  We study a class of zeroth-order distributed optimization problems, where
each agent can control a partial vector and observe a local cost that depends
on the joint vector of all agents, and the agents can communicate with each
other with time delay. We propose and study a gradient descent-based algorithm
using two-point gradient estimators with diminishing smoothing parameters and
diminishing step-size and we establish the convergence rate to a first-order
stationary point for general nonconvex problems. A byproduct of our proposed
method with diminishing step size and smoothing parameters, as opposed to the
fixed-parameter scheme, is that our proposed algorithm does not require any
information regarding the local cost functions. This makes the solution
appealing in practice as it allows for optimizing an unknown (black-box) global
function without prior knowledge of its smoothness parameters. At the same
time, the performance will adaptively match the problem instance parameters.
