Cutting plane methods can be extended into nonconvex optimization

Abstract

We show that it is possible to obtain an $O(\epsilon^{-4/3})$ expected runtime --- including computational cost --- for finding $\epsilon$-stationary points of smooth nonconvex functions using cutting plane methods. This improves on the best-known epsilon dependence achieved by cubic regularized Newton of $O(\epsilon^{-3/2})$ as proved by Nesterov and Polyak (2006). Our techniques utilize the convex until proven guilty principle proposed by Carmon, Duchi, Hinder, and Sidford (2017).