The Lasso with general Gaussian designs with applications to hypothesis testing
一般高斯设计下的 Lasso 及其假设检验应用AI 解读 ↓
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一般高斯设计下的 Lasso 及其假设检验应用
- 研究问题
- 在样本量与变量数同阶、协变量相关的高斯设计下，如何精确刻画 Lasso，并据此构造有效的去偏估计置信区间？
- 方法与思路
- 作者将相关且协方差非奇异的高斯设计联系到更简单的固定设计模型，建立分布距离的非渐近界，并分析去偏 Lasso。
- 材料陈述的贡献
- 摘要称，分布比较界对适当稀疏信号类与正则化参数一致成立，并指出在所研究情形下，有效置信区间需要自由度修正。
- 阅读关注点
- 应核对高斯协方差和稀疏类的限制、固定设计比较模型、分布距离的度量，以及自由度修正与覆盖率结论的适用条件。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；摘要未给出界的形式或修正公式，不能推广到任意非高斯设计或任意稀疏程度。
原文摘要
The Lasso is a method for high-dimensional regression, which is now commonly used when the number of covariates p is of the same order or larger than the number of observations n. Classical asymptotic normality theory does not apply to this model due to two fundamental reasons: (1) The regularized risk is nonsmooth; (2) The distance between the estimator θˆ and the true parameters vector θ∗ cannot be neglected. As a consequence, standard perturbative arguments that are the traditional basis for asymptotic normality fail. On the other hand, the Lasso estimator can be precisely characterized in the regime in which both n and p are large and n/p is of order one. This characterization was first obtained in the case of Gaussian designs with i.i.d. covariates: here we generalize it to Gaussian correlated designs with non-singular covariance structure. This is expressed in terms of a simpler “fixed-design” model. We establish nonasymptotic bounds on the distance between the distribution of various quantities in the two models, which hold uniformly over signals θ∗ in a suitable sparsity class and over values of the regularization parameter. As an application, we study the distribution of the debiased Lasso and show that a degrees-of-freedom correction is necessary for computing valid confidence intervals.