Linear Convergence of Prox-SVRG Method for Separable Non-smooth Convex Optimization Problems under Bounded Metric Subregularity

With the help of bounded metric subregularity which is weaker than strong convexity, we show the linear convergence of proximal stochastic variance-reduced gradient (Prox-SVRG) method for solving a class of separable non-smooth convex optimization problems where the smooth item is a composite of strongly convex function and linear function. We introduce an equivalent characterization for the bounded metric subregularity by taking into account the calmness condition of a perturbed linear system. This equivalent characterization allows us to provide a verifiable sufficient condition to ensure linear convergence of Prox-SVRG and randomized block-coordinate proximal gradient methods. Furthermore, we verify that these sufficient conditions hold automatically when the non-smooth item is the generalized sparse group Lasso regularizer.