Estimation of Distribution
Algorithms (EDA)
Siddhartha K. Shakya
School of Computing.
The Robert Gordon University
Aberdeen, UK
ss@comp.rgu.ac.uk
EDAs
•
A novel paradigm in Evolutionary
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
1
1
0
1
1
0
a
b
d
e
f
c
1
0
0
1
1
1
a
b
c
d
e
f
1
fitness
1
1
1
0
0
0
a
b
d
e
f
c
1
0
1
0
1
0
6
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
1
0
1
1
0
1
0
0
1
0
1
1
1
0
1
0
1
1
0
1
0
0
1
1
Parent population
2
2
4
3
fitness
1
0
1
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
1
0
1
1
0
1
Selected Solution
0
1
1
0
1
1
1
0
0
0
1
1
0
1
0
1
0
1
1
0
1
0
1
1
After Crossover
0
1
1
0
1
1
1
0
0
0
1
0
0
1
0
1
0
1
1
0
1
0
1
1
After mutation
1
2
6
4
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
0
1
1
0
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
0
1
1
0
1
Solution
0
1
0
0
1
1
Simple univariate EDA (UMDA)
1
0
1
1
0
1
0
0
1
0
1
1
1
0
1
0
1
1
0
1
0
0
1
1
Parent population
2
2
4
3
fitness
1
0
1
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
1
0
1
1
0
1
Selected Solution
0
1
1
0
1
1
1
0
0
0
1
1
0
1
0
1
0
1
1
0
1
0
1
1
After mutation
1
2
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
will be added to
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
Links
•
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
. Kluwer Academic Publishers,
2001.
•
Pelikan, M., (2002).
Bayesian optimization algorithm: From single level to
hierarchy
. Ph.D. thesis, University of Illinois at Urbana

Champaign, Urbana,
IL. Also IlliGAL Report No. 2002023.
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