A Reduced-Space Algorithm for Minimizing $\ell_1$-Regularized Convex Functions

Abstract

We present a new method for minimizing the sum of a differentiable convex function and an $\ell_1$-norm regularizer. The main features of the new method include: (i) an evolving set of indices corresponding to variables that are predicted to be nonzero at a solution (i.e., the support); (ii) a reduced-space subproblem defined in terms of the predicted support; (iii) conditions that determine how accurately each subproblem must be solved, which allow for Newton, linear conjugate gradient, and coordinate-descent techniques to be employed; (iv) a computationally practical condition that determines when the predicted support should be updated; and (v) a reduced proximal gradient step that ensures sufficient decrease in the objective function when it is decided that variables should be added to the predicted support. We prove a convergence guarantee for our method and demonstrate its efficiency on a large set of model prediction problems.