Stochastic gradient descent with noise of machine learning type. Part I: Discrete time analysis

TL;DR

This paper analyzes stochastic gradient descent (SGD) with machine learning noise, showing conditions under which it converges rapidly to the global minimum, especially in overparametrized deep learning landscapes.

Contribution

It provides a discrete-time analysis of SGD with realistic noise models, establishing convergence rates and conditions for global optimality in complex energy landscapes.

Findings

SGD can have a uniformly positive learning rate in certain landscapes.

Exponential convergence to the global minimum is possible under Lojasiewicz inequality.

Almost sure convergence occurs even with local minima, from finite energy initializations.

Abstract

Stochastic gradient descent (SGD) is one of the most popular algorithms in modern machine learning. The noise encountered in these applications is different from that in many theoretical analyses of stochastic gradient algorithms. In this article, we discuss some of the common properties of energy landscapes and stochastic noise encountered in machine learning problems, and how they affect SGD-based optimization. In particular, we show that the learning rate in SGD with machine learning noise can be chosen to be small, but uniformly positive for all times if the energy landscape resembles that of overparametrized deep learning problems. If the objective function satisfies a Lojasiewicz inequality, SGD converges to the global minimum exponentially fast, and even for functions which may have local minima, we establish almost sure convergence to the global minimum at an exponential rate from any finite energy initialization. The assumptions that we make in this result concern the behavior where the objective function is either small or large and the nature of the gradient noise, but the energy landscape is fairly unconstrained on the domain where the objective function takes values in an intermediate regime.