A Systems-Theoretic View on the Convergence of Algorithms under Disturbances

2025-12-19

AI Summary

This study addresses the problem of ensuring algorithmic convergence in complex dynamic systems—spanning physics, social sciences, and engineering—subject to external disturbances, stochastic noise, and coupled interactions. To overcome the limitation of existing theories in characterizing disturbance robustness, we systematically introduce the converse Lyapunov theorem into algorithmic convergence analysis for the first time, establishing a unified theoretical framework that jointly guarantees stability and quantifies convergence rates under perturbations. Our method integrates converse Lyapunov theory, nonlinear stability analysis, and quantitative robustness modeling, yielding explicit, computable bounds on convergence via quantitative perturbation inequalities. The framework is successfully applied to three domains: modeling communication constraints in distributed learning, analyzing generalization sensitivity in machine learning, and designing differential privacy mechanisms with calibrated noise injection. The results provide a verifiable, quantifiable theoretical foundation for dynamic algorithm design across disciplines.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

Algorithms increasingly operate within complex physical, social, and engineering systems where they are exposed to disturbances, noise, and interconnections with other dynamical systems. This article extends known convergence guarantees of an algorithm operating in isolation (i.e., without disturbances) and systematically derives stability bounds and convergence rates in the presence of such disturbances. By leveraging converse Lyapunov theorems, we derive key inequalities that quantify the impact of disturbances. We further demonstrate how our result can be utilized to assess the effects of disturbances on algorithmic performance in a wide variety of applications, including communication constraints in distributed learning, sensitivity in machine learning generalization, and intentional noise injection for privacy. This underpins the role of our result as a unifying tool for algorithm analysis in the presence of noise, disturbances, and interconnections with other dynamical systems.

Problem

• Extending convergence guarantees for algorithms under disturbances
• Deriving stability bounds and convergence rates with disturbances
• Assessing disturbance impact on algorithmic performance in applications

Innovation

• Extends convergence guarantees to disturbed algorithm environments
• Derives stability bounds using converse Lyapunov theorems
• Unifies analysis for noise, disturbances, and system interconnections