A Unifying Framework for Parallelizing Sequential Models with Linear Dynamical Systems

2025-09-25

AI Summary

To address the fundamental challenge of limited parallelizability in sequential models, this paper introduces a unified theoretical framework based on linear dynamical systems (LDS). It formally models classical fixed-point algorithms—including Newton’s method, Picard iteration, and the Jacobi method—as approximate linearizations of nonlinear recurrences. From a dynamical systems perspective, the framework characterizes convergence conditions, applicability domains, and inherent parallelization potential, thereby unifying diverse iterative schemes. Methodologically, it integrates LDS modeling, nonlinear recurrence analysis, and approximate linearization techniques to relax the conventional sequential dependency assumption. The results advance the theoretical understanding of existing parallel sequence modeling approaches and provide principled design guidelines for scalable, efficient parallel algorithms. This work establishes a cohesive theoretical foundation for accelerating sequence modeling computations through parallelization.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Natural Language Processing: Learning & Optimization for NLP
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Harnessing parallelism in seemingly sequential models is a central challenge for modern machine learning. Several approaches have been proposed for evaluating sequential processes in parallel using fixed-point methods, like Newton, Picard, and Jacobi iterations. In this work, we show that these methods can be understood within a common framework based on linear dynamical systems (LDSs), where different iteration schemes arise naturally as approximate linearizations of a nonlinear recursion. This unifying view highlights shared principles behind these techniques and clarifies when particular fixed-point methods are most likely to be effective. By bridging diverse algorithms through the language of LDSs, our framework provides a clearer theoretical foundation for parallelizing sequential models and points toward new opportunities for efficient and scalable computation.

Problem

• Develops a unifying framework for parallelizing sequential models
• Connects fixed-point methods through linear dynamical systems theory
• Provides theoretical foundation for scalable parallel computation

Innovation

• Parallelizing sequential models using linear dynamical systems
• Unifying fixed-point methods via approximate linearization framework
• Bridging algorithms through shared LDS principles