Distributed optimization on directed graphs based on inexact ADMM with partial participation

TL;DR

This paper introduces a distributed optimization algorithm for directed networks using inexact ADMM with partial participation, achieving linear convergence and improved efficiency over existing methods.

Contribution

It proposes a novel inexact ADMM-based method for directed graphs that allows partial participation and provides sharper convergence analysis.

Findings

Linear convergence under standard assumptions

Superior convergence rate compared to Push-DIGing

Reduced computation and communication costs

Abstract

We consider the problem of minimizing the sum of cost functions pertaining to agents over a network whose topology is captured by a directed graph (i.e., asymmetric communication). We cast the problem into the ADMM setting, via a consensus constraint, for which both primal subproblems are solved inexactly. In specific, the computationally demanding local minimization step is replaced by a single gradient step, while the averaging step is approximated in a distributed fashion. Furthermore, partial participation is allowed in the implementation of the algorithm. Under standard assumptions on strong convexity and Lipschitz continuous gradients, we establish linear convergence and characterize the rate in terms of the connectivity of the graph and the conditioning of the problem. Our line of analysis provides a sharper convergence rate compared to Push-DIGing. Numerical experiments corroborate the merits of the proposed solution in terms of superior rate as well as computation and communication savings over baselines.