  Deep neural networks are easily fooled by small perturbations known as
adversarial attacks. Adversarial Training (AT) is a technique aimed at learning
features robust to such attacks and is widely regarded as a very effective
defense. However, the computational cost of such training can be prohibitive as
the network size and input dimensions grow. Inspired by the relationship
between robustness and curvature, we propose a novel regularizer which
incorporates first and second order information via a quadratic approximation
to the adversarial loss. The worst case quadratic loss is approximated via an
iterative scheme. It is shown that using only a single iteration in our
regularizer achieves stronger robustness than prior gradient and curvature
regularization schemes, avoids gradient obfuscation, and, with additional
iterations, achieves strong robustness with significantly lower training time
than AT. Further, it retains the interesting facet of AT that networks learn
features which are well-aligned with human perception. We demonstrate
experimentally that our method produces higher quality human-interpretable
features than other geometric regularization techniques. These robust features
are then used to provide human-friendly explanations to model predictions.
