Classifier-independent Lower-Bounds for Adversarial Robustness

Elvis Dohmatob

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arXiv · 2006.09989

Classifier-independent Lower-Bounds for Adversarial Robustness

Elvis Dohmatob

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Abstract

We theoretically analyse the limits of robustness to test-time adversarial and noisy examples in classification. Our work focuses on deriving bounds which uniformly apply to all classifiers (i.e all measurable functions from features to labels) for a given problem. Our contributions are two-fold. (1) We use optimal transport theory to derive variational formulae for the Bayes-optimal error a classifier can make on a given classification problem, subject to adversarial attacks. The optimal adversarial attack is then an optimal transport plan for a certain binary cost-function induced by the specific attack model, and can be computed via a simple algorithm based on maximal matching on bipartite graphs. (2) We derive explicit lower-bounds on the Bayes-optimal error in the case of the popular distance-based attacks. These bounds are universal in the sense that they depend on the geometry of the class-conditional distributions of the data, but not on a particular classifier. Our results are in sharp contrast with the existing literature, wherein adversarial vulnerability of classifiers is derived as a consequence of nonzero ordinary test error.

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stat.ML cs.CV cs.LG cs.NE

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BibTeX RIS

Elvis Dohmatob. 2020-06-17. Classifier-independent Lower-Bounds for Adversarial Robustness. https://arxiv.org/abs/2006.09989

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