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Given a nonconvex function that is an average of $n$ smooth functions, wedesign stochastic first-order methods to find its approximate stationarypoints. The convergence of our new methods depends on the smallest (negative)eigenvalue $-\sigma$ of the Hessian, a parameter that describes how nonconvexthe function is.  Our methods outperform known results for a range of parameter $\sigma$, andcan be used to find approximate local minima. Our result implies an interestingdichotomy: there exists a threshold $\sigma_0$ so that the currently fastestmethods for $\sigma>\sigma_0$ and for $\sigma<\sigma_0$ have differentbehaviors: the former scales with $n^{2/3}$ and the latter scales with$n^{3/4}$.

Natasha: Faster Non-Convex Stochastic Optimization Via Strongly Non-Convex Parameter