American Institute of Mathematical Sciences

Advanced
February  2021, 15(1): 129-145. doi: 10.3934/ipi.2020046

Adversarial defense via the data-dependent activation, total variation minimization, and adversarial training

Bao Wang 1, , Alex Lin 2, , Penghang Yin 2, , Wei Zhu 3, , Andrea L. Bertozzi 2, and  Stanley J. Osher 2,

1. 

Department of Mathematics, Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT, 84112-0090, USA
2. 

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095, USA
3. 

Department of Mathematics, Duke University, Durham, NC, 27708, USA

Please correspond to wangbaonj@gmail.com

Received  November 2019 Revised  April 2020 Published  February 2021 Early access  August 2020

We improve the robustness of Deep Neural Net (DNN) to adversarial attacks by using an interpolating function as the output activation. This data-dependent activation remarkably improves both the generalization and robustness of DNN. In the CIFAR10 benchmark, we raise the robust accuracy of the adversarially trained ResNet20 from $ \sim 46\% $ to $ \sim 69\% $ under the state-of-the-art Iterative Fast Gradient Sign Method (IFGSM) based adversarial attack. When we combine this data-dependent activation with total variation minimization on adversarial images and training data augmentation, we achieve an improvement in robust accuracy by 38.9$ \% $ for ResNet56 under the strongest IFGSM attack. Furthermore, We provide an intuitive explanation of our defense by analyzing the geometry of the feature space.

Keywords: Deep learning, Adversarial defense, Interpolation.
Mathematics Subject Classification: Primary: 68T45; Secondary: 68T01.
Citation: Bao Wang, Alex Lin, Penghang Yin, Wei Zhu, Andrea L. Bertozzi, Stanley J. Osher. Adversarial defense via the data-dependent activation, total variation minimization, and adversarial training. Inverse Problems and Imaging, 2021, 15 (1) : 129-145. doi: 10.3934/ipi.2020046
References:
[1]

N. Akhtar and A. Mian, Threat of adversarial attacks on deep learning in computer vision: A survey, IEEE Access, 6 (2018), 14410–14430, arXiv: 1801.00553. doi: 10.1109/ACCESS.2018.2807385.

[2]

N. Akhtar, J. Liu and A. Mian, Defense Against Universal Adversarial Perturbations, The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. doi: 10.1109/CVPR.2018.00357.

[3]

A. Athalye, N. Carlini and D. Wagner, Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples, International Conference on Machine Learning, 2018.

[4]

A. Athalye, L. Engstrom, A. Ilyas and K. Kwok, Synthesizing Robust Adversarial Examples, International Conference on Machine Learning, 2018.

[5]

Y. Bengio, N. Leonard and A. Courville, Estimating or propagating gradients through stochastic neurons for conditional computation, arXiv preprint, arXiv: 1308.3432, 2013.

[6]

W. Brendel, J. Rauber and M. Bethge, Decision-based adversarial attacks: Reliable attacks against black-box machine learning models, arXiv preprint, arXiv: 1712.04248, 2017.

[7]

N. Carlini and D. A. Wagner, Towards evaluating the robustness of neural networks, IEEE European Symposium on Security and Privacy, (2016), 39–57. doi: 10.1109/SP.2017.49.

[8]

Y. Dong, F. Liao, T. Pang, H. Su, J. Zhu, X. Hu and J. Li, Boosting Adversarial Attacks with Momentum, The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. doi: 10.1109/CVPR.2018.00957.

[9]

I. J. Goodfellow, J. Shlens and C. Szegedy, Explaining and harnessing adversarial examples, arXiv preprint, arXiv: 1412.6275, 2014.

[10]

K. Grosse, N. Papernot, P. Manoharan, M. Backes and P. McDaniel, Adversarial perturbations against deep neural networks for malware classification, arXiv preprint, arXiv: 1606.04435, 2016.

[11]

A. Guisti, J. Guzzi, D. C. Ciresan, F. L. He, J. P. Rodriguez, F. Fontana, M. Faessler, C. Forster, J. Schmidhuber, G. Di Carlo and et al, A machine learning approach to visual perception of forecast trails for mobile robots, IEEE Robotics and Automation Letters, (2016), 661–667.

[12]

C. Guo, M. Rana, M. Cisse and L. van der Maaten, Countering Adversarial Images Using Input Transformations, International Conference on Learning Representations, 2018.

[13]

K. He, X. Zhang, S. Ren and J. Sun, Deep residual learning for image recognition, In CVPR, (2016), 770–778.

[14]

A. Ilyas, L. Engstrom, A. Athalye and J. Lin, Black-box Adversarial Attacks with Limited Queries and Information, International Conference on Machine Learning, 2018.

[15]

D. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint, arXiv: 1412.6980, 2014.

[16]

A. Kurakin, I. J. Goodfellow and S. Bengio, Adversarial examples in the physical world, arXiv preprint, arXiv: 1607.02533, 2016.

[17]

K. Lee, K. Lee, H. Lee and J. Shin, A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks, arXiv e-prints, 2018.

[18]

F. Liao, M. Liang, Y. Dong, T. Pang, X. Hu and J. Zhu, Defense against adversarial attacks using high-level representation guided denoiser, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2018. doi: 10.1109/CVPR.2018.00191.

[19]

Y. Liu, X. Chen, C. Liu and D. Song, Delving into Transferable Adversarial Examples and Black-box Attacks, arXiv preprint, arXiv: 1611.02770, 2016.

[20]

Y. Luo, X. Boix, G. Roig, T. A. Poggio and Q. Zhao, Foveation-based Mechanisms Alleviate Adversarial Examples, CoRR, abs/1511.06292, 2015.

[21]

A. Madry, A. Makelov, L. Schmidt, D. Tsipras and A. Vladu, Towards Deep Learning Models Resistant to Adversarial Attacks, International Conference on Learning Representations, 2018.

[22]

S.-M. Moosavi-Dezfooli, A. Fawzi, O. Fawzi and P. Frossard, Universal adversarial perturbations, 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017. doi: 10.1109/CVPR.2017.17.

[23]

S.-M. Moosavi-Dezfooli, A. Shrivastava and O. Tuzel, Divide, Denoise, and Defend Against Adversarial Attacks, CoRR, abs/1802.06806, 2018.

[24]

T. Na, J. Hwan Ko and S. Mukhopadhyay, Cascade Adversarial Machine Learning Regularized with A Unified Embedding, International Conference on Learning Representations, 2018.

[25]

N. Papernot, P. McDaniel, S. Jha, M. Fredrikson, Z. B. Celik and A. Swami, The limitations of deep learning in adversarial settings, IEEE European Symposium on Security and Privacy, (2016), 372–387. doi: 10.1109/EuroSP.2016.36.

[26]

N. Papernot, P. McDaniel, A. Sinha and M. Wellman, Sok: Towards the science of security and privacy in machien learning, arXiv preprint, arXiv: 1611.03814, 2016.

[27]

N. Papernot, P. McDaniel, X. Wu, S. Jha and A. Swami, Distillation as a Defense to Adversarial Perturbations Against Deep Neural Networks, IEEE European Symposium on Security and Privacy, 2016. doi: 10.1109/SP.2016.41.

[28]

N. Papernot, P. D. McDaniel and I. J. Goodfellow, Transferability in Machine Learning: From Phenomena to Black-box Attacks Using Adversarial Samples, CoRR, abs/1605.07277, 2016.

[29]

A. Prakash, N. Moran, S. Garber, A. DiLillo and J. Storer, Deflecting adversarial attacks with pixel deflection, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2018. doi: 10.1109/CVPR.2018.00894.

[30]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[31]

P. Samangouei, M. Kabkab and R. Chellappa, Defense-GAN: Protecting Classifiers Against Adversarial Attacks Using Generative Models, In International Conference on Learning Representations, 2018.

[32]

Z. Shi, B. Wang and S. J. Osher, Error Estimation of Weighted Nonlocal Laplacian on Random Point Cloud, arXiv preprint, arXiv: 1809.08622, 2014.

[33]

Y. Song, T. Kim, S. Nowozin, S. Ermon and N. Kushman, Pixeldefend: Leveraging Generative Models to Understand and Defend Against Adversarial Examples, In International Conference on Learning Representations, 2018.

[34]

C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan and I. Goodfellow, Intriguing properties of neural networks, arXiv preprint, arXiv: 1312.6199, 2013.

[35]

F. Tramèr, A. Kurakin, N. Papernot, I. Goodfellow, D. Boneh and P. McDaniel, Ensemble Adversarial Training: Attacks and Defenses, International Conference on Learning Representations, 2018.

[36]

B. Wang, X. Luo, Z. Li, W. Zhu, Z. Shi and S. Osher, Deep neural nets with interpolating function as output activation, arXiv preprint, arXiv: 1802.00168, 2018.

[37]

B. Wang, Z. Shi and S. Osher, ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuracies, Advances in Neural Information Processing Systems, 2019.

[38]

B. Wang and S. Osher, Graph interpolating activation improves both natural and robust accuracies in data-efficient deep learning, arXiv preprint, arXiv: 1907.06800, 2019.

[39]

X. Wu, U. Jang, J. Chen, L. Chen and S. Jha, Reinforcing Adversarial Robustness Using Model Confidence Induced by Adversarial Training, International Conference on Machine Learning, 2018.

[40]

C. Xie, J. Wang, Z. Zhang, Z. Ren and A. Yuille, Mitigating Adversarial Effects Through Randomization, International Conference on Learning Representations, 2018.

show all references

References:
[1]

N. Akhtar and A. Mian, Threat of adversarial attacks on deep learning in computer vision: A survey, IEEE Access, 6 (2018), 14410–14430, arXiv: 1801.00553. doi: 10.1109/ACCESS.2018.2807385.

[2]

N. Akhtar, J. Liu and A. Mian, Defense Against Universal Adversarial Perturbations, The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. doi: 10.1109/CVPR.2018.00357.

[3]

A. Athalye, N. Carlini and D. Wagner, Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples, International Conference on Machine Learning, 2018.

[4]

A. Athalye, L. Engstrom, A. Ilyas and K. Kwok, Synthesizing Robust Adversarial Examples, International Conference on Machine Learning, 2018.

[5]

Y. Bengio, N. Leonard and A. Courville, Estimating or propagating gradients through stochastic neurons for conditional computation, arXiv preprint, arXiv: 1308.3432, 2013.

[6]

W. Brendel, J. Rauber and M. Bethge, Decision-based adversarial attacks: Reliable attacks against black-box machine learning models, arXiv preprint, arXiv: 1712.04248, 2017.

[7]

N. Carlini and D. A. Wagner, Towards evaluating the robustness of neural networks, IEEE European Symposium on Security and Privacy, (2016), 39–57. doi: 10.1109/SP.2017.49.

[8]

Y. Dong, F. Liao, T. Pang, H. Su, J. Zhu, X. Hu and J. Li, Boosting Adversarial Attacks with Momentum, The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. doi: 10.1109/CVPR.2018.00957.

[9]

I. J. Goodfellow, J. Shlens and C. Szegedy, Explaining and harnessing adversarial examples, arXiv preprint, arXiv: 1412.6275, 2014.

[10]

K. Grosse, N. Papernot, P. Manoharan, M. Backes and P. McDaniel, Adversarial perturbations against deep neural networks for malware classification, arXiv preprint, arXiv: 1606.04435, 2016.

[11]

A. Guisti, J. Guzzi, D. C. Ciresan, F. L. He, J. P. Rodriguez, F. Fontana, M. Faessler, C. Forster, J. Schmidhuber, G. Di Carlo and et al, A machine learning approach to visual perception of forecast trails for mobile robots, IEEE Robotics and Automation Letters, (2016), 661–667.

[12]

C. Guo, M. Rana, M. Cisse and L. van der Maaten, Countering Adversarial Images Using Input Transformations, International Conference on Learning Representations, 2018.

[13]

K. He, X. Zhang, S. Ren and J. Sun, Deep residual learning for image recognition, In CVPR, (2016), 770–778.

[14]

A. Ilyas, L. Engstrom, A. Athalye and J. Lin, Black-box Adversarial Attacks with Limited Queries and Information, International Conference on Machine Learning, 2018.

[15]

D. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint, arXiv: 1412.6980, 2014.

[16]

A. Kurakin, I. J. Goodfellow and S. Bengio, Adversarial examples in the physical world, arXiv preprint, arXiv: 1607.02533, 2016.

[17]

K. Lee, K. Lee, H. Lee and J. Shin, A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks, arXiv e-prints, 2018.

[18]

F. Liao, M. Liang, Y. Dong, T. Pang, X. Hu and J. Zhu, Defense against adversarial attacks using high-level representation guided denoiser, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2018. doi: 10.1109/CVPR.2018.00191.

[19]

Y. Liu, X. Chen, C. Liu and D. Song, Delving into Transferable Adversarial Examples and Black-box Attacks, arXiv preprint, arXiv: 1611.02770, 2016.

[20]

Y. Luo, X. Boix, G. Roig, T. A. Poggio and Q. Zhao, Foveation-based Mechanisms Alleviate Adversarial Examples, CoRR, abs/1511.06292, 2015.

[21]

A. Madry, A. Makelov, L. Schmidt, D. Tsipras and A. Vladu, Towards Deep Learning Models Resistant to Adversarial Attacks, International Conference on Learning Representations, 2018.

[22]

S.-M. Moosavi-Dezfooli, A. Fawzi, O. Fawzi and P. Frossard, Universal adversarial perturbations, 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017. doi: 10.1109/CVPR.2017.17.

[23]

S.-M. Moosavi-Dezfooli, A. Shrivastava and O. Tuzel, Divide, Denoise, and Defend Against Adversarial Attacks, CoRR, abs/1802.06806, 2018.

[24]

T. Na, J. Hwan Ko and S. Mukhopadhyay, Cascade Adversarial Machine Learning Regularized with A Unified Embedding, International Conference on Learning Representations, 2018.

[25]

N. Papernot, P. McDaniel, S. Jha, M. Fredrikson, Z. B. Celik and A. Swami, The limitations of deep learning in adversarial settings, IEEE European Symposium on Security and Privacy, (2016), 372–387. doi: 10.1109/EuroSP.2016.36.

[26]

N. Papernot, P. McDaniel, A. Sinha and M. Wellman, Sok: Towards the science of security and privacy in machien learning, arXiv preprint, arXiv: 1611.03814, 2016.

[27]

N. Papernot, P. McDaniel, X. Wu, S. Jha and A. Swami, Distillation as a Defense to Adversarial Perturbations Against Deep Neural Networks, IEEE European Symposium on Security and Privacy, 2016. doi: 10.1109/SP.2016.41.

[28]

N. Papernot, P. D. McDaniel and I. J. Goodfellow, Transferability in Machine Learning: From Phenomena to Black-box Attacks Using Adversarial Samples, CoRR, abs/1605.07277, 2016.

[29]

A. Prakash, N. Moran, S. Garber, A. DiLillo and J. Storer, Deflecting adversarial attacks with pixel deflection, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2018. doi: 10.1109/CVPR.2018.00894.

[30]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[31]

P. Samangouei, M. Kabkab and R. Chellappa, Defense-GAN: Protecting Classifiers Against Adversarial Attacks Using Generative Models, In International Conference on Learning Representations, 2018.

[32]

Z. Shi, B. Wang and S. J. Osher, Error Estimation of Weighted Nonlocal Laplacian on Random Point Cloud, arXiv preprint, arXiv: 1809.08622, 2014.

[33]

Y. Song, T. Kim, S. Nowozin, S. Ermon and N. Kushman, Pixeldefend: Leveraging Generative Models to Understand and Defend Against Adversarial Examples, In International Conference on Learning Representations, 2018.

[34]

C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan and I. Goodfellow, Intriguing properties of neural networks, arXiv preprint, arXiv: 1312.6199, 2013.

[35]

F. Tramèr, A. Kurakin, N. Papernot, I. Goodfellow, D. Boneh and P. McDaniel, Ensemble Adversarial Training: Attacks and Defenses, International Conference on Learning Representations, 2018.

[36]

B. Wang, X. Luo, Z. Li, W. Zhu, Z. Shi and S. Osher, Deep neural nets with interpolating function as output activation, arXiv preprint, arXiv: 1802.00168, 2018.

[37]

B. Wang, Z. Shi and S. Osher, ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuracies, Advances in Neural Information Processing Systems, 2019.

[38]

B. Wang and S. Osher, Graph interpolating activation improves both natural and robust accuracies in data-efficient deep learning, arXiv preprint, arXiv: 1907.06800, 2019.

[39]

X. Wu, U. Jang, J. Chen, L. Chen and S. Jha, Reinforcing Adversarial Robustness Using Model Confidence Induced by Adversarial Training, International Conference on Machine Learning, 2018.

[40]

C. Xie, J. Wang, Z. Zhang, Z. Ren and A. Yuille, Mitigating Adversarial Effects Through Randomization, International Conference on Learning Representations, 2018.

Figure 1.  Training and testing procedures of the DNN with softmax and WNLL functions as the output activation layer. (a) and (b) show the training and testing steps for the standard DNN, respectively; (c) and (d) illustrate the training and testing procedure of the WNLL activated DNN, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Samples from CIFAR10. Panel (a): from the top to the last rows show the original, adversarial images by attacking ResNet56 with FGSM and IFGSM ($ \epsilon = 0.02 $); and by attacking ResNet56-WNLL. Panel (b) corresponding to those in panel (a) with $ \epsilon = 0.08 $. Charts (c) and (d) corresponding to the TV minimized images in (a) and (b), respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  $ \epsilon $ v.s. accuracy without defense, and defending by WNLL activation, TVM and augmented training. (a) and (b) plot results for FGSM and IFGSM attack, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Epochs v.s. accuracy of ResNet56 on CIFAR10. (a): without the additional FC layer; (b): with the additional FC layer
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Visualization of the features learned by DNN with softmax ((a), (b), (c), (d)) and WNLL ((e), (f), (g), (h)) activation functions. (a) and (b) plot the 2D features of the original and adversarial testing images; (c) and (d) are the first two principle components of the 64D features for the original and adversarial testing images, respectively. Charts (e), (f) plot the first two components of the training and testing features learned by ResNet56-WNLL; (g) and (h) show the two principle components of the adversarial and TV minimized adversarial images for the test set
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  (a): $ \# $IFGSM iterations v.s. accuracy for the ResNet20 and the ResNet20-WNLL trained with PGD adversarial training. (b):$ \epsilon $ v.s. accuracy for the ResNet20 and the ResNet20-WNLL trained with PGD adversarial training
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Running time and GPU memory for ResNet20 with two different activation functions
Training time Testing time Memory
ResNet20 3925.6 (s) 0.657 (s) 1007 (MB)
ResNet20-WNLL 7378.4 (s) 14.09 (s) 1563 (MB)
Table Options

Download as excel

Training time Testing time Memory
ResNet20 3925.6 (s) 0.657 (s) 1007 (MB)
ResNet20-WNLL 7378.4 (s) 14.09 (s) 1563 (MB)
Table 2.  Mutual classification accuracy on the adversarial images crafted by using FGSM and IFGSM to attack ResNet56 and ResNet56-WNLL. (Unit: $ \% $)
Attack Training data $ \epsilon=0 $ $ \epsilon=0.02 $ $ \epsilon=0.04 $ $ \epsilon=0.06 $ $ \epsilon=0.08 $ $ \epsilon=0.1 $
Accuracy of ResNet56 on adversarial images crafted by attacking ResNet56-WNLL
FGSM Original data 93.0 69.8 56.9 44.6 34.6 28.3
FGSM TVM data 88.3 51.5 37.9 30.1 24.7 20.9
FGSM Original + TVM 93.1 78.5 70.9 64.6 59.8 55.8
IFGSM Original data 93.0 5.22 5.73 6.73 7.55 8.55
IFGSM TVM data 88.3 7.00 6.82 8.30 9.28 10.7
IFGSM Original + TVM 93.1 27.3 28.6 29.5 29.1 29.4
Accuracy of ResNet56-WNLL on adversarial images crafted by attacking ResNet56
FGSM Original data 94.5 65.2 49.0 39.3 32.8 28.3
FGSM TVM data 90.6 45.9 30.9 22.2 16.9 13.8
FGSM Original + TVM data 94.7 78.3 68.2 61.1 56.5 52.5
IFGSM Original data 94.5 3.37 3.71 3.54 4.69 6.41
IFGSM TVM data 90.6 7.88 7.51 7.58 8.07 9.67
IFGSM Original + TVM data 94.7 34.3 33.4 33.1 34.6 35.8
Table Options

Download as excel

Attack Training data $ \epsilon=0 $ $ \epsilon=0.02 $ $ \epsilon=0.04 $ $ \epsilon=0.06 $ $ \epsilon=0.08 $ $ \epsilon=0.1 $
Accuracy of ResNet56 on adversarial images crafted by attacking ResNet56-WNLL
FGSM Original data 93.0 69.8 56.9 44.6 34.6 28.3
FGSM TVM data 88.3 51.5 37.9 30.1 24.7 20.9
FGSM Original + TVM 93.1 78.5 70.9 64.6 59.8 55.8
IFGSM Original data 93.0 5.22 5.73 6.73 7.55 8.55
IFGSM TVM data 88.3 7.00 6.82 8.30 9.28 10.7
IFGSM Original + TVM 93.1 27.3 28.6 29.5 29.1 29.4
Accuracy of ResNet56-WNLL on adversarial images crafted by attacking ResNet56
FGSM Original data 94.5 65.2 49.0 39.3 32.8 28.3
FGSM TVM data 90.6 45.9 30.9 22.2 16.9 13.8
FGSM Original + TVM data 94.7 78.3 68.2 61.1 56.5 52.5
IFGSM Original data 94.5 3.37 3.71 3.54 4.69 6.41
IFGSM TVM data 90.6 7.88 7.51 7.58 8.07 9.67
IFGSM Original + TVM data 94.7 34.3 33.4 33.1 34.6 35.8
Table 3.  Mutual classification accuracy on the adversarial images crafted by using CW-L2 to attack ResNet56 and ResNet56-WNLL. (Unit: $ \% $)
Training data Original data TVM data Original + TVM data
Exp-Ⅰ 52.1 43.2 80.0
Exp-Ⅱ 59.7 41.1 80.1
Table Options

Download as excel

Training data Original data TVM data Original + TVM data
Exp-Ⅰ 52.1 43.2 80.0
Exp-Ⅱ 59.7 41.1 80.1
Table 4.  Testing accuracy on the adversarial/TVM adversarial CIFAR10 dataset. The testing accuracy with no defense is in red italic; and the results with all three defenses are in boldface. (Unit: $ \% $)
Training data Original data TVM data Original + TVM data
ResNet56 4.94/32.2 11.8/54.0 15.1/52.4
ResNet56-WNLL 18.3/35.2 15.0/53.9 28/54.5
Table Options

Download as excel

Training data Original data TVM data Original + TVM data
ResNet56 4.94/32.2 11.8/54.0 15.1/52.4
ResNet56-WNLL 18.3/35.2 15.0/53.9 28/54.5
Table 5.  Testing accuracy on the adversarial/TVM adversarial CIFAR10 dataset. The testing accuracy with no defense is in red italic; and the results with all three defenses are in boldface. (Unit: $ \% $)
Attack Training data $ \epsilon=0 $ $ \epsilon=0.02 $ $ \epsilon=0.04 $ $ \epsilon=0.06 $ $ \epsilon=0.08 $ $ \epsilon=0.1 $
ResNet56
FGSM Original data 93.0 36.9/19.4 29.6/18.9 26.1/18.4 23.1/17.9 20.5/17.1
FGSM TVM data 88.3 27.4/50.4 19.1/47.2 16.6/43.7 15.0/38.9 13.7/35.0
FGSM Original + TVM 93.1 48.6/51.1 42.0/47.6 39.1/44.2 37.1/41.8 35.6/39.1
IFGSM Original data 93.0 0/16.6 0/16.1 0.02/15.9 0.1/15.5 0.25/16.1
IFGSM TVM data 88.3 0.01/43.4 0/42.5 0.02/42.4 0.18/42.7 0.49/42.4
IFGSM Original + TVM 93.1 0.1/38.4 0.09/37.9 0.36/37.9 0.84/37.6 1.04/37.9
ResNet56-WNLL
FGSM Original data 94.5 58.5/26.0 50.1/25.4 42.3/25.5 35.7/24.9 29.2/22.9
FGSM TVM data 90.6 31.5/52.6 24.5/49.6 20.2/45.3 17.3/41.6 14.4/37.5
FGSM Original + TVM 94.7 60.5/ 55.4 56.7/52.0 55.3/48.6 53.2/45.9 50.1/43.7
IFGSM Original data 94.5 0.49/16.7 0.14/17.3 0.3/16.9 1.01/16.6 0.94/16.5
IFGSM TVM data 90.6 0.61/37.3 0.43/36.3 0.63/35.9 0.87/35.9 1.19/35.5
IFGSM Original + TVM 94.7 0.19/38.5 0.3/39.4 0.63/ 40.1 1.26/ 38.9 1.72/ 39.1
Table Options

Download as excel

Attack Training data $ \epsilon=0 $ $ \epsilon=0.02 $ $ \epsilon=0.04 $ $ \epsilon=0.06 $ $ \epsilon=0.08 $ $ \epsilon=0.1 $
ResNet56
FGSM Original data 93.0 36.9/19.4 29.6/18.9 26.1/18.4 23.1/17.9 20.5/17.1
FGSM TVM data 88.3 27.4/50.4 19.1/47.2 16.6/43.7 15.0/38.9 13.7/35.0
FGSM Original + TVM 93.1 48.6/51.1 42.0/47.6 39.1/44.2 37.1/41.8 35.6/39.1
IFGSM Original data 93.0 0/16.6 0/16.1 0.02/15.9 0.1/15.5 0.25/16.1
IFGSM TVM data 88.3 0.01/43.4 0/42.5 0.02/42.4 0.18/42.7 0.49/42.4
IFGSM Original + TVM 93.1 0.1/38.4 0.09/37.9 0.36/37.9 0.84/37.6 1.04/37.9
ResNet56-WNLL
FGSM Original data 94.5 58.5/26.0 50.1/25.4 42.3/25.5 35.7/24.9 29.2/22.9
FGSM TVM data 90.6 31.5/52.6 24.5/49.6 20.2/45.3 17.3/41.6 14.4/37.5
FGSM Original + TVM 94.7 60.5/ 55.4 56.7/52.0 55.3/48.6 53.2/45.9 50.1/43.7
IFGSM Original data 94.5 0.49/16.7 0.14/17.3 0.3/16.9 1.01/16.6 0.94/16.5
IFGSM TVM data 90.6 0.61/37.3 0.43/36.3 0.63/35.9 0.87/35.9 1.19/35.5
IFGSM Original + TVM 94.7 0.19/38.5 0.3/39.4 0.63/ 40.1 1.26/ 38.9 1.72/ 39.1
[1]

Christopher Oballe, David Boothe, Piotr J. Franaszczuk, Vasileios Maroulas. ToFU: Topology functional units for deep learning. Foundations of Data Science, 2021  doi: 10.3934/fods.2021021
[2]

Richard Archibald, Feng Bao, Yanzhao Cao, He Zhang. A backward SDE method for uncertainty quantification in deep learning. Discrete and Continuous Dynamical Systems - S, 2022, 15 (10) : 2807-2835. doi: 10.3934/dcdss.2022062
[3]

Ziju Shen, Yufei Wang, Dufan Wu, Xu Yang, Bin Dong. Learning to scan: A deep reinforcement learning approach for personalized scanning in CT imaging. Inverse Problems and Imaging, 2022, 16 (1) : 179-195. doi: 10.3934/ipi.2021045
[4]

Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009
[5]

Nicholas Geneva, Nicholas Zabaras. Multi-fidelity generative deep learning turbulent flows. Foundations of Data Science, 2020, 2 (4) : 391-428. doi: 10.3934/fods.2020019
[6]

Miria Feng, Wenying Feng. Evaluation of parallel and sequential deep learning models for music subgenre classification. Mathematical Foundations of Computing, 2021, 4 (2) : 131-143. doi: 10.3934/mfc.2021008
[7]

Govinda Anantha Padmanabha, Nicholas Zabaras. A Bayesian multiscale deep learning framework for flows in random media. Foundations of Data Science, 2021, 3 (2) : 251-303. doi: 10.3934/fods.2021016
[8]

Suhua Wang, Zhiqiang Ma, Hongjie Ji, Tong Liu, Anqi Chen, Dawei Zhao. Personalized exercise recommendation method based on causal deep learning: Experiments and implications. STEM Education, 2022, 2 (2) : 157-172. doi: 10.3934/steme.2022011
[9]

Alessandro Scagliotti. Deep Learning approximation of diffeomorphisms via linear-control systems. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022036
[10]

Tudor Barbu. Deep learning-based multiple moving vehicle detection and tracking using a nonlinear fourth-order reaction-diffusion based multi-scale video object analysis. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022083
[11]

Negin Karimi, Ahmad Yousefian Darani, Marcus Greferath. Correcting adversarial errors with generalized regenerating codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022005
[12]

Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695
[13]

Charles Fefferman. Interpolation by linear programming I. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477
[14]

Yanfei Du, Ben Niu, Junjie Wei. A predator-prey model with cooperative hunting in the predator and group defense in the prey. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 5845-5881. doi: 10.3934/dcdsb.2021298
[15]

Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1855-1871. doi: 10.3934/dcds.2021175
[16]

Jean Dolbeault, An Zhang. Parabolic methods for ultraspherical interpolation inequalities. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022080
[17]

H. N. Mhaskar, T. Poggio. Function approximation by deep networks. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4085-4095. doi: 10.3934/cpaa.2020181
[18]

Qi Feng, Man Luo, Zhaoyu Zhang. Deep signature FBSDE algorithm. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022028
[19]

Scott G. McCalla. Paladins as predators: Invasive waves in a spatial evolutionary adversarial game. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1437-1457. doi: 10.3934/dcdsb.2014.19.1437
[20]

Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1

2021 Impact Factor: 1.483

Tools

Metrics

  • PDF downloads (280)
  • HTML views (292)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy