  We propose stochastic optimization algorithms that can find local minima
faster than existing algorithms for nonconvex optimization problems, by
exploiting the third-order smoothness to escape non-degenerate saddle points
more efficiently. More specifically, the proposed algorithm only needs
$\tilde{O}(\epsilon^{-10/3})$ stochastic gradient evaluations to converge to an
approximate local minimum $\mathbf{x}$, which satisfies $\|\nabla
f(\mathbf{x})\|_2\leq\epsilon$ and $\lambda_{\min}(\nabla^2 f(\mathbf{x}))\geq
-\sqrt{\epsilon}$ in the general stochastic optimization setting, where
$\tilde{O}(\cdot)$ hides logarithm polynomial terms and constants. This
improves upon the $\tilde{O}(\epsilon^{-7/2})$ gradient complexity achieved by
the state-of-the-art stochastic local minima finding algorithms by a factor of
$\tilde{O}(\epsilon^{-1/6})$. For nonconvex finite-sum optimization, our
algorithm also outperforms the best known algorithms in a certain regime.
