# Distributed Linearized Alternating Direction Method of Multipliers for Composite Convex Consensus Optimization

arXiv: https://arxiv.org/abs/1512.08122 · HTML: https://mlsanity.com/paper/1512.08122/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Dec 2015.

## Built on by (most cited first, 13 of 13)

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- [Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold](https://mlsanity.com/paper/2308.10547/) (2023)
- [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)
- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)
- [Limited Communications Distributed Optimization via Deep Unfolded Distributed ADMM](https://mlsanity.com/paper/2309.14353/) (2023)
- [Stochastic Constrained Decentralized Optimization for Machine Learning with Fewer Data Oracles: a Gradient Sliding Approach](https://mlsanity.com/paper/2404.02511/) (2024)
- [Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking](https://mlsanity.com/paper/2506.07351/) (2025)
- [Distributed Optimization via Kernelized Multi-armed Bandits](https://mlsanity.com/paper/2312.04719/) (2023)
- [Accelerating Distributed Optimization: A Primal-Dual Perspective on Local Steps](https://mlsanity.com/paper/2407.02689/) (2024)
- [Multi-Timescale Primal Dual Hybrid Gradient with Application to Distributed Optimization](https://mlsanity.com/paper/2506.15387/) (2025)

## Abstract (arXiv)

Given an undirected graph G=(N,E) of agents N=1,ldots,N connected with edges in E, we study how to compute an optimal decision on which there is consensus among agents and that minimizes the sum of agent-specific private convex composite functions Phiii∈N while respecting privacy requirements, where Phiitriangleq xii + fi belongs to agent-i. Assuming only agents connected by an edge can communicate, we propose a distributed proximal gradient method DPGA for consensus optimization over both unweighted and weighted static (undirected) communication networks. In one iteration, each agent-i computes the prox map of xii and gradient of fi, and this is followed by local communication with neighboring agents. We also study its stochastic gradient variant, SDPGA, which can only access to noisy estimates of nabla fi at each agent-i. This computational model abstracts a number of applications in distributed sensing, machine learning and statistical inference. We show ergodic convergence in both sub-optimality error and consensus violation for DPGA and SDPGA with rates O(1/t) and O(1/√t), respectively.