论文标题
通过平方根斜率稳健和无调的稀疏线性回归
Robust and Tuning-Free Sparse Linear Regression via Square-Root Slope
论文作者
论文摘要
我们考虑了高维线性回归模型，并假设测量的一部分是由对手完全了解数据和基础分布的对手改变的。我们对一个浓密添加噪声进行重尾时的情况感兴趣，而测量向量则遵循以下分布。在此框架内，我们为执行任意估计器的性能建立了最小的下限，该估计器的性能取决于损坏的观测值以及添加噪声的尾巴行为。此外，我们设计了所谓的平方根斜率估计器的修改，具有多个理想的特征：（a）事实证明，它对对抗性污染非常健壮，并以较低的误差界限的次数偏差不平等的形式满足性能保证，直到较低的误差范围，直至对数因素； （b）相对于未知的稀疏度和添加噪声的方差完全自适应，并且（c）在计算上可以作为凸优化问题的解决方案进行计算处理。为了分析提出的估计器的性能，我们证明了具有独立关注的矩阵的矩阵的几种特性。
We consider the high-dimensional linear regression model and assume that a fraction of the measurements are altered by an adversary with complete knowledge of the data and the underlying distribution. We are interested in a scenario where dense additive noise is heavy-tailed while the measurement vectors follow a sub-Gaussian distribution. Within this framework, we establish minimax lower bounds for the performance of an arbitrary estimator that depend on the the fraction of corrupted observations as well as the tail behavior of the additive noise. Moreover, we design a modification of the so-called Square-Root Slope estimator with several desirable features: (a) it is provably robust to adversarial contamination, and satisfies performance guarantees in the form of sub-Gaussian deviation inequalities that match the lower error bounds, up to logarithmic factors; (b) it is fully adaptive with respect to the unknown sparsity level and the variance of the additive noise, and (c) it is computationally tractable as a solution of a convex optimization problem. To analyze performance of the proposed estimator, we prove several properties of matrices with sub-Gaussian rows that may be of independent interest.
