Benign population landscapes do not transfer to empirical risk minimization in nonsmooth optimization

Abstract

Benign landscapes provide a geometric explanation for the success of iterative methods on nonconvex problems arising in data science. The landscape is benign if every local minimizer is global and every other critical point is a strict saddle. For methods that avoid strict saddles, convergence to a critical point therefore implies convergence to a global minimizer. For smooth stochastic optimization, a common strategy for establishing such landscapes proceeds in two steps: $(i)$ establish benign geometry for the population objective---which is often easier to analyze---and $(ii)$ transfer this geometry to the empirical objective using concentration of gradients and Hessians. Recent work has extended saddle-point avoidance guarantees to weakly convex nonsmooth optimization through the notion of active strict saddles. We show, however, that benign population geometry need not transfer to empirical objectives in this setting. Specifically, for robust phase retrieval with Gaussian measurements---a well-studied and highly structured problem---the population objective is benign. Nevertheless, the empirical objective has sharp spurious local minima with probability bounded away from zero as the sample size tends to infinity. This asymptotic probability lower bound is independent of the dimension.