  RMSProp and ADAM continue to be extremely popular algorithms for training
neural nets but their theoretical convergence properties have remained unclear.
Further, recent work has seemed to suggest that these algorithms have worse
generalization properties when compared to carefully tuned stochastic gradient
descent or its momentum variants. In this work, we make progress towards a
deeper understanding of ADAM and RMSProp in two ways. First, we provide proofs
that these adaptive gradient algorithms are guaranteed to reach criticality for
smooth non-convex objectives, and we give bounds on the running time.
  Next we design experiments to empirically study the convergence and
generalization properties of RMSProp and ADAM against Nesterov's Accelerated
Gradient method on a variety of common autoencoder setups and on VGG-9 with
CIFAR-10. Through these experiments we demonstrate the interesting sensitivity
that ADAM has to its momentum parameter $\beta_1$. We show that at very high
values of the momentum parameter ($\beta_1 = 0.99$) ADAM outperforms a
carefully tuned NAG on most of our experiments, in terms of getting lower
training and test losses. On the other hand, NAG can sometimes do better when
ADAM's $\beta_1$ is set to the most commonly used value: $\beta_1 = 0.9$,
indicating the importance of tuning the hyperparameters of ADAM to get better
generalization performance.
  We also report experiments on different autoencoders to demonstrate that NAG
has better abilities in terms of reducing the gradient norms, and it also
produces iterates which exhibit an increasing trend for the minimum eigenvalue
of the Hessian of the loss function at the iterates.
