Escaping Saddle Points for Zeroth-order Nonconvex Optimization using Estimated Gradient Descent

TL;DR

This paper introduces a gradient estimation method for zeroth-order non-convex optimization that guarantees convergence to second-order stationary points, expanding the applicability of gradient-based techniques without explicit gradient access.

Contribution

It proposes a model-free algorithm that achieves second-order stationary points in non-convex optimization using estimated gradients, with theoretical query complexity bounds.

Findings

Converges to an $oldsymbol{\e}$-second-order stationary point.

Requires $oldsymbol{ ilde{O}(rac{d^{2+rac{ heta}{2}}}{ ext{ extit{e}}^{8+ heta}})}$ function queries.

Applicable to general non-convex problems without gradient oracle access.

Abstract

Gradient descent and its variants are widely used in machine learning. However, oracle access of gradient may not be available in many applications, limiting the direct use of gradient descent. This paper proposes a method of estimating gradient to perform gradient descent, that converges to a stationary point for general non-convex optimization problems. Beyond the first-order stationary properties, the second-order stationary properties are important in machine learning applications to achieve better performance. We show that the proposed model-free non-convex optimization algorithm returns an $\epsilon$-second-order stationary point with $\widetilde{O}(\frac{d^{2+\frac{\theta}{2}}}{\epsilon^{8+\theta}})$ queries of the function for any arbitrary $\theta>0$.