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Vulnerability to adversarial attacks is one of the principal hurdles to theadoption of deep learning in safety-critical applications. Despite significantefforts, both practical and theoretical, training deep learning models robustto adversarial attacks is still an open problem. In this paper, we analyse thegeometry of adversarial attacks in the large-data, overparameterized limit forBayesian Neural Networks (BNNs). We show that, in the limit, vulnerability togradient-based attacks arises as a result of degeneracy in the datadistribution, i.e., when the data lies on a lower-dimensional submanifold ofthe ambient space. As a direct consequence, we demonstrate that in this limitBNN posteriors are robust to gradient-based adversarial attacks. Crucially, weprove that the expected gradient of the loss with respect to the BNN posteriordistribution is vanishing, even when each neural network sampled from theposterior is vulnerable to gradient-based attacks. Experimental results on theMNIST, Fashion MNIST, and half moons datasets, representing the finite dataregime, with BNNs trained with Hamiltonian Monte Carlo and VariationalInference, support this line of arguments, showing that BNNs can display bothhigh accuracy on clean data and robustness to both gradient-based andgradient-free based adversarial attacks.

On the Robustness of Bayesian Neural Networks to Adversarial Attacks