On the linear convergence of the stochastic gradient method with constant step-size

TL;DR

This paper establishes a weaker necessary condition than the strong growth condition for the linear convergence of stochastic gradient methods with constant step-size, and analyzes convergence behavior under perturbations.

Contribution

It introduces a weaker necessary condition for linear convergence of SGM-CS and studies convergence under additive perturbations for PSGM-CS and proximal stochastic gradient methods.

Findings

Necessary condition for linear convergence weaker than SGC.

Linear convergence to a noise-dominated region under perturbations.

Distance to optimal solution proportional to step-size and noise level.

Abstract

The strong growth condition (SGC) is known to be a sufficient condition for linear convergence of the stochastic gradient method using a constant step-size $\gamma$ (SGM-CS). In this paper, we provide a necessary condition, for the linear convergence of SGM-CS, that is weaker than SGC. Moreover, when this necessary is violated up to a additive perturbation $\sigma$, we show that both the projected stochastic gradient method using a constant step-size (PSGM-CS) and the proximal stochastic gradient method exhibit linear convergence to a noise dominated region, whose distance to the optimal solution is proportional to $\gamma \sigma$.