Robust-to-outliers square-root LASSO, simultaneous inference with a MOM approach
Robust-to-outliers square-root LASSO, simultaneous inference with a MOM approach
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稳健的离群值平方根 LASSO，使用 MOM 方法同时推理
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发表时间：
2021
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通讯作者：
K. Proksch
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作者：
G. Finocchio;A. Derumigny;K. Proksch
本文考虑了噪声方差未知的最小二乘回归问题，其中观测数据点允许被异常值破坏。在已知噪声方差的情况下，基于Lecue和Lerasle Ann.Statist.48（2）：906-931（2020年4月）介绍的均值中值（中位数）方法，我们提出了一种通用的均值中值方法，用于同时推断回归函数和噪声方差，仅需要噪声水平的上限。有趣的是，由于潜在的凹凸优化问题固有的正则性问题，这种推广需要小心。在一般情况下，回归函数属于凸类，我们表明，我们的同时估计实现了高概率相同的收敛速度和类似的风险范围，如果噪声水平是未知的，以及估计的噪声标准差的收敛速度。在高维稀疏线性设置中，我们的估计产生了一个强大的模拟的平方根LASSO。在弱矩条件下，它以高概率联合实现了系数向量的$ell_p$-范数的最小极大估计率$s^{1/p} sqrt{（1/n）log（p/s）}$和噪声标准差的估计率$sqrt{（s/n）log（p/s）}$。这里$n$表示样本大小，$p$表示维度，$s$表示稀疏度。最后，我们提出了一个扩展的情况下，未知的稀疏水平$s$，提供了一个联合自适应估计$（宽波浪线 eta，宽波浪号sigma，宽波浪号s）$。它同时估计系数向量、噪声水平和稀疏水平，并证明这三个分量中的每一个都有很高的概率。
We consider the least-squares regression problem with unknown noise variance, where the observed data points are allowed to be corrupted by outliers. Building on the median-of-means (MOM) method introduced by Lecue and Lerasle Ann.Statist.48(2):906-931(April 2020) in the case of known noise variance, we propose a general MOM approach for simultaneous inference of both the regression function and the noise variance, requiring only an upper bound on the noise level. Interestingly, this generalization requires care due to regularity issues that are intrinsic to the underlying convex-concave optimization problem. In the general case where the regression function belongs to a convex class, we show that our simultaneous estimator achieves with high probability the same convergence rates and a similar risk bound as if the noise level was unknown, as well as convergence rates for the estimated noise standard deviation. In the high-dimensional sparse linear setting, our estimator yields a robust analog of the square-root LASSO. Under weak moment conditions, it jointly achieves with high probability the minimax rates of estimation $s^{1/p} sqrt{(1/n) log(p/s)}$ for the $ell_p$-norm of the coefficient vector, and the rate $sqrt{(s/n) log(p/s)}$ for the estimation of the noise standard deviation. Here $n$ denotes the sample size, $p$ the dimension and $s$ the sparsity level. We finally propose an extension to the case of unknown sparsity level $s$, providing a jointly adaptive estimator $(widetilde eta, widetilde sigma, widetilde s)$. It simultaneously estimates the coefficient vector, the noise level and the sparsity level, with proven bounds on each of these three components that hold with high probability.