[Paper Review] State Space Models as Foundation Models: A Control Theoretic Overview

Carmen Amo Alonso, Jérôme Sieber+1 more|arXiv (Cornell University)|Mar 25, 2024

Fault Detection and Control Systems4 citations

TL;DR

This paper presents a control-theoretic analysis of state space models (SSMs) as foundation models, demonstrating their potential to replace attention mechanisms in sequence modeling. It reveals that linear time-invariant (LTI) SSMs outperform linear time-varying (LTV) SSMs on the Long Range Arena benchmark, despite LTV models being more general, and highlights the critical role of marginal stability in model performance from a systems theory perspective.

ABSTRACT

In recent years, there has been a growing interest in integrating linear state-space models (SSM) in deep neural network architectures of foundation models. This is exemplified by the recent success of Mamba, showing better performance than the state-of-the-art Transformer architectures in language tasks. Foundation models, like e.g. GPT-4, aim to encode sequential data into a latent space in order to learn a compressed representation of the data. The same goal has been pursued by control theorists using SSMs to efficiently model dynamical systems. Therefore, SSMs can be naturally connected to deep sequence modeling, offering the opportunity to create synergies between the corresponding research areas. This paper is intended as a gentle introduction to SSM-based architectures for control theorists and summarizes the latest research developments. It provides a systematic review of the most successful SSM proposals and highlights their main features from a control theoretic perspective. Additionally, we present a comparative analysis of these models, evaluating their performance on a standardized benchmark designed for assessing a model's efficiency at learning long sequences.

Motivation & Objective

• To bridge the gap between foundation models and control theory by analyzing SSMs through a systems-theoretic lens.
• To evaluate the performance of state-of-the-art SSMs on the Long Range Arena (LRA) benchmark, focusing on long-context learning efficiency.
• To investigate why LTI-based SSMs outperform LTV-based SSMs despite the latter's greater generality.
• To identify open research questions in SSM design, particularly regarding eigenvalue dynamics and stability.
• To promote cross-pollination between control theory and foundation model research by highlighting explainability and design opportunities.

Proposed method

• The paper reviews key SSM architectures—S4, S4D, S5, LRU, S6, and RG-LRU—through the lens of linear systems theory.
• It formulates SSMs as discrete-time linear time-invariant (LTI) and linear time-varying (LTV) systems using state-space equations with state transition and output matrices.
• The authors implement and benchmark all SSM variants on the standardized LRA benchmark, including S6 and RG-LRU, which were previously unreported.
• They analyze the impact of eigenvalue initialization on model performance, particularly focusing on marginal stability.
• The study compares SSMs against Transformers and random baselines using consistent hyperparameters and training protocols.
• Theoretical analysis is grounded in control theory, especially stability conditions and system controllability/observability.

Experimental results

Research questions

• RQ1 Why do LTI-based SSMs consistently outperform LTV-based SSMs on the LRA benchmark, despite LTV systems being a more general class?
• RQ2 What is the role of eigenvalue placement, particularly marginal stability, in the performance of SSMs?
• RQ3 How can control-theoretic principles such as stability and observability be leveraged to improve SSM design and explainability?
• RQ4 Can a principled, system-theoretic design of LTV-based SSMs achieve performance comparable to LTI counterparts?
• RQ5 What are the theoretical limitations of input-dependent system matrices in SSMs, and how do they affect learning dynamics?

Key findings

• LTI-based SSMs (S4, S4D, S5, LRU) outperform both LTV-based SSMs (S6, RG-LRU) and Transformers on the LRA benchmark, especially in long-context tasks.
• The LTV-based models S6 and RG-LRU, despite being more general, failed to match the performance of LTI models, even after hyperparameter tuning and eigenvalue initialization for marginal stability.
• Forcing eigenvalues toward marginal stability improved LTI models but degraded or prevented learning in LTV models, indicating a fundamental difference in stability behavior.
• The LRA benchmark results suggest that current LTV parametrizations of SSMs are not yet as effective as LTI variants, despite theoretical advantages.
• The study reveals a critical disconnect: LTV systems are theoretically more expressive, yet in practice, they underperform LTI systems, raising open questions about their design and stability.
• Control theory offers a promising framework for improving SSM explainability and design, particularly through stability and system-theoretic analysis.