Adaptive and Stratified Subsampling for High-Dimensional Robust Estimation

Abstract: We study robust high-dimensional sparse regression under finite-variance heavy-tailed noise, ε-contamination, and α-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$m=\Omega(s\log p)$ achieves the minimax-optimal rate $O(\sqrt{s\log p/m})$. 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 Lecu´e 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 α-mixing extension uses a calendar-time block protocol that guarantees temporal separation (Theorem 4.12). Empirically, AIS achieves 3.1× lower error than uniform subsampling at 20% contamination, and 29.5% lower test MSE on Riboflavin (p=4,088 ≫ n=71).

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Xiaojie_Mao1

Submission Number: 7846