  We study a standard distributed optimization framework where $N$ networked
nodes collaboratively minimize the sum of their local convex costs. The main
body of existing work considers the described problem when the underling
network is either static or deterministically varying, and the distributed
optimization algorithm is of first or second order, i.e., it involves the local
costs' gradients and possibly the local Hessians. In this paper, we consider
the currently understudied but highly relevant scenarios when: 1) only noisy
function values' estimates are available (no gradients nor Hessians can be
evaluated); and 2) the underlying network is randomly varying (according to an
independent, identically distributed process). For the described random
networks-zeroth order optimization setting, we develop a distributed stochastic
approximation method of the Kiefer-Wolfowitz type. Furthermore, under standard
smoothness and strong convexity assumptions on the local costs, we establish
the $O(1/k^{1/2})$ mean square convergence rate for the method -- the rate that
matches that of the method's centralized counterpart under equivalent
conditions.
