自适应与分层抽样用于高维稳健估计
Adaptive and Stratified Subsampling for High-Dimensional Robust Estimation
AI总结：
本文提出自适应重要抽样和分层抽样方法，用于高维稳健估计，在有限方差厚尾噪声和alpha混合依赖下实现最优估计性能。
摘要：
我们研究在有限方差厚尾噪声、epsilon-污染和alpha混合依赖情况下，通过两种抽样估计器：自适应重要抽样（AIS）和分层抽样（SS）进行鲁棒高维稀疏回归。在子高斯设计的范围内，样本量为m的子样本可以达到最小最大最优速率。我们弥合了理论与算法之间的差距：定理4.6适用于终止时的AIS，条件是稳定权重（命题4.1），SS符合Lecue和Lerasle的中位数-均值M估计框架（命题4.3）。去偏差步骤通过新的稀疏精度假设下的nodewise-Lasso精度估计器完全指定，从而产生有效的坐标wise置信区间（定理4.14）。alpha混合扩展使用日历时间块协议，保证时间分离（定理4.12）。实验上，AIS在20%污染情况下比均匀抽样误差低3.10倍，在Riboflavin（p=4,088和n=71）上测试MSE低29.5%。
英文摘要：
We study robust high-dimensional sparse regression under finite-variance heavy-tailed noise, epsilon-contamination, and alpha-mixing dependence via two subsampling estimators: Adaptive Importance Sampling (AIS) and Stratified Sub-sampling (SS). Under sub-Gaussian design whose scopeis precisely delimited and finite-variance noise, a subsample of size m achieves the minimax-optimal rate. We close the theory-algorithm gap: Theorem 4.6 applies to AIS at termination conditional on stabilized weights (Proposition 4.1), and SS fits the median-of-means M-estimation framework of Lecue and Lerasle (Proposition 4.3). The de-biasing step is fully specified via the nodewise-Lasso precision estimator under a new sparse-precision assumption, yielding valid coordinate-wise CIs (Theorem 4.14). The alpha-mixing extension uses a calendar-time block protocol that guarantees temporal separation (Theorem 4.12). Empirically, AIS achieves 3.10 times lower error than uniform subsampling at 20% contamination, and 29.5% lower test MSE on Riboflavin (p=4,088 and n=71).
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