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Sharp propagation of chaos in Rényi divergence

Matthew S. Zhang

TL;DR

The paper proves sharp stationary propagation of chaos in $q$-Rényi divergence for interacting diffusion systems under strong isoperimetric conditions and very weak interaction, obtaining $\mathsf{R}_q(\mu^{[k]} \|\pi^{\otimes k}) \lesssim \widetilde{O}\Bigl(\frac{d k^3 q^2}{N^2}\Bigr)$ for large $N$. Building on the hierarchical framework of Lacker (2023), the authors develop a Renyi-specific analysis that leverages a sub-Gaussian score bound and a Lipschitz recursion for conditional measures to control the Rényi Fisher information and related terms. The results extend prior KL-based propagation of chaos to Rényi divergences, with corollaries giving change-of-measure and sampling guarantees and providing evidence for tight $N^{-2}$ rates (at least for moderate $k$). The work relies on strong isoperimetric (LSI/Talagrand) conditions and a notion of very weak interaction, and it outlines potential extensions to dynamics and broader energy functionals, highlighting practical implications for precise tail control and finite-particle approximations.

Abstract

We establish sharp rates for propagation of chaos in Rényi 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 $\mathsf R_q(μ^1 \,\lVert\, π) = \widetilde O(\frac{d q^2}{N^2})$ bounds on the $q$-Rényi divergence.

Sharp propagation of chaos in Rényi divergence

TL;DR

The paper proves sharp stationary propagation of chaos in -Rényi divergence for interacting diffusion systems under strong isoperimetric conditions and very weak interaction, obtaining for large . Building on the hierarchical framework of Lacker (2023), the authors develop a Renyi-specific analysis that leverages a sub-Gaussian score bound and a Lipschitz recursion for conditional measures to control the Rényi Fisher information and related terms. The results extend prior KL-based propagation of chaos to Rényi divergences, with corollaries giving change-of-measure and sampling guarantees and providing evidence for tight rates (at least for moderate ). The work relies on strong isoperimetric (LSI/Talagrand) conditions and a notion of very weak interaction, and it outlines potential extensions to dynamics and broader energy functionals, highlighting practical implications for precise tail control and finite-particle approximations.

Abstract

We establish sharp rates for propagation of chaos in Rényi 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 bounds on the -Rényi divergence.
Paper Structure (13 sections, 11 theorems, 75 equations, 1 figure)

This paper contains 13 sections, 11 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rényi divergences.
  3. Related work
  4. Preliminaries
  5. Notation
  6. Background
  7. Divergences/distances between probability measures
  8. Isoperimetric inequalities
  9. Main results
  10. Examples
  11. Proofs
  12. Lipschitz recursion
  13. Discussion

Key Result

Theorem 1

Suppose Assumptions as:smoothness, as:isoperimetry and as:weak-interaction hold, and $N = \widetilde{\Omega}(1 \vee \sqrt{d} k^{3/2} q)$ for a sufficiently large implied constant. Then, In particular, for $k = O(1)$, this implies sharp Rényi propagation of chaos rates with order $O(N^{-2})$.

Figures (1)

  • Figure 4.1: Proof roadmap, omitting the arguments in $\mathsf{R}_q$ and $\mathsf{RFI}_q$ for brevity.

Theorems & Definitions (13)

  • Theorem 1: Sharp propagation of chaos under very weak interaction
  • Corollary 2: Change of measure
  • Example 3: Perturbations of strong convex potentials
  • Lemma 4: otto2007new
  • Example 5: Explicit computations for Gaussians
  • Lemma 6: Adapted from VempalaW19
  • Lemma 7: Adapted from chewi2025analysis
  • Lemma 8
  • Lemma 9: $\mathop{\mathrm{\mathsf{KL}}}\nolimits$-propagation of chaos; adapted from kook2024sampling
  • Lemma 10: Exponential moment of $\zeta_k$
  • ...and 3 more