Stagewise Lasso

Many statistical machine learning algorithms minimize either an empirical loss function as in AdaBoost, or a penalized empirical loss as in Lasso or SVM. A single regularization tuning parameter controls the trade-off between fidelity to the data and generalizability, or equivalently between bias and variance. When this tuning parameter changes, a regularization “path” of solutions to the minimization problem is generated, and the whole path is needed to select a tuning parameter to optimize the prediction or interpretation performance. Algorithms such as homotopy-Lasso or LARS-Lasso and Forward Stagewise Fitting (FSF) (aka e-Boosting) are of great interest because of their resulted sparse models for interpretation in addition to prediction. In this paper, we propose the BLasso algorithm that ties the FSF (e-Boosting) algorithm with the Lasso method that minimizes the L1 penalized L2 loss. BLasso is derived as a coordinate descent method with a fixed stepsize applied to the general Lasso loss function (L1 penalized convex loss). It consists of both a forward step and a backward step. The forward step is similar to e-Boosting or FSF, but the backward step is new and revises the FSF (or e-Boosting) path to approximate the Lasso path. In the cases of a finite number of base learners and a bounded Hessian of the loss function, the BLasso path is shown to converge to the Lasso path when the stepsize goes to zero. For cases with a larger number of base learners than the sample size and when the true model is sparse, our simulations indicate that the BLasso model estimates are sparser than those from FSF with comparable or slightly better prediction performance, and that the the discrete stepsize of BLasso and FSF has an additional regularization effect in terms of prediction and sparsity. Moreover, we introduce the Generalized BLasso algorithm to minimize a general convex loss penalized by a general convex function. Since the (Generalized) BLasso relies only on differences not derivatives, we conclude that it provides a class of simple and easy-to-implement algorithms for tracing the regularization or solution paths of penalized minimization problems.