Why do GAs work?

Symbol alphabet : {0, 1, * }

* is a wild card symbol that matches both 0 and 1

A schema is a string with fixed and variable symbols

01*1* template which matches 01010 01011 01110 01111

Notice that the symbol * is never actually manipulated by the GA, it is only a notational device that makes it easier to talk about families of strings

Schemata and Building Blocks

Every evaluated string gives partial information about the fitness of the set of possible schematas of which the string is a member

Under fitness-proportionate replication the number m of individuals in the population belonging to a particular schemata H at time t+1 is related to the same number at time t by:

Where f is the average fitness value of the strings representing schema H, while is the average fitness value over all strings in the population

))(/)()(,()1,( tftftHmtHm H

f

Schemata and Building Blocks

If one assumes that a particular schema remains above the average by a fixed amount for a number t of generations then the solution of the above recirrence is the following exponential growth equation

m(H,t)=m(H,0)(1+c)t

Where m(H,0) stands for the number of schemata H in the population

at time 0, c is a positive constant

Fitness-proportionate reproduction allocates exponentially increasing number of trials to above average schemata

)(tfc

Schemata and Building Blocks

The effect of crossover, which breaks strings apart, is to diminish the exponential increase by a quantity that is proportional to the crossover rate pc and depends on the defining length of a schema and on the string length

1)(

lH

pc

The defining lenth of a given schema is the distance between the firstand the last fixed string positions01*1* is 3 and **1*1010 is 5 therefore short defining length schemata will be less disrupted by a single point crossover…above average schemata with short defining lengths will still be sampled at an exponentially increasing rate

Schemata and Building Blocks

If the mutation probability is pm then the probability of survival of a single bit is 1-pm

Since single bit mutations are independent the total survival probability is thus (1-pm)n where n is the string length

for schemata only the fixed positions matter(1-pm)o(H) where order o(H) of a schema H equals n

minus the number of do not care symbols

Probability of surviving a mutation for pm <<1 can be approximated by 1-o(H)pm

Schemata and Building Blocks

Putting together the effects of reproduction, crossover and mutation we have the schema theorem

This says that the number of short, low-order, above average schemata grows exponentially in subsequent generations of a genetic algorithm

])(1)(

1[)()(

),()1,( mcH pHo

lH

ptftf

tHmtHm

Schemata and Building Blocks

The schema theorem characterizes the successful building blocks of GA chromosome structures

Those schemata with

short defining length,

with little fixed positions and

with instances of above-average fitness

will reproduce exponentially from generation to generation.

Building Block Hypothesis

A genetic algorithm seeks near-optimal performance through the juxtaposition of short, low-order, high-performance schemata, called the building blocks

Consequence: the manner in which we encode a problem is critical for the performance of a GA - it should satisfy the idea of short building blocks

Some Further ResultsSelection

If we only have proportional selection operator, the limitation distribution of chromosome is:

JHJP

HP

JH

HPHP kk

)(

)(

0

)(lim)(

0

0

Some Further ResultsSelection

Tournament selection should allow repetitive choosing. If not, the chromosome would never appear in the new generation

qqN

qqkN

qqN

qqkN

k C

C

C

CxP

1

11

1

11)1()1()(

Some Further ResultsCrossover

One-point crossover

For some common crossover operators

))(

)()(()1)(1(

+

)()1(

)(

211

1

11111

211

21

nk

n

inikik

c

nkc

nk

xxxP

xxPxxNPNn

p

xxxPp

xxxP

)()()(lim 00 11 nn iiiikk

xPxPxxP

)()()(11 nn iiii xPxPxxP

Some Further ResultsCrossover

With the crossover operator, 1) the genes will be independent; 2) the schema will be searched only if the genes it contains exist in the initial population; 3) the limitation probability of the schema equals the product of the initial probability of the genes and has nothing to do with the definition length.

Some Further ResultsMutation

In spite of the value of the mutation probability , any schema will be found by the mutation operator. The limitation probability of the schema with same order is same. And this limitation probability only depends on the order and has nothing to do with the definition length.

niikk n

xxP2

1)(lim

1

Inherently parallel

O(n3) schemata are processed while n chromosomes are processed.

Convergence

The Genetic algorithm with mutation probability pm(0,1), crossover probability pc[0,1] and proportional selection does not converge to the global optimum.

If the genetic algorithm maintaining the best solution found over time before selection converges globally optimum.

Premature convergence

The population converged to an unacceptable poor solution

increase mutation rate

adaptive mutation rate

larger population or lower selection pressure

(but with low selection or high mutation Gas will behave like a random search)

Gray Coding

Desired: points close to each other in representation space also close to each other in problem space

This is not the case when binary numbers represent floating point values

m is number of bits in representation

binary number b = (b1; b2; ; bm)

Gray code number g = (g1; g2; ; gm)

Gray Coding

Binary gray code 000 000 001 001 010 011 011 010 100 110 101 111 110 101 111 100

Gray Coding

PROCEDURE Binary-To-Gray

g1=b1

for k=2 to m do

gk=b k-1 XOR bk

endfor

Gray Coding

PROCEDURE Gray-To-Binary

value = g1 b1 = value for k = 2 to m do if gk = 1 then value = NOT value end if bk =value end for

Real-valued representation

A very natural encoding if the solution we are looking for is a list of real-valued numbers, then encode it as a list of real-valued numbers! (i.e., not as a string of 1’s and 0’s)

Lots of applications, e.g. parameter optimisation

Real valued representationRepresentation of individuals

Individuals are represented as a tuple of n real-valued numbers:

The fitness function maps tuples of real numbers to a single real number:

Rx

x

x

x

X i

n

,2

1

RRf n :

Recombination for real valued representation

Discrete recombination (uniform crossover): given two parents one child is created as follows

a db fc e g h

FD GE HCBAa b C Ed Hgf

Recombination for real valued representation

Intermediate recombination: given two parents one child is created as follows

a db fc e

FD ECBA

(a+A)/2 (b+B)/2 (c+C)/2 (e+E)/2(d+D)/2 (f+F)/2

Recombination for real valued representation

Arithmetic crossover: given two parents one child is created as follows

a db fc e

FD ECBA

xa+(1-x)A xb+(1-x)B xc+(1-x)C xe+(1-x)Exd+(1-x)D xf+(1-x)F

Mutation for real valued representation

Perturb values by adding some random noise

Often, a Gaussian/normal distribution N(0,) is used, where 0 is the mean value is the standard deviation

and

x’i = xi + N(0,i)

for each parameter

Mutation for real valued representation

Perturb values by jumping randomly

if xi[a,b], then x’i = random(a,b)