Brief introduction to genetic algorithms and genetic programming A.E. Eiben Free University...

Brief introduction togenetic algorithms

andgenetic programming

A.E. EibenFree University Amsterdam

A.E. Eiben, GAs and GP 2 EvoNet Summer School 2002

Genetic algorithm(s)

Developed: USA in the 1970’s Early names: J. Holland, K. DeJong, D. Goldberg Typically applied to:

discrete optimization Attributed features:

not too fast good solver for combinatorial problems

Special: many variants, e.g., reproduction models, operators formerly: the GA, nowdays: a GA, GAs


Genotype space = {0,1}LPhenotype space

Encoding (representation)

Decoding(inverse representation)

011101001

010001001

10010010

10010001

Representation


GA: crossover (1)

Crossover is used with probability pc

1-point crossover: Choose a random point on the two parents (same for both) Split parents at this crossover point Create children by exchanging tails

n-point crossover: Choose n random crossover points Split along those points Glue parts, alternating between parents

uniform crossover: Assign 'heads' to one parent, 'tails' to the other Flip a coin for each gene of the first child Make an inverse copy of the gene for the second child


GA: crossover (2)


GA: mutationMutation: Alter each gene independently with a probability pm

Relatively large chance for not being mutated

(exercise: L=100, pm =1/L)


Crossover OR mutation?

Decade long debate: which one is better / necessary / main-background

Answer (at least, rather wide agreement): it depends on the problem, but in general, it is good to have both both have another role mutation-only-EA is possible, xover-only-EA would not work


Exploration: Discovering promising areas in the search space, i.e. gaining information on the problem

Exploitation: Optimising within a promising area, i.e. using information

There is co-operation AND competition between them

Crossover is explorative, it makes a big jump to an area somewhere “in between” two (parent) areas

Mutation is exploitative, it creates random small diversions, thereby staying near (i.e., in the area of ) the parent

Crossover OR mutation? (cont’d)


Only crossover can combine information from two parents

Only mutation can introduce new information (alleles)

Crossover does not change the allele frequencies of the population (thought experiment: 50% 0’s on first bit in the population, ?% after performing n crossovers)

To hit the optimum you often need a ‘lucky’ mutation.

Crossover OR mutation? (cont’d)


Selection Main idea: better individuals get higher chance 2ndary idea: chances proportional to fitness Implementation: roulette wheel technique

Assign to each individual a part of the roulette wheel (“unfair”: size proportional to its fitness)

Spin the wheel n times to select n individuals (fair)

fitness(A) = 3

fitness(B) = 1

fitness(C) = 2

A C

1/6 = 17%

3/6 = 50%

B

2/6 = 33%


Selection (cont’d)

Fitness proportional selection (FPS):

Expected number of times fi is selected for mating is: .

Disadvantages: Outstanding individuals take over the entire population

very quickly danger for premature convergence. Low selection pressure when fitness values are near each

other. Behaves differently on transposed versions of the same

function.

fi

f


Tournament selection: Pick k individuals randomly, without replacement Select the best of these k comparing their fitness values k is called the size of the tournament selection is repeated as many times as necessary

Selection (cont’d)


Generational GA reproduction cycle

1. Select parents for the mating pool

(size of mating pool = population size)

2. Shuffle the mating pool

3. For each consecutive pair apply crossover with probability pc

4. For each new-born apply mutation (bit-flip with probability pm)

5. Replace the whole population by the resulting mating pool


Generational GA reproduction cycle

string1

string2

string3

string4

…

Generation t

string2

string4

string2

string1

…

Mating pool

child1(2,4)

mut(child2(2,4))

string2

mut(string1)

…

Generation t+1

Notes:• Offspring can be: clone, pure mutant, pure crossing, mutated crossing• Generational replacement: whole population deleted/replaced

To be discussed: no survival of the fittest here?


An example after Goldberg ‘89 (1)

Simple problem: max x2 over {0,1,…,31} GA approach:

Representation: binary code, e.g. 01101 13 Population size: 4 1-point xover, no mutation (just an example!) Roulette wheel selection Random initialisation

One generational cycle with the hand shown


The simple GA

Has been subject of many (early) studies Is often used as benchmark for novel GAs Shows many shortcomings, e.g.

Representation is too restrictive Mutation & crossovers only applicable for bit-string & integer

representations Selection mechanism sensitive for converging populations with close

fitness values Generational population model can be improved with explicit survivor

selection


Genetic programming Developed: USA in the 1990’s Early names: J. Koza Typically applied to:

machine learning tasks Attributed features:

competes with neural nets and alike slow needs huge populations (thousands)

Special: non-linear chromosomes: trees, graphs mutation possible but not necessary (disputed!)


MotivationWhy introduce yet another EA?

Reasons: Elements of a search space may vary in length Linear representation may be too ‘unnatural’ Complex variable hierarchy can not be (easily) mapped on linear

structures

Example search space: Graphs without restriction on size and structure

Because fixed length linear representations are too rigid


Credit score example (1)

Given: lot of historical data on: customer profile and creditability index (good/bad).

Needed: a model that classifies good customers. (to be used for evaluating loan applicants)

Data description for customer profiles.


Credit score example (2)

A possible model for classification:IF (NOC = 2) AND (S > 80000) THEN good

In general: IF formula THEN good. Need to find the right formula. Natural representation of formulas is: parse trees

Natural fitness of models: percentage of well classified cases.

AND

S2NOC 80000

>=


GP: representation

Problem domain: modelling (forecasting, regression, classification, data mining, robot control).

Fitness: the performance on a given (training) data set, e.g. the nr. of hits/matches/good predictions

Representation: implied by problem domain, i.e.

individual = model = parse tree parse trees sometimes viewed as LISP expressions

GP = evolving computer programs parse trees sometimes viewed as just-another-genotype

GP = a GA sub-dialect


Replace randomly chosen subtree by a randomly generated (sub)tree

GP: mutation


Exchange randomly selected subtrees in the parents

GP: crossover


GP: selection Standard GA selection is usual

Sometimes overselection to increase efficiency: rank population by fitness and divide it into two groups:

group 1: best c % of population group 2: other 100-c %

when executing selection 80% of selection operations chooses from group 1 20% from group 2

for pop. size = 1000, 2000, 4000, 8000 the portion c is

c = 32%, 16%, 8%, 4% %’s come from rule of thumb


Generating random trees

Given a: Function set F and a terminal set T , both satisfying the closure property.

Trees are randomly generated by: Full method:

Each branch is of length Dmax (pre-specified), nodes with depth < Dmax are from F nodes with depth = Dmax are from T

Grow method: maximum branch length is Dmax (pre-specified)

Ramped half-and-half: for each D Dmax an equal nr. of trees

half of them with full method half of them with grow method


Mutation of trees

Replace randomly chosen subtree by a randomly generated (sub)tree.


Crossover of trees

Exchange randomly selected subtrees in the parents


Standard parameters in GP (1)

Qualitative variables Initialisation: ramped half-and-half.Fitness: adjusted fitness is used.

Selection: fitness proportionate, elitist strategy is not used, over-selection is used for populations of M 1000.

Over selection for population size = 1000: rank population by fitness and divide it into two groups: group 1: best c = 32% of pop, group 2 other 68% 80% of selection operations chooses from group 1, 20% from group 2 for pop. size = 2000, 4000, 8000 the portion c is c = 16%, 8%, 4% motivation: to increase efficiency, %’s come from rule of thumb


Standard parameters in GP (2)

Major numerical parameters Population size M = 500. Maximum number of generations G = 51.

Minor numerical parameters Probability pm of mutation = 0% !!!

Probability pr of reproduction = 10%

Probability pc of crossover = 90%

Probability pip of choosing internal points for xover = 90%

Maximum size Di for initial random S-expressions = 6

Maximum size Dc for S-expressions during the run = 17

… and some “exotic” options usually set at 0 (e.g. permutation, editing, encapsulation, decimation)


Simple symbolic regression (1)

Given a number of sample points in 2:

(x1, y1), (x2, y2), … , (xn, yn)

Find a one-dimensional numerical function f(x):

i {1, …, n} : f(xi) = yi

In the present test 20 sample points are generated by:

x4 + x3 + x2 + x



Specification of the GP for the symbolic regression problem



Graphical representation of an individual (and the benchmark formula x4 + x3 + x2 + x)



Best individual representing a perfect solution:X +(X • (X +(X • (X • (X +(COS(X - X ) - (X - X))))))) =

X + (X • (X + (X • (X • (X + 1))))) =X + X4 + X3 + X2

•

+

+

X

X

•X

• X

X+

X -

-cos

X X-

XX

Brief introduction to genetic algorithms and genetic programming A.E. Eiben Free University...

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