  Trust region and cubic regularization methods have demonstrated good
performance in small scale non-convex optimization, showing the ability to
escape from saddle points. Each iteration of these methods involves computation
of gradient, Hessian and function value in order to obtain the search direction
and adjust the radius or cubic regularization parameter. However, exactly
computing those quantities are too expensive in large-scale problems such as
training deep networks. In this paper, we study a family of stochastic trust
region and cubic regularization methods when gradient, Hessian and function
values are computed inexactly, and show the iteration complexity to achieve
$\epsilon$-approximate second-order optimality is in the same order with
previous work for which gradient and function values are computed exactly. The
mild conditions on inexactness can be achieved in finite-sum minimization using
random sampling. We show the algorithm performs well on training convolutional
neural networks compared with previous second-order methods.
