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