We analyze generalization in XCS with symbolic conditions, based on genetic programming, briefly XCSGP. We start from the results presented in the literature, which showed that XCSGP could not reach optimality in Boolean problems when classifier conditions involved logical disjunctions. We apply a new implementation of XCSGP to the learning of Boolean functions and show that our version can actually reach optimality even when disjunctions are allowed in classifier conditions. We analyze the evolved generalizations and explain why logical disjunctions can make the learning more difficult in XCS models and why our version performs better than the earlier one. Then, we show that in problems that allow many generalizations, so that or clauses are less “convenient”, XCSGP tends to develop solutions that do not exploit logical disjunctions as much as one might expect. However, when the problems allow few generalizations, so that or clauses become an interesting way to introduce simple generalizations, XCSGP exploit them so as to evolve more compact solutions.

1. An Analysis of Generalization in XCS with Symbolic Conditions Pier Luca Lanzi Politecnico di Milano, Italy Illinois Genetic Algorithms Laboratory, University of Illinois at Urbana Champaign, USA CEC 2007, Singapore, September 25-28, 2007

2. One Minute Intro to Classifier Systems Problem Representation Search Evaluate Condition-action rules Genetic Algorithm Online RL

4. Generalization = Evolving maximally accurate, maximally compact solutions Generalization is a major feature of learning classifier systems Generalization relies on the representation The better the representation, the better the generalization

7. Boolean Multiplexer

8. 6-multiplexer: performance

9. 6-multiplexer: number of classifiers

11. Optimal performance reached (thanks to recent new fitness definition) not evolved, why? Overlapping concepts share the total fitness, the more the overlaps, the lower the fitness “ A” has a higher fitness than “A OR B” Only generalization pressure works towards “A OR B” Maximally general solution difficult to evolve, Especially without fitness pressure “ A” has a higher fitness than “A OR B”

12. 11-multiplexer: performance

13. 11-multiplexer: number of classifiers

14. 11-multiplexer: example of population

15. 20-multiplexer: performance

16. 20-multiplexer: number of classifiers

17. 20-multiplexer: example of population

18. EQ5,3

19. Another Boolean Function: EQ 5,3

20. EQ 5,3 : Performance

21. EQ 5,3 : Number of Classifiers

22. EQ 5,3 : example of population

23. Summary

24. Early results showed that XCSGP could not reach optimal performance when “or” clauses are involved Here we showed why disjunctions are more difficult Symbolic conditions are highly overlapping Highly overlapping classifiers have a low fitness When disjunctions are not important, XCS tends to evolve solutions without disjunctions When disjunctions are relevant, XCS tends to use them much more

25. Thank you! Any question?