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

## TL;DR

This paper analyzes the convergence behavior of distributed ADMM algorithms for consensus optimization when nodes experience additive random errors, providing bounds on steady-state error and demonstrating robustness under certain conditions.

## Contribution

It offers new analytical bounds on the mean squared error of noisy distributed ADMM and explores its robustness with bounded errors in convex settings.

## Key findings

- Bounds on steady-state mean squared error for strongly convex functions
- Error remains bounded when local functions are convex with bounded noise
- Numerical results validate theoretical bounds and parameter effects

## Abstract

Alternating Direction Method of Multipliers (ADMM) is a popular convex optimization algorithm, which can be employed for solving distributed consensus optimization problems. In this setting agents locally estimate the optimal solution of an optimization problem and exchange messages with their neighbors over a connected network. The distributed algorithms are typically exposed to different types of errors in practice, e.g., due to quantization or communication noise or loss. We here focus on analyzing the convergence of distributed ADMM for consensus optimization in presence of additive random node error, in which case, the nodes communicate a noisy version of their latest estimate of the solution to their neighbors in each iteration. We present analytical upper and lower bounds on the mean squared steady state error of the algorithm in case that the local objective functions are strongly convex and have Lipschitz continuous gradients. In addition we show that, when the local objective functions are convex and the additive node error is bounded, the estimation error of the noisy ADMM for consensus optimization is also bounded. Numerical results are provided which demonstrate the effectiveness of the presented analyses and shed light on the role of the system and network parameters on performance.