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Adversarial training is a principled approach for training robust neuralnetworks. Despite of tremendous successes in practice, its theoreticalproperties still remain largely unexplored. In this paper, we provide newtheoretical insights of gradient descent based adversarial training by studyingits computational properties, specifically on its inductive bias. We take thebinary classification task on linearly separable data as an illustrativeexample, where the loss asymptotically attains its infimum as the parameterdiverges to infinity along certain directions. Specifically, we show that whenthe adversarial perturbation during training has bounded $\ell_2$-norm, theclassifier learned by gradient descent based adversarial training converges indirection to the maximum $\ell_2$-norm margin classifier at the rate of$\tilde{\mathcal{O}}(1/\sqrt{T})$, significantly faster than the rate$\mathcal{O}(1/\log T)$ of training with clean data. In addition, when theadversarial perturbation during training has bounded $\ell_q$-norm for some$q\ge 1$, the resulting classifier converges in direction to a maximummixed-norm margin classifier, which has a natural interpretation of robustness,as being the maximum $\ell_2$-norm margin classifier under worst-case$\ell_q$-norm perturbation to the data. Our findings provide theoreticalbackups for adversarial training that it indeed promotes robustness againstadversarial perturbation.

Inductive Bias of Gradient Descent based Adversarial Training on Separable Data