Escaping saddle points in zeroth-order optimization: two function evaluations suffice

Zhaolin Ren, Yujie Tang, Na Li

Keywords: zeroth-order optimization, nonconvex optimization, escape saddle points

TL;DR: We provide the first result showing that zeroth-order optimization with constant number of function evaluations per iteration can escape saddle points efficiently.

Abstract: Zeroth-order methods are useful in solving black-box optimization and reinforcement learning problems in unknown environments. It uses function values to estimate the gradient. As optimization problems are often nonconvex, it is a natural question to understand how zeroth-order methods escape saddle points. In this paper, we consider zeroth-order methods, that at each iteration, may freely choose 2$m$ function evaluations where $m$ ranges from 1 to $d$, with $d$ denoting the problem dimension. We show that by adding an appropriate isotropic perturbation at each iteration, a zeroth-order algorithm based on $2m$ function evaluations per iteration can not only find $\epsilon$-second order stationary points polynomially fast, but do so using only $\tilde{O}(\frac{d}{\epsilon^{2.5}})$ function evaluations.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics

Submission Guidelines: Yes

Please Choose The Closest Area That Your Submission Falls Into: Optimization (eg, convex and non-convex optimization)