# Analysis of classifiers’ robustness to adversarial perturbations

- 1k Downloads
- 3 Citations

## Abstract

The goal of this paper is to analyze the intriguing instability of classifiers to adversarial perturbations (Szegedy et al., in: International conference on learning representations (ICLR), 2014). We provide a theoretical framework for analyzing the robustness of classifiers to adversarial perturbations, and show fundamental upper bounds on the robustness of classifiers. Specifically, we establish a general upper bound on the robustness of classifiers to adversarial perturbations, and then illustrate the obtained upper bound on two practical classes of classifiers, namely the linear and quadratic classifiers. In both cases, our upper bound depends on a *distinguishability* measure that captures the notion of *difficulty* of the classification task. Our results for both classes imply that in tasks involving small distinguishability, *no classifier* in the considered set will be robust to adversarial perturbations, even if a good accuracy is achieved. Our theoretical framework moreover suggests that the phenomenon of adversarial instability is due to the low flexibility of classifiers, compared to the difficulty of the classification task (captured mathematically by the distinguishability measure). We further show the existence of a clear distinction between the robustness of a classifier to random noise and its robustness to adversarial perturbations. Specifically, the former is shown to be larger than the latter by a factor that is proportional to \(\sqrt{d}\) (with *d* being the signal dimension) for linear classifiers. This result gives a theoretical explanation for the discrepancy between the two robustness properties in high dimensional problems, which was empirically observed by Szegedy et al. in the context of neural networks. We finally show experimental results on controlled and real-world data that confirm the theoretical analysis and extend its spirit to more complex classification schemes.

## Keywords

Adversarial examples Classification robustness Random noise Instability Deep networks## Notes

### Acknowledgements

We thank the anonymous reviewers for their detailed comments. We thank Hamza Fawzi, Ian Goodfellow for discussions and comments on an early draft of the paper, and Guillaume Aubrun for pointing out a reference for Theorem 4. We also thank Seyed Mohsen Moosavi for his help in preparing experiments.

## References

- Barreno, M., Nelson, B., Sears, R., Joseph, A., & Tygar, D. (2006). Can machine learning be secure? In
*ACM symposium on information, computer and communications security*(pp. 16–25).Google Scholar - Bendale, A., & Boult, T. E. (2016). Towards open set deep networks. In
*Proceedings of the IEEE conference on computer vision and pattern recognition*(pp. 1563–1572).Google Scholar - Bhatia, R. (2013).
*Matrix analysis*(Vol. 169). Berlin: Springer.MATHGoogle Scholar - Biggio, B., Corona, I., Maiorca, D., Nelson, B., Šrndić, N., Laskov, P., et al. (2013). Evasion attacks against machine learning at test time. In
*Joint European conference on machine learning and knowledge discovery in databases*(pp. 387–402). Berlin: Springer.Google Scholar - Biggio, B., Nelson, B., & Laskov, P. (2012). Poisoning attacks against support vector machines. In
*International conference on machine learning (ICML)*.Google Scholar - Bousquet, O., & Elisseeff, A. (2002). Stability and generalization.
*The Journal of Machine Learning Research*,*2*, 499–526.MathSciNetMATHGoogle Scholar - Caramanis, C., Mannor, S., & Xu, H. (2012). Robust optimization in machine learning. In S. Sra, S. Nowozin, & S. J. Wright (Eds.),
*Optimization for machine learning*. Cambridge: MIT Press. chap 14.Google Scholar - Carlini, N., & Wagner, D. (2016).
*Towards evaluating the robustness of neural networks*. arXiv preprint arXiv:1608.04644. - Chalupka, K., Perona, P., & Eberhardt, F. (2014).
*Visual causal feature learning*. arXiv preprint arXiv:1412.2309. - Chang, C. C., & Lin, C. J. (2011). LIBSVM: A library for support vector machines.
*ACM Transactions on Intelligent Systems and Technology*,*2*, 27:1–27:27.CrossRefGoogle Scholar - Chang, Y. W., Hsieh, C. J., Chang, K. W., Ringgaard, M., & Lin, C. J. (2010). Training and testing low-degree polynomial data mappings via linear SVM.
*The Journal of Machine Learning Research*,*11*, 1471–1490.MathSciNetMATHGoogle Scholar - Dalvi, N., Domingos, P., Sanghai, S., & Verma, D. (2004). Adversarial classification. In
*ACM SIGKDD*(pp. 99–108).Google Scholar - Dekel, O., Shamir, O., & Xiao, L. (2010). Learning to classify with missing and corrupted features.
*Machine Learning*,*81*(2), 149–178.MathSciNetCrossRefGoogle Scholar - Fan, R. W., Chang, K. W., Hsieh, C. J., Wang, X. R., & Lin, C. J. (2008). Liblinear: A library for large linear classification.
*The Journal of Machine Learning Research*,*9*, 1871–1874.MATHGoogle Scholar - Fawzi, A., & Frossard, P. (2015) Manitest: Are classifiers really invariant? In
*British machine vision conference (BMVC)*(pp. 106.1–106.13).Google Scholar - Goldberg, Y., & Elhadad, M. (2008). splitsvm: Fast, space-efficient, non-heuristic, polynomial kernel computation for nlp applications. In
*46th Annual meeting of the association for computational linguistics on human language technologies: Short papers*(pp. 237–240).Google Scholar - Goodfellow, I. (2015).
*Adversarial examples*. http://www.iro.umontreal.ca/~memisevr/dlss2015/goodfellow_adv.pdf, presentation at the Deep Learning Summer School, Montreal. - Goodfellow, I., Shlens, J., & Szegedy, C. (2015). Explaining and harnessing adversarial examples. In
*International conference on learning representations*.Google Scholar - Gu, S., & Rigazio, L. (2014).
*Towards deep neural network architectures robust to adversarial examples*. arXiv preprint arXiv:1412.5068. - Krizhevsky, A., & Hinton, G. (2009).
*Learning multiple layers of features from tiny images*. Master’s thesis, Department of Computer Science, University of Toronto.Google Scholar - Lanckriet, G., Ghaoui, L., Bhattacharyya, C., & Jordan, M. (2003). A robust minimax approach to classification.
*The Journal of Machine Learning Research*,*3*, 555–582.MathSciNetMATHGoogle Scholar - LeCun, Y., Bottou, L., Bengio, Y., & Haffner, P. (1998). Gradient-based learning applied to document recognition.
*Proceedings of the IEEE*,*86*(11), 2278–2324.CrossRefGoogle Scholar - Lewis, A., & Pang, J. (1998). Error bounds for convex inequality systems. In J.-P. Crouzeix, J.-E. Martinez-Legaz, & M. Volle (Eds.),
*Generalized convexity, generalized monotonicity: Recent results*(pp. 75–110). Berlin: Springer.Google Scholar - Li, G., Mordukhovich, B. S., & Pham, T. S. (2015). New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors.
*Mathematical Programming*,*153*(2), 333–362.Google Scholar - Łojasiewicz, S. (1961).
*Sur le probleme de la division*(to complete).Google Scholar - Lowe, D. (2004). Distinctive image features from scale-invariant keypoints.
*International Journal of Computer Vision*,*60*(2), 91–110.CrossRefGoogle Scholar - Lugosi, G., & Pawlak, M. (1994). On the posterior-probability estimate of the error rate of nonparametric classification rules.
*IEEE Transactions on Information Theory*,*40*(2), 475–481.MathSciNetCrossRefMATHGoogle Scholar - Luo, X., & Luo, Z. (1994). Extension of Hoffman’s error bound to polynomial systems.
*SIAM Journal on Optimization*,*4*(2), 383–392.MathSciNetCrossRefMATHGoogle Scholar - Luo, Z. Q., & Pang, J. S. (1994). Error bounds for analytic systems and their applications.
*Mathematical Programming*,*67*(1–3), 1–28.MathSciNetCrossRefMATHGoogle Scholar - Matoušek, J. (2002).
*Lectures on discrete geometry*(Vol. 108). New York: Springer.CrossRefMATHGoogle Scholar - Moosavi-Dezfooli, S. M., Fawzi, A., & Frossard, P. (2016). Deepfool: A simple and accurate method to fool deep neural networks. In
*IEEE conference on computer vision and pattern recognition (CVPR)*.Google Scholar - Ng, K., & Zheng, X. (2003). Error bounds of constrained quadratic functions and piecewise affine inequality systems.
*Journal of Optimization Theory and Applications*,*118*(3), 601–618.MathSciNetCrossRefMATHGoogle Scholar - Nguyen, A., Yosinski, J., & Clune, J. (2014).
*Deep neural networks are easily fooled: High confidence predictions for unrecognizable images*. arXiv preprint arXiv:1412.1897. - Pang, J. (1997). Error bounds in mathematical programming.
*Mathematical Programming*,*79*(1–3), 299–332.MathSciNetMATHGoogle Scholar - Simonyan, K., & Zisserman, A. (2014).
*Very deep convolutional networks for large-scale image recognition*. arXiv preprint arXiv:1409.1556. - Srndic, N., & Laskov, P. (2014). Practical evasion of a learning-based classifier: A case study. In
*IEEE symposium on security and privacy*(pp. 197–211). IEEE.Google Scholar - Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I., et al. (2014). Intriguing properties of neural networks. In
*International conference on learning representations (ICLR)*.Google Scholar - Xu, H., Caramanis, C., & Mannor, S. (2009). Robustness and regularization of support vector machines.
*The Journal of Machine Learning Research*,*10*, 1485–1510.MathSciNetMATHGoogle Scholar