Fast Distributed Gradient Methods

TL;DR

This paper introduces two fast distributed gradient algorithms for convex optimization over networks, achieving improved convergence rates with explicit dependence on network parameters, and demonstrates their effectiveness through simulations.

Contribution

The paper proposes two novel distributed Nesterov gradient algorithms with enhanced convergence rates and explicit dependence on network size and topology, advancing distributed optimization methods.

Findings

Achieves convergence rates of O(log K / K) and O(1 / K^2)

Provides explicit constants depending on network size and topology

Demonstrates effectiveness through simulation examples

Abstract

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.