Asymptotic Properties of Stationary Solutions of Coupled Nonconvex Nonsmooth Empirical Risk Minimization

Zhengling Qi, Ying Cui, Yufeng Liu, Jong Shi Pang

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

This paper has two main goals: (a) establish several statistical properties—consistency, asymptotic distributions, and convergence rates—of stationary solutions and values of a class of coupled nonconvex and nonsmooth empirical risk-minimization problems and (b) validate these properties by a noisy amplitude-based phase-retrieval problem, the latter being of much topical interest. Derived from available data via sampling, these empirical risk-minimization problems are the computational workhorse of a population risk model that involves the minimization of an expected value of a random functional. When these minimization problems are nonconvex, the computation of their globally optimal solutions is elusive. Together with the fact that the expectation operator cannot be evaluated for general probability distributions, it becomes necessary to justify whether the stationary solutions of the empirical problems are practical approximations of the stationary solution of the population problem. When these two features, general distribution and nonconvexity, are coupled with nondifferentiability that often renders the problems “non-Clarke regular,” the task of the justification becomes challenging. Our work aims to address such a challenge within an algorithm-free setting. The resulting analysis is, therefore, different from much of the analysis in the recent literature that is based on local search algorithms. Furthermore, supplementing the classical global minimizer-centric analysis, our results offer a promising step to close the gap between computational optimization and asymptotic analysis of coupled, nonconvex, nonsmooth statistical estimation problems, expanding the former with statistical properties of the practically obtained solution and providing the latter with a more practical focus pertaining to computational tractability.

Original languageEnglish (US)
Pages (from-to)2034-2064
Number of pages31
JournalMathematics of Operations Research
Volume47
Issue number3
DOIs
StatePublished - Aug 2022

Bibliographical note

Funding Information:
We wish to thank all the students who participated in the study and the administrative staffs who supported it. We extend our thanks to the Vice President of KMS, S. Ogoshi, and to Professor H. Seguchi, for their financial support.

Publisher Copyright:
Copyright: © 2021 INFORMS.

Keywords

  • asymptotic distribution
  • consistency
  • convergence rates
  • directional stationarity
  • nonconvexity
  • nonsmoothness
  • phase-retrieval problem
  • statistical analysis

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