Faster Gradient-Free Methods for Escaping Saddle Points

Escaping from saddle points has become an important research topic in non-convex optimization. In this paper, we study the case when calculations of explicit gradients are expensive or even infeasible, and only function values are accessible. Currently, there have two types of gradient-free (zeroth-order) methods based on random perturbation and negative curvature finding proposed to escape saddle points efficiently and converge to an $\epsilon$-approximate second-order stationary point. Nesterov's accelerated gradient descent (AGD) method can escape saddle points faster than gradient descent (GD) which have been verified in first-order algorithms. However, whether AGD could accelerate the gradient-free methods is still unstudied. To unfold this mystery, in this paper, we propose two accelerated variants for the two types of gradient-free methods of escaping saddle points. We show that our algorithms can find an $\epsilon$-approximate second-order stationary point with $\tilde{\mathcal{O}}(1/\epsilon^{1.75})$ iteration complexity and $\tilde{\mathcal{O}}(d/\epsilon^{1.75})$ oracle complexity, where $d$ is the problem dimension. Thus, our methods achieve a comparable convergence rate to their first-order counterparts and have fewer oracle complexity compared to prior derivative-free methods for finding second-order stationary points.