Sign Consistency of the Generalized Elastic Net Estimator

Variable selection has become an important and actively used task for understanding or predicting an outcome of interest in many fields such as medicine [2, 3, 4, 5], social media [6, 7, 8], or finance [9, 10, 11]. Through decades, numerous variable selection methods have been developed such as subset selection [12] or regularization techniques [13].

Subset selection methods achieve sparsity by selecting the best subset of relevant variables using the Akaike information criterion [14] or the Bayesian information criterion [15] but are shown to be NP-hard and could be unstable in practice [16, 17].

The regularized variable selection techniques have become popular for their capability to overcome the above difficulties [18, 19, 20, 21]. Among them, the Lasso approach [18] is one of the most popular and can be defined as follows. Let $\displaystyle\mathbf{y}$ satisfy the following linear model

where $\displaystyle\mathbf{y}=(y_{1},\ldots,y_{n})^{{}^{\prime}}\in\mathbb{R}^{n}$ is the response variable, ^′ denoting the transposition, $\displaystyle\mathbf{X}=(\mathbf{X}_{1},\ldots,\mathbf{X}_{p})$ is the design matrix with $\displaystyle n$ rows of observations on $\displaystyle p$ covariates, $\displaystyle\boldsymbol{\beta}^{\star}=(\beta_{1}^{\star},\ldots,\beta_{p}^{\star})^{{}^{\prime}}\in\mathbb{R}^{p}$ is a sparse vector, namely contains a lot of null components, and $\displaystyle\boldsymbol{\epsilon}$ is a Gaussian vector with zero-mean and a covariance matrix equal to $\displaystyle\sigma^{2}\mathbb{I}_{n}$, $\displaystyle\mathbb{I}_{n}$ denoting the identity matrix in $\displaystyle\mathbb{R}^{n}$. The Lasso approach estimates $\displaystyle\boldsymbol{\beta}^{\star}$ with a sparsity enforcing constraint by minimizing the following penalized least-squares criterion:

where $\displaystyle\left\lVert a\right\rVert_{1}=\sum_{k=1}^{p}|a_{k}|$ denotes the $\displaystyle\ell_{1}$ norm of the vector $\displaystyle(a_{1},\dots,a_{p})^{\prime}$, $\displaystyle\left\lVert b\right\rVert_{2}^{2}=\sum_{k=1}^{n}b_{k}^{2}$ denotes the $\displaystyle\ell_{2}$ norm of the vector $\displaystyle(b_{1},\dots,b_{n})^{\prime}$, and $\displaystyle\lambda$ is a positive constant corresponding to the regularization parameter. The Lasso popularity largely comes from the fact that the resulting estimator

is sparse (has only a few nonzero entries), and sparse models are often preferred for their interpretability [22]. Moreover, $\displaystyle\widehat{\boldsymbol{\beta}}^{Lasso}(\lambda)$ can be proved to be sign consistent under some assumptions, namely there exists $\displaystyle\lambda$ such that

where $\displaystyle\textrm{sign}(x)=1$ if $\displaystyle x>0$, -1 if $\displaystyle x<0$ and 0 if $\displaystyle x=0$. Before giving the conditions under which [22] prove the sign consistency of $\displaystyle\widehat{\boldsymbol{\beta}}^{Lasso}$, we first introduce some notations. Without loss of generality, we shall assume as in [22] that the first $\displaystyle q$ components of $\displaystyle\boldsymbol{\beta}^{\star}$ are non null (i.e. the components that are associated to the active variables, and denoted as $\displaystyle\boldsymbol{\beta}_{1}^{\star}$) and the last $\displaystyle p-q$ components of $\displaystyle\boldsymbol{\beta}^{\star}$ are null (i.e. the components that are associated to the non active variables, and denoted as $\displaystyle\boldsymbol{\beta}_{2}^{\star}$). Moreover, we shall denote by $\displaystyle\mathbf{X}_{1}$ (resp. $\displaystyle\mathbf{X}_{2}$) the first $\displaystyle q$ (resp. the last $\displaystyle p-q$) columns of $\displaystyle\mathbf{X}$. Hence, $\displaystyle C_{n}=n^{-1}\mathbf{X}^{\prime}\mathbf{X}$, which is the empirical covariance matrix of the covariates, can be rewritten as follows:

with $\displaystyle C_{11}^{n}=n^{-1}\mathbf{X}_{1}^{{}^{\prime}}\mathbf{X}_{1}$, $\displaystyle C_{12}^{n}=n^{-1}\mathbf{X}_{1}^{{}^{\prime}}\mathbf{X}_{2}$, $\displaystyle C_{21}^{n}=n^{-1}\mathbf{X}_{2}^{{}^{\prime}}\mathbf{X}_{1}$, $\displaystyle C_{22}^{n}=n^{-1}\mathbf{X}_{2}^{{}^{\prime}}\mathbf{X}_{2}$. It is proved by Zhao and Yu in [22] that $\displaystyle\widehat{\boldsymbol{\beta}}^{Lasso}(\lambda)$ is sign consistent when the following Irrepresentable Condition (IC) is satisfied:

where $\displaystyle\alpha$ is a positive constant. In the case where $\displaystyle p\gg n$, Wainwright develops in [23] the necessary and sufficient conditions, for both deterministic and random designs, on $\displaystyle p$, $\displaystyle q$, and $\displaystyle n$ for which it is possible to recover the positions of the null and non null components of $\displaystyle\boldsymbol{\beta}^{\star}$, namely its support, using the Lasso.

When there are high correlations between covariates, especially the active ones, the $\displaystyle C_{11}^{n}$ matrix may not be invertible, and the Lasso estimator fails to be sign consistent. To circumvent this issue, Zou and Hastie [20] introduced the Elastic Net estimator defined by:

where

Yuan and Lin prove in [24] that when the following Elastic Net Condition (EIC) is satisfied the Elastic Net estimator defined by (4) is sign consistent when $\displaystyle p$ and $\displaystyle q$ are fixed: there exist positive $\displaystyle\lambda$ and $\displaystyle\eta$ such that

Moreover, when $\displaystyle p$, $\displaystyle q$, and $\displaystyle n$ go to infinity with $\displaystyle p\gg n$, Jia and Yu prove in [1] that the sign consistency of the Elastic Net estimator holds if additionally to Condition (5) $\displaystyle n$ goes to infinity at a rate faster than $\displaystyle q\log(p-q)$.

In the case where the active and non active covariates are highly correlated, IC (3) and EIC (5) may be violated. To overcome this issue several approaches were proposed: the Standard PArtial Covariance (SPAC) method [25] and preconditioning approaches among others. Xue and Qu [25] developed the so-called SPAC-Lasso which enjoys strong sign consistency in both finite-dimensional ($\displaystyle p<n$) and high-dimensional ($\displaystyle p\gg n$) settings. However, the authors mentioned that the SPAC-Lasso method only selects the active variables that are not highly correlated to the non active ones, which may be a weakness of this approach. The preconditioning approaches consist in transforming the given data $\displaystyle\mathbf{X}$ and $\displaystyle\mathbf{y}$ before applying the Lasso criterion. For example, [26] and [27] proposed to left-multiply $\displaystyle\mathbf{X}$, $\displaystyle\mathbf{y}$ and thus $\displaystyle\boldsymbol{\epsilon}$ in Model (1) by specific matrices to remove the correlations between the columns of $\displaystyle\mathbf{X}$. A major drawback of the latter approach, called HOLP (High dimensional Ordinary Least squares Projection), is that the preconditioning step may increase the variance of the error term and thus may alter the variable selection performance.

Recently, [5] proposed another strategy under the following assumption:

1. (A1)

   $\displaystyle\mathbf{X}$ is assumed to be a random design matrix such that its rows $\displaystyle(\boldsymbol{x}_{i})_{1\leq i\leq n}$ are i.i.d. zero-mean Gaussian random vectors having a covariance matrix equal to $\displaystyle\boldsymbol{\Sigma}$.

More precisely, they propose to rewrite Model (1) in order to remove the correlation existing between the columns of $\displaystyle\mathbf{X}$. Let $\displaystyle\boldsymbol{\Sigma}^{-1/2}:=\boldsymbol{U}\boldsymbol{D}^{-1/2}\boldsymbol{U}^{T}$ where $\displaystyle\boldsymbol{U}$ and $\displaystyle\boldsymbol{D}$ are the matrices involved in the spectral decomposition of the symmetric matrix $\displaystyle\boldsymbol{\Sigma}$ given by: $\displaystyle\boldsymbol{\Sigma}=\boldsymbol{U}\boldsymbol{D}\boldsymbol{U}^{T}$, then, denoting $\displaystyle\widetilde{\mathbf{X}}=\mathbf{X}\boldsymbol{\Sigma}^{-1/2}$, (1) can be rewritten as follows:

where $\displaystyle\widetilde{\boldsymbol{\beta}}^{\star}=\boldsymbol{\Sigma}^{1/2}\boldsymbol{\beta}^{\star}:=\boldsymbol{U}\boldsymbol{D}^{1/2}\boldsymbol{U}^{T}\boldsymbol{\beta}^{\star}$. With such a transformation, the covariance matrix of the $\displaystyle n$ rows of $\displaystyle\widetilde{\mathbf{X}}$ is equal to identity and the columns of $\displaystyle\widetilde{\mathbf{X}}$ are thus uncorrelated. The advantage of such a transformation with respect to the preconditioning approach proposed by [27] is that the error term $\displaystyle\boldsymbol{\epsilon}$ is not modified thus avoiding an increase of the noise which can overwhelm the benefits of a well conditioned design matrix. Their approach then consists in minimizing the following criterion with respect to $\displaystyle\widetilde{\boldsymbol{\beta}}$:

where $\displaystyle\widetilde{\mathbf{X}}=\mathbf{X}\boldsymbol{\Sigma}^{-1/2}$ in order to ensure a sparse estimation of $\displaystyle\boldsymbol{\beta}^{\star}$ thanks to the penalization by the $\displaystyle\ell_{1}$ norm. This criterion actually boils down to the Generalized Lasso proposed by [28]:

and $\displaystyle D=\Sigma^{-1/2}$.

Since, as explained in [28], some problems may occur when the rank of the design matrix is not full, we will consider in this paper the following criterion:

Since it consists in adding an $\displaystyle L_{2}$ penalty part to the Generalized Lasso as in the Elastic Net, we will call it generalized Elastic Net (gEN). We prove in Section 2 that under Assumption (A1) and the Generalized Irrepresentable Condition (GIC) (12) given below among others, $\displaystyle\widehat{\boldsymbol{\beta}}$ is a sign-consistent estimator of $\displaystyle\boldsymbol{\beta}^{\star}$ where $\displaystyle\widehat{\boldsymbol{\beta}}$ is defined by

with

$\displaystyle L_{\lambda,\eta}^{gEN}\left(\widetilde{\boldsymbol{\beta}}\right)$ being defined in Equation (9). The Generalized Irrepresentable Condition (GIC) can be stated as follows: There exist $\displaystyle\lambda,\eta,\alpha,\delta_{4}>0$ such that for all $\displaystyle j$,

Note that GIC coincides with EIC when $\displaystyle\mathbf{X}$ is not random and $\displaystyle\Sigma=\mathbb{I}_{p}$. Moreover, GIC does not require $\displaystyle C_{11}^{n}$ to be invertible. Since EIC and IC are both particular cases of GIC, if the IC or EIC holds, then there exist $\displaystyle\lambda$ or $\displaystyle\eta$ such that the GIC holds.

The rest of the paper is organized as follows. Section 2 is devoted to the theoretical results of the paper. More precisely, we prove that under some mild conditions $\displaystyle\widehat{\boldsymbol{\beta}}$ defined in (10) is a sign-consistent estimator of $\displaystyle\boldsymbol{\beta}^{\star}$. To support our theoretical results, some numerical experiments are presented in Section 3. The proofs of our theoretical results can be found in Section 5.

The goal of this section is to establish the sign consistency of the Generalized Elastic Net estimator defined in (10). To prove this result, we shall use the following lemma.

The proof of Lemma 2.1 is given in Section 5.

The following theorem gives the conditions under which the sign consistency of the generalized Elastic Net estimator $\displaystyle\widehat{\boldsymbol{\beta}}$ defined in (10) holds.

Note that Conditions (16) and (17) are consistent thanks to (15).

The proof of Theorem 2.2 is given in Section 5 and a discussion on the assumptions of Theorem 2.2 is provided in Section 3.

The goal of this section is to discuss the assumptions and illustrate the results of Theorem 2.2. For this, we generated datasets from Model (1) where the matrix $\displaystyle\boldsymbol{\Sigma}$ appearing in (A1) is defined by

In (22), $\displaystyle\boldsymbol{\Sigma}_{11}$ is the correlation matrix of the active variables having its off-diagonal entries equal to $\displaystyle\alpha_{1}$, $\displaystyle\boldsymbol{\Sigma}_{22}$ is the correlation matrix of the non active variables having its off-diagonal entries equal to $\displaystyle\alpha_{3}$ and $\displaystyle\boldsymbol{\Sigma}_{12}$ is the correlation matrix between the active and the non active variables with entries equal to $\displaystyle\alpha_{2}$. In the numerical experiments, $\displaystyle(\alpha_{1},\alpha_{2},\alpha_{3})=(0.3,0.5,0.7)$. Moreover, $\displaystyle\boldsymbol{\beta}^{\star}$ appearing in Model (1) has $\displaystyle q$ non zero components which are equal to $\displaystyle b$ and $\displaystyle\sigma=1$. The number of predictors $\displaystyle p$ is equal to 200, 400, or 600 and the sample size $\displaystyle n$ takes the same values for each value of $\displaystyle p$.

In this paper, we proposed a novel variable selection approach called gEN (generalized Elastic Net) in the framework of linear models where the columns of the design matrix are highly correlated and thus when the standard Lasso criterion usually fails. We proved that under mild conditions, among which the GIC, which is valid when other standard conditions like EIC or IC are not fulfilled, our method provides a sign-consistent estimator of $\displaystyle\boldsymbol{\beta}^{\star}$. For a more thorough discussion regarding the application of our approach in practical situations, we refer the reader to [5].