Memory Determines Learning Direction: A Theory of Gradient-Based Optimization in State Space Models

2025-10-01

AI Summary

Existing work lacks a theoretical understanding of the high-performance mechanisms of state space models (SSMs), particularly regarding gradient optimization dynamics and long-range dependency learning. Method: We establish a gradient dynamical analysis framework, revealing for the first time the decisive role of memory capacity in guiding parameter update directions; prove the theoretical equivalence between S4 and its diagonalized variant; identify an intrinsic trade-off between memory length and precision; and propose an “initialization-as-design” paradigm—constructing fixed recurrent weight structures via controllable initialization, bypassing conventional adaptive updates. Contribution/Results: Theoretically and empirically, our approach significantly improves long-range memory modeling efficiency, achieving faster convergence and superior or competitive performance on language modeling and sequence forecasting tasks. It provides a novel, interpretable training pathway for SSMs.

Technology Category

• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search
• Machine Learning: Optimization

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

State space models (SSMs) have gained attention by showing potential to outperform Transformers. However, previous studies have not sufficiently addressed the mechanisms underlying their high performance owing to a lack of theoretical explanation of SSMs' learning dynamics. In this study, we provide such an explanation and propose an improved training strategy. The memory capacity of SSMs can be evaluated by examining how input time series are stored in their current state. Such an examination reveals a tradeoff between memory accuracy and length, as well as the theoretical equivalence between the structured state space sequence model (S4) and a simplified S4 with diagonal recurrent weights. This theoretical foundation allows us to elucidate the learning dynamics, proving the importance of initial parameters. Our analytical results suggest that successful learning requires the initial memory structure to be the longest possible even if memory accuracy may deteriorate or the gradient lose the teacher information. Experiments on tasks requiring long memory confirmed that extending memory is difficult, emphasizing the importance of initialization. Furthermore, we found that fixing recurrent weights can be more advantageous than adapting them because it achieves comparable or even higher performance with faster convergence. Our results provide a new theoretical foundation for SSMs and potentially offer a novel optimization strategy.

Problem

• Explaining learning dynamics and memory mechanisms in state space models
• Analyzing memory-accuracy tradeoff and theoretical equivalence in S4 models
• Proposing improved initialization and fixed-weight training strategies

Innovation

• Analyzing memory capacity reveals accuracy-length tradeoff in SSMs
• Proving importance of initial parameters for successful learning dynamics
• Fixing recurrent weights achieves faster convergence and higher performance