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

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, under the restricted strong convexity assumption, and the proximal stochastic gradient method, under the strong convexity assumption, exhibit linear convergence to a noise dominated region, whose distance to the optimal solution is proportional to $${\gamma {\sigma }}$$ .