| Abstract: |
| We discuss some qualitative and quantitative results analyzing adversarial training as the adversarial budget vanishes. First, with Bungert, we find that minimizers of the adversarial training problem converge in $L^1$ to a Bayes classifier that has minimal weighted perimeter, showing that adversarial training acts as a selection mechanism for the standard classification problem. Subsequent work by Morris and Murray showed that adversarial minimizers converge to Bayes classifiers in the (much stronger) Hausdorff metric at a rate that degrades with the dimension. Joining forces, we recover the generically optimal rate of convergence and prove that the Hausdorff distance between the adversarial minimizer and the Bayes classifier is $O(\epsilon )$ regardless of the ambient dimension. |
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