Geometric Unification of Evolutionary Algorithms

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Geometric Unification of Evolutionary Algorithms
EvoPhD 2006

• Alberto Moraglio
• amoragn_at_essex.ac.uk

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By Unification I Mean

• EAs Algorithmically irrelevant differences
  name/authorship/solution interpretation/domain of
  application
• EAs Algorithmically relevant differencessolutio
  n representation/genetic operators
• Unification A formal framework that applies to
  all representations

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Contents

• I Geometric Interpretation of Crossover
• II Unification of Major Representations
• III Crossover Principled Design
• IV Unity of Evolutionary Search

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I. Geometric Interpretation of Crossover
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What is crossover?
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Geometric Crossover

• Representation-independent generalization of
  traditional crossover
• Informally all offspring are between parents
• Search space all offspring are on shortest paths
  connecting parents

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Geometric Crossover Distance

• Search Space is a Metric Space d(A,B) length of
  shortest paths between A and B
• Metric space all offspring C are in the segment
  between parents
• C in A,Bd ?? d(A,C)d(C,B)d(A,B)

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Example1 Traditional Crossover

• Traditional Crossover is Geometric Crossover
  under Hamming Distance

Parent1 011101 Parent2 010111 Child
011111
HD(P1,C)HD(C,P2)HD(P1,P2) 1 1
2
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Example2 Blending Crossover

• Blending Crossover for real vectors is geometric
  under Euclidean Distance

ED(P1,C)ED(C,P2)ED(P1,P2)
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Geometric definitions with probability
distributions

• Uniform geometric crossover
• Uniform geometric e-mutation

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Representation independentand formal definition
ofcrossover and mutation in the search space
seen as a geometric space
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II. Unification of Major Representations
Operators
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Minkowski spaces real vectors
Representation real vectors Neighbourhoods
continuous (3 types) Distances Minkowski
distances Implementation algebraic manipulation
of real vector (equation of line passing through
two points) Pre-existing recombination
operators- both blend crossovers and discrete
crossovers fit geometric definition- extended
blend crossovers do not fit
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Hamming spaces binary strings
Representation binary/multary strings Neighbourho
ods bit-flip/site substitution Distances
Hamming distances Implementation symbolic
manipulation of multary strings (mask-based
crossovers) Pre-existing recombination
operators- all binary crossovers fit the
geometric definition
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Cayley spaces - permutations
Representation permutations Neighbourhoods adj.
swap, swap, reversal, insertion Distances
corresponding distances Implementation minimal
permutation sorting by X move algorithms- adj.
swap bubble sort- swap selection sort -
insertion insertion sort - reversal
approximated MPS by reversals (NP-Hard))
Pre-existing recombination operatorsvarious
pre-existing crossover operators are sorting
algorithm in disguise (because sorting
permutations is easier than sorting vectors of
other items)
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Syntactic tree spaces
Representation syntactic tree (lisp
expression) Neighbourhood weighted sub-tree
neighbourhood Distance structural
distance Implementation - sub-tree swap
crossover - common region mask based crossover
Pre-existing recombination operators-
traditional crossover (non-geometric)-
homologous crossover - the geometric framework
can help to clarify what is the landscape and
distance related to homologous crossover and a
distance connected with a geometric crossover
which traditional crossover is an approximation
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Significance of Unification

• Most of the pre-existing crossover operators for
  major representations fit geometric definition
• Established pre-existing operators have emerged
  from experimental work done by generations of
  practitioners over decades
• Geometric crossover compresses in a simple
  formula an empirical phenomenon

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IV. Crossover Principled Design
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Crossover Principled Design

• Domain specific solution representation is
  effective
• Problem for non-standard representations it is
  not clear how crossover should look like
• But given a combinatorial problem you may know
  already a good neighbourhood structure
• Geometric Interpretation of Crossover ? Give me
  your neighbourhood definition and I give you a
  crossover definition

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Crossover Design Example

?
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Non-labelled graph neighbourhood
MOVE Insert/remove an edge Fixed number of
nodes
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Offspring
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V. Is Biological Recombination Geometric?
Yes, come to my other presentation at EuroGP!
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VI. Unity of Evolutionary Search
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Example of evolutionary search
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Abstract convex evolutionary search

• Main result an evolutionary algorithm using
  geometric crossover with any probability
  distribution, any kind of representation, any
  problem, any selection and replacement mechanism,
  does the same search convex search
• Proof based on abstract convexity (axiomatic
  geodesic convexity) and axiomatization of search
  process (abstract search process)

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Nearly Over!
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Summary
Unification (meaning) formally dealing with all
representations at once Geometric Definition
unif. is possible by defining operators
geometrically Unification many interesting
recombinations are geometric Crossover design by
specification of geometric definition to a new
representation General theory using formal
definition only, all EAs do the same search
convex
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Thanks to the Reviewers

• Franz thanks for all your suggestions, Id be
  glad to talk with you over a coffee
• Mario? thanks for the enthusiastic support
• A fan? thanks for warning me that I may be
  victim of a geometric hallucination

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Questions?
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Geometric Crossover Path-relinking

• Meta-heuristic Path-relinking searches on path
  between solutions in the neighbourhood structure
  (not necessarily on a shortest path)
• Geometric crossover can be understood as a
  formalized generalization (to metric spaces) of
  PR that elicits the dual relationship between
  distance and solution representation and gives a
  formal recipe to design new crossover operators
• Formalized it allows theory
• Generalization metric spaces are more general
  than graphs
• Elicits duality syntactic recombination is
  equivalent to neighbourhood search
• Crossover design tells how to build crossovers
  rather than how to search the search space
• Formal recipe it defines exactly what crossover
  is for any representation