Stability, Linear Convergence, and Robustness of the Wang-Elia Algorithm for Distributed Consensus Optimization

TL;DR

This paper analyzes the Wang-Elia algorithm for distributed consensus optimization, proving its stability, linear convergence, and robustness using Lyapunov methods, and revealing connections with gradient tracking and integral control.

Contribution

It provides a Lyapunov-based stability and robustness analysis of the Wang-Elia algorithm, highlighting its linear convergence and fundamental links to other control strategies.

Findings

Proves input-to-state stability of the algorithm.

Establishes linear convergence in the absence of perturbations.

Unveils connections with gradient tracking and distributed integral control.

Abstract

We revisit an algorithm for distributed consensus optimization proposed in 2010 by J. Wang and N. Elia. By means of a Lyapunov-based analysis, we prove input-to-state stability of the algorithm relative to a closed invariant set composed of optimal equilibria and with respect to perturbations affecting the algorithm's dynamics. In the absence of perturbations, this result implies linear convergence of the local estimates and Lyapunov stability of the optimal steady state. Moreover, we unveil fundamental connections with the well-known Gradient Tracking and with distributed integral control. Overall, our results suggest that a control theoretic approach can have a considerable impact on (distributed) optimization, especially when robustness is considered.