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

arXiv: https://arxiv.org/abs/2204.10030 · HTML: https://mlsanity.com/paper/2204.10030/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Apr 2022.

## Built on by (most cited first, 3 of 3)

- [A survey on secure decentralized optimization and learning](https://mlsanity.com/paper/2408.08628/) (2024)
- [Robust Online Learning over Networks](https://mlsanity.com/paper/2309.00520/) (2023)
- [A primal-dual perspective for distributed TD-learning](https://mlsanity.com/paper/2310.00638/) (2023)

## Abstract (arXiv)

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.