Soft computing06

Soft Computing Soft Computing

Lecture 06: Introduction to Genetic Algorithms

“Genetic Algorithms are good at taking large, potentially huge

search spaces and navigating them, looking for optimal combinations of things, solutions you might not

otherwise find in a lifetime.”

- Salvatore Mangano

Computer Design, May 1995

GENETIC ALGORITHMGENETIC ALGORITHM

A biologically inspired model of intelligence and the principles of biological evolution are applied to find solutions to difficult problems

The problems are not solved by reasoning logically about them; rather populations of competing candidate solutions are spawned and then evolved to become better solutions through a process patterned after biological evolution

Less worthy candidate solutions tend to die out, while those that show promise of solving a problem survive and reproduce by constructing new solutions out of their components


GA begin with a population of candidate problem solutions

Candidate solutions are evaluated according to their ability to solve problem instances: only the fittest survive and combine with each other to produce the next generation of possible solutions

Thus increasingly powerful solutions emerge in a Darwinian universe

Learning is viewed as a competition among a population of evolving candidate problem solutions

This method is heuristic in nature and it was introduced by John Holland in 1975


Basic Algorithm

begin set time t = 0; initialise population P(t) = {x1

t, x2t, …, xn

t} of solutions;

while the termination condition is not met do begin evaluate fitness of each member of P(t); select some members of P(t) for creating offspring; produce offspring by genetic operators; replace some members with the new offspring; set time t = t + 1; endend


The Evolutionary Cycle

selection

population evaluation

modification

discard

deleted members

parents

modifiedoffspring

evaluated offspring

initiate & evaluate


Representation of Solutions: The Chromosome

Gene: A basic unit, which represents one characteristic of the individual. The value of each gene is called an allele

Chromosome: A string of genes; it represents an individual i.e. a possible solution of a problem. Each chromosome represents a point in the search space

Population: A collection of chromosomes

An appropriate chromosome representation is important for the efficiency and complexity of the GA



The classical representation scheme for chromosomes is binary vectors of fixed length

In the case of an I-dimensional search space, each chromosome consists of I variables with each variable encoded as a bit string


Example: Cookies Problem

Two parameters sugar and flour (in kgs). The range for both is 0 to 9 kgs. Therefore a chromosome will comprise of two genes called sugar and flour

5 1 Chromosome # 01

2 4 Chromosome # 02


Example: Expression satisfaction Problem

F = (a c) (a c e) (b c d e) (a b c) (e f)

Chromosome: Six binary genes a b c d e f e.g. 100111



Chromosomes have either binary or real valued genes

In binary coded chromosomes, every gene has two alleles

In real coded chromosomes, a gene can be assigned any value from a domain of values

Model Learning

Use GA to learn the concept Yes Reaction from the Food Allergy problem’s data


Chromosomes Encoding

A potential model of the data can be represented as a chromosome with the genetic representation:

Gene # 1 Gene # 2 Gene # 3 Gene # 4Restaurant Meal Day Cost

The alleles of genes are:

Restaurant gene: Sam, Lobdell, Sarah, XMeal gene: breakfast, lunch, XDay gene: Friday, Saturday, Sunday, XCost gene: cheap, expensive, X


Chromosomes Encoding (Hypotheses Representation)

Hypotheses are often represented by bit strings (because they can be easily manipulated by genetic operators), but other numerical and symbolic representations are also possible

Set of if-then rules: Specific sub-strings are allocated for encoding each rule pre-condition and post-condition

Example: Suppose we have an attribute “Outlook” which can take on values: Sunny, Overcast or Rain



We can represent it with 3 bits: 100 would mean the value Sunny, 010 would mean Overcast & 001 would mean Rain

110 would mean Sunny or Overcast111 would mean that we don’t care about its value

The pre-conditions and post-conditions of a rule are encoding by concatenating the individual representation of attributes



Example:

If (Outlook = Overcast or Rain) and Wind = strong then PlayTennis = No

can be encoded as 0111001

Another rule If Wind = Strong

then PlayTennis = Yes

can be encoded as 1111010



An hypothesis comprising of both of these rules can be encoded as a chromosome

01110011111010

Note that even if an attribute does not appear in a rule, we reserve its place in the chromosome, so that we can have fixed length chromosomes


Variable size chromosomes

Sometimes we need a variable size chromosome; e.g. to represent a set of rules

Example: Suppose we are representing a set of rules by a chromosome

If a1 = T and a2 = F then c = TIf a2 = T then c = F

The chromosome would be 10 01 1 11 10 0where a1 = T is represented by 10,

a2 = F by 01, and so on



Evaluation/Fitness Function

It is used to determine the fitness of a chromosome

Creating a good fitness function is one of the challenging tasks of using GA


Example: Cookies Problem

Two parameters sugar and flour (in kgs). The range for both is 0 to 9 kgs. Therefore a chromosome will comprise of two genes called sugar and flour

5 1

2 4

The fitness function for a chromosome is the taste of the resulting cookies; range of 1 to 9


Example: Expression satisfaction Problem

F = (a c) (a c e) (b c d e) (a b c) (e f)

Chromosome: Six binary genes a b c d e f e.g. 100111

Fitness function: No of clauses having truth value of 1e.g. 010010 has fitness 2

Model Learning

Use GA to learn the concept Yes Reaction from the Food Allergy problem’s data


The fitness function can be the number of training samples correctly classified by a chromosome (model)


Population Size

Number of individuals present and competing in an iteration (generation)

If the population size is too large, the processing time is high and the GA tends to take longer to converge upon a solution (because less fit members have to be selected to make up the required population)

If the population size is too small, the GA is in danger of premature convergence upon a sub-optimal solution (all chromosomes will soon have identical traits). This is primarily because there may not be enough diversity in the population to allow the GA to escape local optima


Selection Operators (Algorithms)

They are used to select parents from the current population

The selection is primarily based on the fitness. The better the fitness of a chromosome, the greater its chance of being selected to be a parent


Selection Operators: Random Selection

Individuals are selected randomly with no reference to fitness at all

All the individuals, good or bad, have an equal chance of being selected


Selection Operators: Proportional Selection

Chromosomes are selected based on their fitness relative to the fitness of all other chromosomes

For this all the fitness are added to form a sum S and each chromosome is assigned a relative fitness (which is its fitness divided by the total fitness S)

A process similar to spinning a roulette wheel is adopted to choose a parent; the better a chromosome’s relative fitness, the higher its chances of selection



The selection of only the most fittest chromosomes may result in the loss of a correct gene value which may be present in a less fit member (and then the only chance of getting it back is by mutation)

One way to overcome this risk is to assign probability of selection to each chromosome based on its fitness

In this way even the less fit members have some chance of surviving into the next generationChromosomes are selected based on their fitness relative to the fitness of all other chromosomes



For this all the fitness are added to form a sum S and each chromosome is assigned a relative fitness (which is its fitness divided by the total fitness S)

A process similar to spinning a roulette wheel is adopted to choose a parent; the better a chromosome’s relative fitness, the higher its chances of selection



The probability of selection of a chromosome “i” may be calculated as

pi = fitnessi / j fitnessj

Example

Chromosome Fitness Selection Probability1 7 7/142 4 4/143 2 2/144 1 1/14


Selection Operators: Proportional Selection Algorithm

1. [Sum] Calculate sum of all chromosome fitnesses in population - sum S. 2. [Assign] Assign a range to each chromosome over a line ranging from 0-S.3. [Select] Generate random number from interval (0,S) - r. 4. [Select] Select the chromosome belongs to rOf course, step 1 is performed only once for each population.


Selection Operators: Rank based selection

Rank based selection uses the rank ordering of the fitness values to determine the probability of selection and not the fitness values themselves

This means that the selection probability is independent of the actual fitness value

Ranking therefore has the advantage that a highly fit individual will not dominate in the selection process as a function of the magnitude of its fitness



Proportional Selection have problems when the fitnesses differs very much. For example, if the best chromosome fitness is 90% of all the roulette wheel then the other chromosomes will have very few chances to be selected. Rank selection first ranks the population and then every chromosome receives fitness from this ranking. The worst will have fitness 1, second worst 2 etc. and the best will have fitness N (number of chromosomes in population)You can see in following picture, how the situation changes after changing fitness to order number.

Before ranking After ranking



The population is sorted from best to worst according to the fitness

Each chromosome is then assigned a newfitness based on a linear ranking function

New Fitness = (P – r) + 1

where P = population size, r = fitness rank of the chromosome If P = 11, then a chromosome of rank 1 will have a New Fitness of 10 + 1 = 11 & a chromosome of rank 6 will have 6



A user adjusted slope can also be incorporated

New Fitness = {(P – r) (max - min)/(P – 1)} + min

where max and min are set by the user to determine the slope (max - min)/(P – 1) of the function

Let P = 11, max = 8, min = 3, then a chromosome of rank 1 will have a New fitness of

10*5/10 + 3 = 8& a chromosome of rank 6 will have 5*5/10 + 3 = 5.5



Once the new fitness is assigned, parents are selected by the same roulette wheel procedure used in proportionate selection


Selection Operators: Tournament Selection

Extracts k individuals from the population with uniform probability (without re-insertion) and makes them play a “tournament”, where the probability for an individual to win is generally proportional to its fitness


Reproduction Operators

Genetic operators are applied to chromosomes that are selected to be parents, to create offspring

Basically of two types: Crossover and Mutation

Crossover operators create offspring by recombining the chromosomes of selected parents

Mutation is used to make small random changes to a chromosome in an effort to add diversity to the population


Reproduction Operators: Crossover

Crossover operation takes two candidate solutions and divides them, swapping components to produce two new candidates



Figure illustrates crossover on bit string patterns of length 8

The operator splits them and forms two children whose initial segment comes from one parent and whose tail comes from the other

Input Bit Strings1 1 # 0 1 0 1 # # 1 1 0 # 0 # 1

Resulting Strings1 1 # 0 # 0 # 1 # 1 1 0 1 0 1 #



Two genes sugar and flour (in kgs)Crossover operation on chromosomes

5 1 5 4

2 4 2 1



The place of split in the candidate solution is an arbitrary choice. This split may be at any point in the solution

This splitting point may be randomly chosen or changed systematically during the solution process

Crossover can unite an individual that is doing well in one dimension with another individual that is doing well in the other dimension



Two types: Single point crossover & Uniform crossover

Single type crossoverThis operator takes two parents and randomly selects a single point between two genes to cut both chromosomes into two parts (this point is called cut point)The first part of the first parent is combined with the second part of the second parent to create the first childThe first part of the second parent is combined with the second part of first parent to create the second child

1000010 10000011110001 1110010



Uniform crossoverThe value of each gene of an offspring’s chromosome is randomly taken from either parentThis is equivalent to multiple point crossover

10000101110001 1010010

Reproduction Operators: Crossover (Variable size chromosomes)

Sometimes we need a variable size chromosome; e.g. to represent a set of rules

Example: Suppose we are representing a set of rules by a chromosome

If a1 = T and a2 = F then c = TIf a2 = T then c = F

The chromosome would be 10 01 1 11 10 0where a1 = T is represented by 10,

a2 = F by 01, and so on



The sub-strings can be considered as a single entity during cross-over (i.e. crossover point is not allowed in the middle of the sub-string)

Another way can be to allow all possible crossovers, but assign a very low fitness to resulting chromosomes which have undesirable sub-string meaning(s)

e.g. 01110011111011 would mean, we do not care whether we play tennis or

not



For such chromosomes we use a modified cross-over operator

To perform a crossover operation on two parents, two crossover points are first chosen at random in one of the parents

Example: Let the two parents be 10 01 1 11 10 0

and 01 11 0 10 01 0



Suppose the crossover points chosen randomly for the first parent are after bit position 1 and 8

1st parent 10 01 1 11 10 02nd parent 01 11 0 10 01 0

Let d1 denote the distance from the leftmost of these crossover points to the rule boundary immediately to the left

d1 = 1Let d2 denote the distance from the rightmost of these crossover points to the rule boundary immediately to the left

d2 = 3



The crossover points in the second parent are now randomly chosen, subject to the constraint that they must have the same d1 and d2 values

1st parent 10 01 1 11 10 0 d1 = 1, d2 = 32nd parent 01 11 0 10 01 0

The possible crossover points for the 2nd parent are at bit positions (1, 3), (1, 8), and (6, 8)

2nd parent 01 11 0 10 01 0 d1 = 1, d2 = 301 11 0 10 01 001 11 0 10 01 0



Suppose crossover points (1, 3) happen to be chosen for the 2nd parent

1st parent 10 01 1 11 10 02nd parent 01 11 0 10 01 0

The resulting two offspring would be 11 10 0

and 00 01 1 11 11 0 10 01 0



Reproduction Operators: Mutation

Mutation is another important genetic operator

Mutation takes a single candidate and randomly changes some aspect (gene) of it

For example, mutation may randomly select a bit in the pattern and change it, switching a 1 to a 0 or to # (don’t care)



Mutation is important in that the initial population may exclude an essential component of a solution

For example, if no member of the initial population has a 1 in the first position, then crossover in the middle, cannot produce a child that could become a solution



Each gene of each offspring is mutated with a given mutation rate p (say 0.01)

It is hence possible that no gene may be mutated for many generations. On the other hand more than one gene may be mutated in the same generation (or even in the same chromosome)

For real valued genes, the value is selected randomly from the alleles

If the rate is too low, new traits will appear too slowly in the population. If the rate is too high, each generation will be unrelated to the previous generation


Q: Is it some kind of learning technique like neural networks ?

A: No

Q: Then, what is it ?

Search Techniqes

Calculus Base Techniques

Guided random search techniques

Enumerative Techniques

BFSDFS Dynamic Programming

Tabu Search Hill Climbing

Simulated Annealing

Evolutionary Algorithms

Genetic Programming

Genetic Algorithms

Fibonacci Sort


Figure: Taxonomy of searching techniques


• 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 heuristic for combinatorial problems

• Special Features:– Traditionally emphasizes combining information from good

parents (crossover)– many variants, e.g., reproduction models, operators

GA Quick Overview


The MAXONE problem

Suppose we want to maximize the number of ones in a string of l binary digits

Is it a trivial problem?

It may seem so because we know the answer in advance

However, we can think of it as maximizing the number of correct answers, each encoded by 1, to l yes/no difficult questions`


The MAXONE problem

• An individual is encoded (naturally) as a string of l binary digits

• The fitness f of a candidate solution to the MAXONE problem is the number of ones in its genetic code

• We start with a population of n random strings. Suppose that l = 10 and n = 6


The MAXONE problem (initialization step)

We toss a fair coin 60 times and get the following initial population:

s1 = 1111010101 f (s1) = 7

s2 = 0111000101 f (s2) = 5

s3 = 1110110101 f (s3) = 7

s4 = 0100010011 f (s4) = 4

s5 = 1110111101 f (s5) = 8

s6 = 0100110000 f (s6) = 3


The MAXONE problem (selection step)

Next we apply fitness proportionate selection with the roulette wheel method; the individual i have the probability to chose:

We repeat the extraction as many times as the number of individuals we need to have the same parent population size (6 in our case)

i

if

if

)(

)(

21n

3

Area is Proportional to fitness value

4


The MAXONE problem (selection step)

Suppose that, after performing selection, we get the following population:

s1` = 1111010101 (s1)

s2` = 1110110101 (s3)

s3` = 1110111101 (s5)

s4` = 0111000101 (s2)

s5` = 0100010011 (s4)

s6` = 1110111101 (s5)

Next we mate strings for crossover. For each couple we decide according to crossover probability (for instance 0.6) whether to actually perform crossover or not

Suppose that we decide to actually perform crossover only for couples (s1`, s2`) and (s5`,

s6`). For each couple, we randomly extract a crossover point, for instance 2 for the first and 5 for the second


The MAXONE problem (crossover step)

s1` = 1111010101 s2` = 1110110101

s5` = 0100010011 s6` = 1110111101

Before crossover:

After crossover:

s1`` = 1110110101 s2`` = 1111010101

s5`` = 0100011101 s6`` = 1110110011


The MAXONE problem (crossover step)

The final step is to apply random mutation: for each bit that we are to copy to the new population we allow a small probability of error (for instance 0.1)

Before applying mutation:

s1`` = 1110110101

s2`` = 1111010101

s3`` = 1110111101

s4`` = 0111000101

s5`` = 0100011101

s6`` = 1110110011


The MAXONE problem (mutation step)


The MAXONE problem (mutation step)

After applying mutation:

s1``` = 1110100101 f (s1``` ) = 6

s2``` = 1111110100 f (s2``` ) = 7

s3``` = 1110101111 f (s3``` ) = 8

s4``` = 0111000101 f (s4``` ) = 5

s5``` = 0100011101 f (s5``` ) = 5

s6``` = 1110110001 f (s6``` ) = 6


The MAXONE problem (example end)

In one generation, the total population fitness changed from 34 to 37, thus improved by ~9%

At this point, we go through the same process all over again, until a stopping criterion is met


Short Assignment:

Design a genetic algorithm to learn conjunctive classification rules for the Play-Tennis problem. Describe precisely the bit-string encoding of hypotheses and a set of crossover operators.

Due Date:

24-05-2012

Hint: see Chapter 09, Machine Learning book by Tom. Mitchell


Short Assignment Data:


Major Assignment (10 marks) :

Task: you have to analyse some research paper and give your critical analysis in the form of a report. The report will be critically reviewed and accordingly marked. Deliverables: A report + PresentationRelevant Information: The assignment will be prepared within groups. However, the marks will be assigned based on individual performance.Research Topics: the topics will be uploaded on the group


The New Generation

The new offspring can replace the old population without any fitness comparison

or

only the better ones from the new and old make it to the new generation (more processing needed)


New Generation: Elitism

Elitism is a value between 0 and 1, which represents the fraction of the individuals of a population that will be duplicated to the next generation

Example P = 20 and elitism = 0.1, then 2 individuals of current population do not get replaced

The elite chromosome selection may be based on highest fitness values

If highest fitness valued chromosomes are carried over to the next generation, we ensure that the maximum fitness does not decrease from one generation to next


New Generation: Generation Gap

The number of individuals replaced in the next generation is called generation gap

A generation gap of 100% will mean that whole of the population comprises of new chromosomes

We may have a generation gap which is not fixed. In this method, the fittest P chromosomes will be selected from the set of current population plus new children, and will form the new generation


New Generation: Number of Duplicates Allowed

Duplicates are Chromosomes that are same

If they are allowed then that chromosome has higher probability of producing an offspring, and may probably create many offspring

Eliminating duplicates increases the efficiency of the genetic

search and reduces the danger of premature convergence

Eliminating duplicates means that if an offspring is created which is a duplicate of a chromosome of the current population, we terminate it immediately and create a new one. It increases the processing time in large populations


Termination Requirement

The GA continues until some termination requirement is met, such as

- having a solution whose fitness exceeds some threshold- the fitness of solutions becomes stable & stops improving

References

Engelbrecht Chapter 8 & 9


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