# evolutionary computational intelligence

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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 PresentationTRANSCRIPT

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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