In spite of the wealth of literature on the theoretical properties of the Lasso, there is very little known when the value of the tuning parameter is chosen using the data, even though this is what actually happens in practice. We give a general upper bound on the prediction error of Lasso when the tuning parameter is chosen using a variant of 2-fold cross-validation. No special assumption is made about the structure of the design matrix, and the tuning parameter is allowed to be optimized over an arbitrary data-dependent set of values. The proof is based on a general principle that may extend to other kinds of cross-validation as well as to other penalized regression methods. Based on this result, we propose a new estimate for error variance in high dimensional regression and prove that it has good properties under minimal assumptions.
In this paper, we derive non-asymptotic error bounds for the Lasso estimator when the penalty parameter for the estimator is chosen using $K$-fold cross-validation. Our bounds imply that the cross-validated Lasso estimator has nearly optimal rates of convergence in the prediction, $L^2$, and $L^1$ norms. For example, we show that in the model with the Gaussian noise and under fairly general assumptions on the candidate set of values of the penalty parameter, the estimation error of the cross-validated Lasso estimator converges to zero in the prediction norm with the $\sqrt{s\log p / n}\times \…
The lasso procedure is ubiquitous in the statistical and signal processing literature, and as such, is the target of substantial theoretical and applied research. While much of this research focuses on the desirable properties that lasso possesses---predictive risk consistency, sign consistency, correct model selection---all of it has assumes that the tuning parameter is chosen in an oracle fashion. Yet, this is impossible in practice. Instead, data analysts must use the data twice, once to choose the tuning parameter and again to estimate the model. But only heuristics have ever justified suc…
The lasso and related sparsity inducing algorithms have been the target of substantial theoretical and applied research. Correspondingly, many results are known about their behavior for a fixed or optimally chosen tuning parameter specified up to unknown constants. In practice, however, this oracle tuning parameter is inaccessible so one must use the data to select one. Common statistical practice is to use a variant of cross-validation for this task. However, little is known about the theoretical properties of the resulting predictions with such data-dependent methods. We consider the high-di…
A multiple interval-valued linear regression model considering all the cross-relationships between the mids and spreads of the intervals has been introduced recently. A least-squares estimation of the regression parameters has been carried out by transforming a quadratic optimization problem with inequality constraints into a linear complementary problem and using Lemke's algorithm to solve it. Due to the irrelevance of certain cross-relationships, an alternative estimation process, the LASSO (Least Absolut Shrinkage and Selection Operator), is developed. A comparative study showing the differ…
The adaptive lasso refers to a class of methods that use weighted versions of the $L_1$-norm penalty, with weights derived from an initial estimate of the parameter vector to be estimated. Irrespective of the method chosen to compute this initial estimate, the performance of the adaptive lasso critically depends on the value of a hyperparameter, which controls the magnitude of the weighted $L_1$-norm in the penalized criterion. As for other machine learning methods, cross-validation is very popular for the calibration of the adaptive lasso, that this for the selection of a data-driven optimal …
This paper deals with prediction of anopheles number using environmental and climate variables. The variables selection is performed by an automatic machine learning method based on Lasso and stratified two levels cross validation. Selected variables are debiased while the predictionis generated by simple GLM (Generalized linear model). Finally, the results reveal to be qualitatively better, at selection, the prediction,and the CPU time point of view than those obtained by B-GLM method.
We introduce a novel scheme for choosing the regularization parameter in high-dimensional linear regression with Lasso. This scheme, inspired by Lepski's method for bandwidth selection in non-parametric regression, is equipped with both optimal finite-sample guarantees and a fast algorithm. In particular, for any design matrix such that the Lasso has low sup-norm error under an "oracle choice" of the regularization parameter, we show that our method matches the oracle performance up to a small constant factor, and show that it can be implemented by performing simple tests along a single Lasso …
We propose a new approach to safe variable preselection in high-dimensional penalized regression, such as the lasso. Preselection - to start with a manageable set of covariates - has often been implemented without clear appreciation of its potential bias. Based on sequential implementation of the lasso with increasing lists of predictors, we find a new property of the set of corresponding cross-validation curves, a pattern that we call freezing. It allows to determine a subset of covariates with which we reach the same lasso solution as would be obtained using the full set of covariates. Freez…
We establish a general upper bound for $K$-fold cross-validation ($K$-CV) errors that can be adapted to many $K$-CV-based estimators and learning algorithms. Based on Rademacher complexity of the model and the Orlicz-$Ψ_ν$ norm of the error process, the CV error upper bound applies to both light-tail and heavy-tail error distributions. We also extend the CV error upper bound to $β$-mixing data using the technique of independent blocking. We provide a Python package (\texttt{CVbound}, \url{https://github.com/isaac2math}) for computing the CV error upper bound in $K$-CV-based algorithms. Using t…
We propose a new method of learning a sparse nonnegative-definite target matrix. Our primary example of the target matrix is the inverse of a population covariance or correlation matrix. The algorithm first estimates each column of the target matrix by the scaled Lasso and then adjusts the matrix estimator to be symmetric. The penalty level of the scaled Lasso for each column is completely determined by data via convex minimization, without using cross-validation. We prove that this scaled Lasso method guarantees the fastest proven rate of convergence in the spectrum norm under conditions of w…
One way to solve lasso problems when the dictionary does not fit into available memory is to first screen the dictionary to remove unneeded features. Prior research has shown that sequential screening methods offer the greatest promise in this endeavor. Most existing work on sequential screening targets the context of tuning parameter selection, where one screens and solves a sequence of $N$ lasso problems with a fixed grid of geometrically spaced regularization parameters. In contrast, we focus on the scenario where a target regularization parameter has already been chosen via cross-validated…
We consider a linear regression problem in a high dimensional setting where the number of covariates $p$ can be much larger than the sample size $n$. In such a situation, one often assumes sparsity of the regression vector, \textit i.e., the regression vector contains many zero components. We propose a Lasso-type estimator $\hatβ^{Quad}$ (where '$Quad$' stands for quadratic) which is based on two penalty terms. The first one is the $\ell_1$ norm of the regression coefficients used to exploit the sparsity of the regression as done by the Lasso estimator, whereas the second is a quadratic penalt…
Lasso is a seminal contribution to high-dimensional statistics, but it hinges on a tuning parameter that is difficult to calibrate in practice. A partial remedy for this problem is Square-Root Lasso, because it inherently calibrates to the noise variance. However, Square-Root Lasso still requires the calibration of a tuning parameter to all other aspects of the model. In this study, we introduce TREX, an alternative to Lasso with an inherent calibration to all aspects of the model. This adaptation to the entire model renders TREX an estimator that does not require any calibration of tuning par…
In life sciences, the experts generally use empirical knowledge to recode variables, choose interactions and perform selection by classical approach. The aim of this work is to perform automatic learning algorithm for variables selection which can lead to know if experts can be help in they decision or simply replaced by the machine and improve they knowledge and results. The Lasso method can detect the optimal subset of variables for estimation and prediction under some conditions. In this paper, we propose a novel approach which uses automatically all variables available and all interactions…
Consider the case that we observe $n$ independent and identically distributed copies of a random variable with a probability distribution known to be an element of a specified statistical model. We are interested in estimating an infinite dimensional target parameter that minimizes the expectation of a specified loss function. In \cite{generally_efficient_TMLE} we defined an estimator that minimizes the empirical risk over all multivariate real valued cadlag functions with variation norm bounded by some constant $M$ in the parameter space, and selects $M$ with cross-validation. We referred to …
In this paper, we aim to give a theoretical approximation for the penalty level of $\ell_{1}$-regularization problems. This can save much time in practice compared with the traditional methods, such as cross-validation. To achieve this goal, we develop two Gaussian approximation methods, which are based on a moderate deviation theorem and Stein's method respectively. Both of them give efficient approximations and have good performances in simulations. We apply the two Gaussian approximation methods into three types of ultra-high dimensional $\ell_{1}$ penalized regressions: lasso, square-root …
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Building on a recent framework for distributionally robust optimization, we consider estimation of the inverse covariance matrix for multivariate data. We provide a novel notion of a Wasserstein ambiguity set specifically tailored to this estimation problem, leading to a tractable class of regularized estimators. Special cases include penalized likelihood estimators for Gaussian data, specifically the graphical lasso estimator. As a consequence of this formulation, the radius of the Wasserstein ambiguity set is directly related to the regularization parameter in the estimation problem. Using t…
In some supervised learning settings, the practitioner might have additional information on the features used for prediction. We propose a new method which leverages this additional information for better prediction. The method, which we call the feature-weighted elastic net ("fwelnet"), uses these "features of features" to adapt the relative penalties on the feature coefficients in the elastic net penalty. In our simulations, fwelnet outperforms the lasso in terms of test mean squared error and usually gives an improvement in true positive rate or false positive rate for feature selection. We…
Much theoretical and applied work has been devoted to high-dimensional regression with clean data. However, we often face corrupted data in many applications where missing data and measurement errors cannot be ignored. Loh and Wainwright (2012) proposed a non-convex modification of the Lasso for doing high-dimensional regression with noisy and missing data. It is generally agreed that the virtues of convexity contribute fundamentally the success and popularity of the Lasso. In light of this, we propose a new method named CoCoLasso that is convex and can handle a general class of corrupted data…
There is a clear need for efficient algorithms to tune hyperparameters for statistical learning schemes, since the commonly applied search methods (such as grid search with N-fold cross-validation) are inefficient and/or approximate. Previously existing algorithms that efficiently search for hyperparameters relying on the smoothness of the cost function cannot be applied in problems such as Lasso regression. In this contribution, we develop a hyperparameter optimization method that relies on the structure of proximal gradient methods and does not require a smooth cost function. Such a method i…
In this study, we propose an automatic learning method for variables selection based on Lasso in epidemiology context. One of the aim of this approach is to overcome the pretreatment of experts in medicine and epidemiology on collected data. These pretreatment consist in recoding some variables and to choose some interactions based on expertise. The approach proposed uses all available explanatory variables without treatment and generate automatically all interactions between them. This lead to high dimension. We use Lasso, one of the robust methods of variable selection in high dimension. To …
Regularized regression approaches such as the Lasso have been widely adopted for constructing sparse linear models in high-dimensional datasets. A complexity in fitting these models is the tuning of the parameters which control the level of introduced sparsity through penalization. The most common approach to select the penalty parameter is through $k$-fold cross-validation. While cross-validation is used to minimise the empirical prediction error, approaches such as the $m$-out-of-$n$ paired bootstrap which use smaller training datasets provide consistency in selecting the non-zero coefficien…
Data driven soft sensor design has recently gained immense popularity, due to advances in sensory devices, and a growing interest in data mining. While partial least squares (PLS) is traditionally used in the process literature for designing soft sensors, the statistical literature has focused on sparse learners, such as Lasso and relevance vector machine (RVM), to solve the high dimensional data problem. In the current study, predictive performances of three regression techniques, PLS, Lasso and RVM were assessed and compared under various offline and online soft sensing scenarios applied on …
Logistic lasso regression with nearest neighbors for gradient-based dimension reduction
University of Oslo（奥斯陆大学）；CREST, ENSAI, University of Rennes（法国国家经济统计与分析中心，雷恩高等应用统计学院，雷恩大学）；Univ Angers, CNRS, LAREMA, SFR MATHSTIC（昂热大学，法国国家科学研究中心，拉雷马实验室，数学科学联合研究服务）
Identifying Neural Signatures from fMRI using Hybrid Principal Components Regression
Department of Biostatistics and Informatics, University of Colorado Anschutz Medical Campus（生物统计学与信息学系，科罗拉多大学安施图茨医疗校区）；Department of Biostatistics and Bioinformatics, Emory University（生物统计学与生物信息学系，埃默里大学）；Department of Radiology, University of Colorado Anschutz Medical Campus（放射学系，科罗拉多大学安施图茨医疗校区）；Laureate Institute for Brain Research, Tulsa, OK, USA（脑研究拉奥尔特研究所，俄克拉荷马州塔尔萨市）；Department of Community Medicine, University of Tulsa（社区医学系，塔尔萨大学）；Department of Internal Medicine, Carver College of Medicine, University of Iowa（内科学系，卡弗医学院，爱荷华大学）
Marked point processes intensity estimation using sparse group Lasso method applied to locations of lucrative and cooperative banks in mainland France
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；LJK（拉格朗日数学与计算实验室）；Sciences Po Grenoble, PACTE（格勒诺布尔政治学院PACTE研究所）；Department of Statistics, Institut Teknologi Sepuluh Nopember（十·一理工学院统计学系）
The L1 regularization (Lasso) has proven to be a versatile tool to select relevant features and estimate the model coefficients simultaneously and has been widely used in many research areas such as genomes studies, finance, and biomedical imaging. Despite its popularity, it is very challenging to guarantee the feature selection consistency of Lasso especially when the dimension of the data is huge. One way to improve the feature selection consistency is to select an ideal tuning parameter. Traditional tuning criteria mainly focus on minimizing the estimated prediction error or maximizing the …
We consider estimation of a functional parameter of a realistically modeled data distribution based on observing independent and identically distributed observations. We define an $m$-th order Spline Highly Adaptive Lasso Minimum Loss Estimator (Spline HAL-MLE) of a functional parameter that is defined by minimizing the empirical risk function over an $m$-th order smoothness class of functions. We show that this $m$-th order smoothness class consists of all functions that can be represented as an infinitesimal linear combination of tensor products of $\leq m$-th order spline-basis functions, a…
Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the number of cases. This paper tests two exceptionally fast algorithms for estimating regression coefficients with a lasso penalty. The previously known $\ell_2$ algorithm is based on cyclic coordinate descent. Our new $\ell_1$ algorithm is based on greedy coordinate descent and Edgeworth's algorithm for ordinary $\ell_1$ regression. Each algorithm relies on a tunin…
We consider the estimation and inference in a system of high-dimensional regression equations allowing for temporal and cross-sectional dependency in covariates and error processes, covering rather general forms of weak temporal dependence. A sequence of regressions with many regressors using LASSO (Least Absolute Shrinkage and Selection Operator) is applied for variable selection purpose, and an overall penalty level is carefully chosen by a block multiplier bootstrap procedure to account for multiplicity of the equations and dependencies in the data. Correspondingly, oracle properties with a…
We describe a simple, efficient, permutation based procedure for selecting the penalty parameter in the LASSO. The procedure, which is intended for applications where variable selection is the primary focus, can be applied in a variety of structural settings, including generalized linear models. We briefly discuss connections between permutation selection and existing theory for the LASSO. In addition, we present a simulation study and an analysis of three real data sets in which permutation selection is compared with cross-validation (CV), the Bayesian information criterion (BIC), and a selec…
Variance estimation in the linear model when $p > n$ is a difficult problem. Standard least squares estimation techniques do not apply. Several variance estimators have been proposed in the literature, all with accompanying asymptotic results proving consistency and asymptotic normality under a variety of assumptions. It is found, however, that most of these estimators suffer large biases in finite samples when true underlying signals become less sparse with larger per element signal strength. One estimator seems to be largely neglected in the literature: a residual sum of squares based estima…
In Compressed Sensing and high dimensional estimation, signal recovery often relies on sparsity assumptions and estimation is performed via $\ell_1$-penalized least-squares optimization, a.k.a. LASSO. The $\ell_1$ penalisation is usually controlled by a weight, also called "relaxation parameter", denoted by $λ$. It is commonly thought that the practical efficiency of the LASSO for prediction crucially relies on accurate selection of $λ$. In this short note, we propose to consider the hyper-parameter selection problem from a new perspective which combines the Hedge online learning method by Fre…
High-dimensional prediction typically comprises two steps: variable selection and subsequent least-squares refitting on the selected variables. However, the standard variable selection procedures, such as the lasso, hinge on tuning parameters that need to be calibrated. Cross-validation, the most popular calibration scheme, is computationally costly and lacks finite sample guarantees. In this paper, we introduce an alternative scheme, easy to implement and both computationally and theoretically efficient.
Shandong University（山东大学）；University of Geneva（日内瓦大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
To fit sparse linear associations, a LASSO sparsity inducing penalty with a single hyperparameter provably allows to recover the important features (needles) with high probability in certain regimes even if the sample size is smaller than the dimension of the input vector (haystack). More recently learners known as artificial neural networks (ANN) have shown great successes in many machine learning tasks, in particular fitting nonlinear associations. Small learning rate, stochastic gradient descent algorithm and large training set help to cope with the explosion in the number of parameters pre…
The tuning parameter selection strategy for penalized estimation is crucial to identify a model that is both interpretable and predictive. However, popular strategies (e.g., minimizing average squared prediction error via cross-validation) tend to select models with more predictors than necessary. This paper proposes a simple, yet powerful cross-validation strategy based on maximizing squared correlations between the observed and predicted values, rather than minimizing squared error loss for the purposes of support recovery. The strategy can be applied to all penalized least-squares estimator…
In variable or graph selection problems, finding a right-sized model or controlling the number of false positives is notoriously difficult. Recently, a meta-algorithm called Stability Selection was proposed that can provide reliable finite-sample control of the number of false positives. Its benefits were demonstrated when used in conjunction with the lasso and orthogonal matching pursuit algorithms. In this paper, we investigate the applicability of stability selection to structured selection algorithms: the group lasso and the structured input-output lasso. We find that using stability selec…
Increasingly high-dimensional data sets require that estimation methods do not only satisfy statistical guarantees but also remain computationally feasible. In this context, we consider $ L^{2} $-boosting via orthogonal matching pursuit in a high-dimensional linear model and analyze a data-driven early stopping time $ τ$ of the algorithm, which is sequential in the sense that its computation is based on the first $ τ$ iterations only. This approach is much less costly than established model selection criteria, that require the computation of the full boosting path. We prove that sequential ear…
Estimation of the prediction error of a linear estimation rule is difficult if the data analyst also use data to select a set of variables and construct the estimation rule using only the selected variables. In this work, we propose an asymptotically unbiased estimator for the prediction error after model search. Under some additional mild assumptions, we show that our estimator converges to the true prediction error in $L^2$ at the rate of $O(n^{-1/2})$, with $n$ being the number of data points. Our estimator applies to general selection procedures, not requiring analytical forms for the sele…
