A Unified Model for Large-Scale Inexact Fixed-Point Iteration: A Stochastic Optimization Perspective

Calculating fixed points of a nonlinear function is a central problem in many areas of science and engineering with applications ranging from the study of dynamical systems to optimization and game theory. Fixed-point iteration methods provide a simple way to calculate the fixed point of nonexpansive mappings and they have been studied extensively. Emerging applications, however, necessitate the study of fixed-point calculation under various perturbations. For instance, in the data-driven identification of dynamical systems, the learning is typically erroneous, which in turn impacts the subsequent fixed calculations, which itself is an essential step for control. Motivated by such settings, in this work, we establish a general mathematical modeling framework for the study of inexact fixed-point iteration (FPI) algorithms. In doing so, we leverage and extend the recent advances in the stochastic optimization literature to derive new methods and convergence analysis results. In particular, adopting this view enables us to present a unified mathematical model to study the impact of inexact computations in both expansive and nonexpansive scenarios, a new technical approach for the analysis of inexact FPI methods, and a new inexact FPI method, which under certain assumptions, enjoys a faster convergence rate than traditional FPI algorithms in the expansive case.