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

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 $\gamma \sigma$

Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration

Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal splitting algorithm, establish its convergence, and apply it to an online image restoration problem.Comment: 5 Figure