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 $\gamma \sigma$

A stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions

We propose an inertial forward-backward splitting algorithm to compute the zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal-dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed

A first-order stochastic primal-dual algorithm with correction step

We investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in {\em Y. Drori, S. Sabach, and M. Teboulle, A simple algorithm for a class of nonsmooth convex-concave saddle-point problems, 2015} and {\em I. Loris and C. Verhoeven, On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty, 2011} for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi Fej\'er-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models

Proximity for Sums of Composite Functions

We propose an algorithm for computing the proximity operator of a sum of composite convex functions in Hilbert spaces and investigate its asymptotic behavior. Applications to best approximation and image recovery are described