Penalized logistic regression is extremely useful for binary classification with large number of covariates (higher than the sample size), having several real life applications, including genomic disease classification. However, the existing methods based on the likelihood loss function are sensitive to data contamination and other noise and, hence, robust methods are needed for stable and more accurate inference. In this paper, we propose a family of robust estimators for sparse logistic models utilizing the popular density power divergence based loss function and the general adaptively weighted LASSO penalties. We study the local robustness of the proposed estimators through its influence function and also derive its oracle properties and asymptotic distribution. With extensive empirical illustrations, we demonstrate the significantly improved performance of our proposed estimators over the existing ones with particular gain in robustness. Our proposal is finally applied to analyse four different real datasets for cancer classification, obtaining robust and accurate models, that simultaneously performs gene selection and patient classification.
We consider the problem of estimating a function $f\_{0}$ in logistic regression model. We propose to estimate this function $f\_{0}$ by a sparse approximation build as a linear combination of elements of a given dictionary of $p$ functions. This sparse approximation is selected by the Lasso or Group Lasso procedure. In this context, we state non asymptotic oracle inequalities for Lasso and Group Lasso under restricted eigenvalues assumption as introduced in \cite{BRT}.
High dimensional Poisson regression has become a standard framework for the analysis of massive counts datasets. In this work we estimate the intensity function of the Poisson regression model by using a dictionary approach, which generalizes the classical basis approach, combined with a Lasso or a group-Lasso procedure. Selection depends on penalty weights that need to be calibrated. Standard methodologies developed in the Gaussian framework can not be directly applied to Poisson models due to heteroscedasticity. Here we provide data-driven weights for the Lasso and the group-Lasso derived fr…
The Ising model was originally developed to model magnetisation of solids in statistical physics. As a network of binary variables with the probability of becoming 'active' depending only on direct neighbours, the Ising model appears appropriate for many other processes. For instance, it was recently applied in psychology to model co-occurrences of mental disorders. It has been shown that the connections between the variables (nodes) in the Ising network can be estimated with a series of logistic regressions. This naturally leads to questions of how well such a model predicts new observations …
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）
We apply the network Lasso to solve binary classification and clustering problems for network-structured data. To this end, we generalize ordinary logistic regression to non-Euclidean data with an intrinsic network structure. The resulting "logistic network Lasso" amounts to solving a non-smooth convex regularized empirical risk minimization. The risk is measured using the logistic loss incurred over a small set of labeled nodes. For the regularization, we propose to use the total variation of the classifier requiring it to conform to the underlying network structure. A scalable implementation…
For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) space if and only if $ρ=\lim\frac{p}{n}=0$, where $p$ is the dimension and $n$ is the sample size. Thus, when $p$ is of the same or a higher order of $n$, additional assumptions such as sparsity must be imposed in order to ensure consistency for SIR. By constructing artificial response variables made up from top eigenvectors of the estimated conditional covariance matrix, we introduce a simple Lasso regression method to obtain an e…
We study the asymptotic properties of Lasso+mLS and Lasso+Ridge under the sparse high-dimensional linear regression model: Lasso selecting predictors and then modified Least Squares (mLS) or Ridge estimating their coefficients. First, we propose a valid inference procedure for parameter estimation based on parametric residual bootstrap after Lasso+mLS and Lasso+Ridge. Second, we derive the asymptotic unbiasedness of Lasso+mLS and Lasso+Ridge. More specifically, we show that their biases decay at an exponential rate and they can achieve the oracle convergence rate of $s/n$ (where $s$ is the num…
We propose a self-tuning $\sqrt{\mathrm {Lasso}}$ method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example, perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds for $\sqrt{\mathrm {Lasso}}$ including prediction norm rate …
Department of Computer Science, Aalto University（阿尔托大学计算机科学系）；Aalto University（阿尔托大学）
We apply the network Lasso to classify partially labeled data points which are characterized by high-dimensional feature vectors. In order to learn an accurate classifier from limited amounts of labeled data, we borrow statistical strength, via an intrinsic network structure, across the dataset. The resulting logistic network Lasso amounts to a regularized empirical risk minimization problem using the total variation of a classifier as a regularizer. This minimization problem is a non-smooth convex optimization problem which we solve using a primal-dual splitting method. This method is appeali…
Selection of important covariates and to drop the unimportant ones from a high-dimensional regression model is a long standing problem and hence have received lots of attention in the last two decades. After selecting the correct model, it is also important to properly estimate the existing parameters corresponding to important covariates. In this spirit, Fan and Li (2001) proposed Oracle property as a desired feature of a variable selection method. Oracle property has two parts; one is the variable selection consistency (VSC) and the other one is the asymptotic normality. Keeping VSC fixed an…
In this article we investigate consistency of selection in regression models via the popular Lasso method. Here we depart from the traditional linear regression assumption and consider approximations of the regression function $f$ with elements of a given dictionary of $M$ functions. The target for consistency is the index set of those functions from this dictionary that realize the most parsimonious approximation to $f$ among all linear combinations belonging to an $L_2$ ball centered at $f$ and of radius $r_{n,M}^2$. In this framework we show that a consistent estimate of this index set can …
We consider the least angle regression and forward stagewise algorithms for solving penalized least squares regression problems. In Efron, Hastie, Johnstone & Tibshirani (2004) it is proved that the least angle regression algorithm, with a small modification, solves the lasso regression problem. Here we give an analogous result for incremental forward stagewise regression, showing that it solves a version of the lasso problem that enforces monotonicity. One consequence of this is as follows: while lasso makes optimal progress in terms of reducing the residual sum-of-squares per unit increase i…
We consider the estimation of regression models on strata defined using a categorical covariate, in order to identify interactions between this categorical covariate and the other predictors. A basic approach requires the choice of a reference stratum. We show that the performance of a penalized version of this approach depends on this arbitrary choice. We propose a refined approach that bypasses this arbitrary choice, at almost no additional computational cost. Regarding model selection consistency, our proposal mimics the strategy based on an optimal and covariate-specific choice for the ref…
Least Absolute Shrinkage and Selection Operator or the Lasso, introduced by Tibshirani (1996), is a popular estimation procedure in multiple linear regression when underlying design has a sparse structure, because of its property that it sets some regression coefficients exactly equal to 0. In this article, we develop a perturbation bootstrap method and establish its validity in approximating the distribution of the Lasso in heteroscedastic linear regression. We allow the underlying covariates to be either random or non-random. We show that the proposed bootstrap method works irrespective of t…
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…
We consider a multivariate finite mixture of Gaussian regression models for high-dimensional data, where the number of covariates and the size of the response may be much larger than the sample size. We provide an $\ell_1$-oracle inequality satisfied by the Lasso estimator according to the Kullback-Leibler loss. This result is an extension of the $\ell_1$-oracle inequality established by Meynet in \cite{Meynet} in the multivariate case. We focus on the Lasso for its $\ell_1$-regularization properties rather than for the variable selection procedure, as it was done in Städler in \cite{Stadler}.
In this work, we consider a manufactory process which can be described by a multiple-instance logistic regression model. In order to compute the maximum likelihood estimation of the unknown coefficient, an expectation-maximization algorithm is proposed, and the proposed modeling approach can be extended to identify the important covariates by adding the coefficient penalty term into the likelihood function. In addition to essential technical details, we demonstrate the usefulness of the proposed method by simulations and real examples.
This paper study sparse classification problems. We show that under single-index models, vanilla Lasso could give good estimate of unknown parameters. With this result, we see that even if the model is not linear, and even if the response is not continuous, we could still use vanilla Lasso to train classifiers. Simulations confirm that vanilla Lasso could be used to get a good estimation when data are generated from a logistic regression model.
We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Our Lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the $\ell_1$ estimation loss for regression coefficients. Since the Lasso estimator selects variables simultaneously, we show that oracle inequalities can be established w…
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 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…
After selection with the Group LASSO (or generalized variants such as the overlapping, sparse, or standardized Group LASSO), inference for the selected parameters is unreliable in the absence of adjustments for selection bias. In the penalized Gaussian regression setup, existing approaches provide adjustments for selection events that can be expressed as linear inequalities in the data variables. Such a representation, however, fails to hold for selection with the Group LASSO and substantially obstructs the scope of subsequent post-selective inference. Key questions of inferential interest -- …
University of Science and Technology of China（中国科学技术大学）；Peking University（北京大学）；Central China Normal University（华中师范大学）
When we are interested in high-dimensional system and focus on classification performance, the $\ell_{1}$-penalized logistic regression is becoming important and popular. However, the Lasso estimates could be problematic when penalties of different coefficients are all the same and not related to the data. We proposed two types of weighted Lasso estimates depending on covariates by the McDiarmid inequality. Given sample size $n$ and dimension of covariates $p$, the finite sample behavior of our proposed methods with a diverging number of predictors is illustrated by non-asymptotic oracle inequ…
We show that the two-stage adaptive Lasso procedure (Zou, 2006) is consistent for high-dimensional model selection in linear and Gaussian graphical models. Our conditions for consistency cover more general situations than those accomplished in previous work: we prove that restricted eigenvalue conditions (Bickel et al., 2008) are also sufficient for sparse structure estimation.
We consider the finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. The summands in the negative log partial likelihood function for censored survival data, however, are neither iid nor Lipschitz. We first approximate the negative log partial likelihood function by a sum of iid non-Lipschitz terms, then derive the non-asymptotic oracle inequalitie…
In regression problems where covariates can be naturally grouped, the group Lasso is an attractive method for variable selection since it respects the grouping structure in the data. We study the selection and estimation properties of the group Lasso in high-dimensional settings when the number of groups exceeds the sample size. We provide sufficient conditions under which the group Lasso selects a model whose dimension is comparable with the underlying model with high probability and is estimation consistent. However, the group Lasso is, in general, not selection consistent and also tends to …
Meinshausen and Buhlmann [Ann. Statist. 34 (2006) 1436--1462] showed that, for neighborhood selection in Gaussian graphical models, under a neighborhood stability condition, the LASSO is consistent, even when the number of variables is of greater order than the sample size. Zhao and Yu [(2006) J. Machine Learning Research 7 2541--2567] formalized the neighborhood stability condition in the context of linear regression as a strong irrepresentable condition. That paper showed that under this condition, the LASSO selects exactly the set of nonzero regression coefficients, provided that these coef…
We study a high-dimensional regression model. Aim is to construct a confidence set for a given group of regression coefficients, treating all other regression coefficients as nuisance parameters. We apply a one-step procedure with the square-root Lasso as initial estimator and a multivariate square-root Lasso for constructing a surrogate Fisher information matrix. The multivariate square-root Lasso is based on nuclear norm loss with $\ell_1$-penalty. We show that this procedure leads to an asymptotically $χ^2$-distributed pivot, with a remainder term depending only on the $\ell_1$-error of the…
Regularization aims to improve prediction performance of a given statistical modeling approach by moving to a second approach which achieves worse training error but is expected to have fewer degrees of freedom, i.e., better agreement between training and prediction error. We show here, however, that this expected behavior does not hold in general. In fact, counter examples are given that show regularization can increase the degrees of freedom in simple situations, including lasso and ridge regression, which are the most common regularization approaches in use. In such situations, the regulari…
The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function $ρ$. We then focus on the special cases when either $ρ$ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new pointwise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before; our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error, or a su…
The fused lasso, also known as total-variation denoising, is a locally-adaptive function estimator over a regular grid of design points. In this paper, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to an approach for non-parametric regression. This approach, which we call the $K$-nearest neighbors ($K$-NN) fused lasso, involves (i) computing the $K$-NN graph of the design points; and (ii) performing the fused lasso over this $K$-NN graph. We show that this procedure has a number of theoretical advantages over competing approaches: specificall…
We consider regression problems where the number of predictors greatly exceeds the number of observations. We propose a method for variable selection that first estimates the regression function, yielding a "pre-conditioned" response variable. The primary method used for this initial regression is supervised principal components. Then we apply a standard procedure such as forward stepwise selection or the LASSO to the pre-conditioned response variable. In a number of simulated and real data examples, this two-step procedure outperforms forward stepwise selection or the usual LASSO (applied dir…
Bayesian networks can represent directed gene regulations and therefore are favored over co-expression networks. However, hardly any Bayesian network study concerns the false discovery control (FDC) of network edges, leading to low accuracies due to systematic biases from inconsistent false discovery levels in the same study. We design four empirical tests to examine the FDC of Bayesian networks from three p-value based lasso regression variable selections --- two existing and one we originate. Our method, lassopv, computes p-values for the critical regularization strength at which a predictor…
Lasso is a popular and efficient approach to simultaneous estimation and variable selection in high-dimensional regression models. In this paper, a robust LAD-lasso method for multiple outcomes is presented that addresses the challenges of non-normal outcome distributions and outlying observations. Measured covariate data from space or time, or spectral bands or genomic positions often have natural correlation structure arising from measuring distance between the covariates. The proposed multi-outcome approach includes handling of such covariate blocks by a group fusion penalty, which encourag…
When the design matrix has orthonormal columns, "soft thresholding" the ordinary least squares (OLS) solution produces the Lasso solution [Tibshirani, 1996]. If one uses the Puffer preconditioned Lasso [Jia and Rohe, 2012], then this result generalizes from orthonormal designs to full rank designs (Theorem 1). Theorem 2 refines the Puffer preconditioner to make the Lasso select the same model as removing the elements of the OLS solution with the largest p-values. Using a generalized Puffer preconditioner, Theorem 3 relates ridge regression to the preconditioned Lasso; this result is for the hi…
A new estimator, named S-LASSO, is proposed for the coefficient function of the Function-on-Function linear regression model. The S-LASSO estimator is shown to be able to increase the interpretability of the model, by better locating regions where the coefficient function is zero, and to smoothly estimate non-zero values of the coefficient function. The sparsity of the estimator is ensured by a \textit{functional LASSO penalty}, which pointwise shrinks toward zero the coefficient function, while the smoothness is provided by two roughness penalties that penalize the curvature of the final esti…
In this paper, we propose a majorization-minimization (MM) algorithm for high-dimensional fused lasso regression (FLR) suitable for parallelization using graphics processing units (GPUs). The MM algorithm is stable and flexible as it can solve the FLR problems with various types of design matrices and penalty structures within a few tens of iterations. We also show that the convergence of the proposed algorithm is guaranteed. We conduct numerical studies to compare our algorithm with other existing algorithms, demonstrating that the proposed MM algorithm is competitive in many settings includi…
Regression with the lasso penalty is a popular tool for performing dimension reduction when the number of covariates is large. In many applications of the lasso, like in genomics, covariates are subject to measurement error. We study the impact of measurement error on linear regression with the lasso penalty, both analytically and in simulation experiments. A simple method of correction for measurement error in the lasso is then considered. In the large sample limit, the corrected lasso yields sign consistent covariate selection under conditions very similar to the lasso with perfect measureme…
This paper considers a high-dimensional linear regression problem where there are complex correlation structures among predictors. We propose a graph-constrained regularization procedure, named Sparse Laplacian Shrinkage with the Graphical Lasso Estimator (SLS-GLE). The procedure uses the estimated precision matrix to describe the specific information on the conditional dependence pattern among predictors, and encourages both sparsity on the regression model and the graphical model. We introduce the Laplacian quadratic penalty adopting the graph information, and give detailed discussions on th…
We establish statistical properties of random-weighting methods in LASSO regression under different regularization parameters $λ_n$ and suitable regularity conditions. The random-weighting methods in view concern repeated optimization of a randomized objective function, motivated by the need for computational approximations to Bayesian posterior sampling. In the context of LASSO regression, we repeatedly assign analyst-drawn random weights to terms in the objective function (including the penalty terms), and optimize to obtain a sample of random-weighting estimators. We show that existing appr…
In the high-dimensional regression model a response variable is linearly related to $p$ covariates, but the sample size $n$ is smaller than $p$. We assume that only a small subset of covariates is `active' (i.e., the corresponding coefficients are non-zero), and consider the model-selection problem of identifying the active covariates. A popular approach is to estimate the regression coefficients through the Lasso ($\ell_1$-regularized least squares). This is known to correctly identify the active set only if the irrelevant covariates are roughly orthogonal to the relevant ones, as quantified …
Penalized regression is an attractive framework for variable selection problems. Often, variables possess a grouping structure, and the relevant selection problem is that of selecting groups, not individual variables. The group lasso has been proposed as a way of extending the ideas of the lasso to the problem of group selection. Nonconvex penalties such as SCAD and MCP have been proposed and shown to have several advantages over the lasso; these penalties may also be extended to the group selection problem, giving rise to group SCAD and group MCP methods. Here, we describe algorithms for fitt…
We apply network Lasso to semi-supervised regression problems involving network structured data. This approach lends quite naturally to highly scalable learning algorithms in the form of message passing over an empirical graph which represents the network structure of the data. By using a simple non-parametric regression model, which is motivated by a clustering hypothesis, we provide an analysis of the estimation error incurred by network Lasso. This analysis reveals conditions on the the network structure and the available training data which guarantee network Lasso to be accurate. Remarkabl…
We introduce and study the Group Square-Root Lasso (GSRL) method for estimation in high dimensional sparse regression models with group structure. The new estimator minimizes the square root of the residual sum of squares plus a penalty term proportional to the sum of the Euclidean norms of groups of the regression parameter vector. The net advantage of the method over the existing Group Lasso (GL)-type procedures consists in the form of the proportionality factor used in the penalty term, which for GSRL is independent of the variance of the error terms. This is of crucial importance in models…
In this paper, we investigate a multivariate multi-response (MVMR) linear regression problem, which contains multiple linear regression models with differently distributed design matrices, and different regression and output vectors. The goal is to recover the support union of all regression vectors using $l_1/l_2$-regularized Lasso. We characterize sufficient and necessary conditions on sample complexity \emph{as a sharp threshold} to guarantee successful recovery of the support union. Namely, if the sample size is above the threshold, then $l_1/l_2$-regularized Lasso correctly recovers the s…
Additive isotonic regression attempts to determine the relationship between a multi-dimensional observation variable and a response, under the constraint that the estimate is the additive sum of univariate component effects that are monotonically increasing. In this article, we present a new method for such regression called LASSO Isotone (LISO). LISO adapts ideas from sparse linear modelling to additive isotonic regression. Thus, it is viable in many situations with high dimensional predictor variables, where selection of significant versus insignificant variables are required. We suggest an …
We derive new theoretical results on the properties of the adaptive least absolute shrinkage and selection operator (adaptive lasso) for time series regression models. In particular, we investigate the question of how to conduct finite sample inference on the parameters given an adaptive lasso model for some fixed value of the shrinkage parameter. Central in this study is the test of the hypothesis that a given adaptive lasso parameter equals zero, which therefore tests for a false positive. To this end we construct a simple testing procedure and show, theoretically and empirically through ext…
