  We present and analyze a stochastic distributed method (S-NEAR-DGD) that can
tolerate inexact computation and inaccurate information exchange to alleviate
the problems of costly gradient evaluations and bandwidth-limited communication
in large-scale systems. Our method is based on a class of flexible, distributed
first order algorithms that allow for the trade-off of computation and
communication to best accommodate the application setting. We assume that all
the information exchange between nodes is subject to random distortion and that
only stochastic approximations of the true gradients are available. Our
theoretical results prove that the proposed algorithm converges linearly in
expectation to a neighborhood of the optimal solution for strongly convex
objective functions with Lipschitz gradients. We characterize the dependence of
this neighborhood on algorithm and network parameters, the quality of the
communication channel and the precision of the stochastic gradient
approximations used. Finally, we provide numerical results to evaluate the
empirical performance of our method.
