Geometric Landscape of Homologous Crossover for Syntactic Trees

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CEC 2005
Geometric Landscape of Homologous Crossover
forSyntactic Trees
Alberto Moraglio Riccardo Poli amoragn,rpoli_at_e
ssex.ac.uk
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Contents

• I Abstract Geometric Operators
• II Geometric Crossover for Syntactic Trees
• III Conclusions

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I. Abstract Geometric Operators
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What is crossover?
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Mutation Nearness

• Mutation is naturally interpreted in terms of
  nearness offspring are near the parent
• Example Binary StringP 0 1 0 1 1 1O 0 1 0
  1 0 1
• NEARNESShd(P,O)1

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Crossover Betweenness

• Crossover is naturally interpreted in terms of
  betweenness offspring are between parents
• Example Binary StringP1 0 1 00 1 0P2 1 1
  01 0 1O 0 1 0 1 0 1hd(P1,P2)4hd(P1,O)3
  hd(O,P2)1
• BETWEENNES P1---O-P2

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Geometric Crossover

• DEFINITION Any crossover for which there is at
  least a distance (metric) such as all offspring
  are between parents is a geometric crossover

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Geometric Crossovers across Representations

• Many recombination operators for the most used
  representations are geometric under suitable
  distance
• BINARY one-point, two-points, uniform crossovers
• REAL VECTORS line, arithmetic, discrete
  (non-geometric extended line)
• PERMUTATIONS PMX, Edge Recombination, Cycle
  Crossover, Merge Crossover (non-geometric order
  crossover)
• SYNTACTIC TREES homologous one-point uniform
  crossovers (non-geometric subtree swap
  crossover)

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Geometric Operators Formalization
BALL All points within distance r from x
SEGMENT All points between x and y
UNIFORM ?-MUTATION offspring z are taken
uniformly within the ball of radius ? from the
parent x
UNIFORM CROSSOVER offspring z are taken
uniformly within the segment between parents x
and y
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II. Geometric Crossover for Syntactic Trees

• Homologous Crossover (HC)
• Hyperschema (HS)
• Structural Hamming Distance (SHD)
• HC is geometric under SHD via HS

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One-point (Homologous) Crossover

• Alignment align trees at the root
• Common Region consider common topology
• Common Crossover Point select the same
  crossover point for the two trees within the
  common region
• Subtree Swap
• Restricted restriction of subtree swap crossover

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General Homologous Crossover (HC)

• Alignment align trees at the root
• Common Region common trees topology
• Crossover Mask generate crossover mask over
  common region
• Swap swap nodes within the common region and
  swap subtrees on the boundaries of the common
  region

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HC example - Parent Trees
Blue Parent
Red Parent
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All offspring under HC
0
Common Region black tree structure Crossover
Mask over common region Within Common Region
Node swap (e.g. x2, y2) Boundary Common Region
Subtree swap (e.g. x5. y5)
0
1
1
0
0
0
0
1
1
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Hyperschema
Hyperschema common region tree structure
wildcards Wildcard different nodes same
arity (replace node) Wildcard different
arity (replace subtree)
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Structural Hamming Distance (SHD)

• Recursive Bounded by 1
• Trees have different root arity d1
• Trees have same structure all different nodes
  d1
• SHD is a METRIC

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SHD Hyperschema
PROPERTY SHD is function of the Hyperschema
only d(p1,p2)g(h(p1,p2))
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HC is geometric under SHD

• TO PROVE shd(P1,O)shd(O,P2)shd(P1,P2)
• HYPERSCHEMA set of all offspring
• WILDCARD marginal contribution to total distance
• MARGINAL BETWENNESS for any wildcard an
  offspring equals one parent or the
  other?offsrping are marginally between parents
• WILDCARDS CONTRIBUTIONS ARE INDEPENDENT
  ADDITIVE
• HENCE offsrping are between parents also for the
  total distance

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III. Conclusions
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More Results in the paper!

• TRADITIONAL CROSSOVER subtree swap crossover is
  not geometric
• SPACE STRUCTURE SHD is connected to a fluid
  (non-graphic) neighbourhood structure
• MUTATION SHD is connected with subtree mutation
• LANDSCAPE when trees are interpreted as GP
  programs SHD gives rise to a smooth landscape
  hence homologous crossover is a good choice

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Moral (take home message)

• This result unifies syntactic trees in the
  context of geometric framework, together with
  binary strings, real vectors and permutations.
• Hence, the geometric definition of crossover
  captures in a single formula the notion of
  crossover matured over last two decades of
  research.
• As implications, the geometric unification
• simplifies and clarifies the connection between
  crossover and search space
• gives firm fundations for a general theory of
  evolutinary algorithm
• suggests an automatic way to do crossover
  design for new representations

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Thank you for your attention Questions?