发表机构
The Chinese University of Hong Kong; Southern University of Science and Technology（香港中文大学; 南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对多变量非线性统计量在凸距离下建立两个非渐近Berry-Esseen界，改进维数增长条件，并应用于随机逼近、时间差分学习和U-统计量，给出显式高斯近似误差及收敛充分条件。
AI 中文摘要
本文建立了两个关于凸集上高斯近似的非渐近Berry-Esseen界，适用于多变量非线性统计量。所关注的统计量可以表示为独立中心化随机向量之和加上一个可能依赖于所有观测值的余项。第一个界在独立和的贡献中保留了经典因子$d^{1/4}$，其中$d$为维数，同时通过余项的大小及其对替换单个观测值的敏感性来控制余项。第二个界以四阶矩表示独立和的贡献，并允许维数随样本量增长得更快。对于独立随机向量之和，该界在不施加额外矩假设的情况下，去除了现有四阶矩界中的对数因子。作为应用，我们将这些结果应用于非光滑随机逼近的Polyak--Ruppert平均、线性函数逼近的时间差分学习以及多变量$U$-统计量。所得界提供了显式的高斯近似误差，以及当维数随样本量增长时这些误差收敛到零的充分条件。
英文摘要
In this paper, we establish two nonasymptotic Berry--Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains the classical factor $d^{1/4}$ in the contribution of the independent sum, where $d$ is the dimension, while controlling the remainder through its size and its sensitivity to replacing a single observation. The second bound expresses the contribution of the independent sum in terms of fourth moments and can allow the dimension to grow faster with the sample size. For sums of independent random vectors, it removes the logarithmic factor from an existing fourth moment bound without imposing additional moment assumptions. As applications, we apply these results to Polyak--Ruppert averaging for nonsmooth stochastic approximation, temporal difference learning with linear function approximation, and multivariate $U$-statistics. The resulting bounds provide explicit Gaussian approximation errors and sufficient conditions under which these errors converge to zero as the dimension grows with the sample size.
Comments90 pages
