  Shift invariance is a critical property of CNNs that improves performance on
classification. However, we show that invariance to circular shifts can also
lead to greater sensitivity to adversarial attacks. We first characterize the
margin between classes when a shift-invariant linear classifier is used. We
show that the margin can only depend on the DC component of the signals. Then,
using results about infinitely wide networks, we show that in some simple
cases, fully connected and shift-invariant neural networks produce linear
decision boundaries. Using this, we prove that shift invariance in neural
networks produces adversarial examples for the simple case of two classes, each
consisting of a single image with a black or white dot on a gray background.
This is more than a curiosity; we show empirically that with real datasets and
realistic architectures, shift invariance reduces adversarial robustness.
Finally, we describe initial experiments using synthetic data to probe the
source of this connection.
