Fast Distributed Gradient Methods

• Main Authors: Jakovetic, Dusan, Xavier, Joao, Moura, Jose M. F
• Format: Journal Article
• Language: English
• Published: 13.12.2011
• Subjects: Computer Science - Information Theory; Mathematics - Information Theory
• DOI: 10.48550/arxiv.1112.2972

We study distributed optimization problems when $N$ nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant $L$), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications $\mathcal{K}$ and the per-node gradient evaluations $k$. Our first method, Distributed Nesterov Gradient, achieves rates $O\left({\log \mathcal{K}}/{\mathcal{K}}\right)$ and $O\left({\log k}/{k}\right)$. Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of $L$ and $\mu(W)$ -- the second largest singular value of the $N \times N$ doubly stochastic weight matrix $W$. It achieves rates $O\left({1}/{\mathcal{K}^{2-\xi}}\right)$ and $O\left({1}/{k^2}\right)$ ($\xi>0$ arbitrarily small). Further, we give with both methods explicit dependence of the convergence constants on $N$ and $W$. Simulation examples illustrate our findings.