Distributed Optimization on Riemannian Manifolds for multi-agent networks

TL;DR

This paper introduces a distributed optimization algorithm on Riemannian manifolds for multi-agent networks, extending Euclidean methods to curved spaces and analyzing convergence for convex and non-convex functions.

Contribution

It proposes a novel Riemannian distributed optimization algorithm with detailed convergence analysis applicable to both convex and non-convex functions.

Findings

Algorithm converges for geodesically convex functions

Algorithm performs well on standard applications

Convergence analysis covers non-convex cases

Abstract

We consider the consensual distributed optimization problem in the Riemannian context. Specifically, the minimization of a sum of functions form is studied where each individual function in the sum is located at the node of a network. An algorithm, which is a direct generalization of the Euclidean case, to solve the problem is proposed. The convergence analysis is carried out in full detail for geodesically convex as well as non-convex functions. The algorithm is demonstrated using some standard applications which fit the presented framework.