  We consider a distributed convex optimization problem in a network which is
time-varying and not always strongly connected. The local cost function of each
node is affected by some stochastic process. All nodes of the network
collaborate to minimize the average of their local cost functions. The major
challenge of our work is that the gradient of cost functions is supposed to be
unavailable and has to be estimated only based on the numerical observation of
cost functions. Such problem is known as zeroth-order stochastic convex
optimization (ZOSCO). In this paper we take a first step towards the
distributed optimization problem with a ZOSCO setting. The proposed algorithm
contains two basic steps at each iteration: i) each unit updates a local
variable according to a random perturbation based single point gradient
estimator of its own local cost function; ii) each unit exchange its local
variable with its direct neighbors and then perform a weighted average. In the
situation where the cost function is smooth and strongly convex, our attainable
optimization error is $O(T^{-1/2})$ after $T$ iterations. This result is
interesting as $O(T^{-1/2})$ is the optimal convergence rate in the ZOSCO
problem. We have also investigate the optimization error with the general
Lipschitz convex function, the result is $O(T^{-1/4})$.
