首次给出依赖数据在Wasserstein距离下的最优中心极限定理速率。
Wasserstein-p Central Limit Theorem Rates: From Local Dependence to Markov Chains
- 针对局部依赖和马尔可夫链，提出新分析工具以获得最优收敛率。
- 在W₁下达到O(n⁻¹/²)最优速率，首次实现Wₚ(p≥2)的理论突破。
- 适用于机器学习与运筹学中的依赖数据建模，尤其适合需要精确误差控制的场景。
非渐近中心极限定理（CLT）速率在现代机器学习与运筹学中具有核心作用。本文研究多变量依赖数据在Wasserstein-p（W_p）距离下的CLT速率，适用于一般p≥1。重点分析两类常见依赖结构：局部依赖序列与几何遍历马尔可夫链。在两种情形下，首次建立W₁下的最优O(n⁻¹/²)速率，并在弱矩假设下首次获得W_p（p≥2）的CLT速率，显著优于此前最坏情况下的已知界限。作为局部依赖序列最优W₁速率的应用，我们进一步得到多变量U统计量的首例最优W₁-CLT速率。技术上，推导出适用于依赖数据的可计算的W₁高斯逼近误差辅助界。对马尔可夫链，还证明了与几何遍历链相关的分裂链再生时间具有几何尾部，无需强非周期性等限制条件。这些工具本身具有独立价值，支撑了我们的最优W₁速率及更高阶的W_p结果。
原文摘要 · Abstract (English)
Non-asymptotic central limit theorem (CLT) rates play a central role in modern machine learning and operations research. In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p\ge 1$. We focus on two fundamental dependence structures that commonly arise in practice: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the first optimal $\mathcal O(n^{-1/2})$ 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.
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