  Vulnerability to adversarial attacks is one of the principal hurdles to the
adoption of deep learning in safety-critical applications. Despite significant
efforts, both practical and theoretical, the problem remains open. In this
paper, we analyse the geometry of adversarial attacks in the large-data,
overparametrized limit for Bayesian Neural Networks (BNNs). We show that, in
the limit, vulnerability to gradient-based attacks arises as a result of
degeneracy in the data distribution, i.e., when the data lies on a
lower-dimensional submanifold of the ambient space. As a direct consequence, we
demonstrate that in the limit BNN posteriors are robust to gradient-based
adversarial attacks. Experimental results on the MNIST and Fashion MNIST
datasets with BNNs trained with Hamiltonian Monte Carlo and Variational
Inference support this line of argument, showing that BNNs can display both
high accuracy and robustness to gradient based adversarial attacks.
