Rate of Convergence for Distributed Optimization with Uncertain Communications

TL;DR

This paper establishes the first convergence rate bound of approximately O(1/√t) for a distributed optimization algorithm operating over randomly changing directed communication networks, advancing understanding of convergence under uncertainty.

Contribution

It provides the first convergence rate bound for distributed optimization algorithms over random directed graphs, extending prior convergence results to uncertain communication scenarios.

Findings

Convergence rate of approximately O(1/√t) was established.

The algorithm converges almost surely to an optimizer.

First such bound in the context of random directed communication networks.

Abstract

We consider the distributed optimization problem for the sum of convex functions where the underlying communications network connecting agents at each time is drawn at random from a collection of directed graphs. Building on an earlier work [15], where a modified version of the subgradient-push algorithm is shown to be almost surely convergent to an optimizer on sequences of random directed graphs, we find an upper bound of the order of $\sim O(\frac{1}{\sqrt{t}})$ on the convergence rate of our proposed algorithm, establishing the first convergence bound in such random settings.