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

Hualin Zhang; Huan Xiong; Bin Gu

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arXiv · 2210.01496

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

Hualin Zhang·Huan Xiong·Bin Gu

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Abstract

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.

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Hualin Zhang, Huan Xiong, Bin Gu. 2022-10-04. Zeroth-Order Negative Curvature Finding: Escaping Saddle Points without Gradients. https://arxiv.org/abs/2210.01496

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