Estimation of Distribution Algorithms EDA

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Estimation of Distribution Algorithms (EDA)

• Siddhartha K. Shakya
• School of Computing.
• The Robert Gordon University
• Aberdeen, UK
• ss_at_comp.rgu.ac.uk

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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

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Basic Concept of Solution and Fitness
Graph colouring Problem An Example
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
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Basic concept of a solution and Fitness
Given 2 colour Black 0 White 1
Representation of a solution as a chromosome
Solution
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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.

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GA Iteration

• Initialisation of a parent population
• Evaluation
• Crossover
• Mutation
• Replace parent with child population and go to
  step 2 until termination criteria satisfies

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GA Iteration
Initialization
Evaluation
After Crossover
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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.

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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

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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)

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GA to EDA
Initial Population
Evaluation
Selection
Probabilistic Model Building
Sampling Child Population
EDA framework
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General Notation

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

is a conditional distribution
is a joint probability distribution
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Estimation of Probability distribution
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Simple Univariate Estimation of Distribution
Algorithm
Initial Population
Evaluation
Selection
Calculate univariate marginal probability and
sample Child Population
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Simple univariate EDA (UMDA)
Initialization
Evaluation
Selection
Repeat iteration
Build model
Estimation of Distribution
Calculate Distribution
Sampling
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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

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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

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Univariate EDA Model
x2
x1
x3
x5
x4
x7
x6
Graphical representation of probability model
assuming no dependency among variables. (UMDA,
PBIL, cGA)
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Bivariate EDA Model
a. Chain model (MMIC)
b. Tree model (COMIT)
c. Forest model (BMDA)
Graphical representation of probability model
assuming dependency of order two among variables.
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Multivariate EDA Model
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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.

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Summary

• 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.

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Links

• http//cswww.essex.ac.uk/staff/zhang/MoldeBasedWeb
  /RGroup.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.