Momentum Improves Normalized SGD

We provide an improved analysis of normalized SGD showing that adding momentum provably removes the need for large batch sizes on non-convex objectives. Then, we consider the case of objectives with bounded second derivative and show that in this case a small tweak to the momentum formula allows normalized SGD with momentum to find an $\epsilon$-critical point in $O(1/\epsilon^{3.5})$ iterations, matching the best-known rates without accruing any logarithmic factors or dependence on dimension. We also provide an adaptive method that automatically improves convergence rates when the variance in the gradients is small. Finally, we show that our method is effective when employed on popular large scale tasks such as ResNet-50 and BERT pretraining, matching the performance of the disparate methods used to get state-of-the-art results on both tasks.