SNAP: A semismooth Newton algorithm for pathwise optimization with optimal local convergence rate and oracle properties
This work addresses computational bottlenecks in sparse regression for statisticians and data scientists, offering incremental improvements in speed and accuracy over established algorithms.
The authors tackled the problem of efficiently solving LASSO and Elastic Net optimization in high-dimensional sparse regression by proposing SNAP, a semismooth Newton algorithm that achieves superlinear or optimal local convergence rates, with simulation studies showing it is faster and more accurate than existing methods like LARS and coordinate descent.
We propose a semismooth Newton algorithm for pathwise optimization (SNAP) for the LASSO and Enet in sparse, high-dimensional linear regression. SNAP is derived from a suitable formulation of the KKT conditions based on Newton derivatives. It solves the semismooth KKT equations efficiently by actively and continuously seeking the support of the regression coefficients along the solution path with warm start. At each knot in the path, SNAP converges locally superlinearly for the Enet criterion and achieves an optimal local convergence rate for the LASSO criterion, i.e., SNAP converges in one step at the cost of two matrix-vector multiplication per iteration. Under certain regularity conditions on the design matrix and the minimum magnitude of the nonzero elements of the target regression coefficients, we show that SNAP hits a solution with the same signs as the regression coefficients and achieves a sharp estimation error bound in finite steps with high probability. The computational complexity of SNAP is shown to be the same as that of LARS and coordinate descent algorithms per iteration. Simulation studies and real data analysis support our theoretical results and demonstrate that SNAP is faster and accurate than LARS and coordinate descent algorithms.