Transductive versions of the LASSO and the Dantzig Selector

We consider the linear regression problem, where the number $p$ of covariates is possibly larger than the number $n$ of observations $(x_{i},y_{i})_{i\leq i \leq n}$, under sparsity assumptions. On the one hand, several methods have been successfully proposed to perform this task, for example the LASSO or the Dantzig Selector. On the other hand, consider new values $(x_{i})_{n+1\leq i \leq m}$. If one wants to estimate the corresponding $y_{i}$'s, one should think of a specific estimator devoted to this task, referred by Vapnik as a "transductive" estimator. This estimator may differ from an estimator designed to the more general task "estimate on the whole domain". In this work, we propose a generalized version both of the LASSO and the Dantzig Selector, based on the geometrical remarks about the LASSO in pr\'evious works. The "usual" LASSO and Dantzig Selector, as well as new estimators interpreted as transductive versions of the LASSO, appear as special cases. These estimators are interesting at least from a theoretical point of view: we can give theoretical guarantees for these estimators under hypotheses that are relaxed versions of the hypotheses required in the papers about the "usual" LASSO. These estimators can also be efficiently computed, with results comparable to the ones of the LASSO

On Lasso refitting strategies

A well-know drawback of l_1-penalized estimators is the systematic shrinkage of the large coefficients towards zero. A simple remedy is to treat Lasso as a model-selection procedure and to perform a second refitting step on the selected support. In this work we formalize the notion of refitting and provide oracle bounds for arbitrary refitting procedures of the Lasso solution. One of the most widely used refitting techniques which is based on Least-Squares may bring a problem of interpretability, since the signs of the refitted estimator might be flipped with respect to the original estimator. This problem arises from the fact that the Least-Squares refitting considers only the support of the Lasso solution, avoiding any information about signs or amplitudes. To this end we define a sign consistent refitting as an arbitrary refitting procedure, preserving the signs of the first step Lasso solution and provide Oracle inequalities for such estimators. Finally, we consider special refitting strategies: Bregman Lasso and Boosted Lasso. Bregman Lasso has a fruitful property to converge to the Sign-Least-Squares refitting (Least-Squares with sign constraints), which provides with greater interpretability. We additionally study the Bregman Lasso refitting in the case of orthogonal design, providing with simple intuition behind the proposed method. Boosted Lasso, in contrast, considers information about magnitudes of the first Lasso step and allows to develop better oracle rates for prediction. Finally, we conduct an extensive numerical study to show advantages of one approach over others in different synthetic and semi-real scenarios.Comment: revised versio

Safe Screening With Variational Inequalities and Its Application to LASSO

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 suitable model. In this paper, we propose an approach called "Sasvi" (Safe screening with variational inequalities). Sasvi makes use of the variational inequality that provides the sufficient and necessary optimality condition for the dual problem. Several existing approaches for Lasso screening can be casted as relaxed versions of the proposed Sasvi, thus Sasvi provides a stronger safe screening rule. We further study the monotone properties of Sasvi for Lasso, based on which a sure removal regularization parameter can be identified for each feature. Experimental results on both synthetic and real data sets are reported to demonstrate the effectiveness of the proposed Sasvi for Lasso screening.Comment: Accepted by International Conference on Machine Learning 201

Improved variable selection with Forward-Lasso adaptive shrinkage

Recently, considerable interest has focused on variable selection methods in regression situations where the number of predictors, $p$, is large relative to the number of observations, $n$. Two commonly applied variable selection approaches are the Lasso, which computes highly shrunk regression coefficients, and Forward Selection, which uses no shrinkage. We propose a new approach, "Forward-Lasso Adaptive SHrinkage" (FLASH), which includes the Lasso and Forward Selection as special cases, and can be used in both the linear regression and the Generalized Linear Model domains. As with the Lasso and Forward Selection, FLASH iteratively adds one variable to the model in a hierarchical fashion but, unlike these methods, at each step adjusts the level of shrinkage so as to optimize the selection of the next variable. We first present FLASH in the linear regression setting and show that it can be fitted using a variant of the computationally efficient LARS algorithm. Then, we extend FLASH to the GLM domain and demonstrate, through numerous simulations and real world data sets, as well as some theoretical analysis, that FLASH generally outperforms many competing approaches.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS375 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org