Exact and Inexact Subsampled Newton Methods for Optimization
This work addresses optimization efficiency for machine learning practitioners, but it is incremental as it builds on existing subsampling and Newton methods.
The paper tackles stochastic optimization by using subsampled approximations of the gradient and Hessian, proposing Newton-like methods that achieve superlinear convergence in expectation and analyzing an inexact Newton method with conjugate gradient for solving linear systems, with preliminary numerical results on logistic regression applications.
The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of the paper analyzes an inexact Newton method that solves linear systems approximately using the conjugate gradient (CG) method, and that samples the Hessian and not the gradient (the gradient is assumed to be exact). We provide a complexity analysis for this method based on the properties of the CG iteration and the quality of the Hessian approximation, and compare it with a method that employs a stochastic gradient iteration instead of the CG method. We report preliminary numerical results that illustrate the performance of inexact subsampled Newton methods on machine learning applications based on logistic regression.