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

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

2 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

3 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

4 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

5 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

6 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

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

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

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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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

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

27 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

28 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

29 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

30 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 fitness evaluations Results: average best solution is –8 (very good) 30 / 30


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