  We consider a zeroth-order distributed optimization problem, where the global
objective function is a black-box function and, as such, its gradient
information is inaccessible to the local agents. Instead, the local agents can
only use the values of the objective function to estimate the gradient and
update their local decision variables. In this paper, we also assume that these
updates are done asynchronously. To solve this problem, we propose an
asynchronous zeroth-order distributed optimization method that relies on a
one-point residual feedback to estimate the unknown gradient. We show that this
estimator is unbiased under asynchronous updating, and theoretically analyze
the convergence of the proposed method. We also present numerical experiments
that demonstrate that our method outperforms two-point methods under
asynchronous updating. To the best of our knowledge, this is the first
asynchronous zeroth-order distributed optimization method that is also
supported by theoretical guarantees.
