Adversarial Risk via Optimal Transport and Optimal Couplings

Muni Sreenivas Pydi, Varun Jog
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:7814-7823, 2020.

Abstract

The accuracy of modern machine learning algorithms deteriorates severely on adversarially manipulated test data. Optimal adversarial risk quantifies the best error rate of any classifier in the presence of adversaries, and optimal adversarial classifiers are sought that minimize adversarial risk. In this paper, we investigate the optimal adversarial risk and optimal adversarial classifiers from an optimal transport perspective. We present a new and simple approach to show that the optimal adversarial risk for binary classification with 0 − 1 loss function is completely characterized by an optimal transport cost between the probability distributions of the two classes, for a suitably defined cost function. We propose a novel coupling strategy that achieves the optimal transport cost for several univariate distributions like Gaussian, uniform and triangular. Using the optimal couplings, we obtain the optimal adversarial classifiers in these settings and show how they differ from optimal classifiers in the absence of adversaries. Based on our analysis, we evaluate algorithm-independent fundamental limits on adversarial risk for CIFAR-10, MNIST, Fashion-MNIST and SVHN datasets, and Gaussian mixtures based on them.

Cite this Paper


BibTeX
@InProceedings{pmlr-v119-pydi20a, title = {Adversarial Risk via Optimal Transport and Optimal Couplings}, author = {Pydi, Muni Sreenivas and Jog, Varun}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {7814--7823}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/pydi20a/pydi20a.pdf}, url = {http://proceedings.mlr.press/v119/pydi20a.html}, abstract = {The accuracy of modern machine learning algorithms deteriorates severely on adversarially manipulated test data. Optimal adversarial risk quantifies the best error rate of any classifier in the presence of adversaries, and optimal adversarial classifiers are sought that minimize adversarial risk. In this paper, we investigate the optimal adversarial risk and optimal adversarial classifiers from an optimal transport perspective. We present a new and simple approach to show that the optimal adversarial risk for binary classification with 0 − 1 loss function is completely characterized by an optimal transport cost between the probability distributions of the two classes, for a suitably defined cost function. We propose a novel coupling strategy that achieves the optimal transport cost for several univariate distributions like Gaussian, uniform and triangular. Using the optimal couplings, we obtain the optimal adversarial classifiers in these settings and show how they differ from optimal classifiers in the absence of adversaries. Based on our analysis, we evaluate algorithm-independent fundamental limits on adversarial risk for CIFAR-10, MNIST, Fashion-MNIST and SVHN datasets, and Gaussian mixtures based on them.} }
Endnote
%0 Conference Paper %T Adversarial Risk via Optimal Transport and Optimal Couplings %A Muni Sreenivas Pydi %A Varun Jog %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-pydi20a %I PMLR %P 7814--7823 %U http://proceedings.mlr.press/v119/pydi20a.html %V 119 %X The accuracy of modern machine learning algorithms deteriorates severely on adversarially manipulated test data. Optimal adversarial risk quantifies the best error rate of any classifier in the presence of adversaries, and optimal adversarial classifiers are sought that minimize adversarial risk. In this paper, we investigate the optimal adversarial risk and optimal adversarial classifiers from an optimal transport perspective. We present a new and simple approach to show that the optimal adversarial risk for binary classification with 0 − 1 loss function is completely characterized by an optimal transport cost between the probability distributions of the two classes, for a suitably defined cost function. We propose a novel coupling strategy that achieves the optimal transport cost for several univariate distributions like Gaussian, uniform and triangular. Using the optimal couplings, we obtain the optimal adversarial classifiers in these settings and show how they differ from optimal classifiers in the absence of adversaries. Based on our analysis, we evaluate algorithm-independent fundamental limits on adversarial risk for CIFAR-10, MNIST, Fashion-MNIST and SVHN datasets, and Gaussian mixtures based on them.
APA
Pydi, M.S. & Jog, V.. (2020). Adversarial Risk via Optimal Transport and Optimal Couplings. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:7814-7823 Available from http://proceedings.mlr.press/v119/pydi20a.html.

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