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Evolutionary Computational Intelligence. Lecture 4: Real valued GAs and ES. Ferrante Neri University of Jyvskyl. Real valued problems. Many problems occur as real valued problems, e.g. continuous parameter optimisation f : n Illustration: Ackleys function (often used in EC). - PowerPoint PPT Presentation

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  • Evolutionary Computational IntelligenceLecture 4: Real valued GAs and ES Ferrante NeriUniversity of Jyvskyl

    Real valued GAs and ES

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

    Real valued GAs and ES

  • Mapping real values on bit stringsz [x,y] represented by {a1,,aL} {0,1}L

    [x,y] {0,1}L must be invertible (one phenotype per genotype): {0,1}L [x,y] defines the representation

    Only 2L values out of infinite are representedL determines possible maximum precision of solutionHigh precision long chromosomes (slow evolution)

    Real valued GAs and ES

  • Floating point mutations 1General scheme of floating point mutations

    Uniform mutation:

    Analogous to bit-flipping (binary) or random resetting (integers)

    Real valued GAs and ES

  • Floating point mutations 2Non-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 valueMost common method is to add random deviate to each variable separately, taken from N(0, ) Gaussian distribution and then curtail to rangeStandard deviation controls amount of change (2/3 of deviations will lie in range (- to + )

    Real valued GAs and ES

  • Crossover operators for real valued GAsDiscrete: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 uniformIntermediateexploits 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 crossovervariable (e.g. depend on the age of the population) picked at random every time

    Real valued GAs and ES

  • Single arithmetic crossoverParents: x1,,xn and y1,,ynPick a single gene (k) at random, child1 is:

    reverse for other child. e.g. with = 0.5

    Real valued GAs and ES

  • Simple arithmetic crossoverParents: x1,,xn and y1,,ynPick random gene (k) after this point mix valueschild1 is:

    reverse for other child. e.g. with = 0.5

    Real valued GAs and ES

  • Whole arithmetic crossover

    Parents: x1,,xn and y1,,ynchild1 is:

    reverse for other child. e.g. with = 0.5

    Real valued GAs and ES

  • Box crossoverParents: x1,,xn and y1,,ynchild1 is:

    Where is a VECTOR of numers [0,1]

    Real valued GAs and ES

  • Comparison of the crossoversArithmetic 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

    Real valued GAs and ES

  • Evolution StrategiesFerrante NeriUniversity of Jyvskyl

    Real valued GAs and ES

  • ES quick overviewDeveloped: Germany in the 1970sEarly names: I. Rechenberg, H.-P. SchwefelTypically applied to:numerical optimisationAttributed features:fastgood optimizer for real-valued optimisationrelatively much theorySpecial:self-adaptation of (mutation) parameters standard

    Real valued GAs and ES

  • ES technical summary tableau

    Real valued GAs and ES

  • RepresentationChromosomes consist of three parts:Object variables: x1,,xnStrategy parameters:Mutation step sizes: 1,,nRotation angles: 1,, nNot every component is always presentFull size: x1,,xn, 1,,n ,1,, k where k = n(n-1)/2 (no. of i,j pairs)

    Real valued GAs and ES

  • Parent selectionParents 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 selectedNote that in ES parent means a population member (in GAs: a population member selected to undergo variation)

    Real valued GAs and ES

  • MutationMain mechanism: changing value by adding random noise drawn from normal distributionxi = 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

    Real valued GAs and ES

  • Mutate firstNet mutation effect: x, x, Order is important: first (see later how)then x x = x + N(0,)Rationale: new x , is evaluated twicePrimary: x is good if f(x) is good Secondary: is good if the x it created is goodReversing mutation order this would not work

    Real valued GAs and ES

  • Mutation case 0: 1/5 success rulez 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 = / cif ps > 1/5 = cif ps < 1/5 = if ps = 1/5 where ps is the % of successful mutations, 0.8 c < 1

    Real valued GAs and ES

  • Mutation case 1:Uncorrelated mutation with one Chromosomes: x1,,xn, = exp( N(0,1))xi = xi + N(0,1)Typically the learning rate 1/ nAnd we have a boundary rule < 0 = 0

    Real valued GAs and ES

  • Mutants with equal likelihoodCircle: mutants having the same chance to be created

    Real valued GAs and ES

  • Mutation case 2:Uncorrelated mutation with n sChromosomes: 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) and 1/(2 n) And i < 0 i = 0

    Real valued GAs and ES

  • Mutants with equal likelihoodEllipse: mutants having the same chance to be created

    Real valued GAs and ES

  • Mutation case 3:Correlated mutations Chromosomes: x1,,xn, 1,, n ,1,, kwhere k = n (n-1)/2 and the covariance matrix C is defined as:cii = i2cij = 0 if i and j are not correlated cij = ( i2 - j2 ) tan(2 ij) if i and j are correlatedNote the numbering / indices of the s

    Real valued GAs and ES

  • Correlated mutationsThe 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) and 1/(2 n) and 5 i < 0 i = 0 and | j | > j = j - 2 sign(j)

    Real valued GAs and ES

  • Mutants with equal likelihoodEllipse: mutants having the same chance to be created

    Real valued GAs and ES

  • RecombinationCreates one childActs per variable / position by eitherAveraging parental values, orSelecting one of the parental valuesFrom two or more parents by either:Using two selected parents to make a childSelecting two parents for each position anew

    Real valued GAs and ES

  • Names of recombinations

    Real valued GAs and ES

  • Survivor selectionApplied after creating children from the parents by mutation and recombinationDeterministically chops off the bad stuffBasis of selection is either:The set of children only: (,)-selectionThe set of parents and children: (+)-selection

    Real valued GAs and ES

  • Survivor selection(+)-selection is an elitist strategy(,)-selection can forgetOften (,)-selection is preferred for:Better in leaving local optima Better in following moving optimaUsing the + strategy bad values can survive in x, too long if their host x is very fitSelection pressure in ES is very high

    Real valued GAs and ES

  • Survivor SelectionOn the other hand, (,)-selection can lead to the loss of genotypic informationThe (+)-selection can be preferred when are looking for marginal enhancements

    Real valued GAs and ES

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