Monge blunts Bayes: Hardness Results for Adversarial Training
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:14061415, 2019.
Abstract
The last few years have seen a staggering number of empirical studies of the robustness of neural networks in a model of adversarial perturbations of their inputs. Most rely on an adversary which carries out local modifications within prescribed balls. None however has so far questioned the broader picture: how to frame a resourcebounded adversary so that it can be severely detrimental to learning, a nontrivial problem which entails at a minimum the choice of loss and classifiers. We suggest a formal answer for losses that satisfy the minimal statistical requirement of being proper. We pin down a simple sufficient property for any given class of adversaries to be detrimental to learning, involving a central measure of “harmfulness” which generalizes the wellknown class of integral probability metrics. A key feature of our result is that it holds for all proper losses, and for a popular subset of these, the optimisation of this central measure appears to be independent of the loss. When classifiers are Lipschitz – a now popular approach in adversarial training –, this optimisation resorts to optimal transport to make a lowbudget compression of class marginals. Toy experiments reveal a finding recently separately observed: training against a sufficiently budgeted adversary of this kind improves generalization.
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