Most ReLU Networks Suffer from ℓ2 Adversarial Perturbations

We consider ReLU networks with random weights, in which the dimension decreases at each layer. We show that for most such networks, most examples x admit an adversarial perturbation at an Euclidean distance of Oleft(frac|x|√dright), where d is the input dimension. Moreover, this perturbation can be found via gradient flow, as well as gradient descent with sufficiently small steps. This result can be seen as an explanation to the abundance of adversarial examples, and to the fact that they are found via gradient descent.