Convergence of the Iterates for Momentum and RMSProp for Local Smooth Functions: Adaptation is the Key

TL;DR

This paper extends the Armijo linesearch method to Momentum and RMSProp optimizers using stability theory, providing convergence guarantees for their iterates in non-convex settings without boundedness assumptions.

Contribution

It generalizes the Armijo linesearch to Momentum and RMSProp, establishing convergence results using stability theory and providing the first guarantees for RMSProp in non-convex scenarios.

Findings

Convergence results for Momentum and RMSProp under Lojasiewicz assumption.

First convergence guarantee for RMSProp in non-convex settings.

Extension of linesearch methods to adaptive optimizers.

Abstract

Both accelerated and adaptive gradient methods are among state of the art algorithms to train neural networks. The tuning of hyperparameters is needed to make them work efficiently. For classical gradient descent, a general and efficient way to adapt hyperparameters is the Armijo backtracking. The goal of this work is to generalize the Armijo linesearch to Momentum and RMSProp, two popular optimizers of this family, by means of stability theory of dynamical systems. We establish convergence results, under the Lojasiewicz assumption, for these strategies. As a direct result, we obtain the first guarantee on the convergence of the iterates for RMSProp, in the non-convex setting without the classical bounded assumptions.