# Genetic Algorithms - S. Shakya

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23 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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Estimation of Distribution
Algorithms (EDA)

Siddhartha K. Shakya

School of Computing.

The Robert Gordon University

Aberdeen, UK

ss@comp.rgu.ac.uk

EDAs

Algorithm

Also known as Probabilistic model building
Genetic Algorithms or Iterated density

A probabilistic model based heuristic

Motivated from the GA evolution

More explicit evolution than the GA

Basic Concept of Solution and
Fitness

Given a set of colours, GCP is to try and
assign Colour to each nodes in such the way
that neighbouring nodes will not have same
colour

a

b

d

e

f

c

Graph colouring Problem:
An Example

Basic concept of a solution and Fitness

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fitness

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Solution

Representation
of a solution as
a chromosome

Given 2 colour

Black = 0

White = 1

Chromosome and Fitness in GCP

Chromosome: is a set of colours assigned
to the nodes of graph. (there are other way
of representing GCP in GA, such as order
based representation).

Fitness: is the number of correctly
coloured nodes.

GA Iteration

1.
Initialisation of a “parent” population

2.
Evaluation

3.
Crossover

4.
Mutation

5.
Replace parent with “child” population
and go to step 2 until termination criteria
satisfies

GA Iteration

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Parent population

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fitness

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Selected Solution

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After Crossover

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After mutation

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fitness

Initialization

Evaluation

Selection

Crossover

Mutation

Repeat

iteration

a

b

d

e

f

c

Given 2 colours

(0,1)

GA evolution

Selection

drives evolution towards better
solutions by giving a high pressure to the
selection of high
-
quality solutions

Crossover and mutation (
Variation

operator) together ensures the exploration
of the possible space of the promising
solutions. Maintains the variation in the
population.

Variation in GA Evolution

Has its limitation

Can recombine fit solution to produce
more fit solution

Also can disrupt good solution and
converge in local optimum

Estimation of Distribution Algorithm
(EDA)

To overcome the negative effective of the
crossover and mutation approach of
variation, a probabilistic approach of
variation has been proposed.

Algorithm using such approach is known
as EDA (or PMBGA)

GA to EDA

Simple GA framework

Selection

Crossover

Mutation

Evaluation

Initial Population

Selection

Probabilistic Model
Building

Evaluation

EDA framework

Sampling Child
Population

Initial Population

General Notation

EDA represents a solution as a set of value taken by a
set of random variable.

Chromosome

is a set of value taken by set of random

variables

(Where each

for bit representation)

is a univariate marginal distribution

is a conditional distribution

is a joint probability distribution

1

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1

Solution

0

1

0

0

1

1

Estimation of Probability
distribution

Usually it is not possible to calculate the joint probability distribution, so

it is estimated. For example, assuming all are independent of each
other, the joint probability distribution becomes the product of simple
univariate marginal distribution.

1

0

1

1

0

1

Solution

0

1

0

0

1

1

Simple Univariate Estimation of
Distribution Algorithm

Selection

Evaluation

Calculate univariate
marginal probability
and sample Child
Population

Initial Population

1

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Solution

0

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1

Simple univariate EDA (UMDA)

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Parent population

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fitness

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Selected Solution

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After mutation

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6

4

fitness

Initialization

Evaluation

Selection

Sampling

Repeat

iteration

a

b

d

e

f

c

Given 2 colours

(0,1)

Estimation

of

Distribution

Build model

Calculate Distribution

Note

It is not guaranteed that the above
algorithm will give optimum solution for the
graph colouring problem.

The reason is obvious.

The chromosome representation of GCP has
dependency. i.e. node 1 taking black colour
depends upon the colour of node 2.

But univariate EDAs do not assume any
dependency so it may fail.

However, one could try

Complex Models

To tackle problems where there is dependency
between variables we need to consider more
complex models.

The extra
model building step

univariate EDA.

Different algorithms has been purposed using
different models

They are categorised into three groups

Univariate EDA

Bivariate EDA

Multivariate EDA

Univariate EDA Model

Graphical representation of probability model assuming no
dependency among variables. (UMDA, PBIL, cGA)

x
1

x
2

x
3

x
4

x
6

x
5

x
7

Bivariate EDA Model

Graphical representation of probability model assuming
dependency of order two among variables.

a. Chain model
(MMIC)

b. Tree model

(COMIT)

c. Forest model

(BMDA)

Multivariate EDA Model

Graphical representation of probability model considering
multivariate dependency among variables.

a. Marginal product
model (ECGA)

c. (BOA, EBNA)

b. Triangular model
(FDA)

Finding a probabilistic model

Task of finding a good probabilistic model
(finding the relationship between variable) is a
optimization problem in itself.

Most of the algorithm use Bayesian network to
represent the probabilistic relationship.

Two metric to measure the goodness of
Bayesian Network.

Bayesian Information Criterion (BIC) metric:

Bayesian
-
Dirichlet (BD) metric:

Use greedy heuristic to find a good model.

EDA is an active area of research for GA
community

EDAs are reported to solve GA hard
problems, and also hard optimization
optimisation problems like MAX SAT.

Success and failure of EDAs depends
upon the accuracy of the used
Probabilistic model.

Summary

http://cswww.essex.ac.uk/staff/zhang/MoldeBasedWeb/R
Group.htm

(Research Groups working on EDAs)

http://www.sc.ehu.es/ccwbayes/main.html

(EDA
homepage maintained by Intelligent system group).

Books

Larrañaga
P., and Lozano J. A. (2001)
Estimation of Distribution Algorithms:
A New Tool for Evolutionary Computation