  The study of adversarial robustness has so far largely focused on
perturbations bound in p-norms. However, state-of-the-art models turn out to be
also vulnerable to other, more natural classes of perturbations such as
translations and rotations. In this work, we thoroughly investigate the
vulnerability of neural network--based classifiers to rotations and
translations. While data augmentation offers relatively small robustness, we
use ideas from robust optimization and test-time input aggregation to
significantly improve robustness. Finally we find that, in contrast to the
p-norm case, first-order methods cannot reliably find worst-case perturbations.
This highlights spatial robustness as a fundamentally different setting
requiring additional study. Code available at
https://github.com/MadryLab/adversarial_spatial and
https://github.com/MadryLab/spatial-pytorch.
