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

Zhengling Qi (), Ying Cui (), Yufeng Liu () and Jong-Shi Pang ()
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Zhengling Qi: Department of Decision Sciences, George Washington University, Washington, District of Columbia, 20052
Ying Cui: Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, Minnesota 55455
Yufeng Liu: Department of Statistics and Operations Research, Department of Genetics, Department of Biostatistics, Carolina Center for Genome Sciences, Lineberger Comprehensive Cancer Center, University of North Carolina, Chapel Hill, Chapel Hill, North Carolina 27599
Jong-Shi Pang: The Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, California 90007

Mathematics of Operations Research, 2022, vol. 47, issue 3, 2034-2064

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.

Keywords: Primary: 62F12; 90C26; secondary: 49J52; asymptotic distribution; consistency; convergence rates; directional stationarity; nonconvexity; nonsmoothness; phase-retrieval problem; statistical analysis (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:3:p:2034-2064

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