real valued gas and es 1 evolutionary computational intelligence lecture 4: real valued gas and es...
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![Page 1: Real valued GAs and ES 1 Evolutionary Computational Intelligence Lecture 4: Real valued GAs and ES Ferrante Neri University of Jyväskylä](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d635503460f94a45c52/html5/thumbnails/1.jpg)
Real valued GAs and ES1
Evolutionary Computational Intelligence
Lecture 4:
Real valued GAs and ES
Ferrante Neri
University of Jyväskylä
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Real valued GAs and ES2
Real valued problems
Many problems occur as real valued problems, e.g. continuous parameter optimisation f : n
Illustration: Ackley’s function (often used in EC)
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Real valued GAs and ES3
Mapping real values on bit strings
z [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 represented L determines possible maximum precision of solution High precision long chromosomes (slow evolution)
],[)2(12
),...,(1
01 yxa
xyxaa j
L
jjLLL
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Real valued GAs and ES4
Floating point mutations 1
General scheme of floating point mutations
Uniform mutation:
Analogous to bit-flipping (binary) or random resetting
(integers)
ll xxxx xx ..., , ...,, 11
iiii UBLBxx ,,
iii UBLBx , from (uniform)randomly drawn
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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
nkkk xxyxx ..., ,)1( , ..., ,1
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Real valued GAs and ES8
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
nx
kx
ky
kxx )1(
ny ..., ,
1)1(
1 , ..., ,
1
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• Parents: x1,…,xn and y1,…,yn• child1 is:
• reverse for other child. e.g. with = 0.5
Whole arithmetic crossover
yaxa )1(
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Box crossover
Parents: x1,…,xn and y1,…,yn child1 is:
Where α is a VECTOR of numers [0,1]
yaxa )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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Real valued GAs and ES12
Evolution Strategies
Ferrante Neri
University of Jyväskylä
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Real valued GAs and ES13
ES quick overview
Developed: Germany in the 1970’s 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
Representation Real-valued vectors
Recombination Discrete or intermediary
Mutation Gaussian perturbation
Parent selection Uniform random
Survivor selection (,) or (+)
Specialty Self-adaptation of mutation step sizes
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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 GA’s: a population member selected to undergo variation)
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Real valued GAs and ES17
Mutation
Main mechanism: changing value by adding random noise drawn from normal distribution
x’i = 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 > 1/5
= • c if ps < 1/5
= if ps = 1/5
where ps is the % of successful mutations, 0.8 c < 1
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Real valued GAs and ES20
Mutation case 1:Uncorrelated mutation with one
Chromosomes: x1,…,xn, ’ = • exp( • N(0,1)) x’i = xi + ’ • N(0,1) Typically the “learning rate” 1/ n½
And we have a boundary rule ’ < 0 ’ = 0
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Mutants with equal likelihood
Circle: mutants having the same chance to be created
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Real valued GAs and ES22
Mutation case 2:Uncorrelated mutation with n ’s
Chromosomes: x1,…,xn, 1,…, n ’i = i • exp(’ • N(0,1) + • Ni (0,1)) x’i = 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’ < 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 3:Correlated 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 - j
2 ) • tan(2 ij) if i and j are correlated
Note the numbering / indices of the ‘s
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Real valued GAs and ES25
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’ < 0 i’ = 0 and
| ’j | > ’j = ’j - 2 sign(’j)
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Real valued GAs and ES26
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
Two fixed parents
Two parents selected for each i
zi = (xi + yi)/2 Local intermediary
Global intermediary
zi is xi or yi chosen randomly
Local
discrete
Global
discrete
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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