Improved bounds for Square-Root Lasso and Square-Root Slope
Improved bounds for Square-Root Lasso and Square-Root Slope
复制标题
改进了平方根套索和平方根斜率的界限
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
A. Derumigny
中科院分区：
文献类型：
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作者：
A. Derumigny
将Bellec，Lecue和Tsybakov [1]的结果扩展到稀疏的高度线性回归的设置，我们表明两个估计量，正方形的套索和方形斜坡可以实现最佳的最小值预测速率，这些预测速率是最佳的最小值预测率在设计矩阵上的一些轻度条件下，为（s/n）log（p/s），直至一定的常数。在这里，n是样本大小，p是维度，s是稀疏参数。我们还证明了LQ-Norm中估计误差的最佳性，在平方根套索中，Q在[1，2]中，以及在L2中，以及对方形 - 根斜率的L1标准进行排序。两个估计器都适应噪声的未知方差。平方根斜率也适用于真实参数的稀疏性。接下来，我们证明，任何估计器取决于S达到最小值的s，都承认对仍然达到相同速率的s版本的适应性。我们将此结果应用于方形套索。此外，对于两个估计器，我们都可以获得广泛的置信度范围的有效速率，并像[1]中一样提高了浓度属性。我们的结果是非催化性的。 ;分类 - jel：主要62G08;中学62C20，62G05。
Extending the results of Bellec, Lecue and Tsybakov [1] to the setting of sparse highdimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is (s/n) log (p/s), up to some constant, under some mild conditions on the design matrix. Here, n is the sample size, p is the dimension and s is the sparsity parameter. We also prove optimality for the estimation error in the lq-norm, with q in [1, 2] for the Square-Root Lasso, and in the l2 and sorted l1 norms for the Square-Root Slope. Both estimators are adaptive to the unknown variance of the noise. The Square-Root Slope is also adaptive to the sparsity s of the true parameter. Next, we prove that any estimator depending on s which attains the minimax rate admits an adaptive to s version still attaining the same rate. We apply this result to the Square-root Lasso. Moreover, for both estimators, we obtain valid rates for a wide range of confidence levels, and improved concentration properties as in [1] where the case of known variance is treated. Our results are non-asymptotic. ;Classification-JEL: Primary 62G08; secondary 62C20, 62G05.