Sharp propagation of chaos in R\'enyi divergence

TL;DR

This paper proves precise rates for how quickly the distribution of interacting diffusions converges to a mean-field limit in Re9nyi divergence, under certain geometric and interaction conditions.

Contribution

It extends the entropic hierarchy framework to establish sharp convergence rates in Re9nyi divergence for stationary interacting diffusions.

Findings

Achieves a4(rac{d q^2}{N^2}) bounds on Re9nyi divergence.

Demonstrates sharp propagation of chaos rates under isoperimetric and weak interaction assumptions.

Builds upon and extends previous entropic hierarchy results.

Abstract

We establish sharp rates for propagation of chaos in R\'enyi divergences for interacting diffusion systems at stationarity. Building upon the entropic hierarchy established in Lacker (2023), we show that under strong isoperimetry and weak interaction conditions, one can achieve $\mathsf R_q(\mu^1 \,\lVert\, \pi) = \widetilde O(\frac{d q^2}{N^2})$ bounds on the $q$-R\'enyi divergence.