Nesterov acceleration in benignly non-convex landscapes

While momentum-based optimization algorithms are commonly used in the notoriously non-convex optimization problems of deep learning, their analysis has historically been restricted to the convex and strongly convex setting. In this article, we partially close this gap between theory and practice and demonstrate that virtually identical guarantees can be obtained in optimization problems with a `benign' non-convexity. We show that these weaker geometric assumptions are well justified in overparametrized deep learning, at least locally. Variations of this result are obtained for a continuous time model of Nesterov's accelerated gradient descent algorithm (NAG), the classical discrete time version of NAG, and versions of NAG with stochastic gradient estimates with purely additive noise and with noise that exhibits both additive and multiplicative scaling.

Achieving acceleration despite very noisy gradients

We present a novel momentum-based first order optimization method (AGNES) which provably achieves acceleration for convex minimization, even if the stochastic noise in the gradient estimates is many orders of magnitude larger than the gradient itself. Here we model the noise as having a variance which is proportional to the magnitude of the underlying gradient. We argue, based upon empirical evidence, that this is appropriate for mini-batch gradients in overparameterized deep learning. Furthermore, we demonstrate that the method achieves competitive performance in the training of CNNs on MNIST and CIFAR-10.