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Scaling the weight parameters in Markov logic networks and relational logistic regression models

Felix Weitkämper

TL;DR

This paper investigates how weight parameters in Markov Logic Networks (MLNs) and Relational Logistic Regression (RLRs) scale with domain size and how this affects inference. It derives a probabilistic representation for grounded atoms as $\mathcal{P}_{T,D}(R(\vec{a}))=\int \mathrm{sigmoid}(\delta_{R(\vec{a})}^{T,D})\,d\mu_{T,D}$ and uses this to study asymptotic probabilities as domain size grows. It shows that standard MLNs and their domain-size aware variant (DA-MLNs) can exhibit weight-independent limits for key formulas such as $P\rightarrow R(x)$ and $P\wedge Q(x)\wedge R(x,y)$, motivating a directed alternative. The paper introduces Domain-size Aware Relational Logistic Regression (DA-RLR), proves an algorithm to compute its asymptotic probabilities, and demonstrates weight-dependent limits with domain growth. It discusses when to prefer scaled vs unscaled parameters, the role of random sampling, and outlines future directions including mixed formalisms and projectivity considerations.

Abstract

We consider Markov logic networks and relational logistic regression as two fundamental representation formalisms in statistical relational artificial intelligence that use weighted formulas in their specification. However, Markov logic networks are based on undirected graphs, while relational logistic regression is based on directed acyclic graphs. We show that when scaling the weight parameters with the domain size, the asymptotic behaviour of a relational logistic regression model is transparently controlled by the parameters, and we supply an algorithm to compute asymptotic probabilities. We also show using two examples that this is not true for Markov logic networks. We also discuss using several examples, mainly from the literature, how the application context can help the user to decide when such scaling is appropriate and when using the raw unscaled parameters might be preferable. We highlight random sampling as a particularly promising area of application for scaled models and expound possible avenues for further research.

Scaling the weight parameters in Markov logic networks and relational logistic regression models

TL;DR

This paper investigates how weight parameters in Markov Logic Networks (MLNs) and Relational Logistic Regression (RLRs) scale with domain size and how this affects inference. It derives a probabilistic representation for grounded atoms as and uses this to study asymptotic probabilities as domain size grows. It shows that standard MLNs and their domain-size aware variant (DA-MLNs) can exhibit weight-independent limits for key formulas such as and , motivating a directed alternative. The paper introduces Domain-size Aware Relational Logistic Regression (DA-RLR), proves an algorithm to compute its asymptotic probabilities, and demonstrates weight-dependent limits with domain growth. It discusses when to prefer scaled vs unscaled parameters, the role of random sampling, and outlines future directions including mixed formalisms and projectivity considerations.

Abstract

We consider Markov logic networks and relational logistic regression as two fundamental representation formalisms in statistical relational artificial intelligence that use weighted formulas in their specification. However, Markov logic networks are based on undirected graphs, while relational logistic regression is based on directed acyclic graphs. We show that when scaling the weight parameters with the domain size, the asymptotic behaviour of a relational logistic regression model is transparently controlled by the parameters, and we supply an algorithm to compute asymptotic probabilities. We also show using two examples that this is not true for Markov logic networks. We also discuss using several examples, mainly from the literature, how the application context can help the user to decide when such scaling is appropriate and when using the raw unscaled parameters might be preferable. We highlight random sampling as a particularly promising area of application for scaled models and expound possible avenues for further research.

Paper Structure

This paper contains 27 sections, 17 theorems, 21 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Aims of the Paper
  3. Prior Research on Extrapolation across Domain Sizes
  4. Extrapolation and Scalability
  5. Markov Logic Networks and Relational Logistic Regression
  6. Terminology
  7. Markov Logic Networks
  8. Domain-size Aware Markov Logic Networks
  9. Relational Logistic Regression
  10. Asymptotic Probabilities in MLNs and DA-MLNs
  11. Characterising Probabilities in MLN and DA-MLN
  12. The Formula $P\rightarrow R(x)$
  13. The Formula $P\wedge Q(x)\wedge R(x,y)$
  14. Discussion
  15. Domain-size-Aware RLR
  16. ...and 12 more sections

Key Result

proposition 1

For any MLN $T$, any structure $\mathfrak{X}$ on a choice of domains $D$ and any ground atom $R(\vec{a})$, the $\mu_{T,D}$-weighted mean of $1_{R(\vec{a})}$ on $\mathfrak{X}_{R(\vec{a})}$ and $\mathfrak{X}_{\neg R(\vec{a})}$ is given by $\mathrm{sigmoid}(\delta_{R(\vec{a})}^{T,D}(\mathbb{\mathfrak{X

Theorems & Definitions (40)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • definition 8
  • proposition 1
  • proof
  • ...and 30 more