# Convergence of Message-Passing for Distributed Convex Optimisation with Scaled Diagonal Dominance

## TL;DR

This paper analyzes the convergence of message-passing algorithms for distributed convex optimization, establishing asymptotic convergence and providing tighter, simpler bounds on the convergence rate under scaled diagonal dominance assumptions.

## Contribution

It introduces new convergence results for message-passing algorithms without requiring known convex pairwise components, and offers a simplified, tighter convergence rate bound.

## Key findings

- Proves asymptotic convergence under scaled diagonal dominance.
- Provides a simpler, tighter convergence rate bound.
- Generalizes results for quadratic optimization.

## Abstract

This paper studies the convergence properties the well-known message-passing algorithm for convex optimisation. Under the assumption of pairwise separability and scaled diagonal dominance, asymptotic convergence is established and a simple bound for the convergence rate is provided for message-passing. In comparison with previous results, our results do not require the given convex program to have known convex pairwise components and that our bound for the convergence rate is tighter and simpler. When specialised to quadratic optimisation, we generalise known results by providing a very simple bound for the convergence rate.