We present a statistical perspective on boosting. Special emphasis is given to estimating potentially complex parametric or nonparametric models, including generalized linear and additive models as well as regression models for survival analysis. Concepts of degrees of freedom and corresponding Akaike or Bayesian information criteria, particularly useful for regularization and variable selection in high-dimensional covariate spaces, are discussed as well. The practical aspects of boosting procedures for fitting statistical models are illustrated by means of the dedicated open-source software package mboost. This package implements functions which can be used for model fitting, prediction and variable selection. It is flexible, allowing for the implementation of new boosting algorithms optimizing user-specified loss functions.

Proofs subject to correction. Not to be reproduced without permission. Contributions to the discussion must not exceed 400 words. Contributions longer than 400 words will be cut by the editor. 1 2

We consider the fundamental problem of estimating the mean of a vector y = Xβ + z, where X is an n × p design matrix in which one can have far more variables than observations and z is a stochastic error term—the so-called ‘p> n ’ setup. When β is sparse, or more generally, when there is a sparse subset of covariates providing a close approximation to the unknown mean vector, we ask whether or not it is possible to accurately estimate Xβ using a computationally tractable algorithm. We show that in a surprisingly wide range of situations, the lasso happens to nearly select the best subset of variables. Quantitatively speaking, we prove that solving a simple quadratic program achieves a squared error within a logarithmic factor of the ideal mean squared error one would achieve with an oracle supplying perfect information about which variables should be included in the model and which variables should not. Interestingly, our results describe the average performance of the lasso; that is, the performance one can expect in an vast majority of cases where Xβ is a sparse or nearly sparse superposition of variables, but not in all cases. Our results are nonasymptotic and widely applicable since they simply require that pairs of predictor variables are not too collinear.

Summary. We consider a nonparametric additive model of a conditional mean function in which the number of variables and additive components may be larger than the sample size but the number of non-zero additive components is “small” relative to the sample size. The statistical problem is to determine which additive components are non-zero. The additive components are approximated by truncated series expansions with B-spline bases. With this approximation, the problem of component selection becomes that of selecting the groups of coefficients in the expansion. We apply the adaptive group Lasso to select nonzero components, using the group Lasso to obtain an initial estimator and reduce the dimension of the problem. We give conditions under which the group Lasso selects a model whose number of components is comparable with the underlying model and, the adaptive group Lasso selects the non-zero components correctly with probability approaching one as the sample size increases and achieves the optimal rate of convergence. Following model selection, oracle-efficient, asymptotically normal estimators of the non-zero components can be obtained by using existing methods. The results of Monte Carlo experiments show that the adaptive group Lasso procedure works well with samples of moderate size. A data example is used to illustrate the application of the proposed method. Key words and phrases. Adaptive group Lasso; component selection; highdimensional data; nonparametric regression; selection consistency. Short title. Nonparametric component selection AMS 2000 subject classification. Primary 62G08, 62G20; secondary 62G99 1

We address the problem of sparse selection in linear models. A number of non-convex penalties have been proposed for this purpose, along with a variety of convex-relaxation algorithms for finding good solutions. In this paper we pursue the coordinate-descent approach for optimization, and study its convergence properties. We characterize the properties of penalties suitable for this approach, study their corresponding threshold functions, and describe a df-standardizing reparametrization that assists our pathwise algorithm. The MC+ penalty (Zhang 2010) is ideally suited to this task, and we use it to demonstrate the performance of our algorithm. 1

Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder (2008) which splits the data into two parts. The number of variables is then reduced to a manageable size using the first split, while classical variable selection techniques can be applied to the remaining variables, using the data from the second split. This yields asymptotic error control under minimal conditions. It involves, however, a one-time random split of the data. Results are sensitive to this arbitrary choice: it amounts to a “p-value lottery ” and makes it difficult to reproduce results. Here, we show that inference across multiple random splits can be aggregated, while keeping asymptotic control over the inclusion of noise variables. In addition, the proposed aggregation is shown to improve power, while reducing the number of falsely selected variables substantially. Keywords: High-dimensional variable selection, data splitting, multiple comparisons. 1

Directed acyclic graphs (DAGs) are commonly used to represent causal relationships among random variables in graphical models. Applications of these models arise in the study of physical, as well as biological systems, where directed edges between nodes represent the influence of components of the system on each other. The general problem of estimating DAGs from observed data is computationally NP-hard, Moreover two directed graphs may be observationally equivalent. When the nodes exhibit a natural ordering, the problem of estimating directed graphs reduces to the problem of estimating the structure of the network. In this paper, we propose a penalized likelihood approach that directly estimates the adjacency matrix of DAGs. Both lasso and adaptive lasso penalties are considered and an efficient algorithm is proposed for estimation of high dimensional DAGs. We study variable selection consistency of the two penalties when the number of variables grows to infinity with the sample size. We show that although lasso can only consistently estimate the true network under stringent assumptions, adaptive lasso achieves this task under mild regularity conditions. The performance of the proposed methods are compared to alternative methods in simulated, as well as real, data examples. 1

Abstract: When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use ℓ1-penalized estimators like the Lasso or the Adaptive Lasso which can simultaneously do parameter estimation and model selection. We show that for a time-course of high-dimensional linear models the convergence rates of the Lasso and of the Adaptive Lasso can be improved by combining the different time-points in a suitable way. Moreover, the Adaptive Lasso still enjoys oracle properties and consistent variable selection. The finite sample properties of the proposed methods are illustrated on simulated data and on a real problem of motif finding in DNA sequences.

Methods for ℓ1-type regularization have been widely used in Gaussian graphical model selection tasks to encourage sparse structures. However, often we would like to include more structural information than mere sparsity. In this work, we focus on learning so-called “scale-free ” models, a common feature that appears in many real-work networks. We replace the ℓ1 regularization with a power law regularization and optimize the objective function by a sequence of iteratively reweighted ℓ1 regularization problems, where the regularization coefficients of nodes with high degree are reduced, encouraging the appearance of hubs with high degree. Our method can be easily adapted to improve any existing ℓ1-based methods, such as graphical lasso, neighborhood selection, and JSRM when the underlying networks are believed to be scale free or have dominating hubs. We demonstrate in simulation that our method significantly outperforms the a baseline ℓ1 method at learning scale-free networks and hub networks, and also illustrate its behavior on gene expression data. 1

Abstract: 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 (L1-penalized