  In this letter, we first propose a \underline{Z}eroth-\underline{O}rder
c\underline{O}ordinate \underline{M}ethod~(ZOOM) to solve the stochastic
optimization problem over a decentralized network with only zeroth-order~(ZO)
oracle feedback available. Moreover, we equip a simple mechanism "powerball" to
ZOOM and propose ZOOM-PB to accelerate the convergence of ZOOM. Compared with
the existing methods, we verify the proposed algorithms through two benchmark
examples in the literature, namely the black-box binary classification and the
generating adversarial examples from black-box DNNs in order to compare with
the existing state-of-the-art centralized and distributed ZO algorithms. The
numerical results demonstrate a faster convergence rate of the proposed
algorithms.
