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

Abstract: 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.