CS6665 9 Genetic Alg

8/6/2019 CS6665 9 Genetic Alg

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Golden Rule - 1y Every EA solution should be viewed as based on the

following generic hypothesis

y The optimum value(s) of ----- can be found using a---- evolutionary approach.

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One of the most important parts of an assignmentwill be your analysis and presentation of your results.

EAs are stochastic algorithms, and therefore, yourresults with the same approach may be different fromsomeone elses. Always, clearly and carefully analyze

your results.

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Golden Rule - 2y There must be a reason for every choice made in your

EA

yParameter value choicesyStopping criteria

yDecision to re-run the EA

yDecision to mutate and re-run the EA

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Parametersy If you have chosen a value for a parameter, you

must have a reason for that choice state it.y When in doubt as to a good reason or method to choose

a parameter value, try a few different values and use thebest as determined empirically as the reason. Here isan example of the wrong way to do it:

y I set the normal mutation rate to 0.1.

y W

hat is normal? Is there an abnormal value?W

hy 0.1? Is 0.1a probability per bit or per chromosome for mutation?

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Stopping Criteriay In many (most) cases, when executing an EA, you will

not know the optimum solution. Thus, the questionarises as to why you stopped the EA. Always give your

stopping criteria and why you used it. An example of abad choice is:

y Stopped at 2000 generations and the convergence rate was 0.99 Say what?

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Bottom Line - Summaryy This is a computer SCIENCE class. Many of your home

works will be experiments act like scientists, i.e.,y Be systematic, i.e. have a plan (hypothesis), follow it,

and document what you did (keep a record for the write-up)

y A picture (plot or table) is worth a thousand words (abstraction)

y Everything that happens has a reason. If the occurrence

is of significance, then why did it happen?

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BASICSyEvolutionary Algorithm (EA) vs

Evolutionary Computation (EC)

yEA and EC are somewhat synonymous

yEC is simply a computer-basedimplementation of an EA

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Evolutionaryy What makes an algorithm evolutionary?

y Has a population

y Generates a new population from the old population generations/biological

y The more fit a solution (individual), the more likely itwill be used in or influence the generation of the newpopulation

y Fit implies the presence of a fitness measure

y The search process, while influenced by fitness has acomponent of randomness stochastic/mutation

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REMEMBER!!y Whenever a stochastic process is used to solve a

problem, if the correct or optimal solution is

unknown, you should not report the result of a singlerun. Always report results as an average of several runs,or the best of several runs.

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AI & Evolutionary ComputationyEvolutionary Computation

yMachine learning optimizationand classification paradigms basedon evolutionary (biological)mechanisms

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AI & Evolutionary ComputationyEC is about self-organizationy

Simple processes that lead tocomplex results, e.g. a geneticalgorithm (GA) is simple

yThe whole is > the sum of its parts

yThere is no conservation ofsimplicity Wolfram

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Applications

y Consider the following two examples of optimizationproblems. By consider, I mean think ofy Writing a computer program to solve each

y The time for the program to runy How you will know when you have the optimum

solution?y Optimization of some function of n variables that is

not everywhere differentiable with some set ofconstraints on those n variables

y The traveling salesperson problem

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Application Function Optimizationy Lets say that we have some function

F(x,y,z) = 2x + 3y/x2 - 2z2/(x-y)2

and there are bounds on the ranges of allowedvalues for x, y, and z

An optimization problem might be to find the values ofx, y, and z which gives the minimum value of F.

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Howy do I write the program?

y long will it take to execute?

ywill I know when I have the optimum?y do I determine the relative quality of a solution

y If one set of values for F gives a value less than all ofthe other values that I have tested, is it the optimum?

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Application TSP Optimal pathy For a given table of costs find the minimum cost

transit that visits every city once and only once

y NP- complete

y Grows as O(n!)

y Can be solved using dynamic programming in O(2n n )and spaceO(2n n2 ) (better in time but worse in space

y For 30 cities there are ~2.7X1032 possibilities

y exhaustive search anyone?

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Howy do I write the program?

y long will it take to execute?

ywill I know when I have the optimum?y do I determine the relative quality of a solution

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Examplesy The preceding 2 examples illustrate some of the

problems associated with computerizing analgorithmy Time to solve (TSP)

y

Fitness (F(x,y,z) and TSP)y What is an optimum?

y How do we compare one solution with another?

y Illegal solutions how do we know if a solution isillegal (parameter out of bounds or multiple visits to

the same city) and what next?y How do we code it?

y Are there parts of one coded solution that can be used inanother solution? IMPORTANT!

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Robustnessand Reusabilityy One of the positives of EAs is their robustness.

y An EA used to solve one type of problem, may be useful

in solving a very different problem with little change tothe algorithm

y e.g. Matlabs Optimtool

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Searching the

Search

Space

y Optimization problems can be thought of as searchproblems.y Each solution occupies a position in the search

space.y The parameters and functions of the solution represent

the search space

y The real question is how do we search or travel inthis space?

y Thought of in another way, one can ask, given acurrent location in the space, how do we decidey Are we at the solution?

y If were not there, how do we decide where to go next?

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Searching the

Search

Space

Are we at the solution?

If were not there, how do we decide where to go next?

y There are many approaches to answer thesequestions. We will emphasize the ECapproaches.

y In the search process, we would like a technique

such that each time we try a solution, we gainuseful information that can be used in thechoice of the next solution

y This eliminates a completely random approach

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Example

yFor the problem 4+x = 10, what doesa value of x = 10 tell us?

yT

he error = ?yThe sign of the error tells us whichway to go

yThe magnitude of the error tells ushow far to go

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SGAyAn SGA (Simple Genetic Algorithm) uses a biological

or evolutionary approach to traverse the search space.

y

It was developed in the 70y Chromosomes are encoded as binary strungs

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Define Chromosome Encoding

Scheme

Generate initial population of chromosomes

Compute Chromosome Fitness

Mate Chromosomes

Perform Crossover

Apply Mutation

Stopping Criteria met

No

DoneYes

Define Chromosome Encoding

And Fitness Schemes

Perform Replacement


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Warning!y In order to get people started, we must fix the

techniques used. This is only a very small snapshot of

GAs let alone EC. However, this will give you sufficientbackground to complete the assignment.

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The Problem

y In order to have something tangible to refer to, wewill deal with the following optimization problem

y The problem is to optimize the following function:(-1


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Chromosomesand Encodingy GAs work from an encoding of the parameter space.

y They also work with a population of solutions rather

than a single solution

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Define the ChromosomeE

ncodingScheme

y Chromosome: A sequence of genes in which each gene

represents an encoding for a particular parameter valuey Gene: (Often) a binary string (of bits) which encodes a

particular parameter value

y Encoding Scheme (must be reversible):y It is a translation (formula) such that given a gene (string

of bits), one can determine the represented parametervalue.

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Chromosome Encoding Scheme

y For this problem, since there is only one parameter,thus, the chromosome is made up of a single gene.

y The gene value range is given in the problem statementas -1


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ChromosomeE

ncodingS

chemey Defining an encoding scheme also defines a decoding

scheme

Value(gene) = -1+(gene value)*(3/511)

e.g.

y if gene = 000000111

ygene value = 7 ( i.e. decimal equivalent of binary000000111)

yValue(gene)= -1+7*3/511 = -0.9589041.

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Generate initial population of

chromosomes

yAn initial population is simply a randomlygenerated set of chromosomes (possible solutions)

y Since a chromosome can be viewed as a binarystring, generating a random chromosome simplymeans generating a random string of 1s and 0s foreach chromosome of the population

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Generate initial population ofchromosomes

y IMPORTANT NOTE

y In the flow chart, nothing is said about the size of thepopulation. There is no algorithm for determining thesize of the population, only a few rules of thumb

y Generally

y as the search space size increases, the population sizewill (should) increase

y As the complexity of the fitness calculation functionincreases the population size will (should) decrease

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Generate initial population of

chromosomes

y For binary chromosomes of fixed length n, the sizeof the search space is simply 2n

y In this problem, the search space size is relativelysmall

y Since there are only 9 bits 29 = 512.

y

thus we will use a population size of n=10y The initial population will be generated as 10 random

binary strings, each of length 9

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Compute Chromosome Fitness

y If the first chromosome in the population was

000000111yX=000000111=>7

Value(X)= -1+7*3/511 = -0.9589041

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5130110sin !! xxXF T


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Mate Chromosomesy There are several schemes used to determine which

chromosomes to mate.

y

Generally some stochastic process is used to determinewhich two chromosomes to mate

yA technique called Roulette Wheel is a fairlycommon technique

y

Its positive is that it is simple to implementy Its negative is that it may not maintain diversity in the

population

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Maintaining Population Diversityy With roulette wheel mating, over time, as some

chromosomes get more fit, they will tend tooverwhelm the less fit chromosomes and thus be theonly ones selected for mating.

y In a worst case scenario, you may get to a single orduplicate chromosome for the entire population.When this happens, the only searching that occurs willbe through mutation.

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RouletteWheel Selectiony Compute the relative fitness of each chromosome as a

fraction of the total fitness.

y

e.g. if there are 3 chromosomes of fitness C1=10,C2=20, and C3=30.

y Relative fitness R1=10/60=0.167, R2=20/60 = 0.333,R3=30/60=0.5

y

Note this is sort of like another probability densityfunction

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RouletteWheel Selectiony Using these three relative fitness values, now picture a

roulette wheel with a 1 unit circumference which isdivided as follows

y 0


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RouletteWheel Selectiony With a population of 10, we need 5 mating pairs.

y Each mating pair will generate 2 offspring

yWhat if both mates are the same?ySelect another mate for one of the

duplicates

yOR

yMate these two- mutation may makechanges.

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ApplyMutationy For every bit of every chromosome generate a random

number from 0 to 1

y If the random value is < mutation rate, then flip thebit; otherwise, leave it unchanged

y Note that for a long chromosome (several bits), even asmall mutation rate will generally change at least one

bit

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Questiony If a chromosome is 40 bits in length, and the mutation

rate is 0.02, what is the probability that at least one bitwill change under mutation?

y P( at least 1 bit change) = 1 P(no changes)y P(no changes) = 0.9840 = 0.446

y Thus, the probability of at least one bit change is 0.554

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Perform ReplacementyAt this point we have two populations

y Population A is the original population, i.e. the parents

y Population B is the offspring population, i.e. thechildren

y The goal is to come up with a population thaty Is diverse

y Is an improvement

y

At least keeps the best of population

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Perform ReplacementySome possible approaches

yCombine (sort) population A&B and thenkeep only the best half

yReplace parent(s) only if child is better

ySimply replace parents with children, but

keep best of population to date as aseparate chromosome

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Continuey Now simply continue by going back and computing

the fitness of each chromosome (its probably already

computed, since we did that for mate selection),y With each new generation, continue until the stopping

criteria has been met

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Terminology

y Chromosome is made up of (can be dividedinto)genes. One can think of a gene asencoding a trait. The different possible settingsfor a gene are called alleles. Each gene is

located (generally) at a particular locus(position) on the chromosome. The completecollection of genetic material (all chromosomesfor a biological entity taken together) is called

thegenome. Two individuals that have thesame genomes are said to be of the same

genotype.

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Terminologyy Search space: Collection of candidate solutions

y Similarity or Distance between twochromosomes

y For binary its often the number of changes thatmust occur in A to make it the same as B(Hammingdistance)

y For non-binary Euclidean distance or Euclidean

distance squared

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nnbababaD !


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Whats best?y A GA consists of

y Encoding

y Initial population:

y

Population generation techniquey Population size

y Mate selection algorithm

y Crossover technique (single point, tw0o-point, uniform)

y Mutation scheme and rate

y Replacement scheme

y Fitness function max or min

y Stopping criteria

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Whats best? EncodingyGenerally

yA scheme that does not allow for illegalsolutions

yA scheme that minimizes the searchspace size

yBinary

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yIts all about the encoding

scheme and the fitnessfunction!

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GAExample

Coloring a grid with a GA

y For this problem, there are ~8.47X1011 (i.e. 325 ) possiblecolorings of which about 1000 are correct.

y Whats the probability of randomly generating a legalcoloring?

y ~103 /1011 = 10-8 => like rolling the same number on a rollof the die 10 times in a row.

y What encoding scheme should we use?

y What fitness function should we use?

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OtherMate Selection SchemesyMost of the mate selection schemes try

to:

yReduce stochastic errors, oryRetain best solution(s) to date

(elitist), or

yMaintain a diverse population ofsolutions, or

ySpeed the mate selection process60


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Mate Selection Reduce Stochastic

ErrorsyAre stochastic errors inevitable?

y Consider 4 chromosomes with relative fitness of

.25, .3, .4, and .05y Probabilistically, chromosome 2 (0.3 relative

fitness) should be chosen 1.2 times out of the 4choices.

yObviously, this cant be how can you get .2selections?

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Elitist Schemes

y For some schemes, one could fail to choose the bestchromosome in the population

yHow could this happen with roulette

wheel selection?yHow could this happen with remainder

stochastic sampling?

y

Could not happen

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Elitist Schemes

yTake the best of population to dateand copy it into the next generation

yMeans that over time the

population size will growyOnly replace a parent with a child if

the child is more fityThis is hill climbing because average

fitness of population always increasesor stays the same

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Tournament Selection forDiversity

yA popular reproduction scheme to maintaindiversity is called tournament selection It worksas follows

y Randomly select two (or n) solutions and the bestgoes into the mating pool.

y Selection is systematic, i.e. allow each solution tobe selected exactly twicey

Best solution in population will win twice, etc.y Any solution will have 0, 1, or two copies in the pool

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Tournament Selection

y Once the pool is created

yRandomly select a pair to mate andcontinue until all pairs have been

selectedy Deb and Goldberg claim this technique is at least as

good as any other method in terms ofcomputational complexity and speed of

convergence

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CrossoveryMate selection doesnt create any new

solutions

yMate selection only creates duplicates

yCreation of new solutions (search of

solution space) is done by crossoverand mutation

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Crossover

y Single Point Crossover(SPC) with equal lengthchromosomes

For a binary chromosome of length n bits, generate a

random number x such that1< x = n

For SPC, swap bits x through n of the twochromosomes (bits are numbered 1 through n)

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2-Point Crossovery Like single point except select two points in each

chromosome, e.g. bits 10 and 25. Then swap bits 10-25of the two chromosomes

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Uniform Crossovery In uniform, each bit of the children is a random

sample from the parents

For k=1 to number of bits in chromosome

For each bit k of the two offspring, generate a randomnumber r

if r


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Uniform CrossoveryPro

yBetter maintains diversity

y Generally some elitist scheme is used for childreplacement of parent

yCon

yCan tend to lose good chromosomes

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Mutationy Crossover does most of the searching, even though

mutation does some

y

Mutations main job is to maintain or introducediversity

y Sometimes the mutation rate is increased whenpopulation diversity gets low Stir things up

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Mutation


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Mutation

yMutation is generally performed after crossover,bit-by-bit, on each of the new chromosomes.

Given:chromosome length = n and probability ofmutation of p

m

ForI= 1 to ngenerate a random number Xif X


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Mutation

y

A problem with the bit-by-bit mutation method isthat it requires the generation of a random numberfor every bit of every chromosome

y Goldberg(1989) suggested a mutation clockoperator to reduce the complexity of the mutation

operationy After the first bit is mutated, the location of the next

bit to mutate is created as a random function, i.e.skip some number of bits

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Mutation


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MutationyA question that sometimes arises is what happens to

mutation if rather than a binary chromosome we havesay a ternary chromosome?

yEach chromosome element is a 0, 1, or 2

yWe cant simply complement

yCan we mutate say a 2 to a 2 now?yWhats the difference between randomly

choosing any one of the three possible

values versus only one of the two differentvalues?

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Next Generation

y Once a complete generation has been createdthe old population must in some way bemerged with the new.

y T

o replace all of old with new is calledgenerational replacement

y Remember, this replacement is not done until anew population is created. Dont replace old

chromosomes as new ones are generated.Theother replacement schemes utilize some formof elitism as we have already discussed

CS6665 9 Genetic Alg

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