Asynchronous and Distributed Tracking of Time-Varying Fixed Points

TL;DR

This paper introduces a comprehensive framework for tracking fixed points of time-varying contraction mappings, addressing challenges like imperfect information, asynchrony, and distributed implementation, with applications to online optimization and network feedback systems.

Contribution

It presents a novel analytical framework for tracking fixed points in dynamic, distributed, and imperfect information settings, extending existing methods to more realistic scenarios.

Findings

The framework provides bounds on tracking error under various conditions.

It applies to online gradient methods for time-varying convex problems.

Numerical results validate the effectiveness of the proposed approach.

Abstract

This paper develops an algorithmic framework for tracking fixed points of time-varying contraction mappings. Analytical results for the tracking error are established for the cases where: (i) the underlying contraction self-map changes at each step of the algorithm; (ii) only an imperfect information of the map is available; and, (iii) the algorithm is implemented in a distributed fashion, with communication delays and packet drops leading to asynchronous algorithmic updates. The analytical results are applicable to several classes of problems, including time-varying contraction mappings emerging from online and asynchronous implementations of gradient-based methods for time-varying convex programs. In this domain, the proposed framework can also capture the operating principles of feedback-based online algorithms, where the online gradient steps are suitably modified to accommodate actionable feedback from an underlying physical or logical network. Examples of applications and illustrative numerical results are provided.