Escaping from saddle points with perturbed gradient estimation

Abstract

For non-convex functions where derivative information is difficult to obtain, escaping saddle points remains a significant challenge. Existing zeroth-order optimization algorithms approximate the true gradient using unbiased gradient estimation techniques, employing zero-mean random perturbations, or exploring negative curvature directions to escape saddle points. However, these methods encounter near-zero approximate gradients in the vicinity of saddle points, necessitating multiple small perturbations to escape, thereby consuming a substantial number of function evaluations. In this work, we propose the Two-step Simultaneous Perturbation Stochastic Approximation (2-SPSA) approach, to facilitate saddle point escape, which requires fewer function evaluations. At each iteration, this method requires only 4 function evaluations to estimate the gradients at the current point and its neighboring point, of which their convex combination serves as the descent direction. The randomness inherent in this gradient estimation aids in rapidly jumping out of saddle points. Experimental results indicate that the proposed method can escape saddle points with fewer function evaluations compared to other zeroth-order optimization algorithms.