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Warm Starting State-Space Models with Automata Learning

William Fishell, Sam Nicholas Kouteili, Mark Santolucito

TL;DR

This work learns an adaptive arbitration policy on a suite of arbiters from SYNTCOMP and shows that initializing SSMs with symbolically-learned approximations learn both faster and better.

Abstract

We prove that Moore machines can be exactly realized as state-space models (SSMs), establishing a formal correspondence between symbolic automata and these continuous machine learning architectures. These Moore-SSMs preserve both the complete symbolic structure and input-output behavior of the original Moore machine, but operate in Euclidean space. With this correspondence, we compare the training of SSMs with both passive and active automata learning. In recovering automata from the SYNTCOMP benchmark, we show that SSMs require orders of magnitude more data than symbolic methods and fail to learn state structure. This suggests that symbolic structure provides a strong inductive bias for learning these systems. We leverage this insight to combine the strengths of both automata learning and SSMs in order to learn complex systems efficiently. We learn an adaptive arbitration policy on a suite of arbiters from SYNTCOMP and show that initializing SSMs with symbolically-learned approximations learn both faster and better. We see 2-5 times faster convergence compared to randomly initialized models and better overall model accuracies on test data. Our work lifts automata learning out of purely discrete spaces, enabling principled exploitation of symbolic structure in continuous domains for efficiently learning in complex settings.

Warm Starting State-Space Models with Automata Learning

TL;DR

This work learns an adaptive arbitration policy on a suite of arbiters from SYNTCOMP and shows that initializing SSMs with symbolically-learned approximations learn both faster and better.

Abstract

We prove that Moore machines can be exactly realized as state-space models (SSMs), establishing a formal correspondence between symbolic automata and these continuous machine learning architectures. These Moore-SSMs preserve both the complete symbolic structure and input-output behavior of the original Moore machine, but operate in Euclidean space. With this correspondence, we compare the training of SSMs with both passive and active automata learning. In recovering automata from the SYNTCOMP benchmark, we show that SSMs require orders of magnitude more data than symbolic methods and fail to learn state structure. This suggests that symbolic structure provides a strong inductive bias for learning these systems. We leverage this insight to combine the strengths of both automata learning and SSMs in order to learn complex systems efficiently. We learn an adaptive arbitration policy on a suite of arbiters from SYNTCOMP and show that initializing SSMs with symbolically-learned approximations learn both faster and better. We see 2-5 times faster convergence compared to randomly initialized models and better overall model accuracies on test data. Our work lifts automata learning out of purely discrete spaces, enabling principled exploitation of symbolic structure in continuous domains for efficiently learning in complex settings.
Paper Structure (20 sections, 1 theorem, 17 equations, 16 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 1 theorem, 17 equations, 16 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivating Example
  3. Preliminaries
  4. State Space Models
  5. Moore Machines
  6. Moore-SSMs
  7. Learning Regular Languages from Data
  8. Methodology
  9. SSM Learning
  10. Sample Efficiency of Symbolic vs. Gradient-Based Learning
  11. Symbolic Structure in Gradient-Based Models
  12. Warm-Starting SSMs Using Symbolic Structure
  13. Methodology
  14. Warm-Starting Evaluation
  15. Related Work
  16. ...and 5 more sections

Key Result

lemma 1

Every Moore machine $\mathcal{A} = (S, S_0, \Sigma, \Lambda, T, G)$, whose transition and output maps are defined in equation eq:Moore_Machine, admits an equivalent representation as a discrete-time state--space model of the form where the state vector $x(t)$ encodes the symbolic state $S$, $\mu(t)$ encodes the input, and the matrices $A$, $B$, and $C$ are structured so as to preserve the origina

Figures (16)

  • Figure 1: Warm-starting SSM using symbolic structure VS. learning using traditional random initialization on an dynamic round robin arbitration policy with 4 channels.
  • Figure 2: Highlevel training pipeline visualization for learning regular languages from SYNTCOMP
  • Figure 3: Methodology for generating assumption following traces by randomly picking transitions from states, generating assumption abiding traces
  • Figure 4: Prefix-closed representation of sample trace
  • Figure 5: SSM architecture used to learn SYNTCOMP regular languages
  • ...and 11 more figures

Theorems & Definitions (2)

  • lemma 1: Moore-SSMs
  • proof : lemma \ref{['lemma1:Moore_SSM_Equivalence']}