  We study distributed non-convex optimization on a time-varying multi-agent
network. Each node has access to its own smooth local cost function, and the
collective goal is to minimize the sum of these functions. We generalize the
results obtained previously to the case of non-convex functions. Under some
additional technical assumptions on the gradients we prove the convergence of
the distributed push-sum algorithm to some critical point of the objective
function. By utilizing perturbations on the update process, we show the almost
sure convergence of the perturbed dynamics to a local minimum of the global
objective function. Our analysis shows that this noised procedure converges at
a rate of $O(1/t)$.
