On the Rate of Convergence of Mean-Field Models: Stein's Method Meets the Perturbation Theory

TL;DR

This paper introduces a new framework combining Stein's method and perturbation theory to analyze the convergence rate of Markov chains to mean-field models, providing quantitative error bounds based on stability properties.

Contribution

It presents a novel approach to quantify the convergence rate of CTMCs to mean-field limits using stability analysis, which was not previously established.

Findings

Mean square difference between CTMC steady state and mean-field equilibrium is O(1/M).

Convergence in mean-square sense under stability conditions.

New framework for analyzing finite-size effects in mean-field approximations.

Abstract

This paper studies the rate of convergence of a family of continuous-time Markov chains (CTMC) to a mean-field model. When the mean-field model is a finite-dimensional dynamical system with a unique equilibrium point, an analysis based on Stein's method and the perturbation theory shows that under some mild conditions, the stationary distributions of CTMCs converge (in the mean-square sense) to the equilibrium point of the mean-field model if the mean-field model is globally asymptotically stable and locally exponentially stable. In particular, the mean square difference between the $M$th CTMC in the steady state and the equilibrium point of the mean-field system is $O(1/M),$ where $M$ is the size of the $M$th CTMC. This approach based on Stein's method provides a new framework for studying the convergence of CTMCs to their mean-field limit by mainly looking into the stability of the mean-field model, which is a deterministic system and is often easier to analyze than the CTMCs. More importantly, this approach quantifies the rate of convergence, which reveals the approximation error of using mean-field models for approximating finite-size systems.