On Newton Screening

• URL: http://arxiv.org/abs/2001.10616v3
• Date: Fri, 21 Apr 2023 04:52:05 GMT
• Title: On Newton Screening
• Authors: Jian Huang, Yuling Jiao, Lican Kang, Jin Liu, Yanyan Liu, Xiliang Lu, and Yuanyuan Yang
• Abstract summary: We develop a new screening method called Newton screening (NS) which is a generalized Newton method with a built-in screening mechanism. We show that NS possesses an optimal convergence property in the sense that it achieves one-step local convergence.
• Score: 14.040371216692645
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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