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(PAC-)Learning state machines from data streams: A generic strategy and an improved heuristic (Extended version)

Robert Baumgartner, Sicco Verwer

Abstract

This is an extended version of our publication Learning state machines from data streams: A generic strategy and an improved heuristic, International Conference on Grammatical Inference (ICGI) 2023, Rabat, Morocco. It has been extended with a formal proof on PAC-bounds, and the discussion and analysis of a similar approach has been moved from the appendix and is now a full Section. State machine models are models that simulate the behavior of discrete event systems, capable of representing systems such as software systems, network interactions, and control systems, and have been researched extensively. The nature of most learning algorithms however is the assumption that all data be available at the beginning of the algorithm, and little research has been done in learning state machines from streaming data. In this paper, we want to close this gap further by presenting a generic method for learning state machines from data streams, as well as a merge heuristic that uses sketches to account for incomplete prefix trees. We implement our approach in an open-source state merging library and compare it with existing methods. We show the effectiveness of our approach with respect to run-time, memory consumption, and quality of results on a well known open dataset. Additionally, we provide a formal analysis of our algorithm, showing that it is capable of learning within the PAC framework, and show a theoretical improvement to increase run-time, without sacrificing correctness of the algorithm in larger sample sizes.

(PAC-)Learning state machines from data streams: A generic strategy and an improved heuristic (Extended version)

Abstract

This is an extended version of our publication Learning state machines from data streams: A generic strategy and an improved heuristic, International Conference on Grammatical Inference (ICGI) 2023, Rabat, Morocco. It has been extended with a formal proof on PAC-bounds, and the discussion and analysis of a similar approach has been moved from the appendix and is now a full Section. State machine models are models that simulate the behavior of discrete event systems, capable of representing systems such as software systems, network interactions, and control systems, and have been researched extensively. The nature of most learning algorithms however is the assumption that all data be available at the beginning of the algorithm, and little research has been done in learning state machines from streaming data. In this paper, we want to close this gap further by presenting a generic method for learning state machines from data streams, as well as a merge heuristic that uses sketches to account for incomplete prefix trees. We implement our approach in an open-source state merging library and compare it with existing methods. We show the effectiveness of our approach with respect to run-time, memory consumption, and quality of results on a well known open dataset. Additionally, we provide a formal analysis of our algorithm, showing that it is capable of learning within the PAC framework, and show a theoretical improvement to increase run-time, without sacrificing correctness of the algorithm in larger sample sizes.

Paper Structure

This paper contains 21 sections, 15 theorems, 27 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Methodology
  5. Merge heuristic
  6. Count-Min-Sketches and the heuristic
  7. The runtime problem
  8. Streaming
  9. Streaming the prefix tree
  10. Minimization routine
  11. Experiments on small dataset
  12. The heuristic
  13. Discussion of SpaceSave
  14. Streaming
  15. Analysis
  16. ...and 6 more sections

Key Result

Theorem 1

Given the parameters ($\beta$, $\gamma$)By convention these are be called $\epsilon$ instead of $\beta$ and $\delta$ instead of $\gamma$, however we already used these two letters., and given two CMS whose width $w$ and depth $d$ of a CMS assume $w=\left\lceil{\frac{e}{\beta}}\right\rceil$ and $d=\l

Figures (4)

  • Figure 1: The two solutions to the problem of uniform distributions in the sketches.
  • Figure 2: Boxplots of all heuristics. Due to the difference in magitude in between the SpaceSave heuristic and the other heuristics we separated them into two sub-plots.
  • Figure 3: The function $f(m_0)$ with varying $\mu$ and $\alpha$.
  • Figure :

Theorems & Definitions (25)

  • Definition 1: $\mu$-Distinguishability
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1: Run-time of heuristic
  • proof
  • Lemma 1
  • Theorem 3: Samples processed
  • proof
  • ...and 15 more