On local analysis of distributed optimization

We study distributed non-convex optimization on a time-varying multiagent network. Each node has access to its own smooth local cost function, and the collective goal is to minimize the sum of these functions. The perturbed push-sum algorithm used for convex distributed optimization in [17] is considered and further studied in [8]. We generalize the result obtained in [8] to the case of non-convex functions. Under some additional assumptions on the gradients we prove the convergence of the distributed push-sum algorithm to the set of critical points of the objective function. Moreover, 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.