Non-Convex Distributed Optimization

TL;DR

This paper investigates distributed optimization for non-convex functions over dynamic multi-agent networks, proving convergence to critical points and local minima with a convergence rate of O(1/t).

Contribution

It extends existing distributed optimization results to non-convex functions, demonstrating convergence of the push-sum algorithm and its perturbed variant.

Findings

Convergence to critical points under certain gradient conditions.

Almost sure convergence to local minima with perturbations.

Convergence rate of O(1/t) for the noised algorithm.

Abstract

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 $O(1/t)$.