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Genetic Algorithms in Search, Optimization & Machine Learning by David E. Goldberg

Chapter 1: A Gentle Introduction to Genetic Algorithms



Goals:

o

To abstract and rigorously explain the adaptive processes of natural systems

o

To design
artificial

systems software that retains the important mechanisms of natural
systems



Theme of research on genetic algorithms: robustness

o

Robustness: the balance between efficiency and efficacy necessary for survival in many
different environments



Traditional search
methods: calculus based, enumerative, and random

o

Calculus
-
based methods divide into two classes: indirect and direct



Indirect: set the gradient to zero



Direct (search, or hillclimbing): hop on the function and move in a direction related
to the local gradi
ent



Problem: algorithm requires continuous function and derivative

o

Enumerative schemes: within a finite search space, or a discretized infinite search space, the
search algorithm starts looking at objective function values at every point in the space, one
at a time.



Problem: lack of efficiency

o

Random search: randomly walk and search, saving the best solution along the way



Optimization seeks to improve performance toward some optimal point

o

Note it has two parts: (1) seek to improve according to some (2) opti
mal point



Genetic Algorithms differ from other search and optimization algorithms because:

o

GAs work with a coding of the parameter set, not the parameters themselves

o

GAs search from a population of points, not a single point

o

GAs use payoff (objective funct
ion) information, not derivatives or other auxiliary
knowledge

o

GAs use probabilistic transition rules, not deterministic rules



GAs only require payoff values (objective function values) associated with individual strings in order
to make it a more
canonical method than many search schemes.



To
yield

good results in many practical problems is composed of three operators:

o

Reproduction: Individual strings are copied according to their objective function values. This
can be thought of as some measure of

profit, utility, or goodness that we want to maximize.
Copying strings according
to their

fitness values means that strings with a higher value will
have a higher probability of contributing one or more offspring in the next generation. This
is similar
to Darwin’s natural selection (survival of the fittest)



Implementation detail: Roulette wheel

o

Crossover: 2 steps



Members of the newly reproduced strings in the mating pool are mated at
random



Each pair of strings undergoes crossing over

o

Mutation



Good resul
ts in empirical genetic algorithm studies is on the order of one mutation
per thousand bit (position) transfers



Chromosomes


strings



Genotype


structure



Phenotype


parameter set, solution alternative, or point (in the solution space)



Genes


features or

detectors (or a particular character)



Alleles


values



Locus


position


Chapter 2: Genetic Algorithms Revisited: Mathematical Foundations



Skipped


Chapter 3: Computer Implementation of a Genetic Algorithm



The data structure for a population is an array,
where each element contains the following
information (type individual indexed from 1
-

maxpop):

o

Phenotype (decoded parameter or parameters)

o

Genotype (artificial chromosome or bit string)

o

Fitness (objective function)

o

Other auxiliary information



maxpop: uppe
r bound on the population size



maxstring: upper bound on the string size



individual contains chrom, x, fitness, site, and parent information (see page 61)



global variables defined on page 62



function select is described and implemented on page 63



procedure

crossover is described and implemented on page 64



function mutation is described and implemented on page 65



procedure generation is
described

and implemented on page 66



a fully working version is shown in Appendix C


Chapter 4: Some Applications of Geneti
c Algorithms



Holland’s genetic algorithms are similar to Fraser simulating his epistatic function



The first mention of the words “genetic algorithm” and the first published application of a genetic
algorithm both came in Bagley’s dissertation.

o

Constructed
GAs to search for parameter sets in game evaluation functions

o

He introduced a fitness scaling mechanism to reduce the selection early in a run, thereby
preventing domination of a population by a single super
-
individual, and increase the
selection later in
a run, thereby maintaining appropriate competition among highly fit and
similar strings near population convergence

o

He also introduced self
-
contained controls (GA self
-
regulation). He suggested coding the
crossover and mutation probabilities within the ch
romosomes themselves



Rosenberg simulated a population of single
-
celled organisms with a simple yet rigorous
biochemistry, a permeable membrane, and classical genetic structure (one gene, one enzyme)

o

Important to the development of GAs in artificial applica
tions because of its resemblance to
optimization and root
-
finding



Cavicchio applied GAs to a subroutine selection problem and a pattern recognition problem

o

Pixels and feature detectors



Weinberg proposed the use of a multilayered genetic algorithm to select

a good set of 15 rate
constants that controlled the workings of different simulated E coli cells.



Hollstien’s dissertation was first to apply GAs to a pure problem



Goldberg applied GAs to optimization and machine learning problems in natural gas pipeline
control


Chapter 5: Advanced Operators and Techniques in Genetic Search



Diploidy and dominance permit alternate solutions to be held in abeyance

shielded against
overselection



In Bagley’s dissertation, a diploid chromosome pair mapped to a
particular

phenotype using a
variable dominance map coded as part of the chromosome itself

o

Dominance values tended to fixate early in simulations, leaving dominance determination in
the hands of his complicated and arbitrary tie
-
breaking scheme

o

Prohibited mutation o
perator from processing dominance values, further aggravating his
premature convergence of dominance values



Rosenberg’s biologically oriented study contained a diploid chromosome model; however, since
biochemical interactions were modeled in some detail, d
ominance was not considered as a separate
effect

o

Any dominance effect in this study was the result of the presence or absence of a particular
enzyme, so the enzyme essentially controls the phenotype



Hollstien’s study included diploidy and an evolving domin
ance mechanism

o

Described two simple evolving dominance mechanisms and then put the simplest to use in
his study of function
optimization



Each binary gene was described by two genes, a modifier gene (M or m) and a
functional gene (0 or 1). 0 alleles were d
ominant when there
was

at least
one

M
allele present at one of the homologous modifier loci.



IN this triallelic scheme, Hollstien drew alleles from the 3
-
alphabet {0, 1, 2}. The 2
played the rolde of a dominant “1” and the 1 palyed the role of recessive “
1.”



Brindle performed experiements with a number of dominance schemes in a function optimization
setting, but she ignored previous work in artificial dominance and diploidy, and a number of
schemes she deveoled were without theoretical basis or biological

precedent. The following are her
6 schemes:

o

Random, fixed, global dominance

o

Variable, global dominance

o

Deterministic, variable, global dominance

o

Choose a random chromosome

o

Dominance of the etter chromosome

o

Haploid controls diploid adaptive dominance



A co
mputer implementation of triallelic diploidy and dominance (Holland
-
Hollstien) is presented on
page 162.


Evolutionary Algorithms in Theory and Practice by Thomas Back

Chapter 1: Organinc Evolution and Problem Solving



Each individual in a population repers
ents not onaly a search point in the space of potential
solutions to a given problem, but also may be a temporal contianer of current knowledge about the
“laws” of the environment



Synthetic theory of evolution (neodarwinism) is based on genes as transfer u
nits of heredity



Fitness of an individual is measured only indirectly by its growth rate in comparison to others



Schull claims that individuals themselves by means of an adaptation during their development and
by trying to cope as best as they can with the

situation are also changing the structure of the
adaptive surface



DNA

nucleotide base relationships (adenine


thymine and cytosine


guanine), replication,
transcription, translation are discussed



Meiosis and mitosis are discussed



Artificial Intelligence

is the study of how to make computers do things at which, at the moment,
people are better



Learning is constructing or modifying representations of what is being experienced



Classical AI research concentrates on the use of symbolic representations based u
pon a finite
number of representation primitives and rules for the manipulation of symbols



Currently, the field of AI si starting to spread research into a variety of directions and trieds to
integrate different methods into large
-
scale systems, thus combi
ning their advantages as far as
possible.



Evolutionary Algorithms make use of a subsymbolic representation of knowledge encoded in the
genotypes of individuals



Artificial Neural Networks (ANN) is based upon a simple omodel of the central nervous systems of

higher animals



Optimization problem, where the goal is to find a set of parameters (which might be interpreted as a
“genotype” as well as a “phenotype”) such that a certain quality criterion is maximized or minimized



The problem to determine a memboer of
the level set of an arbitrary global optimization problem
with continuous objective function f on a compact feasible region M within a finite number of steps
is unsolvable



Optimization problems with discrete object variables are called cominatorial optimiz
ation problems



Automatic programming denotes the task of finding a program which calculates a certain input
-
output function

Chapter 2: Specific Evolutionary Algorithms



An evolutionary Algorithm (EA) is defined as an 8 tuple

o

I = A
x

x A
s

is the space of indi
viduals, and A
x
, A
s

denote arbitrary sets

o

Phi: I
-
> Reals denotes a fitness function assigning real values to individuals

o

Omega is a set of probabilistic genetic operators, each of which is controlled by specific
parameters asummarized in the sets

o

S denote
s the selection operator, which may change the number of individuals from lambda
or lambda+mu to mu, where mu, lambda are in the set of integers, and mu=lambda is
permitted

o

Mu is the number of parents individuals

o

Lambda denotes the number of offspring indi
viduals

o

i is a termination criterion for the EA

o

psi describes the complete process of transorming apopulation into a subsequent one by
apllying genetic operators and selection



Evolution strategies: The two membered ES works by creating one n
-
dimensioanl re
al
-
valued vector
of object variables from its parent by applying nmutation with identical tandard deviations to each
object variable. The resulting individual is evaluated and compared to its parent, and the better of
both individuals survives to become p
arent of the next generation, while the other one is discarded

o

The major quality of this strategy is seen in its ability to incorporate the most important
parameters of the strategy (standard deviations and correlation coefficients of normally
distributed
mutations) into the search process, such that optimization not only takes place
on object variables, but also on strategy parameters according to the actual local topology
of the objective function. This capability is termed self
-
adaptation



Evolutionary P
rogramming: The original method (by L. J. Fogel) used uniform random mutations on
discrete underlying alphabets and, in its most elaborated form, a (mu+mu)
-
selection mechanism.
His son D. B. Fogel extended Evolutionary Programming for applications involvi
ng continuous
parameter optimization problems. Five different variants can be identified:

o

Standard EP, which is characterized by the absence of any self
-
adaptation mechanism

o

Continuous standard EP, where the term “continous” indicates that in contrast to
the
generation
-
based working mechanism a newly created individual is immediately evaluated
and inserted into the population

o

Meta
-
EP, which incorporates variances into the genotype in order to allow for their self
-
adaptation

o

Continuous meta
-
EP

o

Rmeta
-
EP, which in addition to standard deviations also incorporates covariances (which are
represented by correlation coefficients) into the genotype for self
-
adaptation



Genetic algorithms:

o

Probably the earliest predecessor of these algorithms emerged in the w
ork of Fraser, a
biologist who wanted to simulate evolution with special emphasis on the interaction of
epistasis with selection

o

Genetic Algorithms in their usual form are a development of Holland, computer scientist
and psychologist at the University of M
ichigan



Summarized his work on adaptive and reproductive plans in 1975

o

Genetic Algorithms need a scaling function to map objective function values to positive real
values, since the standard selection mechanism of Genetic Algorithms requires positive
fitne
ss values and highest fitness values for the best individuals

o