  Large scale optimization problems are ubiquitous in machine learning and data
analysis and there is a plethora of algorithms for solving such problems. Many
of these algorithms employ sub-sampling, as a way to either speed up the
computations and/or to implicitly implement a form of statistical
regularization. In this paper, we consider second-order iterative optimization
algorithms and we provide bounds on the convergence of the variants of Newton's
method that incorporate uniform sub-sampling as a means to estimate the
gradient and/or Hessian. Our bounds are non-asymptotic and quantitative. Our
algorithms are global and are guaranteed to converge from any initial iterate.
  Using random matrix concentration inequalities, one can sub-sample the
Hessian to preserve the curvature information. Our first algorithm incorporates
Hessian sub-sampling while using the full gradient. We also give additional
convergence results for when the sub-sampled Hessian is regularized by
modifying its spectrum or ridge-type regularization. Next, in addition to
Hessian sub-sampling, we also consider sub-sampling the gradient as a way to
further reduce the computational complexity per iteration. We use approximate
matrix multiplication results from randomized numerical linear algebra to
obtain the proper sampling strategy. In all these algorithms, computing the
update boils down to solving a large scale linear system, which can be
computationally expensive. As a remedy, for all of our algorithms, we also give
global convergence results for the case of inexact updates where such linear
system is solved only approximately.
  This paper has a more advanced companion paper, [42], in which we demonstrate
that, by doing a finer-grained analysis, we can get problem-independent bounds
for local convergence of these algorithms and explore trade-offs to improve
upon the basic results of the present paper.
