  We study stochastic inexact Newton methods and consider their application in
nonconvex settings. Building on the work of [R. Bollapragada, R. H. Byrd, and
J. Nocedal, IMA Journal of Numerical
  Analysis, 39 (2018), pp. 545--578] we derive bounds for convergence rates in
expected value for stochastic low rank Newton methods, and stochastic inexact
Newton Krylov methods. These bounds quantify the errors incurred in subsampling
the Hessian and gradient, as well as in approximating the Newton linear solve,
and in choosing regularization and step length parameters. We deploy these
methods in training convolutional autoencoders for the MNIST and CIFAR10 data
sets. Numerical results demonstrate that, relative to first order methods,
these stochastic inexact Newton methods often converge faster, are more
cost-effective, and generalize better.
