Data Rates for Network Linear Equations

TL;DR

This paper develops a distributed quantized algorithm for solving network linear equations under finite data rate constraints, demonstrating exponential convergence and minimal data rate requirements for accurate solutions.

Contribution

It introduces a novel distributed quantized algorithm that guarantees convergence for network linear equations with finite data rates, including cases with unique solutions and least-squares solutions.

Findings

Exponential convergence to the solution under finite data rate constraints.

Larger quantization levels accelerate convergence but are limited by network structure.

Minimal data rate suffices for guaranteed convergence with proper step size selection.

Abstract

In this paper, we study network linear equations subject to digital communications with a finite data rate, where each node is associated with one equation from a system of linear equations. Each node holds a dynamic state and interacts with its neighbors through an undirected connected graph, where along each link the pair of nodes share information. Due to the data-rate constraint, each node builds an encoder-decoder pair, with which it produces transmitted message with a zooming-in finite-level uniform quantizer and also generates estimates of its neighbors' states from the received signals. We then propose a distributed quantized algorithm and show that when the network linear equations admit a unique solution, each node's state is driven to that solution exponentially. We further establish the asymptotic rate of convergence, which shows that a larger number of quantization levels leads to a faster convergence rate but is still fundamentally bounded by the inherent network structure and the linear equations. When a unique least-squares solution exists, we show that the algorithm can compute such a solution with a suitably selected time-varying step size inherited from the encoder and zooming-in quantizer dynamics. In both cases, a minimal data rate is shown to be enough for guaranteeing the desired convergence when the step sizes are properly chosen. These results assure the applicability of various network linear equation solvers in the literature when peer-to-peer communication is digital.