  Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have
received increasing attention due to its improved gradient/Hessian complexities
(i.e., number of queries to stochastic gradient/Hessian oracles) to find local
minima for nonconvex finite-sum optimization. However, it is unclear whether
existing SVRC algorithms can be further improved. Moreover, the semi-stochastic
Hessian estimator adopted in existing SVRC algorithms prevents the use of
Hessian-vector product-based fast cubic subproblem solvers, which makes SVRC
algorithms computationally intractable for high-dimensional problems. In this
paper, we first present a Stochastic Recursive Variance-Reduced Cubic
regularization method (SRVRC) using a recursively updated semi-stochastic
gradient and Hessian estimators. It enjoys improved gradient and Hessian
complexities to find an $(\epsilon, \sqrt{\epsilon})$-approximate local
minimum, and outperforms the state-of-the-art SVRC algorithms. Built upon
SRVRC, we further propose a Hessian-free SRVRC algorithm, namely
SRVRC$_{\text{free}}$, which only requires stochastic gradient and
Hessian-vector product computations, and achieves $\tilde O(dn\epsilon^{-2}
\land d\epsilon^{-3})$ runtime complexity, where $n$ is the number of component
functions in the finite-sum structure, $d$ is the problem dimension, and
$\epsilon$ is the optimization precision. This outperforms the best-known
runtime complexity $\tilde O(d\epsilon^{-3.5})$ achieved by stochastic cubic
regularization algorithm proposed in Tripuraneni et al. 2018.
