# On the Linear Convergence of the ADMM in Decentralized Consensus Optimization

arXiv: https://arxiv.org/abs/1307.5561 · HTML: https://mlsanity.com/paper/1307.5561/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2013.

## Built on by (most cited first, 30 of 57)

- [Decentralized Federated Learning: Fundamentals, State of the Art, Frameworks, Trends, and Challenges](https://mlsanity.com/paper/2211.08413/) (2022)
- [Semi-Decentralized Federated Learning with Cooperative D2D Local Model Aggregations](https://mlsanity.com/paper/2103.10481/) (2021)
- [Decentralized Riemannian Gradient Descent on the Stiefel Manifold](https://mlsanity.com/paper/2102.07091/) (2021)
- [Removing Data Heterogeneity Influence Enhances Network Topology Dependence of Decentralized SGD](https://mlsanity.com/paper/2105.08023/) (2021)
- [Accelerating Gossip SGD with Periodic Global Averaging](https://mlsanity.com/paper/2105.09080/) (2021)
- [Communication-Efficient Topologies for Decentralized Learning with O(1) Consensus Rate](https://mlsanity.com/paper/2210.07881/) (2022)
- [Revisiting Optimal Convergence Rate for Smooth and Non-convex Stochastic Decentralized Optimization](https://mlsanity.com/paper/2210.07863/) (2022)
- [BlueFog: Make Decentralized Algorithms Practical for Optimization and Deep Learning](https://mlsanity.com/paper/2111.04287/) (2021)
- [Networked Signal and Information Processing](https://mlsanity.com/paper/2210.13767/) (2022)
- [Decentralized Nonconvex Optimization with Guaranteed Privacy and Accuracy](https://mlsanity.com/paper/2212.07534/) (2022)
- [Confederated Learning: Federated Learning with Decentralized Edge Servers](https://mlsanity.com/paper/2205.14905/) (2022)
- [Advancements in Federated Learning: Models, Methods, and Privacy](https://mlsanity.com/paper/2302.11466/) (2023)
- [Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold](https://mlsanity.com/paper/2308.10547/) (2023)
- [Stochastic Unrolled Federated Learning](https://mlsanity.com/paper/2305.15371/) (2023)
- [DSGD-CECA: Decentralized SGD with Communication-Optimal Exact Consensus Algorithm](https://mlsanity.com/paper/2306.00256/) (2023)
- [Federated Stochastic Primal-dual Learning with Differential Privacy](https://mlsanity.com/paper/2204.12284/) (2022)
- [DR-DSGD: A Distributionally Robust Decentralized Learning Algorithm over Graphs](https://mlsanity.com/paper/2208.13810/) (2022)
- [DISA: A Dual Inexact Splitting Algorithm for Distributed Convex Composite Optimization](https://mlsanity.com/paper/2209.01850/) (2022)
- [Decentralized Inexact Proximal Gradient Method With Network-Independent Stepsizes for Convex Composite Optimization](https://mlsanity.com/paper/2302.03238/) (2023)
- [From Noisy Fixed-Point Iterations to Private ADMM for Centralized and Federated Learning](https://mlsanity.com/paper/2302.12559/) (2023)
- [Communication Efficient Federated Learning via Ordered ADMM in a Fully Decentralized Setting](https://mlsanity.com/paper/2202.02580/) (2022)
- [Fast and Robust Sparsity Learning over Networks: A Decentralized Surrogate Median Regression Approach](https://mlsanity.com/paper/2202.05498/) (2022)
- [Fully Decentralized, Scalable Gaussian Processes for Multi-Agent Federated Learning](https://mlsanity.com/paper/2203.02865/) (2022)
- [Personalized Graph Federated Learning with Differential Privacy](https://mlsanity.com/paper/2306.06399/) (2023)
- [Distributed Conformal Prediction via Message Passing](https://mlsanity.com/paper/2501.14544/) (2025)
- [From Centralized to Decentralized Federated Learning: Theoretical Insights, Privacy Preservation, and Robustness Challenges](https://mlsanity.com/paper/2503.07505/) (2025)
- [Decentralized Multi-Task Stochastic Optimization With Compressed Communications](https://mlsanity.com/paper/2112.12373/) (2021)
- [Mobilizing Personalized Federated Learning in Infrastructure-Less and Heterogeneous Environments via Random Walk Stochastic ADMM](https://mlsanity.com/paper/2304.12534/) (2023)
- [Transient growth of accelerated optimization algorithms](https://mlsanity.com/paper/2103.08017/) (2021)
- [A Penalty-Based Method for Communication-Efficient Decentralized Bilevel Programming](https://mlsanity.com/paper/2211.04088/) (2022)

## Abstract (arXiv)

In decentralized consensus optimization, a connected network of agents collaboratively minimize the sum of their local objective functions over a common decision variable, where their information exchange is restricted between the neighbors. To this end, one can first obtain a problem reformulation and then apply the alternating direction method of multipliers (ADMM). The method applies iterative computation at the individual agents and information exchange between the neighbors. This approach has been observed to converge quickly and deemed powerful. This paper establishes its linear convergence rate for decentralized consensus optimization problem with strongly convex local objective functions. The theoretical convergence rate is explicitly given in terms of the network topology, the properties of local objective functions, and the algorithm parameter. This result is not only a performance guarantee but also a guideline toward accelerating the ADMM convergence.