Necessary and sufficient conditions for variable selection consistency of the Lasso in high dimensions
Consider the multivariate linear regression model \[ y_i=\mathbf{X}^\prime_i\boldsymbol{\beta}+\epsilon_i\,, \] for \(i=1,\ldots,n\), where \(\mathbf{X}_i\) and \(\boldsymbol{\beta}\) are \(p\)-dimensional vectors containing the covariates and regression parameters, respectively, and the \(\epsilon_i\) are IID errors with mean zero. Assume that only \(p_0\) of the \(p\) parameters are nonzero, where both \(p\) and \(p_0\) may grow with the sample size \(n\). In the context of the Lasso method of penalized regression, and under some regularity and boundedness conditions, the author provides necessary and sufficient conditions for Lasso to be variable selection consistent (i.e., to choose correctly which parameters are nonzero with probability tending to 1 as \(n\to\infty\)), which show that this property continues to hold for a wide range of penalty terms and \(p\) potentially very large compared to \(n\). They also show that in many cases the Lasso estimators cannot be both variable selection consistent and \(\sqrt{n}\)-consistent.
scientific article; zbMATH DE number 7367863
scientific article