Abstract
The use of ML technology to design safety-critical systems requires a complete understanding of the neural network’s properties. Among the relevant properties in an industrial context, the verification of partial monotony may become mandatory. This paper proposes a method to evaluate the monotony property using a Mixed Integer Linear Programming (MILP) solver. Contrary to the existing literature, this monotony analysis provides a lower and upper bound of the space volume where the property does not hold, that we denote “Non-Monotonic Space Coverage”. This work has several advantages: (i) our formulation of the monotony property works on discrete inputs, (ii) the iterative nature of our algorithm allows for refining the analysis as needed, and (iii) from an industrial point of view, the results of this evaluation are valuable to the aeronautical domain where it can support the certification demonstration. We applied this method to an avionic case study (braking distance estimation using a neural network) where the verification of the monotony property is of paramount interest from a safety perspective.
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Notes
- 1.
Note that we simplify the crosshatched area’s shape in order to know the omega value for the explanation.
References
Biannic, J., Hardier, G., Roos, C., Seren, C., Verdier, L.: Surrogate models for aircraft flight control: some off-line and embedded applications. Aerospace Lab (12), 1 (2016)
Carlini, N., Wagner, D.A.: Towards evaluating the robustness of neural networks. In: IEEE SP, pp. 39–57. IEEE Computer Society (2017)
Cheng, C.-H., Nührenberg, G., Ruess, H.: Maximum resilience of artificial neural networks. In: D’Souza, D., Narayan Kumar, K. (eds.) ATVA 2017. LNCS, vol. 10482, pp. 251–268. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68167-2_18
Damour, M., et al.: Towards certification of a reduced footprint ACAS-Xu system: a hybrid ML-based solution. In: Habli, I., Sujan, M., Bitsch, F. (eds.) SAFECOMP 2021. LNCS, vol. 12852, pp. 34–48. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-83903-1_3
EASA: Concept paper first usable guidance for level 1 machine learning applications (2021). https://www.easa.europa.eu/downloads/134357/en
Feelders, A.J.: Prior knowledge in economic applications of data mining. In: Zighed, D.A., Komorowski, J., Żytkow, J. (eds.) PKDD 2000. LNCS (LNAI), vol. 1910, pp. 395–400. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45372-5_42
Gauffriau, A., Malgouyres, F., Ducoffe, M.: Overestimation learning with guarantees. arXiv preprint arXiv:2101.11717 (2021)
Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3(3), 227–252 (2002)
Gupta, A., Shukla, N., Marla, L., Kolbeinsson, A., Yellepeddi, K.: How to incorporate monotonicity in deep networks while preserving flexibility? arXiv preprint arXiv:1909.10662 (2019)
Gurobi Optimization, LLC: Gurobi Optimizer Reference Manual (2022). https://www.gurobi.com
Hao, J., Ye, W., Jia, L., Wang, G., Allen, J.: Building surrogate models for engineering problems by integrating limited simulation data and monotonic engineering knowledge. Adv. Eng. Inform. 49, 101342 (2021)
Jian, Z.D., Chang, H.J., Hsu, T.S., Wang, D.W.: Learning from simulated world - surrogates construction with deep neural network. In: SIMULTECH 2017: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications. SCITEPRESS (2017)
Karpf, J.: Inductive modelling in law: example based expert systems in administrative law. In: Proceedings of the 3rd International Conference on Artificial Intelligence and Law, pp. 297–306 (1991)
Katz, G., et al.: The marabou framework for verification and analysis of deep neural networks. In: Dillig, I., Tasiran, S. (eds.) CAV 2019. LNCS, vol. 11561, pp. 443–452. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-25540-4_26
Liu, X., Han, X., Zhang, N., Liu, Q.: Certified monotonic neural networks. Adv. Neural. Inf. Process. Syst. 33, 15427–15438 (2020)
Madry, A., Makelov, A., Schmidt, L., Tsipras, D., Vladu, A.: Towards deep learning models resistant to adversarial attacks. In: ICLR. OpenReview.net (2018)
Mamalet, F., et al.: White paper machine learning in certified systems. IRT Saint Exupéry - ANITI (2021)
Müller, M.N., Makarchuk, G., Singh, G., Püschel, M., Vechev, M.: PRIMA: general and precise neural network certification via scalable convex hull approximations. Proc. ACM Program. Lang. 6(POPL), 1–33 (2022)
Nguyen, A.P., Martínez, M.R.: Mononet: towards interpretable models by learning monotonic features. arXiv preprint arXiv:1909.13611 (2019)
Peterson, E., DeVore, M., Cooper, J., Carr, G.: Run time assurance as an alternate concept to contemporary development assurance processes. NASA/CR-2020-220586 (2020)
Raghunathan, A., Steinhardt, J., Liang, P.S.: Semidefinite relaxations for certifying robustness to adversarial examples. In: Advances in Neural Information Processing Systems, pp. 10877–10887 (2018)
Sudakov, O., Koroteev, D., Belozerov, B., Burnaev, E.: Artificial neural network surrogate modeling of oil reservoir: a case study. In: Lu, H., Tang, H., Wang, Z. (eds.) ISNN 2019. LNCS, vol. 11555, pp. 232–241. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-22808-8_24
Tjeng, V., Xiao, K.Y., Tedrake, R.: Evaluating robustness of neural networks with mixed integer programming. In: ICLR (2019)
Tsuzuku, Y., Sato, I., Sugiyama, M.: Lipschitz-margin training: scalable certification of perturbation invariance for deep neural networks. In: NeurIPS, pp. 6542–6551 (2018)
Urban, C., Christakis, M., Wüstholz, V., Zhang, F.: Perfectly parallel fairness certification of neural networks. Proc. ACM Program. Lang. 4(OOPSLA), 1–30 (2020)
Urban, C., Miné, A.: A review of formal methods applied to machine learning. arXiv preprint arXiv:2104.02466 (2021)
Wang, S., Pei, K., Whitehouse, J., Yang, J., Jana, S.: Formal security analysis of neural networks using symbolic intervals. In: 27th USENIX Security Symposium (USENIX Security 2018), Baltimore, MD, pp. 1599–1614. USENIX Association, August 2018
Wang, S., et al.: Beta-CROWN: efficient bound propagation with per-neuron split constraints for neural network robustness verification. In: Advances in Neural Information Processing Systems (2021)
Weng, T.W., et al.: Towards fast computation of certified robustness for ReLU networks. arXiv preprint arXiv:1804.09699 (2018)
Xiang, W., Tran, H.D., Johnson, T.T.: Output reachable set estimation and verification for multilayer neural networks. IEEE Trans. Neural Netw. Learn. Syst. 29(11), 5777–5783 (2018)
Xu, K., et al.: Automatic perturbation analysis for scalable certified robustness and beyond. In: NeurIPS (2020)
Zhang, H., Weng, T.W., Chen, P.Y., Hsieh, C.J., Daniel, L.: Efficient neural network robustness certification with general activation functions. In: Advances in Neural Information Processing Systems, pp. 4939–4948 (2018)
Zhang, H., Zhang, P., Hsieh, C.J.: Recurjac: an efficient recursive algorithm for bounding jacobian matrix of neural networks and its applications. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 33, pp. 5757–5764 (2019)
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Vidot, G., Ducoffe, M., Gabreau, C., Ober, I., Ober, I. (2022). Formal Monotony Analysis of Neural Networks with Mixed Inputs: An Asset for Certification. In: Groote, J.F., Huisman, M. (eds) Formal Methods for Industrial Critical Systems. FMICS 2022. Lecture Notes in Computer Science, vol 13487. Springer, Cham. https://doi.org/10.1007/978-3-031-15008-1_3
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