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Syntactic vs Semantic Linear Abstraction and Refinement of Neural Networks

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Automated Technology for Verification and Analysis (ATVA 2023)


Abstraction is a key verification technique to improve scalability. However, its use for neural networks is so far extremely limited. Previous approaches for abstracting classification networks replace several neurons with one of them that is similar enough. We can classify the similarity as defined either syntactically (using quantities on the connections between neurons) or semantically (on the activation values of neurons for various inputs). Unfortunately, the previous approaches only achieve moderate reductions, when implemented at all. In this work, we provide a more flexible framework, where a neuron can be replaced with a linear combination of other neurons, improving the reduction. We apply this approach both on syntactic and semantic abstractions, and implement and evaluate them experimentally. Further, we introduce a refinement method for our abstractions, allowing for finding a better balance between reduction and precision.

This research was funded in part by the German Research Foundation (DFG) project 427755713 GoPro, the German Federal Ministry of Education and Research (BMBF) within the project SEMECO Q1 (03ZU1210AG), and the DFG research training group ConVeY (GRK 2428).

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  1. 1.

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  2. 2.


  1. Altschuler, J., et al.: Greedy column subset selection: new bounds and distributed algorithms. In: Balcan, M., Weinberger, K.Q. (eds.) Proceedings of the 33nd International Conference on Machine Learning, ICML, New York City, NY, USA, vol. 48, pp. 2539–2548. JMLR Workshop and Conference Proceedings. (2016)

    Google Scholar 

  2. Ashok, P., Hashemi, V., Křetínský, J., Mohr, S.: DeepAbstract: neural network abstraction for accelerating verification. In: Hung, D.V., Sokolsky, O. (eds.) ATVA 2020. LNCS, vol. 12302, pp. 92–107. Springer, Cham (2020).

    Chapter  Google Scholar 

  3. Bishop, C.M.: Pattern Recognition and Machine Learning. Information Science and Statistics, 5th edn. Springer, Cham (2007)

    MATH  Google Scholar 

  4. Brix, C., et al.: First three years of the international verification of neural networks competition (VNN-COMP). Int. J. Softw. Tools Technol. Transfer 1–11 (2023).

  5. Caruana, R., Lawrence, S., Giles, C.: Overfitting in neural nets: backpropagation, conjugate gradient, and early stopping. In: Leen, T., Dietterich, T., Tresp, V. (eds.) Advances in Neural Information Processing Systems, vol. 13. MIT Press (2000)

    Google Scholar 

  6. Chau, C., Křetínskỳ, J., Mohr, S.: Syntactic vs semantic linear abstraction and refinement of neural networks (2023)

    Google Scholar 

  7. Cheng, Y., Wang, D., Zhou, P., Zhang, T.: A survey of model compression and acceleration for deep neural networks. Preprint arXiv:1710.09282 (2017)

  8. Elboher, Y.Y., Gottschlich, J., Katz, G.: An abstraction-based framework for neural network verification. In: Lahiri, S.K., Wang, C. (eds.) CAV 2020. LNCS, vol. 12224, pp. 43–65. Springer, Cham (2020).

    Chapter  MATH  Google Scholar 

  9. Farahat, A.K., Ghodsi, A., Kamel, M.S.: A fast greedy algorithm for generalized column subset selection. Preprint arXiv:1312.6820 (2013)

  10. Farahat, A.K., Ghodsi, A., Kamel, M.S.: An efficient greedy method for unsupervised feature selection. In: 11th International Conference on Data Mining, Vancouver, BC, Canada, pp. 161–170. IEEE (2011)

    Google Scholar 

  11. Fazlyab, M., et al.: Efficient and accurate estimation of Lipschitz constants for deep neural networks. In: Wallach, H., et al. (eds.) Advances in Neural Information Processing Systems, vol. 32. Curran Associates Inc. (2019)

    Google Scholar 

  12. Gong, Y., Liu, L., Yang, M., Bourdev, L.: Compressing deep convolutional networks using vector quantization. Preprint arXiv:1412.6115 (2014)

  13. Hinton, G., Vinyals, O., Dean, J., et al.: Distilling the knowledge in a neural network. In: NeurIPS Deep Learning Workshop (2014)

    Google Scholar 

  14. Huang, G., Liu, Z., van der Maaten, L., Weinberger, K.Q.: Densely connected convolutional networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4700–4708 (2017)

    Google Scholar 

  15. Jian, X., Jinyu, L., Yifan, G.: Restructuring of deep neural network acoustic models with singular value decomposition. In: Interspeech, pp. 2365–2369 (2013).

  16. 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).

    Chapter  Google Scholar 

  17. Kirkwood, J.R., Kirkwood, B.H.: Elementary Linear Algebra. Chapman and Hall/CRC (2017)

    Google Scholar 

  18. Krizhevsky, A., Hinton, G., et al.: Learning multiple layers of features from tiny images (2009)

    Google Scholar 

  19. Lawrence, S., Giles, C., Tsoi, A.: Lessons in neural network training: overfitting may be harder than expected. In: Anon (ed.) Proceedings of the National Conference on Artificial Intelligence, pp. 540–545. AAAI (1997)

    Google Scholar 

  20. LeCun, Y.: The MNIST database of handwritten digits (1998).

  21. Maas, A.L., Hannun, A.Y., Ng, A.Y.: Rectifier nonlinearities improve neural network acoustic models.

  22. Prabhakar, P.: Bisimulations for neural network reduction. In: Finkbeiner, B., Wies, T. (eds.) VMCAI 2022. LNCS, vol. 13182, pp. 285–300. Springer, Cham (2022).

    Chapter  Google Scholar 

  23. Prabhakar, P., Rahimi Afzal, Z.: Abstraction based output range analysis for neural networks. In: Wallach, H., et al. (eds.) Advances in Neural Information Processing Systems, vol. 32. Curran Associates Inc. (2019)

    Google Scholar 

  24. Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015).

    Article  Google Scholar 

  25. Shitov, Y.: Column subset selection is NP-complete. Linear Algebra Appl. 610, 52–58 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  26. Singh, G., Gehr, T., Püschel, M., Vechev, M.: An abstract domain for certifying neural networks. Proc. ACM Program. Lang. 3(POPL) (2019).

  27. Singh, G., Gehr, T., Püschel, M., Vechev, M.T.: Boosting robustness certification of neural networks. In: 7th International Conference on Learning Representations, ICLR, New Orleans, LA, USA. (2019)

    Google Scholar 

  28. Sotoudeh, M., Thakur, A.V.: Abstract neural networks. In: Pichardie, D., Sighireanu, M. (eds.) SAS 2020. LNCS, vol. 12389, pp. 65–88. Springer, Cham (2020).

    Chapter  Google Scholar 

  29. Tran, H.-D., et al.: Robustness verification of semantic segmentation neural networks using relaxed reachability. In: Silva, A., Leino, K.R.M. (eds.) CAV 2021. LNCS, vol. 12759, pp. 263–286. Springer, Cham (2021).

    Chapter  Google Scholar 

  30. Virmaux, A., Scaman, K.: Lipschitz regularity of deep neural networks: analysis and efficient estimation. In: Bengio, S., et al. (eds.) Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems, NeurIPS, Montréal, Canada, pp. 3839–3848 (2018)

    Google Scholar 

  31. Wang, S., et al.: Beta-CROWN: efficient bound propagation with per-neuron split constraints for neural network robustness verification. In: Ranzato, M., et al. (eds.) Advances in Neural Information Processing Systems, vol. 34, pp. 29909–29921. Curran Associates Inc. (2021)

    Google Scholar 

  32. Xiao, H., Rasul, K., Vollgraf, R.: Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. Preprint arXiv:1708.07747 (2017)

  33. Xu, K., et al.: Fast and complete: enabling complete neural network verification with rapid and massively parallel incomplete verifiers. In: International Conference on Learning Representations (2021)

    Google Scholar 

  34. Zhang, C., et al.: Understanding deep learning requires rethinking generalization. CoRR, abs/1611.03530 (2016).

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Chau, C., Křetínský, J., Mohr, S. (2023). Syntactic vs Semantic Linear Abstraction and Refinement of Neural Networks. In: André, É., Sun, J. (eds) Automated Technology for Verification and Analysis. ATVA 2023. Lecture Notes in Computer Science, vol 14215. Springer, Cham.

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