Distributed Seeking for Fixed Points of Biased Stochastic Operators: A Communication-Efficient Approach

TL;DR

This paper presents a communication-efficient distributed algorithm for fixed point seeking of stochastic operators over multi-agent networks, accommodating biased and variance-affected conditions with theoretical convergence guarantees.

Contribution

It introduces a novel inexact Krasnosel'skifi--Mann iteration method with integrated compression and skipping, unifying theories with distributed non-convex optimization.

Findings

Convergence guarantees established for the proposed algorithm.

Effective communication compression and skipping improve efficiency.

Numerical simulations confirm theoretical results.

Abstract

This paper investigates the distributed fixed point seeking problem of sum-separable stochastic operators over the multi-agent network. Based on inexact Krasnosel'ski\u{\i}--Mann iterations, the communication-efficient distributed algorithm is proposed under the relaxed growth bias and variance conditions, generalizing traditional unbiased and bounded additive variance assumptions. To enhance communication efficiency, we integrate communication compression and dynamic period skipping techniques, particularly adopting a unified compressor that allows both relative and absolute compression errors. By introducing a surrogate function for general non-contractive and contractive operators, we establish convergence guarantees of the distributed fixed point iteration, achieving among the first theoretical unifications with distributed non-convex optimization algorithms. Finally, numerical simulations validate the effectiveness of the theoretical results.