Inbreeding Properties of Geometric Crossover and Nongeometric Recombinations

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

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

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I. Geometric Crossover
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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

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

• The traditional n-point crossover is geometric
  under the Hamming distance.

H(A,X) H(X,B) H(A,B)
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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!

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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!

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II. Non-geometricity
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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

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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!

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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

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III. Inbreeding Properties
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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

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Purity
Theorem When both parents are the same P1, their
child must be P1.
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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.
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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.
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IV. Non-geometric Recombinations
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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.

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Extended line crossover
C
P1
P2
Theorem Extended line crossover is
non-geometric. Proof the converge property does
not hold.
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Kozas subtree swap crossover
P1
P2P1
C1
C2
Theorem Kozas crossover is non-geometric. Proof
the property of purity does not hold.
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Daviss order crossover
Theorem Daviss order crossover is
non-geometric. Proof the converge property does
not hold.
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V. Implications
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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

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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

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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

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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

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Thank you for your attention!Questions?