Linearized Bregman for l 1-regularized Logistic Regression

Sparse logistic regression is an important linear classifier in statistical learning, providing an attractive route for feature selection. A popular approach is based on minimizing an l1-regularization term with a regularization parameter λ that affects the solution sparsity. To determine an appropriate value for the regularization parameter, one can apply the grid search method or the Bayesian approach. The grid search method requires constructing a regularization path, by solving a sequence of minimization problems with varying values of the regularization parameter, which is typically time consuming. In this paper, we introduce a fast procedure that generates a new regularization path without tuning the regularization parameter. We first derive the direct Bregman method by replacing the l1-norm by Bregman divergence, and contrast it with the grid search method. For faster path computation, we further derive the linearized Bregman algorithm, which is algebraically simple and computationally efficient. Finally we demonstrate some empirical results for the linearized Bregman algorithm on benchmark data and study feature selection as an inverse problem. Compared with the grid search method, the linearized Bregman algorithm generates a different regularization path with comparable classification performance, in a much more computationally efficient manner. Proceedings of the 30 th International Conference on Machine Learning, Atlanta, Georgia, USA, 2013. JMLR: W&CP volume 28. Copyright 2013 by the author(s).