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Inbreeding Properties of Geometric Crossover and Nongeometric Recombinations

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Title: Inbreeding Properties of Geometric Crossover and Nongeometric 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
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
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
Kozas subtree swap crossover
P1
P2P1
C1
C2
Theorem Kozas crossover is non-geometric. Proof
the property of purity does not hold.
22
Daviss order crossover
Theorem Daviss 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, Kozas subtree swap crossover,
    Daviss 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?
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