Last-iterate convergence analysis of stochastic momentum methods for neural networks

TL;DR

This paper proves the last-iterate convergence of stochastic momentum methods for non-convex neural network optimization, using fixed momentum factors, and validates results on MNIST and CIFAR-10 datasets.

Contribution

It provides the first unified analysis of last-iterate convergence for stochastic momentum methods with fixed momentum factors in non-convex settings.

Findings

Last-iterate convergence is established for stochastic heavy ball and Nesterov methods.

Convergence results hold with fixed, constant momentum factors.

Empirical validation on MNIST and CIFAR-10 datasets confirms theoretical findings.

Abstract

The stochastic momentum method is a commonly used acceleration technique for solving large-scale stochastic optimization problems in artificial neural networks. Current convergence results of stochastic momentum methods under non-convex stochastic settings mostly discuss convergence in terms of the random output and minimum output. To this end, we address the convergence of the last iterate output (called last-iterate convergence) of the stochastic momentum methods for non-convex stochastic optimization problems, in a way conformal with traditional optimization theory. We prove the last-iterate convergence of the stochastic momentum methods under a unified framework, covering both stochastic heavy ball momentum and stochastic Nesterov accelerated gradient momentum. The momentum factors can be fixed to be constant, rather than time-varying coefficients in existing analyses. Finally, the last-iterate convergence of the stochastic momentum methods is verified on the benchmark MNIST and CIFAR-10 datasets.