IS53024A: Artificial Intelligence
Genetic Learning Algorithms
1. Genetic Algorithms and Natural Selection
Genetic Algorithms (GA) are alternative to the traditional optimization
techniques by using directed random searches to locate optimal solutions.
The GA search methods are rooted in the mechanisms of evolution and natural
genetics. In nature, individuals best suited to competition for scanted resources
survive. The various features that uniquely characterize an individual are
determined by its genetic content.
Only the fittest individuals survive and reproduce, a natural phenomenon called
"the survival of the fittest". The reproduction process generates diversity in the
gene pool. Evolution is initiated when the genetic chromosomes from two
parents recombine during reproduction. New combinations of genes are
generated from previous ones and a new gene pool results. Specifically the
exchange of genetic material among the chromosomes is called crossover.
Segments of the two parent chromosomes are exchanged during crossover,
creating the possibility of the right combination of genes for better individuals.
The genetic algorithms manipulate a population of potential solutions to an
optimization ( or search ) problem. They operate on encoded representations of
the solutions, equivalent to the genetic material of individuals in the nature, and
not directly on the solutions themselves. Usually the solutions are strings of bits
from a binary alphabet.
As in nature, selection provides the necessary driving mechanism for better
solutions to survive. Each solution is associated with a fitness value that
reflects how good it is, compared with other solutions in the population. The
higher the fitness value of an individual, the higher its chances for survival and
reproduction.
Recombination of genetic material in the genetic algorithms is simulated
through a crossover mechanism that exchanges portions between strings.
Another operation, called mutation, causes sporadic and random alteration of
the bits of strings. Mutation too has a direct analogy from nature and plays the
role of regenerating lost genetic material.
2. Genetic Algorithm Structure
The genetic operators- crossover and mutation, generate, promote and
juxtapose building blocks to form optimal strings. The Crossover tends to
conserve the genetic information present in the strings to be crossed. Mutation
generates radically new building blocks.
A Simple Genetic Algorithm
procedure simple-GA
begin
t  0
Initialize P( T )
Evaluate P( T )
while NOT( termination-condition ) do
begin
t  t + 1
Select v, v1, v2  P( T - 1 )
Cross-over( v1, v2 )
Mutate( v )
Evaluate( v, v1, v2 )
end
end.
Selection provides the favorable bias toward building blocks with higher fitness
values and ensures that they increase in representation from generation to
generation. The crucial operation is juxtaposing the building blocks achieved
during crossover, and this is the cornerstone of the GA mechanics.
The Schemata Theorem
A genetic algorithm searches for the optimal string schemata in a competition
with other ones from the current population in order to increase the number of
the instances of the schemata in the next population. The notion that strings
with high fitness values can be located by sampling schemata with high fitness
values, called building blocks, and further combining these building blocks
effectively is the building block hypothesis. The building block hypothesis
assumes that the juxtaposition of good building blocks gives good strings.
When the effects of selection, crossover and mutation on the rate at which
instances of schema increase from generation to generation are considered, one
can see that proportionate selection increases or decreases the number in
relation to the average fitness value of the schema.
A schema must have a short defining rate too. Because crossover is disruptive,
the higher the defining rate of a schema, the higher the probability that the
crossover point will fall between its fixed positions and an instance will be
destroyed. Thus schemata with high fitness values and small defining lengths
grow exponentially with time. This is the essence of the schema theorem,
which is the fundamental theorem of genetic algorithms.
The fundamental schemata theorem of genetic algorithms states that schemata
with high fitness values and small defining lengths grow exponentially in time.
The following equation is a formal statement of the schemata theorem:
N ( h, t + 1 ) >= N ( h, t )*fi ( h, t ) / f ( t ) [ 1 - pc* ( h ) / ( l -1 ) - pm*o ( h ) ]
where: fi ( h, t ) : average fitness value of schema h in generation t ;
f(t)
: average fitness value of the population in generation t ;
pc
: crossover probability ;
pm
: mutation probability ;
 ( h ) : defining length of the schema h ;
o ( h ) : order of the schema h ;
N( h, t ) : expected number of instances of schema h in generation t ;
l
: number of bit positions in a string.
3. Components of A Genetic Algorithm
A simple genetic algorithm has the following components:






encoding mechanism
fitness function
selection schemes
genetic operators ( crossover and mutation )
reproduction
control parameters
3.1. Encoding Mechanism
The encoding mechanism depends on the nature of the problem variables. A
large number of optimization problems have real-value continuous variables. A
common method for encoding them uses their integer representation. Each
variable is first lineary mapped to an integer defined in the prespecified range,
and second the integer is encoded using a fixed number of binary bits. The
binary codes of all the variables are then concatenated to obtain a binary string.
3.2. Fitness Function
The objective function provides the mechanism for evaluating each string.
However, its range of values varies from problem to problem. To maintain
uniformity over various problem domains it is introduced a fitness function that
normalizes the objective function to a convenient range of 0 and 1. The
normalized value of the objective function is the fitness of the string, which the
selection mechanism uses to evaluate the strings of the population.
3.3. Selection Schemes
Selection models nature's survival-of-the-fittest mechanism. Fitter solutions
survive while weaker ones perish. In the genetic algorithm, a fitter string
receives a higher number of offspring and thus has a higher chance of surviving
in the subsequent generation.
There are two basic types of selection schemes commonly used today:
proportionate selection and ranking ( ordinal ) selection. Proportionate-based
selection selects individuals based on their fitness values relative to the fitness
of the other individuals in the population.
Some common proportionate selection mechanisms are: canonical proportional
selection, and stochastic universal selection.
Rank-based selection schemes select individuals upon their rank within the
population. This entails that the selection pressure is independent of the fitness
distribution and is solely based upon the relative ranking of the population.
Common ranking selection schemes are: tournament selection, - selection,
truncation selection, and linear ranking.
Canonical Proportional Selection Algorithm
Input: the population P( I ), N number of individuals
/ M denotes average fitness /
Output: the population after selection P( I )'
Proportional ( I1, I2, ..., IN ):
s0  0
for i  1 to N do
si  si - 1 + ( fi / M )
for i  1 to N do
r  random[ 0, sN ]
Ii' Ii such that sl - 1 <= r < sl
return ( I1', I2', ..., IN' )
Tournament Selection Algorithm
Input:
the population P( I ), N number of individuals,
and tournament size t { 1, 2, ..., N }
Output: the population after selection P( I )'
Tournament ( t, I1, I2, ..., IN ):
for i  1 to N do
Ii' best fit individual out of
t randomly picked individuals from ( I1, I2, ..., IN )
return ( I1', I2', ..., IN' )
Linear Ranking Selection Algorithm
Input: the population P( I ), N number of individuals, and the
reproduction rate of the worst individual -  [ 0, 1 ]
Output: the population after selection P( I )'
Linear_ranking ( -, I1, I2, ..., IN ):
sort( I1, I2, ..., IN ) according fitness
s0  0
for i  1 to N do
si  si - 1 + pi / computed with the equation below /
for i  1 to N do
r  random[ 0, sN ]
Ii' Ii such that sl - 1 <= r < sl
return ( I1', I2', ..., IN' )
where the selection probability lineary assigned to the individuals according to
their rank is defined:
pi = 1 / N ( - + ( + - - )*( i - 1 ) / ( N - 1 ) )
Here - is the probability of the worst individual to be selected and + is the
probability of the best individual to be selected. In this way all the individuals
get a different rank, i.e. a different selection probability even if they have the
same fitness value.
Stochastic Universal Sampling
Input: the population P( I ), N number of individuals and the
reproduction rate for each fitness value Ri  [0, N ]
Output: the population after selection P( I )'
SUS ( R1, R2, ..., RN , I1, I2, ..., IN ):
sum  0
j1
ptr  random[ 0, 1 ]
for i  1 to N do
sum  sum + Ri
where Ri is the reproduction rate of individual Ii
while ( sum > ptr ) do
Ij '  Ii
jj+1
ptr  ptr + 1
return ( I1', I2', ..., IN' )
3.4. Genetic Operators - Crossover and Mutation
The most crucial operation in the genetic algorithms is crossover.
Crossover is the process of picking at random from the current population pairs
of strings and exchanging portions of them. The simplest approach is singlepoint crossover. Assuming that the string length is l, a crossover point is chosen
randomly from the range between 1 and l-1. The portions of the two parent
strings beyond this crossover point are exchanged to form two new strings.
The crossover point may assume any of the l-1 possible values with equal
probability. After choosing a pair of strings , the algorithm invokes crossover
only if a randomly generated number in the range 0 to 1 is greater than pc, the
crossover rate (or probability of crossover).
The crossover rate is usually from 0.5 to 1.0. Otherwise the strings remain
unaltered. The value of pc lies in the range from 0 to 1. In a large population pc
gives the fraction of strings actually crossed.
Uniform Crossover Algorithm
Uniform_crossover ( I1, I2 ):
n  length( I1 )
for i  1 to n do
if random_real( 0,1 ) <= p
then
a  I1 [ i ]
I1 [ i ]  I2 [ i ]
I2 [ i ]  a
return ( I1', I2' )
where: p is the probability for crossover
k-point Crossover Algorithm
k-point_crossover ( k, I1, I2 ):
n  length( I1 )
if k < 1 and k > n + 1
then
return Error
for i  0 to N do
I[i]0
i1
while i <= k do
K[ i ]  random_integer( 0,n )
if I[ K[ i ]] = 0
then
I[ K[ i ]]  1
ii+1
for i  1 to k do
for j  K[ i ] to n do
a  I1 [ j ]
I1 [ j ]  I2 [ j ]
I2 [ j ]  a
return ( I1', I2' )
After crossover, strings are subjected to mutation. Mutation of a bit involves
flipping it: changing 0 to 1 and vice versa. Just as pc controls the probability of
a crossover, another parameter pm ( mutation rate ) gives the probability that a
bit will be flipped. The mutation rate is usually from 0.001 to 0.05. The bits of
a string are independently mutated -- that is, the mutation of a bit does not
affect the probability of mutation of other bits.
Bitflip Mutation Algorithm ( one-point mutation )
Bitflip_mutation ( I ):
n  length( I )
k  random_integer( 1, n )
I [ k ]  new value from the alphabet
return ( I )
Mutation Algorithm ( uniform mutation )
Mutation ( I ):
n  length( I )
for i  1 to n do
if random_real( 0,1 ) <= p the probability for mutation
then
I [ i ]  new value from the alphabet
return ( I )
Genetic algorithms treat the mutation as a secondary operator with the role of
restoring lost genetic material.
3.5. Reproduction
Genetic algorithms modify a population of potential solutions during the course
of a run, using both the application of operators such as crossover and
mutation, and the application of a reproductive technique.
There are two main reproductive techniques in general use. The first which is
the most widely used is called generational reproduction, and the second
steady-state reproduction.
Generational reproduction replaces the entire population with a new population.
This is done by repeatedly selecting an individual from the old population,
according fitness and with replacement, and adding that member to the new
population. Since selection is biased by fitness, individuals having a fitness
value greater than the population average will on average be represented in the
new population to a greater extent. Likewise, individuals with below-average
fitness will decrease in the population according to the same ratio.
The steady-state technique replaces only a few individuals during a generation,
usually whatever number is required for an operator. Very often one individual
is reproduced at a generation. This is done by selecting an individual from the
population according to its fitness and making a copy.
However, in order to insert the copy in the population, room must be made for
it, which requires also that an individual for deletion is selected. Sometimes,
the members for deletion are selected randomly. More often, the least-fit
individual is chosen for deletion.
The recent implementations use steady-state reproduction because of several
advantages:
- First, the steady-state reproduction automatically grants elitist status to all
good individuals in the population ;
- Second, When a new individual is created in the steady-state algorithm it is
ready for immediate use, while in the
generational algorithms a good new individual is not used until the next
generation which may be many steps ahead;
- Third, one can impose the condition that only one copy of an individual exists
in the population at any one time.
This simple change has shown for some problems large improvements in the
speed at which the genetic algorithm converges on the correct solution.
A Hill-climbing Algorithm
procedure hill-climb
begin
t  0
repeat
local = FALSE
select current individual vc at random
evaluate vc
repeat
select v  Neighborhs( vc )
by flipping bits of vc
select this vn  v which has largest
value of objective function f
if f(vc ) < f(vn )
then
vc  vn
else
local = TRUE
until local
t  t + 1
until t = MAX
end.
A Simulated Annealing Algorithm
procedure simulated-annealing
begin
t  0
initialize temperature T
select current individual vc at random
evaluate vc
repeat
repeat
select a vn  Neighborhs( vc )
by flipping bits of vc
if f(vc ) < f(vn )
then
vc  vn
else
if (rand[0,1)<exp{(f(vn)-f(vc))/T})
vc  vn
until terminating condition
T  g( T,t )
t  t + 1
until stop criterion
end.