Recently dictionary screening has been proposed as an effective way to improve the computational efficiency of solving the lasso problem, which is one of the most commonly used method for learning sparse representations. To address today's ever increasing large dataset, effective screening relies on a tight region bound on the solution to the dual lasso. Typical region bounds are in the form of an intersection of a sphere and multiple half spaces. One way to tighten the region bound is using more half spaces, which however, adds to the overhead of solving the high dimensional optimization problem in lasso screening. This paper reveals the interesting property that the optimization problem only depends on the projection of features onto the subspace spanned by the normals of the half spaces. This property converts an optimization problem in high dimension to much lower dimension, and thus sheds light on reducing the computation overhead of lasso screening based on tighter region bounds.
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…
Recently, to solve large-scale lasso and group lasso problems, screening rules have been developed, the goal of which is to reduce the problem size by efficiently discarding zero coefficients using simple rules independently of the others. However, screening for overlapping group lasso remains an open challenge because the overlaps between groups make it infeasible to test each group independently. In this paper, we develop screening rules for overlapping group lasso. To address the challenge arising from groups with overlaps, we take into account overlapping groups only if they are inclusive …
Recent computational strategies based on screening tests have been proposed to accelerate algorithms addressing penalized sparse regression problems such as the Lasso. Such approaches build upon the idea that it is worth dedicating some small computational effort to locate inactive atoms and remove them from the dictionary in a preprocessing stage so that the regression algorithm working with a smaller dictionary will then converge faster to the solution of the initial problem. We believe that there is an even more efficient way to screen the dictionary and obtain a greater acceleration: insid…
This paper is a survey of dictionary screening for the lasso problem. The lasso problem seeks a sparse linear combination of the columns of a dictionary to best match a given target vector. This sparse representation has proven useful in a variety of subsequent processing and decision tasks. For a given target vector, dictionary screening quickly identifies a subset of dictionary columns that will receive zero weight in a solution of the corresponding lasso problem. These columns can be removed from the dictionary prior to solving the lasso problem without impacting the optimality of the solut…
Lasso is a widely used regression technique to find sparse representations. When the dimension of the feature space and the number of samples are extremely large, solving the Lasso problem remains challenging. To improve the efficiency of solving large-scale Lasso problems, El Ghaoui and his colleagues have proposed the SAFE rules which are able to quickly identify the inactive predictors, i.e., predictors that have $0$ components in the solution vector. Then, the inactive predictors or features can be removed from the optimization problem to reduce its scale. By transforming the standard Lass…
This paper focusses on "safe" screening techniques for the LASSO problem. Motivated by the need for low-complexity algorithms, we propose a new approach, dubbed "joint" screening test, allowing to screen a set of atoms by carrying out one single test. The approach is particularized to two different sets of atoms, respectively expressed as sphere and dome regions. After presenting the mathematical derivations of the tests, we elaborate on their relative effectiveness and discuss the practical use of such procedures.
Université Rouen Normandie（鲁昂诺曼底大学）；Criteo AI Lab（Criteo人工智能实验室）；INSA Rouen Normandie（鲁昂国立应用科学学院）；Universitéé de Montpellier（蒙彼利埃大学）
Leveraging on the convexity of the Lasso problem , screening rules help in accelerating solvers by discarding irrelevant variables, during the optimization process. However, because they provide better theoretical guarantees in identifying relevant variables, several non-convex regularizers for the Lasso have been proposed in the literature. This work is the first that introduces a screening rule strategy into a non-convex Lasso solver. The approach we propose is based on a iterative majorization-minimization (MM) strategy that includes a screening rule in the inner solver and a condition for …
SAS Institute Inc.（赛仕软件研究所有限公司）；Arizona State University（亚利桑那州立大学）
Sparse learning techniques have been routinely used for feature selection as the resulting model usually has a small number of non-zero entries. Safe screening, which eliminates the features that are guaranteed to have zero coefficients for a certain value of the regularization parameter, is a technique for improving the computational efficiency. Safe screening is gaining increasing attention since 1) solving sparse learning formulations usually has a high computational cost especially when the number of features is large and 2) one needs to try several regularization parameters to select a su…
Univ Rennes（雷恩大学）；CentraleSupelec（中央理工-高等电力学院）；Univ. de Bretagne Occidentale（西布列塔尼大学）；Inria Rennes - Bretagne Atlantique（法国国家信息与自动化研究所雷恩-布列塔尼大西洋中心）
In this paper, we propose a novel safe screening test for Lasso. Our procedure is based on a safe region with a dome geometry and exploits a canonical representation of the set of half-spaces (referred to as "dual cutting half-spaces" in this paper) containing the dual feasible set. The proposed safe region is shown to be always included in the state-of-the-art "GAP Sphere" and "GAP Dome" proposed by Fercoq et al. (and strictly so under very mild conditions) while involving the same computational burden. Numerical experiments confirm that our new dome enables to devise more powerful screening …
In high dimensional settings, sparse structures are crucial for efficiency, either in term of memory, computation or performance. In some contexts, it is natural to handle more refined structures than pure sparsity, such as for instance group sparsity. Sparse-Group Lasso has recently been introduced in the context of linear regression to enforce sparsity both at the feature level and at the group level. We adapt to the case of Sparse-Group Lasso recent safe screening rules that discard early in the solver irrelevant features/groups. Such rules have led to important speed-ups for a wide range o…
Convex sparsity-promoting regularizations are ubiquitous in modern statistical learning. By construction, they yield solutions with few non-zero coefficients, which correspond to saturated constraints in the dual optimization formulation. Working set (WS) strategies are generic optimization techniques that consist in solving simpler problems that only consider a subset of constraints, whose indices form the WS. Working set methods therefore involve two nested iterations: the outer loop corresponds to the definition of the WS and the inner loop calls a solver for the subproblems. For the Lasso …
Texas A&M University（得克萨斯农工大学）；University of Washington（华盛顿大学）
We present safe active incremental feature selection~(SAIF) to scale up the computation of LASSO solutions. SAIF does not require a solution from a heavier penalty parameter as in sequential screening or updating the full model for each iteration as in dynamic screening. Different from these existing screening methods, SAIF starts from a small number of features and incrementally recruits active features and updates the significantly reduced model. Hence, it is much more computationally efficient and scalable with the number of features. More critically, SAIF has the safe guarantee as it has t…
In high dimensional regression settings, sparsity enforcing penalties have proved useful to regularize the data-fitting term. A recently introduced technique called screening rules propose to ignore some variables in the optimization leveraging the expected sparsity of the solutions and consequently leading to faster solvers. When the procedure is guaranteed not to discard variables wrongly the rules are said to be safe. In this work, we propose a unifying framework for generalized linear models regularized with standard sparsity enforcing penalties such as $\ell_1$ or $\ell_1/\ell_2$ norms. O…
Screening rules allow to early discard irrelevant variables from the optimization in Lasso problems, or its derivatives, making solvers faster. In this paper, we propose new versions of the so-called $\textit{safe rules}$ for the Lasso. Based on duality gap considerations, our new rules create safe test regions whose diameters converge to zero, provided that one relies on a converging solver. This property helps screening out more variables, for a wider range of regularization parameter values. In addition to faster convergence, we prove that we correctly identify the active sets (supports) of…
Convex sparsity-inducing regularizations are ubiquitous in high-dimensional machine learning, but solving the resulting optimization problems can be slow. To accelerate solvers, state-of-the-art approaches consist in reducing the size of the optimization problem at hand. In the context of regression, this can be achieved either by discarding irrelevant features (screening techniques) or by prioritizing features likely to be included in the support of the solution (working set techniques). Duality comes into play at several steps in these techniques. Here, we propose an extrapolation technique …
Practical considerations for variable screening in the super learner
Biostatistics Division, Kaiser Permanente Washington Health Research Institute（Kaiser Permanente华盛顿健康研究机构生物统计学部）；Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center（弗雷德 Hutchinson癌症研究中心疫苗与传染病部）；Department of Biostatistics, University of Washington（华盛顿大学生物统计学系）；Seattle Central College（西雅图中央学院）
Feature screening for ultra high dimensional feature spaces plays a critical role in the analysis of data sets whose predictors exponentially exceed the number of observations. Such data sets are becoming increasingly prevalent in areas such as bioinformatics, medical imaging, and social network analysis. Frequently, these data sets have both categorical response and categorical covariates, yet extant feature screening literature rarely considers such data types. We propose a new screening procedure rooted in the Cochran-Armitage trend test. Our method is specifically applicable for data where…
We investigate the relation of two fundamental tools in machine learning and signal processing, that is the support vector machine (SVM) for classification, and the Lasso technique used in regression. We show that the resulting optimization problems are equivalent, in the following sense. Given any instance of an $\ell_2$-loss soft-margin (or hard-margin) SVM, we construct a Lasso instance having the same optimal solutions, and vice versa. As a consequence, many existing optimization algorithms for both SVMs and Lasso can also be applied to the respective other problem instances. Also, the equ…
We introduce a computationally effective algorithm for a linear model selection consisting of three steps: screening--ordering--selection (SOS). Screening of predictors is based on the thresholded Lasso that is l_1 penalized least squares. The screened predictors are then fitted using least squares (LS) and ordered with respect to their t statistics. Finally, a model is selected using greedy generalized information criterion (GIC) that is l_0 penalized LS in a nested family induced by the ordering. We give non-asymptotic upper bounds on error probability of each step of the SOS algorithm in te…
In high dimensional settings, sparse structures are crucial for efficiency, both in term of memory, computation and performance. It is customary to consider $\ell_1$ penalty to enforce sparsity in such scenarios. Sparsity enforcing methods, the Lasso being a canonical example, are popular candidates to address high dimension. For efficiency, they rely on tuning a parameter trading data fitting versus sparsity. For the Lasso theory to hold this tuning parameter should be proportional to the noise level, yet the latter is often unknown in practice. A possible remedy is to jointly optimize over t…
We propose a new framework for deriving screening rules for convex optimization problems. Our approach covers a large class of constrained and penalized optimization formulations, and works in two steps. First, given any approximate point, the structure of the objective function and the duality gap is used to gather information on the optimal solution. In the second step, this information is used to produce screening rules, i.e. safely identifying unimportant weight variables of the optimal solution. Our general framework leads to a large variety of useful existing as well as new screening rul…
We describe a regularized regression model for the selection of gene-environment (GxE) interactions. The model focuses on a single environmental exposure and induces a main-effect-before-interaction hierarchical structure. We propose an efficient fitting algorithm and screening rules that can discard large numbers of irrelevant predictors with high accuracy. We present simulation results showing that the model outperforms existing joint selection methods for (GxE) interactions in terms of selection performance, scalability and speed, and provide a real data application. Our implementation is a…
We propose a generalization of the lasso that allows the model coefficients to vary as a function of a general set of modifying variables. These modifiers might be variables such as gender, age or time. The paradigm is quite general, with each lasso coefficient modified by a sparse linear function of the modifying variables $Z$. The model is estimated in a hierarchical fashion to control the degrees of freedom and avoid overfitting. The modifying variables may be observed, observed only in the training set, or unobserved overall. There are connections of our proposal to varying coefficient mod…
We propose a Multi-step Screening Procedure (MSP) for the recovery of sparse linear models in high-dimensional data. This method is based on a repeated small penalty strategy that quickly converges to an estimate within a few iterations. Specifically, in each iteration, an adaptive lasso regression with a small penalty is fit within the reduced feature space obtained from the previous step, rendering its computational complexity roughly comparable with the Lasso. MSP is shown to select the true model under complex correlation structures among the predictors and response, even when the irrepres…
Arizona State University（亚利桑那州立大学）；University of Minnesota（明尼苏达大学）
In this paper, we consider the problem of estimating multiple graphical models simultaneously using the fused lasso penalty, which encourages adjacent graphs to share similar structures. A motivating example is the analysis of brain networks of Alzheimer's disease using neuroimaging data. Specifically, we may wish to estimate a brain network for the normal controls (NC), a brain network for the patients with mild cognitive impairment (MCI), and a brain network for Alzheimer's patients (AD). We expect the two brain networks for NC and MCI to share common structures but not to be identical to ea…
1 School of Computing, Informatics；2 Imaging Genetics Center, Institute for Neuroimaging
Genome-wide association studies (GWAS) have achieved great success in the genetic study of Alzheimer's disease (AD). Collaborative imaging genetics studies across different research institutions show the effectiveness of detecting genetic risk factors. However, the high dimensionality of GWAS data poses significant challenges in detecting risk SNPs for AD. Selecting relevant features is crucial in predicting the response variable. In this study, we propose a novel Distributed Feature Selection Framework (DFSF) to conduct the large-scale imaging genetics studies across multiple institutions. To…
Institute of Computer Science, Polish Academy of Sciences（波兰科学院计算机科学研究所）；Warsaw University of Technology（华沙理工大学）
We consider selection of random predictors for high-dimensional regression problem with binary response for a general loss function. Important special case is when the binary model is semiparametric and the response function is misspecified under parametric model fit. Selection for such a scenario aims at recovering the support of the minimizer of the associated risk with large probability. We propose a two-step selection procedure which consists of screening and ordering predictors by Lasso method and then selecting a subset of predictors which minimizes Generalized Information Criterion on t…
Variable selection plays an important role in high dimensional statistical modeling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality $p$, estimation accuracy and computational cost are two top concerns. In a recent paper, Candes and Tao (2007) propose the Dantzig selector using $L_1$ regularization and show that it achieves the ideal risk up to a logarithmic factor $\log p$. Their innovative procedure and remarkable result are challenged when the dimensionality is ultra high as the factor $\log p$ can be large and …
We propose a new measure of variable importance in high-dimensional regression based on the change in the LASSO solution path when one covariate is left out. The proposed procedure provides a novel way to calculate variable importance and conduct variable screening. In addition, our procedure allows for the construction of P-values for testing whether each coefficient is equal to zero as well as for testing hypotheses involving multiple regression coefficients simultaneously; bootstrap techniques are used to construct the null distribution. For low-dimensional linear models, our method can ach…
We propose a new variable selection algorithm, subsample-ordered least-angle regression (solar), and its coordinate descent generalization, solar-cd. Solar re-constructs lasso paths using the $L_0$ norm and averages the resulting solution paths across subsamples. Path averaging retains the ranking information of the informative variables while averaging out sensitivity to high dimensionality, improving variable selection stability, efficiency, and accuracy. We prove that: (i) with a high probability, path averaging perfectly separates informative variables from redundant variables on the avera…
The graphical lasso (glasso) is a widely-used fast algorithm for estimating sparse inverse covariance matrices. The glasso solves an L1 penalized maximum likelihood problem and is available as an R library on CRAN. The output from the glasso, a regularized covariance matrix estimate a sparse inverse covariance matrix estimate, not only identify a graphical model but can also serve as intermediate inputs into multivariate procedures such as PCA, LDA, MANOVA, and others. The glasso indeed produces a covariance matrix estimate which solves the L1 penalized optimization problem in a dual sense; ho…
In high dimensional regression, feature clustering by their effects on outcomes is often as important as feature selection. For that purpose, clustered Lasso and octagonal shrinkage and clustering algorithm for regression (OSCAR) are used to make feature groups automatically by pairwise $L_1$ norm and pairwise $L_\infty$ norm, respectively. This paper proposes efficient path algorithms for clustered Lasso and OSCAR to construct solution paths with respect to their regularization parameters. Despite too many terms in exhaustive pairwise regularization, their computational costs are reduced by u…
High dimensional regression benefits from sparsity promoting regularizations. Screening rules leverage the known sparsity of the solution by ignoring some variables in the optimization, hence speeding up solvers. When the procedure is proven not to discard features wrongly the rules are said to be \emph{safe}. In this paper we derive new safe rules for generalized linear models regularized with $\ell_1$ and $\ell_1/\ell_2$ norms. The rules are based on duality gap computations and spherical safe regions whose diameters converge to zero. This allows to discard safely more variables, in particul…
We introduce a new approach to variable selection, called Predictive Correlation Screening, for predictor design. Predictive Correlation Screening (PCS) implements false positive control on the selected variables, is well suited to small sample sizes, and is scalable to high dimensions. We establish asymptotic bounds for Familywise Error Rate (FWER), and resultant mean square error of a linear predictor on the selected variables. We apply Predictive Correlation Screening to the following two-stage predictor design problem. An experimenter wants to learn a multivariate predictor of gene express…
Screening is the problem of finding a superset of the set of non-zero entries in an unknown p-dimensional vector β* given n noisy observations. Naturally, we want this superset to be as small as possible. We propose a novel framework for screening, which we refer to as Multiple Grouping (MuG), that groups variables, performs variable selection over the groups, and repeats this process multiple number of times to estimate a sequence of sets that contains the non-zero entries in β*. Screening is done by taking an intersection of all these estimated sets. The MuG framework can be used in conjunct…
We propose a novel application of the Simultaneous Orthogonal Matching Pursuit (S-OMP) procedure for sparsistant variable selection in ultra-high dimensional multi-task regression problems. Screening of variables, as introduced in \cite{fan08sis}, is an efficient and highly scalable way to remove many irrelevant variables from the set of all variables, while retaining all the relevant variables. S-OMP can be applied to problems with hundreds of thousands of variables and once the number of variables is reduced to a manageable size, a more computationally demanding procedure can be used to iden…
High-dimensional data are commonly seen in modern statistical applications, variable selection methods play indispensable roles in identifying the critical features for scientific discoveries. Traditional best subset selection methods are computationally intractable with a large number of features, while regularization methods such as Lasso, SCAD and their variants perform poorly in ultrahigh-dimensional data due to low computational efficiency and unstable algorithm. Sure screening methods have become popular alternatives by first rapidly reducing the dimension using simple measures such as m…
Consider a linear regression model where the design matrix X has n rows and p columns. We assume (a) p is much large than n, (b) the coefficient vector beta is sparse in the sense that only a small fraction of its coordinates is nonzero, and (c) the Gram matrix G = X'X is sparse in the sense that each row has relatively few large coordinates (diagonals of G are normalized to 1). The sparsity in G naturally induces the sparsity of the so-called graph of strong dependence (GOSD). We find an interesting interplay between the signal sparsity and the graph sparsity, which ensures that in a broad co…
Variable selection in high dimensional space has challenged many contemporary statistical problems from many frontiers of scientific disciplines. Recent technology advance has made it possible to collect a huge amount of covariate information such as microarray, proteomic and SNP data via bioimaging technology while observing survival information on patients in clinical studies. Thus, the same challenge applies to the survival analysis in order to understand the association between genomics information and clinical information about the survival time. In this work, we extend the sure screening…
It is known that for a certain class of single index models (SIMs) $Y = f(\boldsymbol{X}_{p \times 1}^\intercal\boldsymbolβ_0, \varepsilon)$, support recovery is impossible when $\boldsymbol{X} \sim \mathcal{N}(0, \mathbb{I}_{p \times p})$ and a model complexity adjusted sample size is below a critical threshold. Recently, optimal algorithms based on Sliced Inverse Regression (SIR) were suggested. These algorithms work provably under the assumption that the design $\boldsymbol{X}$ comes from an i.i.d. Gaussian distribution. In the present paper we analyze algorithms based on covariance screeni…
Variable selection is a widely studied problem in high dimensional statistics, primarily since estimating the precise relationship between the covariates and the response is of great importance in many scientific disciplines. However, most of theory and methods developed towards this goal for the linear model invoke the assumption of iid sub-Gaussian covariates and errors. This paper analyzes the theoretical properties of Sure Independence Screening (SIS) (Fan and Lv [J. R. Stat. Soc. Ser. B Stat. Methodol. 70 (2008) 849-911]) for high dimensional linear models with dependent and/or heavy tail…
Taking into account high-order interactions among covariates is valuable in many practical regression problems. This is, however, computationally challenging task because the number of high-order interaction features to be considered would be extremely large unless the number of covariates is sufficiently small. In this paper, we propose a novel efficient algorithm for LASSO-based sparse learning of such high-order interaction models. Our basic strategy for reducing the number of features is to employ the idea of recently proposed safe feature screening (SFS) rule. An SFS rule has a property t…
This paper explores the following question: what kind of statistical guarantees can be given when doing variable selection in high-dimensional models? In particular, we look at the error rates and power of some multi-stage regression methods. In the first stage we fit a set of candidate models. In the second stage we select one model by cross-validation. In the third stage we use hypothesis testing to eliminate some variables. We refer to the first two stages as "screening" and the last stage as "cleaning." We consider three screening methods: the lasso, marginal regression, and forward stepwi…
We study the problem of high-dimensional regression when there may be interacting variables. Approaches using sparsity-inducing penalty functions such as the Lasso can be useful for producing interpretable models. However, when the number variables runs into the thousands, and so even two-way interactions number in the millions, these methods may become computationally infeasible. Typically variable screening based on model fits using only main effects must be performed first. One problem with screening is that important variables may be missed if they are only useful for prediction when certa…
We consider the problem of selective inference after solving a (randomized) convex statistical learning program in the form of a penalized or constrained loss function. Our first main result is a change-of-measure formula that describes many conditional sampling problems of interest in selective inference. Our approach is model-agnostic in the sense that users may provide their own statistical model for inference, we simply provide the modification of each distribution in the model after the selection. Our second main result describes the geometric structure in the Jacobian appearing in the ch…
