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We consider escaping saddle points of nonconvex problems where only thefunction evaluations can be accessed. Although a variety of works have beenproposed, the majority of them require either second or first-orderinformation, and only a few of them have exploited zeroth-order methods,particularly the technique of negative curvature finding with zeroth-ordermethods which has been proven to be the most efficient method for escapingsaddle points. To fill this gap, in this paper, we propose two zeroth-ordernegative curvature finding frameworks that can replace Hessian-vector productcomputations without increasing the iteration complexity. We apply the proposedframeworks to ZO-GD, ZO-SGD, ZO-SCSG, ZO-SPIDER and prove that these ZOalgorithms can converge to $(\epsilon,\delta)$-approximate second-orderstationary points with less query complexity compared with prior zeroth-orderworks for finding local minima.

Zeroth-Order Negative Curvature Finding: Escaping Saddle Points without Gradients