# Explicit Convergence Rate of a Distributed Alternating Direction Method of Multipliers

arXiv: https://arxiv.org/abs/1312.1085 · HTML: https://mlsanity.com/paper/1312.1085/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2013.

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

- [Accelerated Gradient Tracking over Time-varying Graphs for Decentralized Optimization](https://mlsanity.com/paper/2104.02596/) (2021)
- [Provably Accelerated Decentralized Gradient Method Over Unbalanced Directed Graphs](https://mlsanity.com/paper/2107.12065/) (2021)
- [Optimized Gradient Tracking for Decentralized Online Learning](https://mlsanity.com/paper/2306.06375/) (2023)

## Abstract (arXiv)

Consider a set of N agents seeking to solve distributively the minimization problem infx sumn = 1N fn(x) where the convex functions fn are local to the agents. The popular Alternating Direction Method of Multipliers has the potential to handle distributed optimization problems of this kind. We provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures. The rate of convergence of our method is considered. It is assumed that the infimum of the problem is reached at a point x_star, the functions fn are twice differentiable at this point and sum nabla2 fn(x_star) > 0 in the positive definite ordering of symmetric matrices. With these assumptions, it is shown that the convergence to the consensus x_star is linear and the exact rate is provided. Application examples where this rate can be optimized with respect to the ADMM free parameter ρ are also given.