Cutting plane methods can be extended into nonconvex optimization

We show that it is possible to obtain an O(ε-4/3) expected runtime --- including computational cost --- for finding ε-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(ε-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).