Distributed Stochastic Approximation: Weak Convergence and Network Design

TL;DR

This paper analyzes distributed stochastic approximation algorithms on digraph networks, proving weak convergence, linking it to network design, and improving convergence rates through stochastic differential equations, supported by simulations.

Contribution

It introduces a methodology for network design to achieve desired consensus behavior and connects convergence properties with network and objective function characteristics.

Findings

Proved weak convergence of broadcast gossip algorithms.

Formulated an ODE linking convergence points with network properties.

Enhanced convergence rate analysis using stochastic differential equations.

Abstract

This paper studies distributed stochastic approximation algorithms based on broadcast gossip on communication networks represented by digraphs. Weak convergence of these algorithms is proved, and an associated ordinary differential equation (ODE) is formulated connecting convergence points with local objective functions and network properties. Using these results, a methodology is proposed for network design, aimed at achieving the desired asymptotic behavior at consensus. Convergence rate of the algorithm is also analyzed and further improved using an attached stochastic differential equation. Simulation results illustrate the theoretical concepts.