1. Graph-Based Concept Learning Jesus A. Gonzalez, Lawrence B. Holder, and Diane J. Cook Department of Computer Science and Engineering University of Texas at Arlington Box 19015, Arlington, TX 76019-0015 {gonzalez,holder,cook}@cse.uta.edu http://cygnus.uta.edu/subdue/

2. MOTIVATION AND GOAL l Need for non-logic-based relational concept learner l Empirical and theoretical comparisons of relational learners l Logic-based relational learners (ILP) l FOIL [Quinlan et al.] l Progol [Muggleton et al.] l Graph-based relational learner l SUBDUE

3. SUBDUE KNOWLEDGE DISCOVERY SYSTEM l SUBDUE discovers patterns (substructures) in structural data sets l SUBDUE represents data as a labeled graph. l Vertices represent objects or attributes l Edges represent relationships between objects l Input: Labeled graph l Output: Discovered patterns and instances

4. SUBDUE EXAMPLE object triangle object square on shape 4 instances of InputOutput

5. l Starts with a single vertex and repeatedly expands by one edge l Computationally-constrained beam search l Polynomially-constrained inexact graph matching l Search space is all sub-graphs of input graph l Guided by compression heuristic l Minimum description length SUBDUE’S SEARCH

6. EVALUATION CRITERION MINIMUM DESCRIPTION LENGTH l Minimum Description Length (MDL) principle l The best theory to describe a set of data is the one that minimizes the DL of the entire data set. l DL of the graph: the number of bits necessary to completely describe the graph. l Search for the substructure that results in the maximum compression.

7. CONCEPT LEARNING SUBDUE l Modify Subdue for concept learning (SubdueCL) l Accept positive and negative graphs as input examples l Find substructure describing positive examples, but not negative examples l Learn multiple rules (DNF)

8. CONCEPT LEARNING SUBDUE l Evaluation criteria based on number of positive examples covered without covering negative examples l Substructure value = 1 - Error

9. l Examples in graph format (chess domain): WK = White King WR = White Rook BK = Black King lt = less than adj = adjacent pos = position eq = equal CONCEPT LEARNING SUBDUE EXAMPLE a) Board Configuration b) Graph Representation

10. PRELIMINARY RESULTS Comparison with FOIL and Progol Significance test p for the Vote domain Significance test p for the Chess domain SUBDUE:error = 0.004600 +/- 0.006186 FOIL:error = 0.006600 +/- 0.007183 PROGOL:error = 0.002600 +/- 0.002675 SUBDUE - FOIL =-0.002000 +/- 0.007542 (p=0.211723) SUBDUE - PROGOL =0.002000 +/- 0.004989 (p=0.118354) FOIL - PROGOL =0.004000 +/- 0.007242 (p=0.057322) ANOVA: 0.306232

11. RELATED THEORY l Galois lattice [reference?] l Subdue’s search space is similar to the Galois lattice l Polynomial convergence results for the Galois lattice apply to Subdue l PAC analysis of conceptual graphs [reference?] l Subdue’s representation is a superset of conceptual graphs l PAC sample complexity results for conceptual graphs apply to Subdue

12. CONCLUSIONS l Empirical results indicate Subdue is competitive with ILP systems l More empirical comparisons are necessary l Theoretical results on Galois lattice and conceptual graphs apply to Subdue l Need to identify specific components of the theory directly applicable to Subdue l Expand theories where needed