Abstract

In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth $2$ nets -- for arbitrary data with any number of gates with adequately smooth and bounded activations, like sigmoid and tanh, and for a class of distributions from which the initial weight is sampled. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show that the logistic loss function on any size neural net can be Frobenius norm regularized by a width-independent parameter such that the regularized loss is a ``Villani function'' -- and thus be able to build on recent progress with analyzing SGD on such objectives.