Statistical consistency and asymptotic normality for high-dimensional robust $M$-estimators
高维稳健 M 估计量的一致性与渐近正态性AI 解读 ↓
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高维稳健 M 估计量的一致性与渐近正态性
- 研究问题
- 在误差或协变量受到重尾与异常值污染的稀疏高维线性模型中，正则化稳健估计何时兼具统计保证与可计算性？
- 方法与思路
- 作者在损失导数有界和局部受限曲率等条件下研究驻点，结合非凸惩罚与复合梯度下降，并以凸估计提供第二步的初值。
- 材料陈述的贡献
- 摘要称真值邻域内驻点达到所述极小极大速率，适当非凸惩罚下等于局部 oracle 解，并给出局部线性算法收敛结果。
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原文摘要
We study theoretical properties of regularized robust $M$-estimators, applicable when data are drawn from a sparse high-dimensional linear model and contaminated by heavy-tailed distributions and/or outliers in the additive errors and covariates. We first establish a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution: When the derivative of the loss function is bounded and satisfies a local restricted curvature condition, all stationary points within a constant radius of the true regression vector converge at the minimax rate enjoyed by the Lasso with sub-Gaussian errors. When an appropriate nonconvex regularizer is used in place of an $\ell_{1}$-penalty, we show that such stationary points are in fact unique and equal to the local oracle solution with the correct support; hence, results on asymptotic normality in the low-dimensional case carry over immediately to the high-dimensional setting. This has important implications for the efficiency of regularized nonconvex $M$-estimators when the errors are heavy-tailed. Our analysis of the local curvature of the loss function also has useful consequences for optimization when the robust regression function and/or regularizer is nonconvex and the objective function possesses stationary points outside the local region. We show that as long as a composite gradient descent algorithm is initialized within a constant radius of the true regression vector, successive iterates will converge at a linear rate to a stationary point within the local region. Furthermore, the global optimum of a convex regularized robust regression function may be used to obtain a suitable initialization. The result is a novel two-step procedure that uses a convex $M$-estimator to achieve consistency and a nonconvex $M$-estimator to increase efficiency. We conclude with simulation results that corroborate our theoretical findings.