Convergence guarantees for RMSProp and ADAM in non-convex optimization and an empirical comparison to Nesterov acceleration

TL;DR

This paper provides theoretical convergence guarantees for RMSProp and ADAM in non-convex optimization and empirically compares their performance and generalization to Nesterov acceleration across various neural network training tasks.

Contribution

It offers the first convergence proofs for RMSProp and ADAM on smooth non-convex functions and empirically evaluates their convergence and generalization against Nesterov's method.

Findings

ADAM with high momentum ($eta_1=0.99$) outperforms Nesterov in training and test loss.

Nesterov often generalizes better when ADAM's momentum is set to $eta_1=0.9$.

Nesterov reduces gradient norms more effectively and influences Hessian eigenvalues.

Abstract

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.