  This paper is concerned with distributed stochastic multi-agent constrained
optimization problem over time-varying network with a class of communication
noise. This paper considers the problem in composite optimization setting which
is more general in the literature of noisy network optimization. It is
noteworthy that the mainstream existing methods for noisy network optimization
are Euclidean projection based. Based on Bregman projection-based mirror
descent scheme, we present a non-Euclidean method and investigate their
convergence behavior. This method is the distributed stochastic composite
mirror descent type method (DSCMD-N) which provides a more general algorithm
framework. Some new error bounds for DSCMD-N are obtained.
  To the best of our knowledge, this is the first work to analyze and derive
convergence rates of optimization algorithm in noisy network optimization. We
also show that an optimal rate of $O(1/\sqrt{T})$ in nonsmooth convex
optimization can be obtained for the proposed method under appropriate
communication noise condition. Moreover, novel convergence results are
comprehensively derived in expectation convergence, high probability
convergence, and almost surely sense.
