Inbreeding Properties of Geometric Crossover and Non-geometric Recombinations

1. FOGA 2007 Inbreeding Properties of Geometric Crossover and Non-geometric Recombinations Alberto Moraglio & Riccardo Poli

2. Contents • Geometric Crossover • Non-geometricity • Inbreeding Properties of Geometric Crossover • Non-geometric Recombinations • Implications • Conclusions

3. I. Geometric Crossover

4. y x Geometric Crossover • Metric line segment • A binary operator GX is a geometric crossover under d if all offspring are in the d-segment between its parents • Geometric crossover is function of the metric of the search space

5. Geometric Uniform Crossover All points in the d-segment between parents have the same probability of being offspring

6. B 1 0 1 1 0 1 X A 2 B 1 1 0 1 1 3 A X 0 1 0 1 1 Geometric Crossover • The traditional n-point crossover is geometric under the Hamming distance. H(A,X) + H(X,B) = H(A,B)

7. Many Recombinations are Geometric • Traditional Crossover extended to multary strings (Hamming distance) • Recombinations for real vectors (Euclidean, Manhattan distances) • PMX, Cycle Crossovers for permutations (Swap distance) • Homologous Crossover for GP trees (Structural Hamming distance) • Homologous Crossover for sequences (Edit distance) • Ask me for more examples over a coffee!

8. Geometric crossover is important because…. • Unifies EAs with any solution representation • Simplifies relationship between crossover and fitness landscape • Can be used to design effective crossovers for any problem/representation • Is the starting point for a truly general theory of evolutionary algorithms • These are strong claims: you are welcome to discuss them with me later during the excursion!

9. II. Non-geometricity

10. The non-geometricity question • Many recombination operators are geometric and we do not have an example of non-geometric crossover… • Is any recombination a geometric crossover given a suitable distance? • This is a very important question because on its answer depends the possibility of a general theory of geometric crossover

11. Is proving non-geometricity possible? • Proving Geometricity: by trial and error: select a promising metric d and prove it fits the recombination RX. • If it does, RX is geometric. • If it does not, RX may be geometric under some other metric. So this does not imply that RX is non-geometric. Try with a new distance. • Proving Non-geometricity: it requires to show that it is not definable as geometric crossover for any distance. We cannot use the previous procedure to prove non-geometricity because there are infinitely many distances to rule out!

12. Axiomatic interpretation of the definition of geometric crossover • Without specifying the distance d, the definition of geometric crossover can be treated as an axiomatic object because it is based on d that is an axiomatic object • Properties of geometric crossover deriving from its axiomatic definition are valid for all geometric crossover with any distance d • Proving non-geometricity: if a recombination operator does not respect one or more axiomatic properties of geometric crossover is non-geometric

13. III. Inbreeding Properties

14. Properties Requirements • Implicit distance: Metric properties of geometric crossover must be testable without making explicit use of the distance. We want to test if a distance exist, so we cannot assume its existence a priori. • Generality: Must be usable to test geometricity of a recombination for any solution representation • Partial segment: Must be usable with crossover with any probability distribution and also with crossover whose offspring cover only part of the segment • Do properties respecting these requirements exist? Yes, inbreeding properties based on breading between close relatives

15. Purity Theorem: When both parents are the same P1, their child must be P1.

16. Convergence Theorem: C is the child of P1 and P2 and C is not P1. Then the recombination of C and P2 cannot produce P1.

17. Partition Theorem: C is the child of P1 and P2. Then the recombination of P1 and C and the recombination of C and P2 cannot produce the same offspring unless the offspring is C.

18. IV. Non-geometric Recombinations

19. Non-geometricity and Inbreeding properties It is possible to prove non-geometricity of a recombination operator under any distance, any probability distribution and any represenation producing a single counter-example to any inbreeding property because they must hold for all geometric crossovers. Then if they do not hold, the operator is non-geometric.

20. Extended line crossover C P1 P2 Theorem: Extended line crossover is non-geometric. Proof: the converge property does not hold.

21. Koza’s subtree swap crossover P1 P2=P1 C1 C2 Theorem: Koza’s crossover is non-geometric. Proof: the property of purity does not hold.

22. Davis’s order crossover Theorem: Davis’s order crossover is non-geometric. Proof: the converge property does not hold.

23. V. Implications

24. Knowing the non-geometricity of an operator is good… • Geometricity: Knowing that an operator is non-geometric we are not tempted to prove that it is geometric with one more distance • Fitness landscape: It does not have a simple interpretation in the fitness landscape • Problem match: If you know a “good” distance for a problem the geometric crossover associated with this distance is likely to be good. This analysis cannot be done for non-geometric crossover • Class separation: the mere existence of a single non-geometric recombination implies that there are two distinct classes of recombination operators separated by their metric properties

25. Class Separation and Theory of Everything • the performance of an EAs derives from how its way of searching the search space is matched with some properties of the fitness landscape • if geometric crossover without specifying a distance is synonym of all recombinations • a general theory of geometric crossover would be a theory of random search in disguise because there would be no common condition on the landscape to be found common to all operators to make them work in average better than random search (for NFL) • so the condition on which a specific geometric crossover works well would depend on specific aspects of its specific underlying distance and all geometric crossovers would not work for the same reason! • Since there are non-geometric crossovers, in principle there may exist a general condition on the fitness landscape that does not depend on the specific characteristics of the underlying distance, but only on the fact that it is a metric, that makes them work on average better than random search

26. Toward a general theory • It can be shown using the language of abstract convexity that all EAs with geometric crossovers do a form of abstract convex search • As a rule-of-thumb, the general statistical condition on the fitness landscape that makes convex search better than random search is that of positive spatial autocorrelation of the landscape: closer solutions have stronger fitness correlation. This can be studied rigorously and in full generality using Gaussian random fields over generic metric spaces

27. Summary • Geometric crossover: offspring are in the segment between parents under a suitable distance • Proving non-geometricity is difficult: need to prove non-geometricity under all distances! • Inbreeding properties of crossover (purity, convergence, partition): hold for all geometric crossovers, follow logically from axiomatic definition of crossover only • Imbreeding properties allow us to prove non-geometricity in a very simple way: producing a simple counter-example • Non-geometric recombinations: Extended-line recombination, Koza’s subtree swap crossover, Davis’s order crossover • Foundational implications: • there are two classes of recombination operators separated by metric properties • a general theory of all geometric crossovers makes sense • unification is not a tautology

28. Thank you for your attention!Questions?