  We consider the standard model of distributed optimization of a sum of
functions $F(\bz) = \sum_{i=1}^n f_i(\bz)$, where node $i$ in a network holds
the function $f_i(\bz)$. We allow for a harsh network model characterized by
asynchronous updates, message delays, unpredictable message losses, and
directed communication among nodes. In this setting, we analyze a modification
of the Gradient-Push method for distributed optimization, assuming that
\begin{enumerate*}[label=(\roman*)] \item node $i$ is capable of generating
gradients of its function $f_i(\bz)$ corrupted by zero-mean bounded-support
additive noise at each step, \item $F(\bz)$ is strongly convex, and \item each
$f_i(\bz)$ has Lipschitz gradients. We show that our proposed method
asymptotically performs as well as the best bounds on centralized gradient
descent that takes steps in the direction of the sum of the noisy gradients of
all the functions $f_1(\bz), \ldots, f_n(\bz)$ at each step.
