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.

翻译：有限时间中心极限定理（CLT）收敛速率在现代机器学习中具有核心地位。本文研究多元相依数据在Wasserstein-$p$（$W_p$）距离下的CLT收敛速率，其中$p \geq 1$为一般参数。我们聚焦于机器学习中常见的两种基本依赖结构：局部依赖序列与几何遍历马尔可夫链。在这两种设定下，我们首次在$W_1$距离下建立了最优的$O(n^{-1/2})$收敛速率，并在温和矩假设下首次获得了$W_p$（$p\ge 2$）距离的CLT收敛速率，显著改进了现有相依数据体系中的最佳已知界。作为局部依赖序列最优$W_1$收敛速率的应用，我们进一步得到了多元$U$-统计量的首个最优$W_1$-CLT收敛速率。在技术层面，我们推导出一个适用于研究相依数据的、易于处理的$W_1$高斯近似误差辅助界。对于马尔可夫链，我们进一步证明了几何遍历链对应的分裂链的再生时间具有几何尾部，且无需假设强非周期性或其他限制性条件。这些工具可能具有独立的研究价值，不仅支撑了我们获得最优$W_1$收敛速率，也为$W_p$（$p\ge 2$）距离下的结果奠定了理论基础。