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.

本文研究了基于 Hessian 矩阵近似的非凸优化中信任域和立方正则化方法的变体。通过对不精确 Hessian 矩阵的渐进解和相应子问题的近似解，提供了迭代复杂度，以实现达到二阶最优条件的近似解，并且在现有文献中条件松弛。