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We derive bounds on the sample complexity of empirical risk minimization(ERM) in the context of minimizing non-convex risks that admit the strictsaddle property. Recent progress in non-convex optimization has yieldedefficient algorithms for minimizing such functions. Our results imply thatthese efficient algorithms are statistically stable and also generalize well.In particular, we derive fast rates which resemble the bounds that are oftenattained in the strongly convex setting. We specify our bounds to PrincipalComponent Analysis and Independent Component Analysis. Our results andtechniques may pave the way for statistical analyses of additional strictsaddle problems.

Fast Rates for Empirical Risk Minimization of Strict Saddle Problems