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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Chapter 4 1

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing 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 2 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing ES technical summary tableau RepresentationReal-valued vectors RecombinationDiscrete or intermediary MutationGaussian perturbation Parent selectionUniform random Survivor selection ( , ) or ( + ) SpecialtySelf-adaptation of mutation step sizes 3 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Introductory example Task: minimimise f : R n R Algorithm: “two-membered ES” using Vectors from R n directly as chromosomes Population size 1 Only mutation creating one child Greedy selection 4 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Introductory example: pseudocde Set t = 0 Create initial point x t = x 1 t,…,x n t REPEAT UNTIL (TERMIN.COND satisfied) DO Draw z i from a normal distr. for all i = 1,…,n y i t = x i t + z i IF f(x t ) < f(y t ) THEN x t+1 = x t ELSE x t+1 = y t FI Set t = t+1 OD 5 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Introductory example: mutation mechanism 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 = / cif p s > 1/5 = cif p s < 1/5 = if p s = 1/5 where p s is the % of successful mutations, 0.8 c 1 6 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Illustration of normal distribution 7 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Another historical example: the jet nozzle experiment 8 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing The famous jet nozzle experiment (movie) 9 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Representation Chromosomes consist of three parts: Object variables: x 1,…,x n Strategy parameters: Mutation step sizes: 1,…, n Rotation angles: 1,…, n Not every component is always present Full size: x 1,…,x n, 1,…, n, 1,…, k where k = n(n-1)/2 (no. of i,j pairs) 10 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Mutation Main mechanism: changing value by adding random noise drawn from normal distribution x’ i = x i + N(0, ) Key idea: is part of the chromosome x 1,…,x n, is also mutated into ’ (see later how) Thus: mutation step size is coevolving with the solution x 11 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing 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 Step-size only survives through “hitch-hiking” Reversing mutation order this would not work 12 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Mutation case 1: Uncorrelated mutation with one Chromosomes: x 1,…,x n, ’ = exp( N(0,1)) x’ i = x i + ’ N(0,1) Typically the “learning rate” 1/ n ½ And we have a boundary rule ’ < 0 ’ = 0 13 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Mutants with equal likelihood Circle: mutants having the same chance to be created 14 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Mutation case 2: Uncorrelated mutation with n ’s Chromosomes: x 1,…,x n, 1,…, n ’ i = i exp( ’ N(0,1) + N i (0,1)) x’ i = x i + ’ i N i (0,1) Two learning rate parameters: ’ overall learning rate coordinate wise learning rate 1/(2 n) ½ and 1/(2 n ½ ) ½ Boundary rule: i ’ < 0 i ’ = 0 15 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Mutants with equal likelihood Ellipse: mutants having the same chance to be created 16 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Mutation case 3: Correlated mutations Chromosomes: x 1,…,x n, 1,…, n, 1,…, k where k = n (n-1)/2 Covariance matrix C is defined as: c ii = i 2 c ij = 0 if i and j are not correlated c ij = ½ ( i 2 - j 2 ) tan(2 ij ) if i and j are correlated Note the numbering / indices of the ‘s 17 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Correlated mutations cont’d The mutation mechanism is then: ’ i = i exp( ’ N(0,1) + N i (0,1)) ’ j = j + N (0,1) x ’ = x + N(0,C’) x stands for the vector x 1,…,x n C’ is the covariance matrix C after mutation of the values 1/(2 n) ½ and 1/(2 n ½ ) ½ and 5° i ’ < 0 i ’ = 0 and | ’ j | > ’ j = ’ j - 2 sign( ’ j ) NB Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is probably the best EA for numerical optimisation, cf. CEC-2005 competition 18 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Mutants with equal likelihood Ellipse: mutants having the same chance to be created 19 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing 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 20 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Names of recombinations Two fixed parents Two parents selected for each i z i = (x i + y i )/2Local intermediaryGlobal intermediary z i is x i or y i chosen randomly Local discreteGlobal discrete 21 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing 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) 22 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Survivor selection Applied after creating children from the parents by mutation and recombination Deterministically chops off the “bad stuff” Two major variants, distinguished by the basis of selection: ( , )-selection based on the set of children only ( + )-selection based on the set of parents and children: 23 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Survivor selection cont’d ( + )-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 Selective pressure in ES is high compared with GAs, 7 is a traditionally good setting (decreasing over the last couple of years, 3 seems more popular lately) 24 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Self-adaptation illustrated Given a dynamically changing fitness landscape (optimum location shifted every 200 generations) Self-adaptive ES is able to follow the optimum and adjust the mutation step size after every shift ! 25 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Self-adaptation illustrated cont’d 26 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Prerequisites for self-adaptation > 1 to carry different strategies > to generate offspring surplus Not “too” strong selection, e.g., 7 ( , )-selection to get rid of misadapted ‘s Mixing strategy parameters by (intermediary) recombination on them 27 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Example application: the cherry brandy experiment Task: to create a colour mix yielding a target colour (that of a well known cherry brandy) Ingredients: water + red, yellow, blue dye Representation: w, r, y,b no self-adaptation! Values scaled to give a predefined total volume (30 ml) Mutation: lo / med / hi values used with equal chance Selection: (1,8) strategy 28 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Example application: cherry brandy experiment cont’d Fitness: students effectively making the mix and comparing it with target colour Termination criterion: student satisfied with mixed colour Solution is found mostly within 20 generations Accuracy is very good 29 / 30

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Evolution Strategies A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Example application: the Ackley function (Bäck et al ’93) The Ackley function (here used with n =30): Evolution strategy: Representation: -30 < x i < 30 (coincidence of 30’s!) 30 step sizes (30,200) selection Termination : after 200000 fitness evaluations Results: average best solution is 7.48 10 –8 (very good) 30 / 30

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