# Convergence Rate of Distributed ADMM over Networks

arXiv: https://arxiv.org/abs/1601.00194 · HTML: https://mlsanity.com/paper/1601.00194/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2016.

## Built on by (most cited first, 16 of 16)

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- [Fully Decentralized, Scalable Gaussian Processes for Multi-Agent Federated Learning](https://mlsanity.com/paper/2203.02865/) (2022)
- [Communication-Efficient Stochastic Distributed Learning](https://mlsanity.com/paper/2501.13516/) (2025)
- [Learning to accelerate distributed ADMM using graph neural networks](https://mlsanity.com/paper/2509.05288/) (2025)
- [Limited Communications Distributed Optimization via Deep Unfolded Distributed ADMM](https://mlsanity.com/paper/2309.14353/) (2023)
- [Provably Accelerated Decentralized Gradient Method Over Unbalanced Directed Graphs](https://mlsanity.com/paper/2107.12065/) (2021)
- [Distributed optimization: designed for federated learning](https://mlsanity.com/paper/2508.08606/) (2025)
- [Moreau Envelope ADMM for Decentralized Weakly Convex Optimization](https://mlsanity.com/paper/2308.16752/) (2023)
- [Fast networked data selection via distributed smoothed quantile estimation](https://mlsanity.com/paper/2406.01929/) (2024)
- [Accelerating Distributed Optimization: A Primal-Dual Perspective on Local Steps](https://mlsanity.com/paper/2407.02689/) (2024)
- [Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems](https://mlsanity.com/paper/2408.15969/) (2024)
- [Modular Distributed Nonconvex Learning with Error Feedback](https://mlsanity.com/paper/2503.14055/) (2025)

## Abstract (arXiv)

We propose a distributed algorithm based on Alternating Direction Method of Multipliers (ADMM) to minimize the sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate O(frac1T), where T is the number of iterations. We then show that if the functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an ε-optimal solution can be computed with O(√kappaf log (1/ε)) iterations, where kappaf is the condition number of the problem. Our analysis also highlights the effect of network structure on the convergence rate through maximum and minimum degree of nodes as well as the algebraic connectivity of the network.