We consider minimization of a smooth nonconvex function with inexact oracle
access to gradient and Hessian (but not the function value) to achieve
$(\epsilon_{g}, \epsilon_{H})$-approximate second-order optimality. A novel
feature of our method is that if an approximate direction of negative curvature
is chosen as the step, we choose its sense to be positive or negative with
equal probability. We also use relative inexactness measures on gradient and
Hessian and relax the coupling between the first- and second-order tolerances
$\epsilon_{g}$ and $\epsilon_{H}$. Our convergence analysis includes both an
expectation bound based on martingale analysis and a high-probability bound
based on concentration inequalities. We apply our algorithm to empirical risk
minimization problems and obtain gradient sample complexity.

非凸函数的最小化，利用近似正负曲率方向步长，相对不精确度度量梯度和 Hessian 矩阵，松弛一阶和二阶精度的耦合，通过马丁格尔分析和浓度不等式得到收敛性分析，并将算法应用于经验风险最小化问题。