A Method for Large-Scale L1-Regularized Logistic Regression

Abstract

Logistic regression with l1 regularization has been proposed as a promising method for feature selection in classification problems. Several specialized solution methods have been proposed for l1-regularized logistic regression problems (LRPs). However, existing methods do not scale well to large problems that arise in many practical settings. In this paper we describe an efficient interior-point method for solving l1-regularized LRPs. Small problems with up to a thousand or so features and examples can be solved in seconds on a PC. A variation on the basic method, that uses a preconditioned conjugate gradient method to compute the search step, can solve large sparse problems, with a million features and examples (e.g., the 20 Newsgroups data set), in a few tens of minutes, on a PC. Numerical experiments show that our method outperforms standard methods for solving convex optimization problems as well as other methods specifically designed for l1regularized LRPs. Introduction Logistic regression Let x ∈ R denote a vector of feature variables, and b ∈ −1,+1 denote the associated binary output. In the logistic model, the conditional probability of b, given x, has the form Prob(b|x) = 1/(1 + exp (