  The recently developed Distributed Block Proximal Method, for solving
stochastic big-data convex optimization problems, is studied in this paper
under the assumption of constant stepsizes and strongly convex (possibly
non-smooth) local objective functions. This class of problems arises in many
learning and classification problems in which, for example, strongly-convex
regularizing functions are included in the objective function, the decision
variable is extremely high dimensional, and large datasets are employed. The
algorithm produces local estimates by means of block-wise updates and
communication among the agents. The expected distance from the (global)
optimum, in terms of cost value, is shown to decay linearly to a constant value
which is proportional to the selected local stepsizes. A numerical example
involving a classification problem corroborates the theoretical results.
