Evolutionary Computational Intelligence Lecture 4: Real valued GAs and ES Ferrante Neri University of Jyväskylä 1 Real valued GAs and ES Real valued problems 2 Many problems occur as real valued problems, e.g. continuous parameter optimisation f : n Illustration: Ackley’s function (often used in EC) Real valued GAs and ES 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 ( a 1 ,..., a L ) x 3 y x 2 1 L L 1 ( a L j 2 ) [ x, y ] j j0 Only 2L values out of infinite are represented L determines possible maximum precision of solution High precision long chromosomes (slow evolution) Real valued GAs and ES Floating point mutations 1 General scheme of floating point mutations x x1 , ..., x l x x1 , ..., x l x i , x i LB i , UB i Uniform mutation: x i drawn randomly 4 (uniform) from LB i , UB i Analogous to bit-flipping (binary) or random resetting (integers) Real valued GAs and ES Floating point mutations 2 Non-uniform mutations: – – – – 5 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 + ) Real valued GAs and ES 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 6 Real valued GAs and ES Single arithmetic crossover 7 • • • Parents: x1,…,xn and y1,…,yn Pick a single gene (k) at random, child1 is: x1 , ..., x k , y k (1 ) x k , ..., x n • reverse for other child. e.g. with = 0.5 Real valued GAs and ES Simple arithmetic crossover • • • Parents: x1,…,xn and y1,…,yn Pick random gene (k) after this point mix values child1 is: x , ..., x , y (1 ) x , ..., y (1 ) x 1 k k 1 k 1 n n • 8 reverse for other child. e.g. with = 0.5 Real valued GAs and ES Whole arithmetic crossover 9 • • Parents: x1,…,xn and y1,…,yn child1 is: • reverse for other child. e.g. with = 0.5 a x (1 a ) y Real valued GAs and ES Box crossover Parents: x1,…,xn and y1,…,yn child1 is: a x (1 a ) y 10 Where α is a VECTOR of numers [0,1] Real valued GAs and ES Comparison of the crossovers 11 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 Real valued GAs and ES Evolution Strategies Ferrante Neri University of Jyväskylä 12 Real valued GAs and ES ES quick overview Developed: Germany in the 1970’s Early names: I. Rechenberg, H.-P. Schwefel Typically applied to: – Attributed features: – – – fast good optimizer for real-valued optimisation relatively much theory Special: – 13 numerical optimisation self-adaptation of (mutation) parameters standard Real valued GAs and ES ES technical summary tableau 14 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 Real valued GAs and ES Representation Chromosomes consist of three parts: – – 15 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) Real valued GAs and ES Parent selection 16 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) Real valued GAs and ES Mutation Main mechanism: changing value by adding random noise drawn from normal distribution x’i = xi + N(0,) Key idea: – – 17 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 first Net mutation effect: x, x’, ’ Order is important: – – Rationale: new x’ ,’ is evaluated twice – – 18 first ’ (see later how) then x x’ = x + N(0,’) 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 Real valued GAs and ES Mutation case 0: 1/5 success rule z values drawn from normal distribution N(,) – – is varied on the fly by the “1/5 success rule”: This rule resets after every k iterations by – – – 19 mean is set to 0 variation is called mutation step size = / 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 Real valued GAs and ES Mutation case 1: Uncorrelated mutation with one 20 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 Real valued GAs and ES Mutants with equal likelihood Circle: mutants having the same chance to be created 21 Real valued GAs and ES 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: – – 22 ’ overall learning rate coordinate wise learning rate 1/(2 n)½ and 1/(2 n½) ½ And i’ < 0 i’ = 0 Real valued GAs and ES Mutants with equal likelihood 23 Ellipse: mutants having the same chance to be created Real valued GAs and ES Mutation case 3: Correlated mutations 24 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 Real valued GAs and ES 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’) – – 1/(2 n)½ and 1/(2 n½) ½ and 5° i’ < 0 i’ = 0 and | ’j | > ’j = ’j - 2 sign(’j) 25 x stands for the vector x1,…,xn C’ is the covariance matrix C after mutation of the values Real valued GAs and ES Mutants with equal likelihood 26 Ellipse: mutants having the same chance to be created Real valued GAs and ES Recombination Creates one child Acts per variable / position by either – – From two or more parents by either: – – 27 Averaging parental values, or Selecting one of the parental values Using two selected parents to make a child Selecting two parents for each position anew Real valued GAs and ES Names of recombinations zi = (xi + yi)/2 Two fixed parents Two parents selected for each i Local intermediary Global intermediary Local zi is xi or yi chosen randomly discrete 28 Global discrete Real valued GAs and ES Survivor selection Applied after creating children from the parents by mutation and recombination Deterministically chops off the “bad stuff” Basis of selection is either: – – 29 The set of children only: (,)-selection The set of parents and children: (+)-selection Real valued GAs and ES Survivor selection (+)-selection is an elitist strategy (,)-selection can “forget” Often (,)-selection is preferred for: – – – 30 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 Real valued GAs and ES Survivor Selection 31 On the other hand, (,)-selection can lead to the loss of genotypic information The (+)-selection can be preferred when are looking for “marginal” enhancements Real valued GAs and ES