A linear algorithm for optimization over directed graphs with geometric convergence

TL;DR

This paper introduces a linear distributed optimization algorithm for directed graphs that converges geometrically to the global minimum, reducing complexity compared to existing methods based on push-sum techniques.

Contribution

The paper proposes a novel linear algorithm using inexact gradient and gradient estimation, avoiding the need for eigenvector learning in directed graph optimization.

Findings

Algorithm converges geometrically under strong convexity and Lipschitz conditions.

Reduces computational and communication complexity compared to push-sum based methods.

Simulation results validate theoretical convergence rates.

Abstract

In this letter, we study distributed optimization, where a network of agents, abstracted as a directed graph, collaborates to minimize the average of locally-known convex functions. Most of the existing approaches over directed graphs are based on push-sum (type) techniques, which use an independent algorithm to asymptotically learn either the left or right eigenvector of the underlying weight matrices. This strategy causes additional computation, communication, and nonlinearity in the algorithm. In contrast, we propose a linear algorithm based on an inexact gradient method and a gradient estimation technique. Under the assumptions that each local function is strongly-convex with Lipschitz-continuous gradients, we show that the proposed algorithm geometrically converges to the global minimizer with a sufficiently small step-size. We present simulations to illustrate the theoretical findings.