Low complexity convergence rate bounds for push-sum algorithms with homogeneous correlation structure

TL;DR

This paper derives low-complexity bounds on the convergence rate of push-sum algorithms with homogeneous correlation, enabling efficient algorithm tuning and demonstrating significant runtime improvements in large networks.

Contribution

It extends existing convergence bounds for push-sum algorithms, introduces a parametric analysis of message weights, and provides practical methods for algorithm optimization.

Findings

Convergence bounds are computed with low complexity.

Optimal message weights are convex and explicitly differentiable.

Numerical results show over 10,000-fold runtime reduction for large graphs.

Abstract

The objective of this work is to establish an upper bound for the almost sure convergence rate for a class of push-sum algorithms. The current work extends the methods and results of the authors on a similar low-complexity bound on push-sum algorithms with some particular synchronous message passing schemes and complements the general approach of Gerencs\'er and Gerencs\'er from 2022 providing an exact, but often less accessible description. Furthermore, a parametric analysis is presented on the ``weight'' of the messages, which is found to be convex with an explicit expression for the gradient. This allows the fine-tuning of the algorithm used for improved efficiency. Numerical results confirm the speedup in evaluating the computable bounds without deteriorating their performance, for a graph on 120 vertices the runtime drops by more than 4 orders of magnitude.