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Symbolic Parameter Learning in Probabilistic Answer Set Programming

Damiano Azzolini, Elisabetta Gentili, Fabrizio Riguzzi

TL;DR

The paper tackles parameter learning in Probabilistic Answer Set Programming under Credal Semantics by creating two symbolically driven algorithms that extract nonlinear equations from NNFs and solve for learnable probabilities. One approach uses constrained nonlinear optimization with off-the-shelf solvers, the other employs Expectation Maximization, both leveraging Second Level Algebraic Model Counting to generate the symbolic equations. Empirical results across four datasets show the constrained-optimization method often outperforms EM and PASTA in both speed and final log-likelihood, while EM remains competitive but memory-intensive. The work demonstrates a scalable, open-source pipeline for learning PASP parameters and points to future work on broader semantics and equation-simplification techniques to further improve scalability and applicability.

Abstract

Parameter learning is a crucial task in the field of Statistical Relational Artificial Intelligence: given a probabilistic logic program and a set of observations in the form of interpretations, the goal is to learn the probabilities of the facts in the program such that the probabilities of the interpretations are maximized. In this paper, we propose two algorithms to solve such a task within the formalism of Probabilistic Answer Set Programming, both based on the extraction of symbolic equations representing the probabilities of the interpretations. The first solves the task using an off-the-shelf constrained optimization solver while the second is based on an implementation of the Expectation Maximization algorithm. Empirical results show that our proposals often outperform existing approaches based on projected answer set enumeration in terms of quality of the solution and in terms of execution time. The paper has been accepted at the ICLP2024 conference and is under consideration in Theory and Practice of Logic Programming (TPLP).

Symbolic Parameter Learning in Probabilistic Answer Set Programming

TL;DR

The paper tackles parameter learning in Probabilistic Answer Set Programming under Credal Semantics by creating two symbolically driven algorithms that extract nonlinear equations from NNFs and solve for learnable probabilities. One approach uses constrained nonlinear optimization with off-the-shelf solvers, the other employs Expectation Maximization, both leveraging Second Level Algebraic Model Counting to generate the symbolic equations. Empirical results across four datasets show the constrained-optimization method often outperforms EM and PASTA in both speed and final log-likelihood, while EM remains competitive but memory-intensive. The work demonstrates a scalable, open-source pipeline for learning PASP parameters and points to future work on broader semantics and equation-simplification techniques to further improve scalability and applicability.

Abstract

Parameter learning is a crucial task in the field of Statistical Relational Artificial Intelligence: given a probabilistic logic program and a set of observations in the form of interpretations, the goal is to learn the probabilities of the facts in the program such that the probabilities of the interpretations are maximized. In this paper, we propose two algorithms to solve such a task within the formalism of Probabilistic Answer Set Programming, both based on the extraction of symbolic equations representing the probabilities of the interpretations. The first solves the task using an off-the-shelf constrained optimization solver while the second is based on an implementation of the Expectation Maximization algorithm. Empirical results show that our proposals often outperform existing approaches based on projected answer set enumeration in terms of quality of the solution and in terms of execution time. The paper has been accepted at the ICLP2024 conference and is under consideration in Theory and Practice of Logic Programming (TPLP).
Paper Structure (13 sections, 10 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 10 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Probabilistic Answer Set Programming
  4. Parameter Learning In Probabilistic Answer Set Programs
  5. Algorithms for Parameter Learning
  6. Extracting Equations from a NNF
  7. Solving Parameter Learning with Constrained Optimization
  8. Solving Parameter Learning with Expectation Maximization
  9. Related Work
  10. Experiments
  11. Datasets Descriptions
  12. Results
  13. Conclusions

Figures (2)

  • Figure 1: Execution times for EM, constrained optimization solved withCOBYLA and SLSQP, and PASTA, by increasing the number of interpretations. For path15 and coloring5 the line for EM is missing since it cannot solve any instance. The initial probabilities for learnable facts are set to 0.5.
  • Figure 2: Execution times of constrained optimization solved withCOBYLA and SLSQP on coloring5, path15, shop12, and smoke6 with 0.1, 0.5, and 0.9 as initial values for the learnable facts.

Theorems & Definitions (5)

  • Example 1
  • Definition 1: Parameter Learning in probabilistic answer set programs
  • Example 2
  • Example 3
  • Example 4