  (Non-)robustness of neural networks to small, adversarial pixel-wise
perturbations, and as more recently shown, to even random spatial
transformations (e.g., translations, rotations) entreats both theoretical and
empirical understanding. Spatial robustness to random translations and
rotations is commonly attained via equivariant models (e.g., StdCNNs, GCNNs)
and training augmentation, whereas adversarial robustness is typically achieved
by adversarial training. In this paper, we prove a quantitative trade-off
between spatial and adversarial robustness in a simple statistical setting. We
complement this empirically by showing that: (a) as the spatial robustness of
equivariant models improves by training augmentation with progressively larger
transformations, their adversarial robustness worsens progressively, and (b) as
the state-of-the-art robust models are adversarially trained with progressively
larger pixel-wise perturbations, their spatial robustness drops progressively.
Towards achieving pareto-optimality in this trade-off, we propose a method
based on curriculum learning that trains gradually on more difficult
perturbations (both spatial and adversarial) to improve spatial and adversarial
robustness simultaneously.
