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We propose stochastic optimization algorithms that can find local minimafaster than existing algorithms for nonconvex optimization problems, byexploiting the third-order smoothness to escape non-degenerate saddle pointsmore efficiently. More specifically, the proposed algorithm only needs$\tilde{O}(\epsilon^{-10/3})$ stochastic gradient evaluations to converge to anapproximate local minimum $\mathbf{x}$, which satisfies $\|\nablaf(\mathbf{x})\|_2\leq\epsilon$ 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. Thisimproves upon the $\tilde{O}(\epsilon^{-7/2})$ gradient complexity achieved bythe state-of-the-art stochastic local minima finding algorithms by a factor of$\tilde{O}(\epsilon^{-1/6})$. For nonconvex finite-sum optimization, ouralgorithm also outperforms the best known algorithms in a certain regime.

Third-order Smoothness Helps: Even Faster Stochastic Optimization Algorithms for Finding Local Minima