AI 中文总结
该研究在比例维 regime 下，针对满足特定条件的正则化M估计量，证明了其线性泛函的定量中心极限定理，确定了波动尺度并给出了收敛速率。
AI 中文摘要
我们针对比例维 regime（即p=O(n)）下的正则化经验风险最小化器的线性泛函，证明了一个定量中心极限定理。数据列相互独立、不一定同分布，且满足逐列均匀庞加莱不等式。在均匀曲率与光滑性假设下，针对二次正则化项，我们证明：每个非退化统计量√n u^T θ̂（以其期望为中心、标准差归一化）在Wasserstein距离下收敛到标准正态随机变量，收敛速率为O((log n)^7 n^{-1/4})。该证明基于最小化器的矩界与稳定性界、二阶留一展开式，以及针对独立变量函数的扰动正态近似论证。我们还证明了方差上界Var(u^T θ̂) ≤ C ||u||_2^2 / n，确定了√n波动尺度。
英文摘要
We prove a quantitative central limit theorem for linear functionals of regularized empirical-risk minimizers in the proportional-dimensional regime \(p=O(n)\). The data columns are independent, not necessarily identically distributed, and satisfy a uniform columnwise Poincaré inequality. Under uniform curvature and smoothness assumptions, and for a quadratic regularizer, we show that every nondegenerate statistic \(\sqrt n\,u^\top\hatθ\), centered by its expectation and normalized by its standard deviation, converges to a standard normal random variable in Wasserstein distance, with rate \(O((\log n)^7n^{-1/4})\). The proof is based on moment and stability bounds for the minimizer, a second-order leave-one-out expansion, and a perturbative normal-approximation argument for functions of independent variables. We also prove the variance upper bound \(\Var(u^\top\hatθ)\le C\norm{u}_2^2/n\), identifying the \(\sqrt n\) fluctuation scale.
Comments62 pages
