Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Propagation of chaos in path spaces via information theory

Lei Li, Yuelin Wang, Yuliang Wang

Abstract

Propagation of chaos for interacting particle systems has been an active research topic over decades. We propose an alternative approach to study the mean-field limit of the stochastic interacting particle systems via tools from information theory. In our framework, the propagation of chaos is reduced to the space for driving processes with possible lower dimension. Indeed, after applying the data processing inequality, one only needs to estimate the difference between the drifts of the particle system and the mean-field Mckean stochastic differential equation. This point is particularly useful in situations where the discrepancy in the driving processes is more apparent than the investigated processes. We will take the second order system as well as other examples for the illustration of how our framework could be used. This approach allows us to focus on probability measures in path spaces for the driving processes, avoiding using the usual hypocoercivity technique or taking the pseudo-inverse of the diffusion matrix, which might be more stable for numerical computation. Our framework is different from current approaches in literature and could provide new insight into the study of interacting particle systems.

Propagation of chaos in path spaces via information theory

Abstract

Propagation of chaos for interacting particle systems has been an active research topic over decades. We propose an alternative approach to study the mean-field limit of the stochastic interacting particle systems via tools from information theory. In our framework, the propagation of chaos is reduced to the space for driving processes with possible lower dimension. Indeed, after applying the data processing inequality, one only needs to estimate the difference between the drifts of the particle system and the mean-field Mckean stochastic differential equation. This point is particularly useful in situations where the discrepancy in the driving processes is more apparent than the investigated processes. We will take the second order system as well as other examples for the illustration of how our framework could be used. This approach allows us to focus on probability measures in path spaces for the driving processes, avoiding using the usual hypocoercivity technique or taking the pseudo-inverse of the diffusion matrix, which might be more stable for numerical computation. Our framework is different from current approaches in literature and could provide new insight into the study of interacting particle systems.
Paper Structure (14 sections, 12 theorems, 115 equations)

This paper contains 14 sections, 12 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. The main idea of the new framework
  3. The application to the second order systems
  4. The well-posedness of the mean field McKean SDE.
  5. Propagation of chaos in path space and the corollaries.
  6. Some auxiliary lemmas.
  7. Other applications
  8. Application in neural networks.
  9. Application in numerical analysis.
  10. More discussions
  11. Discussion on the reversed relative entropy.
  12. Discussion on the mass-independence.
  13. Basics on path measure and Girsanov's transform
  14. Proof of Lemma \ref{['lmm:exp_estimate']}

Key Result

lemma 1

Consider a given conditional probability $P_{Y \mid X}$ and that $Y$ is produced by $P_{Y \mid X}$ given $X$. If $P_Y$ is the distribution of $Y$ when $X$ is generated by $P_X$, and $Q_Y$ is the distribution of $Y$ when $X$ is generated by $Q_X$, then for any convex function $f: \mathbb{R}^{+} \righ where the $f$-divergence $D_f(\cdot \| \cdot)$ is defined by

Theorems & Definitions (20)

  • lemma 1: data processing inequality
  • remark 1
  • remark 2
  • remark 3
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • theorem 1
  • proof
  • ...and 10 more