  We consider a distributed multi-agent network system where the goal is to
minimize a sum of convex objective functions of the agents subject to a common
convex constraint set. Each agent maintains an iterate sequence and
communicates the iterates to its neighbors. Then, each agent combines weighted
averages of the received iterates with its own iterate, and adjusts the iterate
by using subgradient information (known with stochastic errors) of its own
function and by projecting onto the constraint set.
  The goal of this paper is to explore the effects of stochastic subgradient
errors on the convergence of the algorithm. We first consider the behavior of
the algorithm in mean, and then the convergence with probability 1 and in mean
square. We consider general stochastic errors that have uniformly bounded
second moments and obtain bounds on the limiting performance of the algorithm
in mean for diminishing and non-diminishing stepsizes. When the means of the
errors diminish, we prove that there is mean consensus between the agents and
mean convergence to the optimum function value for diminishing stepsizes. When
the mean errors diminish sufficiently fast, we strengthen the results to
consensus and convergence of the iterates to an optimal solution with
probability 1 and in mean square.
