  Adversarial training is a common approach to improving the robustness of deep
neural networks against adversarial examples. In this work, we propose a novel
regularization approach as an alternative. To derive the regularizer, we
formulate the adversarial robustness problem under the robust optimization
framework and approximate the loss function using a second-order Taylor series
expansion. Our proposed second-order adversarial regularizer (SOAR) is an upper
bound based on the Taylor approximation of the inner-max in the robust
optimization objective. We empirically show that the proposed method
significantly improves the robustness of networks against the $\ell_\infty$ and
$\ell_2$ bounded perturbations generated using cross-entropy-based PGD on
CIFAR-10 and SVHN.
