# Analysis of Distributed ADMM Algorithm for Consensus Optimization in Presence of Error

## TL;DR

This paper analyzes how distributed ADMM algorithms for consensus optimization behave when nodes introduce errors, showing linear convergence under certain conditions and providing numerical validation.

## Contribution

It offers a novel convergence analysis of distributed ADMM in the presence of additive node errors, including conditions for linear convergence and insights into convergence points.

## Key findings

- ADMM converges linearly despite node errors under specific conditions
- The convergence point is characterized in the presence of additive noise
- Numerical results confirm the theoretical analysis

## Abstract

ADMM is a popular algorithm for solving convex optimization problems. Applying this algorithm to distributed consensus optimization problem results in a fully distributed iterative solution which relies on processing at the nodes and communication between neighbors. Local computations usually suffer from different types of errors, due to e.g., observation or quantization noise, which can degrade the performance of the algorithm. In this work, we focus on analyzing the convergence behavior of distributed ADMM for consensus optimization in presence of additive node error. We specifically show that (a noisy) ADMM converges linearly under certain conditions and also examine the associated convergence point. Numerical results are provided which demonstrate the effectiveness of the presented analysis.