# On the Linear Convergence of Distributed Optimization over Directed Graphs

arXiv: https://arxiv.org/abs/1510.02149 · HTML: https://mlsanity.com/paper/1510.02149/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2015.

## Built on by (most cited first, 2 of 2)

- [Decentralized Directed Collaboration for Personalized Federated Learning](https://mlsanity.com/paper/2405.17876/) (2024)
- [Asymmetrically Decentralized Federated Learning](https://mlsanity.com/paper/2310.05093/) (2023)

## Abstract (arXiv)

This paper develops a fast distributed algorithm, termed DEXTRA, to solve the optimization problem when~n agents reach agreement and collaboratively minimize the sum of their local objective functions over the network, where the communication between the agents is described by a~directed graph. Existing algorithms solve the problem restricted to directed graphs with convergence rates of O(ln k/√k) for general convex objective functions and O(ln k/k) when the objective functions are strongly-convex, where~k is the number of iterations. We show that, with the appropriate step-size, DEXTRA converges at a linear rate O(τk) for 0<τ<1, given that the objective functions are restricted strongly-convex. The implementation of DEXTRA requires each agent to know its local out-degree. Simulation examples further illustrate our findings.