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

2025-07-22

AI Summary

This paper addresses the convergence rate analysis of push-sum–type distributed averaging algorithms over networks with homogeneous correlation structures, for which no computable almost-sure convergence rate bound existed previously. Method: We propose a differentiable convex parametrization framework grounded in stochastic recursive sequence modeling and probabilistic tail estimation, wherein message weight optimization is intrinsically embedded into theoretical bound derivation—ensuring both mathematical rigor and implementability. Extending the Gerencsér & Gerencsér (2022) framework, our approach incorporates a gradient-driven weight tuning mechanism to substantially accelerate bound computation. Contribution/Results: Experiments on a 120-node network demonstrate over four orders-of-magnitude speedup in convergence bound computation, without sacrificing bound tightness. To the best of our knowledge, this work provides the first low-complexity, high-accuracy, and deployable theoretical tool for evaluating convergence rates of distributed cooperative algorithms.

Technology Category

Search and Optimization: Distributed SearchConstraint Satisfaction and Optimization: Distributed CSP/OptimizationReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

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 Gerencsér and Gerencsér 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.

Problem

• Establish upper bound for push-sum algorithm convergence rate
• Analyze message weight impact on algorithm efficiency
• Demonstrate runtime speedup for large graph evaluations

Innovation

• Establishes upper bound for push-sum convergence rate
• Presents convex parametric analysis on message weights
• Achieves runtime speedup without performance loss