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We propose a reduction for non-convex optimization that can (1) turn anstationary-point finding algorithm into an local-minimum finding one, and (2)replace the Hessian-vector product computations with only gradientcomputations. It works both in the stochastic and the deterministic settings,without hurting the algorithm's performance.  As applications, our reduction turns Natasha2 into a first-order methodwithout hurting its performance. It also converts SGD, GD, SCSG, and SVRG intoalgorithms finding approximate local minima, outperforming some best knownresults.

Neon2: Finding Local Minima via First-Order Oracles