  This paper deals with the minimization of large sum of convex functions by
Inexact Newton (IN) methods employing subsampled functions, gradients and
Hessian approximations. The Conjugate Gradient method is used to compute the
inexact Newton step and global convergence is enforced by a nonmonotone line
search procedure. The aim is to obtain methods with affordable costs and fast
convergence. Assuming strongly convex functions, R-linear convergence and
worst-case iteration complexity of the procedure are investigated when
functions and gradients are approximated with increasing accuracy. A set of
rules for the forcing parameters and subsample Hessian sizes are derived that
ensure local q-linear/superlinear convergence of the proposed method.
  The random choice of the Hessian subsample is also considered and convergence
in the mean square, both for finite and infinite sums of functions, is proved.
Finally, global convergence with asymptotic R-linear rate of IN methods is
extended to the case of sum of convex function and strongly convex objective
function. Numerical results on well known binary classification problems are
also given. Adaptive strategies for selecting forcing terms and Hessian
subsample size, streaming out of the theoretical analysis, are employed and the
numerical results showed that they yield effective IN methods.
