  Cubic regularization (CR) is an optimization method with emerging popularity
due to its capability to escape saddle points and converge to second-order
stationary solutions for nonconvex optimization. However, CR encounters a high
sample complexity issue for finite-sum problems with a large data size.
%Various inexact variants of CR have been proposed to improve the sample
complexity. In this paper, we propose a stochastic variance-reduced
cubic-regularization (SVRC) method under random sampling, and study its
convergence guarantee as well as sample complexity. We show that the iteration
complexity of SVRC for achieving a second-order stationary solution within
$\epsilon$ accuracy is $O(\epsilon^{-3/2})$, which matches the state-of-art
result on CR types of methods. Moreover, our proposed variance reduction scheme
significantly reduces the per-iteration sample complexity. The resulting total
Hessian sample complexity of our SVRC is ${\Oc}(N^{2/3} \epsilon^{-3/2})$,
which outperforms the state-of-art result by a factor of $O(N^{2/15})$. We also
study our SVRC under random sampling without replacement scheme, which yields a
lower per-iteration sample complexity, and hence justifies its practical
applicability.
