Efficient Approximate Solution Path Algorithm for Order Weight L_1Norm with Accuracy Guarantee Variable selection is a challenging problem in highdimensional linear regression problems with a large number of predictors. Thus, sparsityinducing and clusteringinducing regularization methods are widely used to identify highly correlated covariates. Ordered Weight L_1 (OWL) family of regularizers for linear regression perform well to identify precise clusters of correlated covariates and interpret the effect of each variable. Solution path algorithms are helpful to select hyperparameters to tune the OWL model. Due to overcomplex representation of the penalty, so far the OWL model has no solution path algorithms for hyperparameter selection. To address this challenge, in this paper, we propose an efficient approximate solution path algorithm (OWLAGPath) to solve the OWL model with accuracy guarantee. For a given accuracy bound e, OWLAGPath can find the corresponding solutions for the OWL model with numerous hyperparameters while keeping the sparsity and precise features grouping properties. Theoretically, we prove that all the solutions produced by OWLAGPath can strictly satisfy the given accuracy bound e. The experimental results on three benchmark datasets not only confirm the effectiveness and efficiency of our OWLAGPath algorithm, but also show the advantages of OWLAGPath for model selection than the existing algorithms.

Selection Consistency of LassoBased Procedures for Misspecified HighDimensional Binary Model and Random Regressors We consider selection of random predictors for a highdimensional regression problem with a binary response for a general loss function. An important special case is when the binary model is semiparametric and the response function is misspecified under a parametric model fit. When the true response coincides with a postulated parametric response for a certain value of parameter, we obtain a common framework for parametric inference. Both cases of correct specification and misspecification are covered in this contribution. Variable selection for such a scenario aims at recovering the support of the minimizer of the associated risk with large probability. We propose a twostep selection ScreeningSelection (SS) procedure which consists of screening and ordering predictors by Lasso method and then selecting the subset of predictors which minimizes the Generalized Information Criterion for the corresponding nested family of models. We prove consistency of the proposed selection method under conditions that allow for a much larger number of predictors than the number of observations. For the semiparametric case when distribution of random predictors satisfies linear regressions condition, the true and the estimated parameters are collinear and their common support can be consistently identified. This partly explains robustness of selection procedures to the response function misspecification.