Convergence guarantees for RMSProp and ADAM in non-convex optimization and their comparison to Nesterov acceleration on autoencoders

RMSProp and ADAM continue to be extremely popular algorithms for training neural nets but their theoretical foundations have remained unclear. In this work we make progress towards that by giving proofs that these adaptive gradient algorithms are guaranteed to reach criticality for smooth non-convex objectives and we give bounds on the running time. We then design experiments to compare the performances of RMSProp and ADAM against Nesterov Accelerated Gradient method on a variety of autoencoder setups. Through these experiments we demonstrate the interesting sensitivity that ADAM has to its momentum parameter $\beta_1$. We show that in terms of getting lower training and test losses, at very high values of the momentum parameter ($\beta_1 = 0.99$) (and large enough nets if using mini-batches) ADAM outperforms NAG at any momentum value tried for the latter. On the other hand, NAG can sometimes do better when ADAM's $\beta_1$ is set to the most commonly used value: $\beta_1 = 0.9$. We also report experiments on different autoencoders to demonstrate that NAG has better abilities in terms of reducing the gradient norms and finding weights which increase the minimum eigenvalue of the Hessian of the loss function.