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We propose a semismooth Newton algorithm for pathwise optimization (SNAP) forthe LASSO and Enet in sparse, high-dimensional linear regression. SNAP isderived from a suitable formulation of the KKT conditions based on Newtonderivatives. It solves the semismooth KKT equations efficiently by actively andcontinuously seeking the support of the regression coefficients along thesolution path with warm start. At each knot in the path, SNAP converges locallysuperlinearly for the Enet criterion and achieves an optimal local convergencerate for the LASSO criterion, i.e., SNAP converges in one step at the cost oftwo matrix-vector multiplication per iteration. Under certain regularityconditions on the design matrix and the minimum magnitude of the nonzeroelements of the target regression coefficients, we show that SNAP hits asolution with the same signs as the regression coefficients and achieves asharp estimation error bound in finite steps with high probability. Thecomputational complexity of SNAP is shown to be the same as that of LARS andcoordinate descent algorithms per iteration. Simulation studies and real dataanalysis support our theoretical results and demonstrate that SNAP is fasterand accurate than LARS and coordinate descent algorithms.

SNAP: A semismooth Newton algorithm for pathwise optimization with optimal local convergence rate and oracle properties