Finite-sample and asymptotic analysis of generalization ability with an application to penalized regression

In this paper, we study the generalization ability (GA)---the ability of a model to predict outcomes in new samples from the same population---of the extremum estimators. By adapting the classical concentration inequalities, we propose upper bounds for the empirical out-of-sample prediction error for extremum estimators, which is a function of the in-sample error, the severity of heavy tails, the sample size of in-sample data and model complexity. The error bounds not only serve to measure GA, but also to illustrate the trade-off between in-sample and out-of-sample fit, which is connected to the traditional bias-variance trade-off. Moreover, the bounds also reveal that the hyperparameter K, the number of folds in $K$-fold cross-validation, cause the bias-variance trade-off for cross-validation error, which offers a route to hyperparameter optimization in terms of GA. As a direct application of GA analysis, we implement the new upper bounds in penalized regression estimates for both n>p and n<p cases. We show that the L2 norm difference between penalized and un-penalized regression estimates can be directly explained by the GA of the regression estimates and the GA of empirical moment conditions. Finally, we show that all the penalized regression estimates are L2 consistent based on GA analysis

Support vector machine for functional data classification

In many applications, input data are sampled functions taking their values in infinite dimensional spaces rather than standard vectors. This fact has complex consequences on data analysis algorithms that motivate modifications of them. In fact most of the traditional data analysis tools for regression, classification and clustering have been adapted to functional inputs under the general name of functional Data Analysis (FDA). In this paper, we investigate the use of Support Vector Machines (SVMs) for functional data analysis and we focus on the problem of curves discrimination. SVMs are large margin classifier tools based on implicit non linear mappings of the considered data into high dimensional spaces thanks to kernels. We show how to define simple kernels that take into account the unctional nature of the data and lead to consistent classification. Experiments conducted on real world data emphasize the benefit of taking into account some functional aspects of the problems.Comment: 13 page

Regularization and Model Selection with Categorial Predictors and Effect Modifiers in Generalized Linear Models

Varying-coefficient models with categorical effect modifiers are considered within the framework of generalized linear models. We distinguish between nominal and ordinal effect modifiers, and propose adequate Lasso-type regularization techniques that allow for (1) selection of relevant covariates, and (2) identification of coefficient functions that are actually varying with the level of a potentially effect modifying factor. We investigate large sample properties, and show in simulation studies that the proposed approaches perform very well for finite samples, too. In addition, the presented methods are compared with alternative procedures, and applied to real-world medical data

Pivotal estimation via square-root Lasso in nonparametric regression

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 and sparsity. Our analysis is based on new impact factors that are tailored for bounding prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely on moderate deviation theory for self-normalized sums to achieve Gaussian-like results under weak conditions. Moreover, we derive bounds on the performance of ordinary least square (ols) applied to the model selected by $\sqrt{\mathrm {Lasso}}$ accounting for possible misspecification of the selected model. Under mild conditions, the rate of convergence of ols post $\sqrt{\mathrm {Lasso}}$ is as good as $\sqrt{\mathrm {Lasso}}$'s rate. As an application, we consider the use of $\sqrt{\mathrm {Lasso}}$ and ols post $\sqrt{\mathrm {Lasso}}$ as estimators of nuisance parameters in a generic semiparametric problem (nonlinear moment condition or $Z$-problem), resulting in a construction of $\sqrt{n}$-consistent and asymptotically normal estimators of the main parameters.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1204 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fixed effects selection in the linear mixed-effects model using adaptive ridge procedure for L0 penalty performance

This paper is concerned with the selection of fixed effects along with the estimation of fixed effects, random effects and variance components in the linear mixed-effects model. We introduce a selection procedure based on an adaptive ridge (AR) penalty of the profiled likelihood, where the covariance matrix of the random effects is Cholesky factorized. This selection procedure is intended to both low and high-dimensional settings where the number of fixed effects is allowed to grow exponentially with the total sample size, yielding technical difficulties due to the non-convex optimization problem induced by L0 penalties. Through extensive simulation studies, the procedure is compared to the LASSO selection and appears to enjoy the model selection consistency as well as the estimation consistency