_2 safe-region estimation · Model_MN_2 | safe rate: 100% -> 100% | No clear effect |
| Use ε-ProVe instead of Monte Carlo sampling | φ₂ ACAS Xu_2.1 safe-region estimation · φ₂ ACAS Xu_2.1 | safe rate: 99.25% -> 97.47% | Hurt |
| Use H1 instead of H5 split heuristic (ablation) | ε-ProVe · Model_2_20 | safe rate: 78.5% -> 78.91% | Helped |
| Use H2 instead of H5 split heuristic (ablation) | ε-ProVe · Model_2_20 | safe rate: 78.5% -> 79.01% | Helped |
| Use H3 instead of H5 split heuristic (ablation) | ε-ProVe · Model_2_20 | safe rate: 78.5% -> 78.63% | Helped |
| Use H4 instead of H5 split heuristic (ablation) | ε-ProVe · Model_2_20 | safe rate: 78.5% -> 78.8% | Helped |
| Use H1 instead of H5 split heuristic (ablation) | ε-ProVe · Model_MN_3 | safe rate: 52.95% -> 53.14% | Helped |
| Use H2 instead of H5 split heuristic (ablation) | ε-ProVe · Model_MN_3 | safe rate: 52.95% -> 53.08% | Helped |
| Use H1 instead of H5 split heuristic (ablation) | ε-ProVe · φ₂ ACAS Xu_2.1 | safe rate: 97.47% -> 97.32% | Hurt |
Surprising findings
- Sampling-based underestimation (Monte Carlo style) is orders of magnitude faster than exact counting while remaining within about 1-2% of exact safe rate.
Limitations
- Guarantees depend on the assumption that safe regions can be partitioned into k·ε-bounded hyperrectangles; the coverage guarantee is weaker otherwise.
- Results are underestimates and cannot certify that a region is fully safe; only probabilistic safety bounds are given.
- Evaluated only on small models and ACAS Xu / navigation networks; split heuristic choice strongly affects performance and the best heuristic is not clear for larger inputs.
Experimental setup
Datasets: ACAS Xu property specifications (phi_2, phi_3) synthetic 2D ReLU networks mapless navigation DNNs
Benchmarks: ACAS Xu (phi_2, phi_3) 2D ReLU toy models Mapless navigation DNNs
Metrics: number of safe regions safe rate runtime percentage distance from exact/MC safe rate
Baselines: exact count (ProVe) Monte Carlo sampling VeriNet
Builds on: iterative refinement / branch-and-bound verification statistical prediction of tolerance limits #DNN-Verification
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