On Uniform Boundedness Properties of SGD and its Momentum Variants

TL;DR

This paper investigates conditions under which stochastic gradient descent (SGD) and its momentum variants maintain bounded iterates and function values, providing theoretical guarantees for a variety of practical machine learning problems.

Contribution

It establishes uniform boundedness properties of SGD and momentum methods under broad conditions, extending to generalized dissipativity and various step-size schedules.

Findings

Boundedness holds for common step-sizes like decay and cosine with restarts.

Applications include phase retrieval, Gaussian mixtures, neural networks, and logistic regression.

Extends to functions with sub-quadratic tail growth.

Abstract

A theoretical, and potentially also practical, problem with stochastic gradient descent is that trajectories may escape to infinity. In this note, we investigate uniform boundedness properties of iterates and function values along the trajectories of the stochastic gradient descent algorithm and its important momentum variant. Under smoothness and $R$-dissipativity of the loss function, we show that broad families of step-sizes, including the widely used step-decay and cosine with (or without) restart step-sizes, result in uniformly bounded iterates and function values. Several important applications that satisfy these assumptions, including phase retrieval problems, Gaussian mixture models, and some neural network classifiers, are discussed in detail. We further extend the uniform boundedness of SGD and its momentum variant under the generalized dissipativity for the functions whose tails grow slower than quadratic functions. This includes some interesting applications, for example, Bayesian logistic regression and logistic regression with $\ell_1$ regularization.