  Screening and working set techniques are important approaches to reducing the
size of an optimization problem. They have been widely used in accelerating
first-order methods for solving large-scale sparse learning problems. In this
paper, we develop a new screening method called Newton screening (NS) which is
a generalized Newton method with a built-in screening mechanism. We derive an
equivalent KKT system for the Lasso and utilize a generalized Newton method to
solve the KKT equations. Based on this KKT system, a built-in working set with
a relatively small size is first determined using the sum of primal and dual
variables generated from the previous iteration, then the primal variable is
updated by solving a least-squares problem on the working set and the dual
variable updated based on a closed-form expression. Moreover, we consider a
sequential version of Newton screening (SNS) with a warm-start strategy. We
show that NS possesses an optimal convergence property in the sense that it
achieves one-step local convergence. Under certain regularity conditions on the
feature matrix, we show that SNS hits a solution with the same signs as the
underlying true target and achieves a sharp estimation error bound with high
probability. Simulation studies and real data analysis support our theoretical
results and demonstrate that SNS is faster and more accurate than several
state-of-the-art methods in our comparative studies.
