  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.
