Abstract
Deep Neural Networks (DNNs) are being deployed in a wide range of settings today, from safety-critical applications like autonomous driving to commercial applications involving image classifications. However, recent research has shown that DNNs can be brittle to even slight variations of the input data. Therefore, rigorous testing of DNNs has gained widespread attention.
While DNN robustness under norm-bound perturbation got significant attention over the past few years, our knowledge is still limited when natural variants of the input images come. These natural variants, e.g., a rotated or a rainy version of the original input, are especially concerning as they can occur naturally in the field without any active adversary and may lead to undesirable consequences. Thus, it is important to identify the inputs whose small variations may lead to erroneous DNN behaviors. The very few studies that looked at DNN’s robustness under natural variants, however, focus on estimating the overall robustness of DNNs across all the test data rather than localizing such error-producing points. This work aims to bridge this gap.
To this end, we study the local per-input robustness properties of the DNNs and leverage those properties to build a white-box (DeepRobust-W) and a black-box (DeepRobust-B) tool to automatically identify the non-robust points. Our evaluation of these methods on three DNN models spanning three widely used image classification datasets shows that they are effective in flagging points of poor robustness. In particular, DeepRobust-W and DeepRobust-B are able to achieve an F1 score of up to 91.4% and 99.1%, respectively. We further show that DeepRobust-W can be applied to a regression problem in a domain beyond image classification. Our evaluation on three self-driving car models demonstrates that DeepRobust-W is effective in identifying points of poor robustness with F1 score up to 78.9%.
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References
Chauffeur model. https://github.com/udacity/self-driving-car/tree/master/steering-models/community-models/chauffeur (2016)
Epoch model. https://github.com/udacity/self-driving-car/tree/master/steering-models/community-models/cg23 (2016)
Amershi, S., Begel, A., Bird, C., DeLine, R., Gall, H., Kamar, E., Nagappan, N., Nushi, B., Zimmermann, T.: Software engineering for machine learning: A case study. In: Proceedings of the 41st International Conference on Software Engineering: Software Engineering in Practice. pp. 291–300. ICSE-SEIP ’19, IEEE Press (2019). https://doi.org/10.1109/ICSE-SEIP.2019.00042, https://doi.org/10.1109/ICSE-SEIP.2019.00042
Balunovic, M., Baader, M., Singh, G., Gehr, T., Vechev, M.: Certifying geometric robustness of neural networks. In: Advances in Neural Information Processing Systems. pp. 15287–15297 (2019)
Bojarski, M., Del Testa, D., Dworakowski, D., Firner, B., Flepp, B., Goyal, P., Jackel, L.D., Monfort, M., Muller, U., Zhang, J., et al.: End to end learning for self-driving cars. arXiv preprint \({\rm arXiv{:}1604.07316}\) (2016)
Bojarski, M., Testa, D.D., Dworakowski, D., Firner, B., Flepp, B., Goyal, P., Jackel, L.D., Monfort, M., Muller, U., Zhang, J., Zhang, X., Zhao, J., Zieba, K.: End to end learning for self-driving cars. CoRR abs/1604.07316 (2016), http://arxiv.org/abs/1604.07316
Bunel, R., Turkaslan, I., Torr, P.H., Kohli, P., Kumar, M.P.: A unified view of piecewise linear neural network verification. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems. p. 4795–4804. NIPS’18, Curran Associates Inc., Red Hook, NY, USA (2018)
Byun, T., Sharma, V., Vijayakumar, A., Rayadurgam, S., Cofer, D.: Input prioritization for testing neural networks (01 2019)
Carlini, N., Wagner, D.: Towards evaluating the robustness of neural networks. In: Security and Privacy (SP), 2017 IEEE Symposium on. pp. 39–57. IEEE (2017)
Cohen, J.: Statistical Power Analysis for the Behavioral Sciences. Lawrence Erlbaum Associates (1988)
Du, X., Xie, X., Li, Y., Ma, L., Liu, Y., Zhao, J.: Deepstellar: Model-based quantitative analysis of stateful deep learning systems. In: Proceedings of the 2019 27th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering. p. 477–487. ESEC/FSE 2019, Association for Computing Machinery, New York, NY, USA (2019). https://doi.org/10.1145/3338906.3338954, https://doi.org/10.1145/3338906.3338954
Ehlers, R.: Formal verification of piece-wise linear feed-forward neural networks. In: International Symposium on Automated Technology for Verification and Analysis. pp. 269–286. Springer (2017)
Engstrom, L., Tran, B., Tsipras, D., Schmidt, L., Madry, A.: A rotation and a translation suffice: Fooling cnns with simple transformations. arXiv preprint \({\rm arXiv{:}1712.02779}\) (2017)
Engstrom, L., Tran, B., Tsipras, D., Schmidt, L., Madry, A.: Exploring the landscape of spatial robustness. In: International Conference on Machine Learning. pp. 1802–1811 (2019)
Engstrom, L., Tran, B., Tsipras, D., Schmidt, L., MÄ…dry, A.: A rotation and a translation suffice: Fooling cnns with simple transformations. In: Proceedings of the 36th international conference on machine learning (ICML) (2019)
Eniser, H.F., Gerasimou, S., Sen, A.: Deepfault: Fault localization for deep neural networks. In: Hähnle, R., van der Aalst, W. (eds.) Fundamental Approaches to Software Engineering. pp. 171–191. Springer International Publishing, Cham (2019)
Feinman, R., Curtin, R.R., Shintre, S., Gardner, A.B.: Detecting adversarial samples from artifacts. arXiv preprint \({\rm arXiv{:}1703.00410}\) (2017)
Gal, Y.: Uncertainty in Deep Learning (2016)
Gal, Y., Ghahramani, Z.: Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In: Balcan, M.F., Weinberger, K.Q. (eds.) Proceedings of The 33rd International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 48, pp. 1050–1059. PMLR, New York, New York, USA (20–22 Jun 2016), http://proceedings.mlr.press/v48/gal16.html
Gao, X., Saha, R., Prasad, M., Roychoudhury, A.: Fuzz testing based data augmentation to improve robustness of deep neural networks. In: Proceedings of the 42nd International Conference on Software Engineering. ICSE 2020, ACM (2020)
Gerasimou, S., Eniser, H.F., Sen, A., Çakan, A.: Importance-driven deep learning system testing. In: International Conference of Software Engineering (ICSE) (2020)
Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair,S., Courville, A., Bengio, Y.: Generative adversarial nets. In: Advances in neural information processing systems. pp. 2672–2680 (2014)
Goodfellow, I.J., Shlens, J., Szegedy, C.: Explaining and harnessing adversarial examples. In: International Conference on Learning Representations (ICLR) (2015)
Gross, D., Jansen, N., Pérez, G.A., Raaijmakers, S.: Robustness verification for classifier ensembles. In: Hung, D.V., Sokolsky, O. (eds.) Automated Technology for Verification and Analysis. pp. 271–287. Springer International Publishing, Cham (2020)
Gu, S., Rigazio, L.: Towards deep neural network architectures robust to adversarial examples. In: International Conference on Learning Representations (ICLR) (2015)
Guo, C., Gardner, J., You, Y., Wilson, A.G., Weinberger, K.: Simple black-box adversarial attacks. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 97, pp. 2484–2493. PMLR, Long Beach, California, USA (09–15 Jun 2019), http://proceedings.mlr.press/v97/guo19a.html
Guo, C., Pleiss, G., Sun, Y., Weinberger, K.Q.: On calibration of modern neural networks. In: Precup, D., Teh, Y.W. (eds.) Proceedings of the 34th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 70, pp. 1321–1330. PMLR, International Convention Centre, Sydney, Australia (06–11 Aug 2017), http://proceedings.mlr.press/v70/guo17a.html
He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 770–778 (2016)
He, P., Meister, C., Su, Z.: Structure-invariant testing for machine translation. In: International Conference of Software Engineering (ICSE) (2020)
Huang, X., Kwiatkowska, M., Wang, S., Wu, M.: Safety verification of deep neural networks. In: International Conference on Computer Aided Verification. pp. 3–29. Springer (2017)
Ilyas, A., Santurkar, S., Tsipras, D., Engstrom, L., Tran, B., Madry, A.: Adversarial examples are not bugs, they are features (2019), http://arxiv.org/abs/1905.02175
Islam, M.J., Nguyen, G., Pan, R., Rajan, H.: A comprehensive study on deep learning bug characteristics. In: Proceedings of the 2019 27th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering. pp. 510–520. ESEC/FSE 2019, Association for Computing Machinery, New York, NY, USA (2019). https://doi.org/10.1145/3338906.3338955, https://doi.org/10.1145/3338906.3338955
Jha, S., Raj, S., Fernandes, S., Jha, S.K., Jha, S., Jalaian, B., Verma, G., Swami, A.: Attribution-based confidence metric for deep neural networks. In: Advances in Neural Information Processing Systems. pp. 11826–11837 (2019)
Jiang, H., Kim, B., Gupta, M.: To trust or not to trust a classifier. In: Advances in Neural Information Processing Systems. pp. 5541–5552 (2018)
Katz, G., Barrett, C., Dill, D.L., Julian, K., Kochenderfer, M.J.: Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks, pp. 97–117. Springer International Publishing, Cham (2017)
Kim, J., Feldt, R., Yoo, S.: Guiding deep learning system testing using surprise adequacy. In: Proceedings of the 41st International Conference on Software Engineering. pp. 1039–1049. IEEE Press (2019)
Krizhevsky, A.: Learning multiple layers of features from tiny images. University of Toronto (05 2012)
Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. In: Advances in neural information processing systems. pp. 1097–1105 (2012)
Kurakin, A., Goodfellow, I., Bengio, S.: Adversarial examples in the physical world. arXiv preprint \({\rm arXiv{:}1607.02533}\) (2016)
Li, Z., Ma, X., Xu, C., Cao, C., Xu, J., Lü, J.: Boosting operational dnn testing efficiency through conditioning. In: Proceedings of the 2019 27th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering. p. 499–509. ESEC/FSE 2019, Association for Computing Machinery, New York, NY, USA (2019). https://doi.org/10.1145/3338906.3338930, https://doi.org/10.1145/3338906.3338930
Ma, L., Juefei-Xu, F., Sun, J., Chen, C., Su, T., Zhang, F., Xue, M., Li, B., Li, L., Liu, Y., et al.: Deepgauge: Comprehensive and multi-granularity testing criteria for gauging the robustness of deep learning systems. arXiv preprint \({\rm arXiv{:}1803.07519}\) (2018)
Ma, S., Liu, Y., Lee, W.C., Zhang, X., Grama, A.: Mode: automated neural network model debugging via state differential analysis and input selection. In: Proceedings of the 2018 26th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering. pp. 175–186. ACM (2018)
Ma, X., Li, B., Wang, Y., Erfani, S.M., Wijewickrema, S., Schoenebeck, G., Song, D., Houle, M.E., Bailey, J.: Characterizing adversarial subspaces using local intrinsic dimensionality. In: International Conference on Learning Representations (ICLR) (2018)
van der Maaten, L., Hinton, G.: Visualizing data using t-SNE. Journal of Machine Learning Research 9, 2579–2605 (2008), http://www.jmlr.org/papers/v9/vandermaaten08a.html
Madry, A., Makelov, A., Schmidt, L., Tsipras, D., Vladu, A.: Towards deep learning models resistant to adversarial attacks. In: International Conference on Learning Representations (ICLR) (2018)
Madry, A., Makelov, A., Schmidt, L., Tsipras, D., Vladu, A.: Towards deep learning models resistant to adversarial attacks. In: International Conference on Learning Representations (ICLR) (2018)
Mann, H.B., Whitney, D.R.: On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics 18(1), 50–60 (1947)
Mao, C., Zhong, Z., Yang, J., Vondrick, C., Ray, B.: Metric learning for adversarial robustness. In: Advances in Neural Information Processing Systems. pp. 478–489 (2019)
Metzen, J.H., Genewein, T., Fischer, V., Bischoff, B.: On detecting adversarial perturbations. In: International Conference on Learning Representations (ICLR) (2017)
Mirman, M., Gehr, T., Vechev, M.: Differentiable abstract interpretation for provably robust neural networks. In: International Conference on Machine Learning. pp. 3575–3583 (2018)
Moon, S., An, G., Song, H.O.: Parsimonious black-box adversarial attacks via efficient combinatorial optimization. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 97, pp. 4636–4645. PMLR, Long Beach, California, USA (09–15 Jun 2019), http://proceedings.mlr.press/v97/moon19a.html
Ozdag, M., Raj, S., Fernandes, S., Velasquez, A., Pullum, L., Jha, S.K.: On the susceptibility of deep neural networks to natural perturbations. In: AISafety@IJCAI (2019)
Papernot, N., McDaniel, P., Jha, S., Fredrikson, M., Celik, Z.B., Swami, A.: The limitations of deep learning in adversarial settings. In: 2016 IEEE European Symposium on Security and Privacy (EuroS&P). pp. 372–387. IEEE (2016)
Papernot, N., McDaniel, P., Wu, X., Jha, S., Swami, A.: Distillation as a defense to adversarial perturbations against deep neural networks. In: Security and Privacy (SP), 2016 IEEE Symposium on. pp. 582–597. IEEE (2016)
Pei, K., Cao, Y., Yang, J., Jana, S.: Deepxplore: Automated whitebox testing of deep learning systems. In: Proceedings of the 26th Symposium on Operating Systems Principles. pp. 1–18. ACM (2017)
Pei, K., Cao, Y., Yang, J., Jana, S.: Towards practical verification of machine learning: The case of computer vision systems. arXiv preprint \({\rm arXiv{:}1712.01785}\) (2017)
Pham, H.V., Lutellier, T., Qi, W., Tan, L.: Cradle: Cross-backend validation to detect and localize bugs in deep learning libraries. In: Proceedings of the 41st International Conference on Software Engineering. p. 1027–1038. ICSE ’19, IEEE Press (2019). https://doi.org/10.1109/ICSE.2019.00107, https://doi.org/10.1109/ICSE.2019.00107
Qiu, X., Meyerson, E., Miikkulainen, R.: Quantifying point-prediction uncertainty in neural networks via residual estimation with an i/o kernel. In: International Conference on Learning Representations (2020), https://openreview.net/forum?id=rkxNh1Stvr
Sawilowsky, S.: New effect size rules of thumb. Journal of Modern Applied Statistical Methods 8, 597–599 (11 2009). https://doi.org/10.22237/jmasm/1257035100
Saxena, U.: Automold. https://github.com/UjjwalSaxena/Automold--Road-Augmentation-Library/
Sen, K., Marinov, D., Agha, G.: CUTE: A concolic unit testing engine for C. In: FSE (2005)
Seshia, S.A., Desai, A., Dreossi, T., Fremont, D.J., Ghosh, S., Kim, E., Shivakumar, S., Vazquez-Chanlatte, M., Yue, X.: Formal specification for deep neural networks. In: International Symposium on Automated Technology for Verification and Analysis. pp. 20–34. Springer (2018)
Shaham, U., Yamada, Y., Negahban, S.: Understanding adversarial training: Increasing local stability of neural nets through robust optimization. arXiv preprint \({\rm arXiv{:}1511.05432}\) (2015)
Shankar, V., Dave, A., Roelofs, R., Ramanan, D., Recht, B., Schmidt, L.: A systematic framework for natural perturbations from videos (06 2019)
Silver, D., Huang, A., Maddison, C.J., Guez, A., Sifre, L., van den Driessche,G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., Dieleman, S., Grewe, D., Nham, J., Kalchbrenner, N., Sutskever, I., Lillicrap, T., Leach, M., Kavukcuoglu, K., Graepel, T., Hassabis, D.: Mastering the game of go with deep neural networks and tree search. Nature 529, 484–503 (2016), http://www.nature.com/nature/journal/v529/n7587/full/nature16961.html
Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. In: International Conference on Learning Representations (ICLR) (2015)
SIMPSON, E.H.: Measurement of diversity. Nature 163(4148), 688–688 (1949), https://doi.org/10.1038/163688a0
Stocco, A., Weiss, M., Calzana, M., Tonella, P.: Misbehaviour prediction for autonomous driving systems. In: Proceedings of 42nd International Conference on Software Engineering. p. 12 pages. ICSE ’20, ACM (2020)
Stocco, A., Weiss, M., Calzana, M., Tonella, P.: Misbehaviour prediction for autonomous driving systems. In: International Conference of Software Engineering (ICSE) (2020)
Sun, Y., Wu, M., Ruan, W., Huang, X., Kwiatkowska, M., Kroening, D.: Concolic testing for deep neural networks (2018)
Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I., Fergus, R.: Intriguing properties of neural networks. In: International Conference on Learning Representations (ICLR) (2014)
Teye, M., Azizpour, H., Smith, K.: Bayesian uncertainty estimation for batch normalized deep networks. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 4907–4916. PMLR, Stockholmsmässan, Stockholm Sweden (10–15 Jul 2018), http://proceedings.mlr.press/v80/teye18a.html
Tian, Y., Pei, K., Jana, S., Ray, B.: Deeptest: Automated testing of deep-neural-network-driven autonomous cars. In: International Conference of Software Engineering (ICSE), 2018 IEEE conference on. IEEE (2018)
Tian, Y., Zhong, Z., Ordonez, V., Kaiser, G., Ray, B.: Testing dnn image classifier for confusion & bias errors. In: International Conference of Software Engineering (ICSE) (2020)
Tramèr, F., Kurakin, A., Papernot, N., Goodfellow, I., Boneh, D., McDaniel, P.: Ensemble adversarial training: Attacks and defenses. arXiv preprint \({\rm arXiv{:}1705.07204}\) (2017)
Tsipras, D., Santurkar, S., Engstrom, L., Turner, A., Madry, A.: Robustness may be at odds with accuracy. In: International Conference on Learning Representations (ICLR) (2019)
Udacity: A self-driving car simulator built with Unity. https://github.com/udacity/self-driving-car-sim (2017), online; accessed 18 August 2019
Udeshi, S., Jiang, X., Chattopadhyay, S.: Callisto: Entropy-based test generation and data quality assessment for machine learning systems. In: 2020 IEEE 13th International Conference on Software Testing, Validation and Verification (ICST). pp. 448–453 (2020)
Wang, F., Jiang, M., Qian, C., Yang, S., Li, C., Zhang, H., Wang, X., Tang, X.: Residual attention network for image classification. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 3156–3164 (2017)
Wang, J., Dong, G., Sun, J., Wang, X., Zhang, P.: Adversarial sample detection for deep neural network through model mutation testing. In: Proceedings of the 41st International Conference on Software Engineering. p. 1245–1256. ICSE ’19, IEEE Press (2019). https://doi.org/10.1109/ICSE.2019.00126, https://doi.org/10.1109/ICSE.2019.00126
Wang, S., Chen, Y., Abdou, A., Jana, S.: Mixtrain: Scalable training of formally robust neural networks. arXiv preprint \({\rm arXiv{:}1811.02625}\) (2018)
Wang, S., Pei, K., Whitehouse, J., Yang, J., Jana, S.: Efficient formal safety analysis of neural networks. In: Proceedings of the 32Nd International Conference on Neural Information Processing Systems. pp. 6369–6379. NIPS’18, Curran Associates Inc., USA (2018), http://dl.acm.org/citation.cfm?id=3327345.3327533
Wang, S., Pei, K., Whitehouse, J., Yang, J., Jana, S.: Formal security analysis of neural networks using symbolic intervals. USENIX Security Symposium (2018)
Wong, E., Schmidt, F., Metzen, J.H., Kolter, J.Z.: Scaling provable adversarial defenses. In: Advances in Neural Information Processing Systems. pp. 8400–8409 (2018)
Xiao, C., Li, B., Zhu, J.Y., He, W., Liu, M., Song, D.: Generating adversarial examples with adversarial networks. In: 27th International Joint Conference on Artificial Intelligence (IJCAI) (2018)
Xiao, C., Zhu, J.Y., Li, B., He, W., Liu, M., Song, D.: Spatially transformed adversarial examples. In: International Conference on Learning Representations (ICLR) (2018)
Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms (2017)
Yang, F., Wang, Z., Heinze-Deml, C.: Invariance-inducing regularization using worst-case transformations suffices to boost accuracy and spatial robustness. In: Advances in Neural Information Processing Systems 32. pp. 14757–14768 (2019)
Yuval Netzer, T.W., Coates, A., Bissacco, A., Wu, B., Ng, A.Y.: Reading digits in natural images with unsupervised feature learning. In: NIPS Workshop on Deep Learning and Unsupervised Feature Learning (2011)
Zagoruyko, S., Komodakis, N.: Wide residual networks. In: BMVC (2016)
Zhang, H., Chan, W.K.: Apricot: A weight-adaptation approach to fixing deep learning models. In: 2019 34th IEEE/ACM International Conference on Automated Software Engineering (ASE). pp. 376–387 (Nov 2019). https://doi.org/10.1109/ASE.2019.00043
Zhang, M., Zhang, Y., Zhang, L., Liu, C., Khurshid, S.: Deeproad: Gan-based metamorphic autonomous driving system testing. arXiv preprint \({\rm arXiv{:}1802.02295}\) (2018)
Zhao, Z., Dua, D., Singh, S.: Generating natural adversarial examples. In: International Conference on Learning Representations (ICLR) (2018)
Zhou, H., Li, W., Kong, Z., Guo, J., Zhang, Y., Zhang, L., Yu, B., Liu, C.: Deepbillboard: Systematic physical-world testing of autonomous driving systems. In: International Conference of Software Engineering (ICSE) (2020)
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Zhong, Z., Tian, Y., Ray, B. (2021). Understanding Local Robustness of Deep Neural Networks under Natural Variations. In: Guerra, E., Stoelinga, M. (eds) Fundamental Approaches to Software Engineering. FASE 2021. Lecture Notes in Computer Science(), vol 12649. Springer, Cham. https://doi.org/10.1007/978-3-030-71500-7_16
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