  This paper considers a consensus optimization problem, where all the nodes in
a network, with access to the zeroth-order information of its local objective
function only, attempt to cooperatively achieve a common minimizer of the sum
of their local objectives. To address this problem, we develop ZoPro, a
zeroth-order proximal algorithm, which incorporates a zeroth-order oracle for
approximating Hessian and gradient into a recently proposed, high-performance
distributed second-order proximal algorithm. We show that the proposed ZoPro
algorithm, equipped with a dynamic stepsize, converges linearly to a
neighborhood of the optimum in expectation, provided that each local objective
function is strongly convex and smooth. Extensive simulations demonstrate that
ZoPro converges faster than several state-of-the-art distributed zeroth-order
algorithms and outperforms a few distributed second-order algorithms in terms
of running time for reaching given accuracy.
