  This paper investigates the stochastic distributed nonconvex optimization
problem of minimizing a global cost function formed by the summation of $n$
local cost functions. We solve such a problem by involving zeroth-order (ZO)
information exchange. In this paper, we propose a ZO distributed primal-dual
coordinate method (ZODIAC) to solve the stochastic optimization problem. Agents
approximate their own local stochastic ZO oracle along with coordinates with an
adaptive smoothing parameter. We show that the proposed algorithm achieves the
convergence rate of $\mathcal{O}(\sqrt{p}/\sqrt{T})$ for general nonconvex cost
functions. We demonstrate the efficiency of proposed algorithms through a
numerical example in comparison with the existing state-of-the-art centralized
and distributed ZO algorithms.
