Sharp oracle inequalities for stationary points of nonconvex penalized M-estimators

TL;DR

This paper extends the theoretical framework of oracle inequalities to stationary points of nonconvex penalized M-estimators, providing sharp bounds that measure their statistical performance relative to the best model approximation.

Contribution

It introduces a novel framework for deriving sharp oracle inequalities for stationary points in nonconvex optimization, applicable to various estimation problems.

Findings

Derived sharp oracle inequalities for stationary points

Extended convex optimization techniques to nonconvex problems

Demonstrated applicability to multiple estimation scenarios

Abstract

Many statistical estimation procedures lead to nonconvex optimization problems. Algorithms to solve these are often guaranteed to output a stationary point of the optimization problem. Oracle inequalities are an important theoretical instrument to asses the statistical performance of an estimator. Oracle results have focused on the theoretical properties of the uncomputable (global) minimum or maximum. In the present work a general framework used for convex optimization problems to derive oracle inequalities for stationary points is extended. A main new ingredient of these oracle inequalities is that they are sharp: they show closeness to the best approximation within the model plus a remainder term. We apply this framework to different estimation problems.