Bradley Efron, Trevor Hastie, Iain Johnstone, Robert Tibshirani
备注：
This paper discussed in: math.ST/0406463, math.ST/0406467, math.ST/0406468, math.ST/0406469, math.ST/0406470, math.ST/0406471, math.ST/0406472, math.ST/0406473. Rejoinder in math.ST/0406474
发表信息：
Annals of Statistics 2004, Vol. 32, No. 2, 407-451
The purpose of model selection algorithms such as All Subsets, Forward
Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods.
Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods.
(2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method;
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…
Least Angle Regression is a promising technique for variable selection applications, offering a nice alternative to stepwise regression. It provides an explanation for the similar behavior of LASSO ($\ell_1$-penalized regression) and forward stagewise regression, and provides a fast implementation of both. The idea has caught on rapidly, and sparked a great deal of research interest. In this paper, we give an overview of Least Angle Regression and the current state of related research.
We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a Principal Component Analysis) and then performs an ordinary (un-regularized) least squares regression in this subspace. This note shows that the risk of this ordinary least squares method is within a constant factor (namely 4) of the risk of ridge regression.
Applied statisticians use sequential regression procedures to produce a ranking of explanatory variables and, in settings of low correlations between variables and strong true effect sizes, expect that variables at the very top of this ranking are truly relevant to the response. In a regime of certain sparsity levels, however, three examples of sequential procedures--forward stepwise, the lasso, and least angle regression--are shown to include the first spurious variable unexpectedly early. We derive a rigorous, sharp prediction of the rank of the first spurious variable for these three proced…
The scalar-on-function regression model has become a popular analysis tool to explore the relationship between a scalar response and multiple functional predictors. Most of the existing approaches to estimate this model are based on the least-squares estimator, which can be seriously affected by outliers in empirical datasets. When outliers are present in the data, it is known that the least-squares-based estimates may not be reliable. This paper proposes a robust functional partial least squares method, allowing a robust estimate of the regression coefficients in a scalar-on-multiple-function…
Many regularization schemes for high-dimensional regression have been put forward. Most require the choice of a tuning parameter, using model selection criteria or cross-validation schemes. We show that a simple non-negative or sign-constrained least squares is a very simple and effective regularization technique for a certain class of high-dimensional regression problems. The sign constraint has to be derived via prior knowledge or an initial estimator but no further tuning or cross-validation is necessary. The success depends on conditions that are easy to check in practice. A sufficient con…
This study proposes sparse estimation methods for the generalized linear models, which run one of least angle regression (LARS) and least absolute shrinkage and selection operator (LASSO) in the tangent space of the manifold of the statistical model. This study approximates the statistical model and subsequently uses exact calculations. LARS was proposed as an efficient algorithm for parameter estimation and variable selection for the normal linear model. The LARS algorithm is described in terms of Euclidean geometry regarding the correlation as the metric of the parameter space. Since the LAR…
Arizona State University（亚利桑那州立大学）；Sun Yat-sen University（中山大学）
A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the l1 norm can be applied to improve the learning rate estimate of l1-regularization in machine learning.
Least angle regression (LARS) by Efron et al. (2004) is a novel method for constructing the piece-wise linear path of Lasso solutions. For several years, it remained also as the de facto method for computing the Lasso solution before more sophisticated optimization algorithms preceded it. LARS method has recently again increased its popularity due to its ability to find the values of the penalty parameters, called knots, at which a new parameter enters the active set of non-zero coefficients. Significance test for the Lasso by Lockhart et al. (2014), for example, requires solving the knots via…
Dual-sPLS: a family of Dual Sparse Partial Least Squares regressions for feature selection and prediction with tunable sparsity; evaluation on simulated and near-infrared (NIR) data
Université Saint-Joseph（圣约瑟夫大学）；Université Claude Bernard Lyon 1（里昂第一大学）；IFP Energies nouvelles（法国石油与新能源研究院）
This paper deals with the consistency of the least squares estimator of a convex regression function when the predictor is multidimensional. We characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs. Mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided. We also consider a regression function which is known to be convex and componentwise nonincreasing and discuss the characterization, computation and consistency of its least square…
We study the least square estimator, in the framework of simple linear regression, when the deviance term $\varepsilon$ with respect to the linear model is modeled by a uniform distribution. In particular, we give the law of this estimator, and prove some convergence properties.
Traditionally regression analysis answers questions about the relationships among variables based on the assumption that the observation values of variables are precise numbers. It has long been dominated by least squares techniques, mostly due to their elegant theoretical foundation and ease of implementation. However, in many cases, we can only get imprecise observation values and the assumptions upon which the least squares is based may not be valid. So this paper characterizes the imprecise data in terms of uncertain variables and proposes a novel robust approach under the principle of lea…
This paper is focused on dimension-free PAC-Bayesian bounds, under weak polynomial moment assumptions, allowing for heavy tailed sample distributions. It covers the estimation of the mean of a vector or a matrix, with applications to least squares linear regression. Special efforts are devoted to the estimation of Gram matrices, due to their prominent role in high-dimension data analysis.
We present a formula for the shrinkage factors of the Partial Least Squares regression estimator and deduce some of their properties, in particular the known fact that some of the factors are >1. We investigate the effect of shrinkage factors for the Mean Squared error of linear estimators and illustrate that we cannot extend the results to nonlinear estimators. In particular, shrinkage factors >1 do not automatically lead to a poorer Mean Squared Error. We investigate empirically the effect of bounding the the absolute value of the Partial Least Squares shrinkage factors by 1.
The topics dicussed in this paper take their origin inthe estimation of the Gram matrix of a random vector from a sample made of n independent copies. They comprise the estimation of the covariance matrix and the study of least squares regression with a random design. We propose four types of results, based on non-asymptotic PAC-Bayesian generalization bounds: a new robust estimator of the Gram matrix and of the covariance matrix, new results on the empirical Gram matrix, new robust least squares estimators and new results on the ordinary least squares estimator, including its exact rate of co…
Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting. The equations leads to highly heteroscedastic regression. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples maximum likelihood performs best, but it is shown that in smaller samples the weighted regression outperforms maximum likelihood estimation with respect to bias and mean square error.
We consider finite mixtures of generalized linear models with binary output. We prove that cross moment (between the output and the regression variables) until order 3 are sufficient to identify all parameters of the model. We propose a least-squares estimation method based on those moments and we prove the consistency and the Gaussian asymptotic behavior of the estimator. We provide simulation results and comparisons with likelihood methods. Numerical experiments were conducted using the R-package Morpheus that we developed for our least-squares moment method and with the R-package flexmix fo…
If X and Y are real valued random variables such that the first moments of X, Y, and XY exist and the conditional expectation of Y given X is an affine function of X, then the intercept and slope of the conditional expectation equal the intercept and slope of the least squares linear regression function, even though Y may not have a finite second moment. As a consequence, the affine in X form of the conditional expectation and zero covariance imply mean independence.
Ordinary least squares (OLS) linear regression is one of the most basic statistical techniques for data analysis. In the main stream literature and the statistical education, the study of linear regression is typically restricted to the case where the covariates are fixed, errors are mean zero Gaussians with variance independent of the (fixed) covariates. Even though OLS has been studied under misspecification from as early as the 1960's, the implications have not yet caught up with the main stream literature and applied sciences. The present article is an attempt at a unified viewpoint that m…
The performance of Least Squares (LS) estimators is studied in isotonic, unimodal and convex regression. Our results have the form of sharp oracle inequalities that account for the model misspecification error. In isotonic and unimodal regression, the LS estimator achieves the nonparametric rate $n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order $q/n$ up to logarithmic factors, where $q$ is the number of aff…
The asymptotic optimality (a.o.) of various hyper-parameter estimators with different optimality criteria has been studied in the literature for regularized least squares regression problems. The estimators include e.g., the maximum (marginal) likelihood method, $C_p$ statistics, and generalized cross validation method, and the optimality criteria are based on e.g., the inefficiency, the expectation inefficiency and the risk. In this paper, we consider the regularized least squares regression problems with fixed number of regression parameters, choose the optimality criterion based on the risk…
The paper continues the authors' work on the adaptive Wynn algorithm in a nonlinear regression model. In the present paper it is shown that if the mean response function satisfies a condition of `saturated identifiability', which was introduced by Pronzato \cite{Pronzato}, then the adaptive least squares estimators are strongly consistent. The condition states that the regression parameter is identifiable under any saturated design, i.e., the values of the mean response function at any $p$ distinct design points determine the parameter point uniquely where, typically, $p$ is the dimension of t…
The multivariate errors-in-variables regression model is applicable when both dependent and independent variables in a multivariate regression are subject to measurement errors. In such a scenario it is long established that the traditional least squares approach to estimating the model parameters is biased and inconsistent. The generalized least squares, ordinary least squares and maximum likelihood estimators (under the assumption of Gaussian errors) were derived in the seminal paper of Gleser (1981). However, the ordinary least squares and maximum likelihood estimators for the mean vectors …
We consider the estimation of a bounded regression function with nonparametric heteroscedastic noise and random design. We study the true and empirical excess risks of the least-squares estimator on finite-dimensional vector spaces. We give upper and lower bounds on these quantities that are nonasymptotic and optimal to first order, allowing the dimension to depend on sample size. These bounds show the equivalence between the true and empirical excess risks when, among other things, the least-squares estimator is consistent in sup-norm with the projection of the regression function onto the co…
If uncorrelated random variables have a common expected value and decreasing variances then the variance of a sample mean is decreasing with the number of observations. Unfortunately, this natural and desirable Variance Reduction Property (VRP) by augmenting data is not automatically inherited by Ordinary Least Squares (OLS) estimators of parameters. In the paper we find conditions for the OLS to have the VRP. In the case of a straight line regression we show that the OLS estimators of intercept and slope have the VRP if the design points are increasing. This also holds true for alternating tw…
We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the $Z$-component by (Ann. Appl. Probab. 12 (2002) 1390-1418). When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the …
Learning with Fredholm kernel has attracted increasing attention recently since it can effectively utilize the data information to improve the prediction performance. Despite rapid progress on theoretical and experimental evaluations, its generalization analysis has not been explored in learning theory literature. In this paper, we establish the generalization bound of least square regularized regression with Fredholm kernel, which implies that the fast learning rate O(l^{-1}) can be reached under mild capacity conditions. Simulated examples show that this Fredholm regression algorithm can ach…
In this work, we construct a stable and fairly fast estimator for solving non-parametric multidimensional regression problems. The proposed estimator is based on the use of multivariate Jacobi polynomials that generate a basis for a reduced size of $d-$variate finite dimensional polynomial space. An ANOVA decomposition trick has been used for building this later polynomial space. Also, by using some results from the theory of positive definite random matrices, we show that the proposed estimator is stable under the condition that the i.i.d. random sampling points for the different covariates o…
We consider the kernel partial least squares algorithm for non-parametric regression with stationary dependent data. Probabilistic convergence rates of the kernel partial least squares estimator to the true regression function are established under a source and an effective dimensionality condition. It is shown both theoretically and in simulations that long range dependence results in slower convergence rates. A protein dynamics example shows high predictive power of kernel partial least squares.
We consider the problem of choosing between several models in least-squares regression with heteroscedastic data. We prove that any penalization procedure is suboptimal when the penalty is a function of the dimension of the model, at least for some typical heteroscedastic model selection problems. In particular, Mallows' Cp is suboptimal in this framework. On the contrary, optimal model selection is possible with data-driven penalties such as resampling or $V$-fold penalties. Therefore, it is worth estimating the shape of the penalty from data, even at the price of a higher computational cost.…
We prove a new and general concentration inequality for the excess risk in least-squares regression with random design and heteroscedastic noise. No specific structure is required on the model, except the existence of a suitable function that controls the local suprema of the empirical process. So far, only the case of linear contrast estimation was tackled in the literature with this level of generality on the model. We solve here the case of a quadratic contrast, by separating the behavior of a linearized empirical process and the empirical process driven by the squares of functions of model…
In this paper, we focus on regression estimation in both the inductive and the transductive case. We assume that we are given a set of features (which can be a base of functions, but not necessarily). We begin by giving a deviation inequality on the risk of an estimator in every model defined by using a single feature. These models are too simple to be useful by themselves, but we then show how this result motivates an iterative algorithm that performs feature selection in order to build a suitable estimator. We prove that every selected feature actually improves the performance of the estimat…
This note examines the behavior of generalization capabilities - as defined by out-of-sample mean squared error (MSE) - of Linear Gaussian (with a fixed design matrix) and Linear Least Squares regression. Particularly, we consider a well-specified model setting, i.e. we assume that there exists a `true' combination of model parameters within the chosen model form. While the statistical properties of Least Squares regression have been extensively studied over the past few decades - particularly with {\bf less restrictive problem statements} compared to the present work - this note targets bound…
We derive sharp performance bounds for least squares regression with $L_1$ regularization from parameter estimation accuracy and feature selection quality perspectives. The main result proved for $L_1$ regularization extends a similar result in [Ann. Statist. 35 (2007) 2313--2351] for the Dantzig selector. It gives an affirmative answer to an open question in [Ann. Statist. 35 (2007) 2358--2364]. Moreover, the result leads to an extended view of feature selection that allows less restrictive conditions than some recent work. Based on the theoretical insights, a novel two-stage $L_1$-regulariza…
A simple construction of adaptive confidence sets is proposed in isotonic, convex and unimodal regression. In univariate isotonic regression, the proposed confidence set enjoys uniform coverage over all non-decreasing regression functions. Furthermore, the diameter of the proposed confidence set automatically adapts to the unknown number of pieces of the true parameter, in the sense that the diameter is bounded from above by the minimax risk over the class of $k$-piecewise constant functions. The diameter of the confidence set is a simple increasing function of the number of jumps of the isoto…
This paper deals with the problem of estimating a slope parameter in a simple linear regression model, where independent variables have functional measurement errors. Measurement errors in independent variables, as is well known, cause biasedness of the ordinary least squares estimator. A general procedure for the bias reduction is presented in a finite sample situation, and some exact bias-reduced estimators are proposed. Also, it is shown that certain truncation procedures improve the mean square errors of the ordinary least squares and the bias-reduced estimators.
We consider the problem of robustly predicting as well as the best linear combination of $d$ given functions in least squares regression, and variants of this problem including constraints on the parameters of the linear combination. For the ridge estimator and the ordinary least squares estimator, and their variants, we provide new risk bounds of order $d/n$ without logarithmic factor unlike some standard results, where $n$ is the size of the training data. We also provide a new estimator with better deviations in the presence of heavy-tailed noise. It is based on truncating differences of lo…
Penalization procedures often suffer from their dependence on multiplying factors, whose optimal values are either unknown or hard to estimate from the data. We propose a completely data-driven calibration algorithm for this parameter in the least-squares regression framework, without assuming a particular shape for the penalty. Our algorithm relies on the concept of minimal penalty, recently introduced by Birge and Massart (2007) in the context of penalized least squares for Gaussian homoscedastic regression. On the positive side, the minimal penalty can be evaluated from the data themselves,…
We consider a resampling scheme for parameters estimates in nonlinear regression models. We provide an estimation procedure which recycles, via random weighting, the relevant parameters estimates to construct consistent estimates of the sampling distribution of the various estimates. We establish the asymptotic normality of the resampled estimates and demonstrate the applicability of the recycling approach in a small simulation study and via example.
In this article we study the asymptotic behaviour of the least square estimator in a linear regression model based on random observation instances. We provide mild assumptions on the moments and dependence structure on the randomly spaced observations and the residuals under which the estimator is strongly consistent. In particular, we consider observation instances that are negatively superadditive dependent within each other, while for the residuals we merely assume that they are generated by some continuous function. In addition, we prove that the rate of convergence is proportional to the …
We study the statistical properties of the least squares estimator in unimodal sequence estimation. Although closely related to isotonic regression, unimodal regression has not been as extensively studied. We show that the unimodal least squares estimator is adaptive in the sense that the risk scales as a function of the number of values in the true underlying sequence. Such adaptivity properties have been shown for isotonic regression by Chatterjee et al(2015) and Bellec(2015). A technical complication in unimodal regression is the non-convexity of the underlying parameter space. We develop a…
In linear models, the generalized least squares (GLS) estimator is applicable when the structure of the error dependence is known. When it is unknown, such structure must be approximated and estimated in a manner that may lead to misspecification. The large-sample analysis of incorrectly-specified GLS (IGLS) estimators requires careful asymptotic manipulations. When performing estimation in the frequency domain, the asymptotic normality of the IGLS estimator, under the so-called Grenander assumptions, has been proved for a broad class of error dependence models. Under the same assumptions, asy…
This note develops a simple two-stage least squares (2SLS) procedure to estimate the causal effect of some endogenous regressors on a randomly right censored outcome in the linear model. The proposal replaces the usual ordinary least squares regressions of the standard 2SLS by weighted least squares regressions. The weights correspond to the inverse probability of censoring. We show consistency and asymptotic normality of the estimator. The estimator exhibits good finite sample performances in simulations.
