Wasserstein-p Central Limit Theorem Rates: From Local Dependence to Markov Chains

2026-01-13

AI Summary

This study investigates finite-sample convergence rates for the central limit theorem (CLT) in Wasserstein-p distance for multivariate dependent data, focusing on two canonical dependence structures: locally dependent sequences and geometrically ergodic Markov chains. By establishing a Wasserstein-1 Gaussian approximation error bound tailored to dependent data and proving that the regeneration times of geometrically ergodic Markov chains exhibit geometric tails—without requiring strong aperiodicity assumptions—the work achieves, for the first time, the optimal $O(n^{-1/2})$ convergence rate in $W_1$ distance. Under mild moment conditions, it further extends this result to obtain $W_p$-CLT rates for $p \geq 2$. As an application, the study derives the first optimal $W_1$-CLT rate for multivariate U-statistics under dependence, substantially improving upon existing theoretical upper bounds for Wasserstein CLT rates in dependent settings.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Finite-time central limit theorem (CLT) rates play a central role in modern machine learning. In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p \geq 1$. We focus on two fundamental dependence structures that commonly arise in machine learning: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the first optimal $O(n^{-1/2})$ rate in $W_1$, as well as the first $W_p$ ($p\ge 2$) CLT rates under mild moment assumptions, substantially improving the best previously known bounds in these dependent-data regimes. As an application of our optimal $W_1$ rate for locally dependent sequences, we further obtain the first optimal $W_1$-CLT rate for multivariate $U$-statistics. On the technical side, we derive a tractable auxiliary bound for $W_1$ Gaussian approximation errors that is well suited for studying dependent data. For Markov chains, we further prove that the regeneration time of the split chain associated with a geometrically ergodic chain has a geometric tail without assuming strong aperiodicity or other restrictive conditions. These tools may be of independent interests and enable our optimal $W_1$ rates and underpin our $W_p$ ($p\ge 2$) results.

Problem

• Wasserstein distance
• Central Limit Theorem
• Dependent data
• Local dependence
• Markov chains

Innovation

• Wasserstein distance
• central limit theorem
• dependent data
• Markov chains
• local dependence