An investigation of Newton-Sketch and subsampled Newton methods

• Published in: Optimization methods & software Vol. 35; no. 4; pp. 661 - 680
• Main Authors: Berahas, Albert S., Bollapragada, Raghu, Nocedal, Jorge
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 03.07.2020; Taylor & Francis Ltd
• Subjects: Approximation; Conjugate gradient method; Data points; Iterative methods; Linear systems; machine learning; MATHEMATICS AND COMPUTING; Newton methods; Newton's method; Optimization; Reduction; Sketching; stochastic optimization; subsampling
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2020.1725751

Sketching, a dimensionality reduction technique, has received much attention in the statistics community. In this paper, we study sketching in the context of Newton's method for solving finite-sum optimization problems in which the number of variables and data points are both large. We study two forms of sketching that perform dimensionality reduction in data space: Hessian subsampling and randomized Hadamard transformations. Each has its own advantages, and their relative tradeoffs have not been investigated in the optimization literature. Our study focuses on practical versions of the two methods in which the resulting linear systems of equations are solved approximately, at every iteration, using an iterative solver. The advantages of using the conjugate gradient method vs. a stochastic gradient iteration are revealed through a set of numerical experiments, and a complexity analysis of the Hessian subsampling method is presented.