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In recent years, a rich variety of regularization procedures have beenproposed for high dimensional regression problems. However, tuning parameterchoice and computational efficiency in ultra-high dimensional problems remainvexing issues. The routine use of $\ell_1$ regularization is largelyattributable to the computational efficiency of the LARS algorithm, but similarefficiency for better behaved penalties has remained elusive. In this article,we propose a highly efficient path following procedure for combination of anyconvex loss function and a broad class of penalties. From a Bayesianperspective, this algorithm rapidly yields maximum a posteriori estimates atdifferent hyper-parameter values. To bypass the inefficiency and potentialinstability of cross validation, we propose an empirical Bayes procedure forrapidly choosing the optimal model and corresponding hyper-parameter value.This approach applies to any penalty that corresponds to a proper priordistribution on the regression coefficients. While we mainly focus on sparseestimation of generalized linear models, the method extends to more generalregularizations such as polynomial trend filtering after reparameterization.The proposed algorithm scales efficiently to large $p$ and/or $n$. Solutionpaths of 10,000 dimensional examples are computed within one minute on a laptopfor various generalized linear models (GLM). Operating characteristics areassessed through simulation studies and the methods are applied to several realdata sets.

Path Following and Empirical Bayes Model Selection for Sparse Regression