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.

提出了一种 Stochastic Variance-Reduced Cubic regularization (SVRC) 算法的改进型，叫做 Stochastic Recursive Variance-Reduced Cubic regularization (SRVRC) 算法，该算法结合递归更新的半随机梯度和 Hessian 估计器，可以更快更好地找到一个（ε，sqrt（ε））- 近似局部极小值点。并且在这个基础上，我们进一步提出了一种基于 SRVRC 的无 Hessian 的算法：SRVRC$_{ext {free}}$，它只需要随机梯度和 Hessian 向量积运算，并且达到更好的运算速度。