Robustness and Equivariance of Neural Networks

Amit Deshpande, Sandesh Kamath, K.V.Subrahmanyam

Abstract:

Neural networks models are known to be vulnerable to geometric transformations
as well as small pixel-wise perturbations of input. Convolutional Neural Networks
(CNNs) are translation-equivariant but can be easily fooled using rotations and
small pixel-wise perturbations. Moreover, CNNs require sufficient translations in
their training data to achieve translation-invariance. Recent work by Cohen &
Welling (2016), Worrall et al. (2016), Kondor & Trivedi (2018), Cohen & Welling
(2017), Marcos et al. (2017), and Esteves et al. (2018) has gone beyond translations,
and constructed rotation-equivariant or more general group-equivariant
neural network models. In this paper, we do an extensive empirical study of various
rotation-equivariant neural network models to understand how effectively they
learn rotations. This includes Group-equivariant Convolutional Networks (GCNNs)
by Cohen & Welling (2016), Harmonic Networks (H-Nets) by Worrall et al.
(2016), Polar Transformer Networks (PTN) by Esteves et al. (2018) and Rotation
equivariant vector field networks by Marcos et al. (2017). We empirically compare
the ability of these networks to learn rotations efficiently in terms of their
number of parameters, sample complexity, rotation augmentation used in training.
We compare them against each other as well as Standard CNNs. We observe
that as these rotation-equivariant neural networks learn rotations, they instead become
more vulnerable to small pixel-wise adversarial attacks, e.g., Fast Gradient
Sign Method (FGSM) and Projected Gradient Descent (PGD), in comparison with
Standard CNNs. In other words, robustness to geometric transformations in these
models comes at the cost of robustness to small pixel-wise perturbations.

Keywords: robust, adversarial, equivariance, rotations, GCNNs, CNNs, steerable, neural networks

TL;DR: Robustness to rotations comes at the cost of robustness of pixel-wise adversarial perturbations.