  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
nonsmoothempirical 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 which 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 the
much of the analysis in the recent literature that is based on local search
algorithms. Furthermore, supplementing the classical minimizer-centric
analysis, our results offer a first 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.
