We augment adversarial training (AT) with worst case adversarial training
 (WCAT) which improves adversarial robustness by 11% over the current state-
 of-the-art result in the `2-norm on CIFAR-10. We interpret adversarial training as
 Total Variation Regularization, which is a fundamental tool in mathematical im-
 age processing, and WCAT as Lipschitz regularization, which appears in Image
 Inpainting. We obtain verifiable worst and average case robustness guarantees,
 based on the expected and maximum values of the norm of the gradient of the
 loss.
Improvements to adversarial robustness, as well as provable robustness guarantees, are obtained by augmenting adversarial training with a tractable Lipschitz regularization Explores augmenting the training loss with an additional gradient regularization term to improve robustness of models against adversarial examples Uses a trick to simplify the adversarial loss by one in which the adversarial perturbation appears in closed form.