Part of Advances in Neural Information Processing Systems 34 (NeurIPS 2021)
Sebastien Bubeck, Yeshwanth Cherapanamjeri, Gauthier Gidel, Remi Tachet des Combes
Daniely and Schacham recently showed that gradient descent finds adversarial examples on random undercomplete two-layers ReLU neural networks. The term “undercomplete” refers to the fact that their proof only holds when the number of neurons is a vanishing fraction of the ambient dimension. We extend their result to the overcomplete case, where the number of neurons is larger than the dimension (yet also subexponential in the dimension). In fact we prove that a single step of gradient descent suffices. We also show this result for any subexponential width random neural network with smooth activation function.