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State Space Models as Foundation Models: A Control Theoretic Overview

Carmen Amo Alonso, Jerome Sieber, Melanie N. Zeilinger

TL;DR

A systematic review of the most successful SSM proposals and highlights their main features from a control theoretic perspective is provided, and a comparative analysis of these models is presented, evaluating their performance on a standardized benchmark designed for assessing a model’s efficiency at learning long sequences.

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.

State Space Models as Foundation Models: A Control Theoretic Overview

TL;DR

A systematic review of the most successful SSM proposals and highlights their main features from a control theoretic perspective is provided, and a comparative analysis of these models is presented, evaluating their performance on a standardized benchmark designed for assessing a model’s efficiency at learning long sequences.

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.
Paper Structure (54 sections, 1 theorem, 13 equations, 4 figures, 2 tables)

This paper contains 54 sections, 1 theorem, 13 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. State Space Models
  3. Learning setup
  4. Parametrization
  5. Discretization
  6. Structure and Initialization
  7. Implementation
  8. Scaffolding and Layers
  9. Review of existing methods
  10. Structured State Space Sequence Model (S4)
  11. Parametrization
  12. Discretization
  13. Structure and Initialization
  14. Implementation
  15. Scaffolding
  16. ...and 39 more sections

Key Result

Lemma 2.2

(Informal) A dynamical system with dynamics eqn:dynamics_discrete has long-range memory, i.e., captures information from past inputs, if the eigenvalues of $A$ are inside the unit circle and very close to the unit circumference, i.e. $\vert eig(A) \vert \leq 1$ and $\vert eig(A) \vert \approx 1$$\fo

Figures (4)

  • Figure 1: A. General scaffolding of a SSM. The dynamical model \ref{['eqn:dynamics_discrete']} is represented in green. The input to the SSM is pre-processed and forked off in a skip connection (lower signal). The nature of the pre-processing map (linear or nonlinear) depends on the specific scaffolding. The output of the recursion is then post-processed with a nonlinear gate. B. Overall architecture of a SSM. Each of the SSMs including its scaffolding (Fig. 1.A.) is structured in a layered fashion, where the output from one layer is the input to the next.
  • Figure 1: Overview of the model features for the different SSM models considered. Accronyms used are as follows: Linear Time-Invariant (LTI), Linear Time-Varying (LTV), Single Input Single Output (SISO), Multiple Input Multiple Output (MIMO). Details on the scaffolding can be found in MLP MLP, H3 Gu2020, Mamba mamba, Hawk and Griffin griffin.
  • Figure 2: Complex plane representation of the unit disk and the eigenvalues of discrete-time dynamics matrix $\bar{A}$\ref{['eqn:dynamics_discrete']} resulting from the initialization method in each of the models S4, S4D, S5, LRU, S6, and RG-LRU. Since the initialization of S6 and RG-LRU are input dependent, we plot the initialization for two sample inputs (blue and orange).
  • Figure 2: Model performance in terms of test accuracy on the LRA benchmark. The first entry (Random) represents the performance of random guessing on the task, i.e., indicating the baseline above which a model is considered to have learned a meaningful representation. Models failing to exceed this baseline on a task are marked as FAIL. The best model on each task is highlighted in bold.

Theorems & Definitions (2)

  • Lemma 2.2
  • Definition 2.3