A unified framework for distributed optimization algorithms over time-varying directed graphs

TL;DR

This paper introduces a unified framework for various distributed optimization algorithms over time-varying directed graphs, providing a common convergence analysis and deriving a new scheme.

Contribution

It unifies multiple existing algorithms under a single framework and offers a weaker convergence condition, also proposing a novel distributed optimization method.

Findings

Unified convergence proof under weaker algebraic conditions

Derivation of a new distributed optimization scheme

Broader applicability to time-varying directed graphs

Abstract

In this paper, we propose a framework under which the decentralized optimization algorithms suggested in \cite{JKJJ,MA, NO,NO2} can be treated in a unified manner. More precisely, we show that the distributed subgradient descent algorithms \cite{JKJJ, NO}, the subgradient-push algorithm \cite{NO2}, and the distributed algorithm with row-stochastic matrix \cite{MA} can be derived by making suitable choices of consensus matrices, step-size and subgradient from the decentralized subgradient descent proposed in \cite{NO}. As a result of such unified understanding, we provide a convergence proof that covers the algorithms in \cite{JKJJ,MA, NO,NO2} under a novel algebraic condition that is strictly weaker than the conventional graph-theoretic condition in \cite{NO}. This unification also enables us to derive a new distributed optimization scheme.