Abstract
We present POLAR (The source code can be found at https://github.com/ChaoHuang2018/POLAR_Tool. The full version of this paper can be found at https://arxiv.org/abs/2106.13867.), a POLynomial ARithmetic-based framework for efficient time-bounded reachability analysis of neural-network controlled systems. Existing approaches leveraging the standard Taylor Model (TM) arithmetic for approximating the neural-network controller cannot deal with non-differentiable activation functions and suffer from rapid explosion of the remainder when propagating TMs. POLAR overcomes these shortcomings by integrating TM arithmetic with Bernstein polynomial interpolation and symbolic remainders. The former enables TM propagation across non-differentiable activation functions and local refinement of TMs, and the latter reduces error accumulation in the TM remainder for linear mappings in the neural network. Experimental results show POLAR significantly outperforms the state-of-the-art tools on both efficiency and tightness of the reachable set overapproximation.
C. Huang—Part of the work was done when the author was in Northwestern University, US.
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Notes
- 1.
The results of ReachNN* are based on GPU acceleration.
- 2.
These are lower bounds on the improvements since other tools terminated early for certain settings due to explosion of their computed flowpipes.
References
Althoff, M.: An introduction to CORA 2015. In: International Workshop on Applied veRification for Continuous and Hybrid Systems (ARCH). EPiC Series in Computing, vol. 34, pp. 120–151 (2015)
Alur, R., Dill, D.L.: A theory of timed automata. Theoret. Comput. Sci. 126(2), 183–235 (1994)
Beard, R.: Quadrotor dynamics and control rev 0.1 (2008)
Berz, M., Makino, K.: Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliable Comput. 4, 361–369 (1998). https://doi.org/10.1023/A:1024467732637
Chen, X.: Reachability analysis of non-linear hybrid systems using taylor models. Ph.D. thesis, RWTH Aachen University (2015)
Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_18
Chen, X., Sankaranarayanan, S.: Decomposed reachability analysis for nonlinear systems. In: Proceedings of RTSS 2016, pp. 13–24 (2016)
Dutta, S., Chen, X., Sankaranarayanan, S.: Reachability analysis for neural feedback systems using regressive polynomial rule inference. In: Proceedings of HSCC 2019, pp. 157–168. ACM (2019)
Fan, J., Huang, C., Chen, X., Li, W., Zhu, Q.: ReachNN*: a tool for reachability analysis of neural-network controlled systems. In: Hung, D.V., Sokolsky, O. (eds.) ATVA 2020. LNCS, vol. 12302, pp. 537–542. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-59152-6_30
Frehse, G., Le Guernic, C., Donzé, A., Cotton, S., Ray, R., Lebeltel, O., Ripado, R., Girard, A., Dang, T., Maler, O.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_30
Huang, C., Fan, J., Li, W., Chen, X., Zhu, Q.: ReachNN: reachability analysis of neural-network controlled systems. ACM Trans. Embed. Comput. Syst. 18(5s), 106:1-106:22 (2019)
Huang, C., Fan, J., Li, W., Chen, X., Zhu, Q.: Divide and slide: layer-wise refinement for output range analysis of deep neural networks. IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. (TCAD) 39(11), 3323–3335 (2020)
Ivanov, R., Carpenter, T., Weimer, J., Alur, R., Pappas, G., Lee, I.: Verisig 2.0: verification of neural network controllers using Taylor model preconditioning. In: Silva, A., Leino, K.R.M. (eds.) CAV 2021. LNCS, vol. 12759, pp. 249–262. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-81685-8_11
Ivanov, R., Weimer, J., Alur, R., Pappas, G.J., Lee, I.: Verisig: verifying safety properties of hybrid systems with neural network controllers. In: Proceedings of HSCC 2018, pp. 169–178. ACM (2019)
Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Interval Analysis. Applied Interval Analysis, Springer, Cham (2001). https://doi.org/10.1007/978-1-4471-0249-6_2
Katz, G., Barrett, C., Dill, D.L., Julian, K., Kochenderfer, M.J.: Reluplex: an efficient SMT solver for verifying deep neural networks. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10426, pp. 97–117. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63387-9_5
Levine, S., Finn, C., Darrell, T., Abbeel, P.: End-to-end training of deep visuomotor policies. J. Mach. Learn. Res. 17(1), 1334–1373 (2016)
Lorentz, G.G.: Bernstein Polynomials. American Mathematical Society (2013)
Makino, K., Berz, M.: Taylor models and other validated functional inclusion methods. Int. J. Pure Appl. Math. 4(4), 379–456 (2003)
Meiss, J.D.: Differential Dynamical Systems. SIAM publishers (2007)
Mnih, V., et al.: Human-level control through deep reinforcement learning. Nature 518(7540), 529–533 (2015)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM (2009)
Prajna, S., Parrilo, P.A., Rantzer, A.: Nonlinear control synthesis by convex optimization. IEEE Trans. Autom. Control 49(2), 310–314 (2004)
Singh, G., Ganvir, R., Püschel, M., Vechev, M.T.: Beyond the single neuron convex barrier for neural network certification. In: Proceedings of NeurIPS 2019, pp. 15072–15083 (2019)
Tran, H.-D., et al.: NNV: the neural network verification tool for deep neural networks and learning-enabled cyber-physical systems. In: Lahiri, S.K., Wang, C. (eds.) CAV 2020. LNCS, vol. 12224, pp. 3–17. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-53288-8_1
Wang, Z., Huang, C., Zhu, Q.: Efficient global robustness certification of neural networks via interleaving twin-network encoding. In: Proceedings of DAT2 2022 (2022)
Weng, T.W., et al.: Towards fast computation of certified robustness for relu networks. In: Proceedings of ICML 2018 (2018)
Zhang, H., Weng, T.W., Chen, P.Y., Hsieh, C.J., Daniel, L.: Efficient neural network robustness certification with general activation functions. In: Proceedings of NeurIPS 2018, pp. 4944–4953 (2018)
Zhu, Q., et al.: Safety-assured design and adaptation of learning-enabled autonomous systems. In: Proceedings of ASPDAC 2021 (2021)
Zhu, Q., et al.: Know the unknowns: addressing disturbances and uncertainties in autonomous systems. In: Proceedings of ICCAD 2020 (2020)
Acknowledgement
We gratefully acknowledge the support from the National Science Foundation awards CCF-1646497, CCF-1834324, CNS-1834701, CNS-1839511, IIS-1724341, CNS-2038853, ONR grant N00014-19-1-2496, and the US Air Force Research Laboratory (AFRL) under contract number FA8650-16-C-2642.
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Huang, C., Fan, J., Chen, X., Li, W., Zhu, Q. (2022). POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems. In: Bouajjani, A., Holík, L., Wu, Z. (eds) Automated Technology for Verification and Analysis. ATVA 2022. Lecture Notes in Computer Science, vol 13505. Springer, Cham. https://doi.org/10.1007/978-3-031-19992-9_27
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