  This paper proposes a framework to study the convergence of stochastic
optimization and learning algorithms. The framework is modeled over the
different challenges that these algorithms pose, such as (i) the presence of
random additive errors (e.g. due to stochastic gradients), and (ii) random
coordinate updates (e.g. due to asynchrony in distributed set-ups). The paper
covers both convex and strongly convex problems, and it also analyzes online
scenarios, involving changes in the data and costs. The paper relies on
interpreting stochastic algorithms as the iterated application of stochastic
operators, thus allowing us to use the powerful tools of operator theory. In
particular, we consider operators characterized by additive errors with
sub-Weibull distribution (which parameterize a broad class of errors by their
tail probability), and random updates. In this framework we derive convergence
results in mean and in high probability, by providing bounds to the distance of
the current iteration from a solution of the optimization or learning problem.
The contributions are discussed in light of federated learning applications.
