Geometric Landscape of Homologous Crossover for Syntactic Trees Alberto Moraglio & Riccardo Poli...
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![Page 1: Geometric Landscape of Homologous Crossover for Syntactic Trees Alberto Moraglio & Riccardo Poli {amoragn,rpoli}@essex.ac.uk CEC 2005.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d2a5503460f949ff70d/html5/thumbnails/1.jpg)
Geometric Landscape of Homologous Crossover for
Syntactic Trees
Alberto Moraglio & Riccardo Poli
{amoragn,rpoli}@essex.ac.uk
CEC 2005
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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?
CrossoverIs there any
common
aspect ?
Is it possible to give a
representation-
independent definition
of crossover and mutation?
100000011101000
100111100011100
100110011101000
100001100011100
Binary Strings
Permutations
Real Vectors
Syntactic Trees
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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
• NEARNESS:hd(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 0|0 1 0P2 = 1 1 0|1 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
|),(|
)),((}|Pr{)|(
xB
xBzxPzUMxzfUM
}),(|{);( ryxdSyrxB
)},(),(),(|{];[ yxdyzdzxdSzyx
BALL: All points within distance r from x
SEGMENT: All points between x and y
|],[|
]),[(}2,1|Pr{),|(
yx
yxzyPxPzUXyxzfUX
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 HCCommon 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
10
0 1 0
1 0 01
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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 d=1 • Trees have same structure & all different nodes d=1• SHD is a METRIC
mqarityparityiftsdistqphdm
qarityparityif
qarityparityifqphd
TTdist
miii )()( ),(),(
1
1
)()( 1
0)()( ),(
),(
,1
21
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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 otheroffsrping 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?