Convergence guarantees for RMSProp and ADAM in non-convex optimization and an empirical comparison to Nesterov acceleration
This work addresses the problem of understanding and improving adaptive gradient algorithms for researchers and practitioners in machine learning, offering incremental theoretical and empirical insights.
The paper tackles the unclear theoretical convergence and generalization properties of RMSProp and ADAM in non-convex optimization, providing proofs that these algorithms reach criticality with runtime bounds and showing empirically that ADAM with high momentum outperforms Nesterov's Accelerated Gradient in training and test losses on autoencoders and VGG-9 with CIFAR-10.
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 $β_1$. We show that at very high values of the momentum parameter ($β_1 = 0.99$) 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 $β_1$ is set to the most commonly used value: $β_1 = 0.9$, 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.