Evolutionary Computational Intelligence

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Evolutionary Computational Intelligence

• Lecture 4
• Real valued GAs and ES

Ferrante Neri University of Jyväskylä
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Real valued problems

• Many problems occur as real valued problems, e.g.
  continuous parameter optimisation f ? n ? ?
• Illustration Ackleys function (often used in
  EC)

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Mapping real values on bit strings

• z ? x,y ? ? represented by a1,,aL ? 0,1L
• x,y ? 0,1L must be invertible (one phenotype
  per genotype)
• ? 0,1L ? x,y defines the representation
• Only 2L values out of infinite are represented
• L determines possible maximum precision of
  solution
• High precision ? long chromosomes (slow evolution)

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Floating point mutations 1

• General scheme of floating point mutations
• Uniform mutation
• Analogous to bit-flipping (binary) or random
  resetting (integers)

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Floating point mutations 2

• Non-uniform mutations
• Many methods proposed,such as time-varying range
  of change etc.
• Most schemes are probabilistic but usually only
  make a small change to value
• Most common method is to add random deviate to
  each variable separately, taken from N(0, ?)
  Gaussian distribution and then curtail to range
• Standard deviation ? controls amount of change
  (2/3 of deviations will lie in range (- ? to ?)

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Crossover operators for real valued GAs

• Discrete
• each allele value in offspring z comes from one
  of its parents (x,y) with equal probability zi
  xi or yi
• Could use n-point or uniform
• Intermediate
• exploits idea of creating children between
  parents (hence a.k.a. arithmetic recombination)
• zi ? xi (1 - ?) yi where ? 0 ? ? ? 1.
• The parameter ? can be
• constant uniform arithmetical crossover
• variable (e.g. depend on the age of the
  population)
• picked at random every time

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Single arithmetic crossover

• Parents ?x1,,xn ? and ?y1,,yn?
• Pick a single gene (k) at random,
• child1 is
• reverse for other child. e.g. with ? 0.5

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Simple arithmetic crossover

• Parents ?x1,,xn ? and ?y1,,yn?
• Pick random gene (k) after this point mix values
• child1 is
• reverse for other child. e.g. with ? 0.5

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Whole arithmetic crossover

• Parents ?x1,,xn ? and ?y1,,yn?
• child1 is
• reverse for other child. e.g. with ? 0.5

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Box crossover

• Parents ?x1,,xn ? and ?y1,,yn?
• child1 is
• Where a is a VECTOR of numers 0,1

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Comparison of the crossovers

• Arithmetic crossover works on a line which
  connects the two parents
• Box crossover works in a hyper-rectangular where
  the two parents are located in the vertexes

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Evolution Strategies
Ferrante Neri University of Jyväskylä
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ES quick overview

• Developed Germany in the 1970s
• Early names I. Rechenberg, H.-P. Schwefel
• Typically applied to
• numerical optimisation
• Attributed features
• fast
• good optimizer for real-valued optimisation
• relatively much theory
• Special
• self-adaptation of (mutation) parameters standard

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ES technical summary tableau
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Representation

• Chromosomes consist of three parts
• Object variables x1,,xn
• Strategy parameters
• Mutation step sizes ?1,,?n?
• Rotation angles ?1,, ?n?
• Not every component is always present
• Full size ? x1,,xn, ?1,,?n ,?1,, ?k ?
• where k n(n-1)/2 (no. of i,j pairs)

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Parent selection

• Parents are selected by uniform random
  distribution whenever an operator needs one/some
• Thus ES parent selection is unbiased - every
  individual has the same probability to be
  selected
• Note that in ES parent means a population
  member (in GAs a population member selected to
  undergo variation)

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Mutation

• Main mechanism changing value by adding random
  noise drawn from normal distribution
• xi xi N(0,?)
• Key idea
• ? is part of the chromosome ? x1,,xn, ? ?
• ? is also mutated into ? (see later how)
• Thus mutation step size ? is coevolving with the
  solution x

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Mutate ? first

• Net mutation effect ? x, ? ? ? ? x, ? ?
• Order is important
• first ? ? ? (see later how)
• then x ? x x N(0,?)
• Rationale new ? x ,? ? is evaluated twice
• Primary x is good if f(x) is good
• Secondary ? is good if the x it created is
  good
• Reversing mutation order this would not work

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Mutation case 0 1/5 success rule

• z values drawn from normal distribution N(?,?)
• mean ? is set to 0
• variation ? is called mutation step size
• ? is varied on the fly by the 1/5 success rule
• This rule resets ? after every k iterations by
• ? ? / c if ps gt 1/5
• ? ? c if ps lt 1/5
• ? ? if ps 1/5
• where ps is the of successful mutations, 0.8 ?
  c lt 1

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Mutation case 1Uncorrelated mutation with one ?

• Chromosomes ? x1,,xn, ? ?
• ? ? exp(? N(0,1))
• xi xi ? N(0,1)
• Typically the learning rate ? ? 1/ n½
• And we have a boundary rule ? lt ?0 ? ? ?0

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Mutants with equal likelihood

• Circle mutants having the same chance to be
  created

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Mutation case 2Uncorrelated mutation with n ?s

• Chromosomes ? x1,,xn, ?1,, ?n ?
• ?i ?i exp(? N(0,1) ? Ni (0,1))
• xi xi ?i Ni (0,1)
• Two learning rate parameters
• ? overall learning rate
• ? coordinate wise learning rate
• ? ? 1/(2 n)½ and ? ? 1/(2 n½) ½
• And ?i lt ?0 ? ?i ?0

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Mutants with equal likelihood

• Ellipse mutants having the same chance to be
  created

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Mutation case 3Correlated mutations

• Chromosomes ? x1,,xn, ?1,, ?n ,?1,, ?k?
• where k n (n-1)/2
• and the covariance matrix C is defined as
• cii ?i2
• cij 0 if i and j are not correlated
• cij ½ ( ?i2 - ?j2 ) tan(2 ?ij) if i and
  j are correlated
• Note the numbering / indices of the ?s

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Correlated mutations

• The mutation mechanism is then
• ?i ?i exp(? N(0,1) ? Ni (0,1))
• ?j ?j ? N (0,1)
• x x N(0,C)
• x stands for the vector ? x1,,xn ?
• C is the covariance matrix C after mutation of
  the ? values
• ? ? 1/(2 n)½ and ? ? 1/(2 n½) ½ and ? ? 5
• ?i lt ?0 ? ?i ?0 and
• ?j gt ? ? ?j ?j - 2 ? sign(?j)

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Mutants with equal likelihood

• Ellipse mutants having the same chance to be
  created

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Recombination

• Creates one child
• Acts per variable / position by either
• Averaging parental values, or
• Selecting one of the parental values
• From two or more parents by either
• Using two selected parents to make a child
• Selecting two parents for each position anew

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Names of recombinations
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Survivor selection

• Applied after creating ? children from the ?
  parents by mutation and recombination
• Deterministically chops off the bad stuff
• Basis of selection is either
• The set of children only (?,?)-selection
• The set of parents and children (??)-selection

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Survivor selection

• (??)-selection is an elitist strategy
• (?,?)-selection can forget
• Often (?,?)-selection is preferred for
• Better in leaving local optima
• Better in following moving optima
• Using the strategy bad ? values can survive in
  ?x,?? too long if their host x is very fit
• Selection pressure in ES is very high

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Survivor Selection

• On the other hand, (?,?)-selection can lead to
  the loss of genotypic information
• The (??)-selection can be preferred when are
  looking for marginal enhancements