Distributed Linear Equations over Random Networks

08/22/2020
by   Peng Yi, et al.
0

Distributed linear algebraic equation over networks, where nodes hold a part of problem data and cooperatively solve the equation via node-to-node communications, is a basic distributed computation task receiving an increasing research attention. Communications over a network have a stochastic nature, with both temporal and spatial dependence due to link failures, packet dropouts or node recreation, etc. In this paper, we study the convergence and convergence rate of distributed linear equation protocols over a ∗-mixing random network, where the temporal and spatial dependencies between the node-to-node communications are allowed. When the network linear equation admits exact solutions, we prove the mean-squared exponential convergence rate of the distributed projection consensus algorithm, while the lower and upper bound estimations of the convergence rate are also given for independent and identically distributed (i.i.d.) random graphs. Motivated by the randomized Kaczmarz algorithm, we also propose a distributed randomized projection consensus algorithm, where each node randomly selects one row of local linear equations for projection per iteration, and establish an exponential convergence rate for this algorithm. When the network linear equation admits no exact solution, we prove that a distributed gradient-descent-like algorithm with diminishing step-sizes can drive all nodes' states to a least-squares solution at a sublinear rate. These results collectively illustrate that distributed computations may overcome communication correlations if the prototype algorithms enjoy certain contractive properties or are designed with suitable parameters.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

12/29/2020

Consensus-Based Distributed Computation of Link-Based Network Metrics

Average consensus algorithms have wide applications in distributed compu...
12/20/2018

Using First Hitting Times to Find Sets that Maximize the Convergence Rate to Consensus

In a model of communication in a social network described by a simple co...
03/20/2018

Frank-Wolfe with Subsampling Oracle

We analyze two novel randomized variants of the Frank-Wolfe (FW) or cond...
08/25/2019

Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus

In this paper we discuss the convergence rate for Galerkin approximation...
05/07/2018

Convergence Rate Analysis for Periodic Gossip Algorithms in Wireless Sensor Networks

Periodic gossip algorithms have generated a lot of interest due to their...
11/12/2021

Solving A System Of Linear Equations By Randomized Orthogonal Projections

Randomization has shown catalyzing effects in linear algebra with promis...
01/03/2019

Finite rate distributed weight-balancing and average consensus over digraphs

This paper proposes and analyzes the first distributed algorithm that so...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.