  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, Newton-CG, 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.
