Title: Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information,
Abstract: We consider variants of trust-region and cubic regularization methods for
non-convex optimization, in which the Hessian matrix is approximated. Under
mild conditions on the inexact Hessian, and using approximate solution of the
corresponding sub-problems, we provide iteration complexity to achieve $
\epsilon $-approximate second-order optimality which have shown to be tight.
Our Hessian approximation conditions constitute a major relaxation over the
existing ones in the literature. Consequently, we are able to show that such
mild conditions allow for the construction of the approximate Hessian through
various random sampling methods. In this light, we consider the canonical
problem of finite-sum minimization, provide appropriate uniform and non-uniform
sub-sampling strategies to construct such Hessian approximations, and obtain
optimal iteration complexity for the corresponding sub-sampled trust-region and
cubic regularization methods.