抗对抗污染的迭代硬阈值算法，实现信号自适应与最优估计。
Adversarial Contamination Meets Hard Thresholding: An Iterative Algorithm with Signal Adaptivity and Minimax Optimality
- 分两阶段迭代更新系数与污染向量，用不同阈值尺度提升鲁棒性。
- 在高维回归中达到近似极小极大最优估计，支持恢复更准确。
- 适用于信号强度适中的场景，适合需要理论保障的统计推断任务。
普遍存在的数据污染（源于测量误差、异常值或对抗性篡改）推动了稳健统计方法的发展。本文提出一种两阶段对抗污染抵抗型迭代硬阈值（AC-IHT）算法，用于存在污染的高维回归。该非凸算法通过迭代更新系数向量和污染向量，采用不同的阈值尺度，实现了近似极小极大最优（仅差对数因子）的估计。进一步证明，该AC-IHT估计器具有信号自适应性：在合适的信号条件下，可自适应获得更优的估计速率和更精确的支持恢复。此外，其具备强奥兰多性质，为渐近推断提供了理论基础。数值实验验证了其在有限样本下的优越性能。最后，我们讨论了该方法在广义线性模型及重尾噪声设定下的理论拓展。
原文摘要 · Abstract (English)
Pervasive data contamination -- stemming from measurement errors, outliers, or adversarial corruption -- has motivated the development of robust statistical methods. In this context, we propose a two-stage Adversarial Contamination-resistant Iterative Hard Thresholding (AC-IHT) algorithm for high-dimensional regression with contamination. Our nonconvex algorithm achieves minimax near-optimal (up to logarithmic terms) estimation by iteratively updating the coefficient vector and the contamination vector with different thresholding scales. We further demonstrate that our AC-IHT estimator is signal-adaptive: under proper signal conditions, it adaptively attains a sharper estimation rate and more accurate support recovery. Moreover, it enjoys the strong oracle property, laying a theoretical foundation for asymptotic inference. Numerical experiments confirm its superior finite-sample performance. Finally, we discuss theoretical extensions of the proposed procedure to generalized linear models and to heavy-tailed noise settings.
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