  We propose two algorithms that can find local minima faster than the
state-of-the-art algorithms in both finite-sum and general stochastic nonconvex
optimization. At the core of the proposed algorithms is
$\text{One-epoch-SNVRG}^+$ using stochastic nested variance reduction (Zhou et
al., 2018a), which outperforms the state-of-the-art variance reduction
algorithms such as SCSG (Lei et al., 2017). In particular, for finite-sum
optimization problems, the proposed
$\text{SNVRG}^{+}+\text{Neon2}^{\text{finite}}$ algorithm achieves
$\tilde{O}(n^{1/2}\epsilon^{-2}+n\epsilon_H^{-3}+n^{3/4}\epsilon_H^{-7/2})$
gradient complexity to converge to an $(\epsilon, \epsilon_H)$-second-order
stationary point, which outperforms $\text{SVRG}+\text{Neon2}^{\text{finite}}$
(Allen-Zhu and Li, 2017) , the best existing algorithm, in a wide regime. For
general stochastic optimization problems, the proposed
$\text{SNVRG}^{+}+\text{Neon2}^{\text{online}}$ achieves
$\tilde{O}(\epsilon^{-3}+\epsilon_H^{-5}+\epsilon^{-2}\epsilon_H^{-3})$
gradient complexity, which is better than both
$\text{SVRG}+\text{Neon2}^{\text{online}}$ (Allen-Zhu and Li, 2017) and
Natasha2 (Allen-Zhu, 2017) in certain regimes. Furthermore, we explore the
acceleration brought by third-order smoothness of the objective function.
