Wasserstein-p Central Limit Theorem Rates: From Local Dependence to Markov Chains
Yixuan Zhang, Qiaomin Xie
TL;DR
The paper tackles finite-time central limit theorem rates in Wasserstein distances for multivariate dependent data, focusing on locally dependent sequences and geometrically ergodic Markov chains. It develops a tractable auxiliary bound for $\mathcal{W}_1$ Gaussian approximation under dependence and extends Rai\v{c}'s framework to dependent data, enabling optimal $\mathcal{W}_1$ rates and new $\mathcal{W}_p$ rates. The locally dependent results yield an optimal $O(n^{-1/2})$ rate in $\mathcal{W}_1$ (and $O(n^{- frac{\delta}{2}})$ for $\delta\in(0,1]$) as well as a $\mathcal{W}_p$ rate for $M$-dependent data, with an application to multivariate U-statistics. For Markov chains, the paper proves an optimal $O(n^{-1/2})$ rate in $\mathcal{W}_1$ for geometrically ergodic chains (under a $V$-dominated moment with $\delta>1$) and establishes $\mathcal{W}_p$ rates ($p\ge2$) by leveraging a regeneration decomposition to reduce to $1$-dependent data. The results substantially improve the state of knowledge on dependence-induced CLT rates and provide tools that are broadly applicable to uncertainty quantification in ML settings involving dependent data.
Abstract
Finite-time central limit theorem (CLT) rates play a central role in modern machine learning (ML). In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($\mathcal W_p$) distance, for general $p\ge 1$. We focus on two fundamental dependence structures that commonly arise in ML: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the \textit{first optimal} $\mathcal O(n^{-1/2})$ rate in $\mathcal W_1$, as well as the first $\mathcal W_p$ ($p\ge 2$) CLT rates under mild moment assumptions, substantially improving the best previously known bounds in these dependent-data regimes. As an application of our optimal $\mathcal W_1$ rate for locally dependent sequences, we further obtain the first optimal $\mathcal W_1$--CLT rate for multivariate $U$-statistics. On the technical side, we derive a tractable auxiliary bound for $\mathcal W_1$ Gaussian approximation errors that is well suited to 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 $\mathcal W_1$ rates and underpin our $\mathcal W_p$ ($p\ge 2$) results.