Design Principles for Sequence Models via Coefficient Dynamics

2025-10-10

AI Summary

This paper addresses the lack of a unified theoretical foundation for design principles of deep sequence models—including Transformers, State Space Models (SSMs), and gated linear RNNs. We propose the first unified modeling framework grounded in **coefficient dynamics**, wherein output combination coefficients are modeled as the state trajectory of an autonomous linear dynamical system driven by impulse inputs. Our framework mathematically uncovers, for the first time, a shared impulse-response–linear-attention coupling mechanism across these three major architectures. It systematically characterizes intrinsic trade-offs among expressive power, computational efficiency, input selectivity, and training stability. By integrating linear systems theory, softmax-based attention, and impulse response analysis, our work not only explains the empirical success of modern sequence models but also yields interpretable, scalable design principles. These principles provide theoretically grounded guidance for task-aware architectural innovation in sequence modeling.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: (Large) Language Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Deep sequence models, ranging from Transformers and State Space Models (SSMs) to more recent approaches such as gated linear RNNs, fundamentally compute outputs as linear combinations of past value vectors. To draw insights and systematically compare such architectures, we develop a unified framework that makes this output operation explicit, by casting the linear combination coefficients as the outputs of autonomous linear dynamical systems driven by impulse inputs. This viewpoint, in spirit substantially different from approaches focusing on connecting linear RNNs with linear attention, reveals a common mathematical theme across diverse architectures and crucially captures softmax attention, on top of RNNs, SSMs, and related models. In contrast to new model proposals that are commonly evaluated on benchmarks, we derive design principles linking architectural choices to model properties. Thereby identifying tradeoffs between expressivity and efficient implementation, geometric constraints on input selectivity, and stability conditions for numerically stable training and information retention. By connecting several insights and observations from recent literature, the framework both explains empirical successes of recent designs and provides guiding principles for systematically designing new sequence model architectures.

Problem

• Unified framework for comparing sequence model architectures
• Deriving design principles linking architecture to model properties
• Identifying tradeoffs between expressivity and efficient implementation

Innovation

• Unified framework models coefficients as linear dynamics
• Derives design principles linking architecture to properties
• Explains empirical success and guides new model design