Improved Convergence Bounds For Operator Splitting Algorithms With Rare Extreme Errors

TL;DR

This paper develops probabilistic convergence bounds for operator splitting algorithms under rare extreme approximation errors, improving understanding of their robustness and performance in practical applications.

Contribution

It introduces new probabilistic bounds for convergence of AxPGD, AxAPGD, and AxWLM-ADMM considering rare extreme errors, extending prior deterministic analyses.

Findings

Derived probabilistic convergence bounds as functions of error bounds and variance.

Validated bounds through a spacecraft system control example.

Showed improved accuracy over previous bounds in sparse model predictive control.

Abstract

In this paper, we improve upon our previous work[24,22] and establish convergence bounds on the objective function values of approximate proximal-gradient descent (AxPGD), approximate accelerated proximal-gradient descent (AxAPGD) and approximate proximal ADMM (AxWLM-ADMM) schemes. We consider approximation errors that manifest rare extreme events and we propagate their effects through iterations. We establish probabilistic asymptotic and non-asymptotic convergence bounds as functions of the range (upper/lower bounds) and variance of approximation errors. We use the derived bound to assess AxPGD in a sparse model predictive control of a spacecraft system and compare its accuracy with previously derived bounds.