Introduction%20to%20Genetic%20Algorithms%20and%20Evolutionary%20Computing

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Introduction to Genetic Algorithms and
Evolutionary Computing

• Randy Ribler
• ribler_at_lynchburg.edu
• Computer Science Department
• Lynchburg College, Lynchburg, VA, USA

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Evolutionary Computing

• Consider the spectacular results of biological
  evolution
• Designs better than any other known
• Adaptive to changing requirements
• Natural Selection works the same way regardless
  of the application
• Algorithm is relatively unaware of application
• Making it more aware may improve performance

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Evolutionary Computing (Limitations)

• Simulation of Natural Selection On a Smaller
  Scale
• Cant wait millions of years for evolution
• Computer are fast and getting faster
• EC works well on distributed computer
  architectures
• Population sizes are much more limited
• Actual evolutionary process is very complex we
  can only approximate the process
• Process is not fully understood
• More accurate approximation requires additional
  computation resources

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Simple Genetic Algorithms

• General Technique for multi-variable function
  optimization and machine learning
• Example Design an airplane wing
• Variables
• length, width, rotation in x-axis, rotation in
  y-axis

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Example Design an airplane wing

• Variables
• Wing length (l)
• Wing Width (w)
• Sweep Angle (s)
• Pitch Angle (p)

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Fitness Function for Airplane Wing

• Fitness(l,w,s,p) Efficiency(l,w,s,p)
  lift(l,w,s,p) / drag(l,w,s,p)
• This is an extreme simplification, and I dont
  know anything about designing airplane wings, but
  an aerospace engineer can give us the proper
  input variables and efficiency functions and that
  is what we would use.
• Real case would have many more variables and a
  complex fitness function perhaps a wind tunnel
  simulation.

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What is the Range of Each Variable?

• Length
• Range 0 to 16 meters
• 4-bits provides a 1 meter resolution
• 5-bits would provides a .5 meter resolution
• Width
• Range 0 to 8 feet
• 3-bits provides a 1 foot resolution
• Sweep angle
• Range 0 to 180 Degrees
• 4-bits provides a 180/16 degree resolution
• Pitch angle
• Range -90 to 90 degrees
• 5-bits provides a 180/32 degree resolution

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Genetic Representation of Airplane Wing

• 4 bits for length
• 3 bits for width
• 4 bits for sweep angle
• 5 bits for pitch angle
• Total of 16 bits to describe all 4 variables
• Any random 16 bit value completely describes
  particular values for an airplane wing.

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Create a population of random airplane wing
strings

• Select a population size.
• Lets use 500 for our airplane wing.
• Create 500 random 16-bit values.
• Each 16-bit value represents the string for an
  experimental aircraft wing
• The fitness function can provide a measure of the
  quality of each of or 500 experimental wings
• Because the values were generated randomly, most
  wings will be pretty bad and have low fitness
  values, but some will be better than others

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Terminology

• String - Sequence of genetic information fully
  describing an individual in the population.
• Gene location of a bit on a string
• Allele value of a gene

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Natural Selection

• As in the natural world
• The more fit members of the population have a
  greater probability of propagating their genes
  into the next generation

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Creating the Gene Pool for the Next Generation

• More fit members should be more prominent in the
  gene pool
• Less fit members should not be completely
  excluded
• Excluding the less fit might remove important
  gene subsequences from the gene pool
• Converging too quickly to a solution can provide
  suboptimal results

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Reproduction

• Select 500 members of the gene pool where the
  probability of the ith member of the population
  being selected is

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Reproduction (continued)

• Once the 500 members of the gene pool for the
  next generation is established, pairs of parents
  are selected randomly. To distinguish between
  the two parents we will designate one as the
  father and the other as the mother, although
  their functions are identical.

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Reproduction (continued)

• Each pair of parents creates two children using
  one of two methods
• Cloning
• One child is an exact copy of the father
• One child is an exact copy of the mother
• Crossover
• Some bits are copied from the mother, some from
  the father
• Copy from one parent until you reach a crossover
  point, then copy from the other parent.

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Crossover with Single Crossover Point
?

• Father
• Mother
• Child 1
• Child 2

0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1
1 1 1 1 0 0 0 0
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Crossover with Three Crossover Points
?
?
?

• Father
• Mother
• Child 1
• Child 2

0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
0 1 1 0 0 0 1 1
1 0 0 1 1 1 0 0
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Mutation

• Each bit copied to a child has a probability of
  being changed (from 1 to 0, or 0 to 1).
• Usually the probability of mutation is relatively
  low, but enough to encourage diversity.
• In other models, mutation is the principal method
  of genetic change.

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Summary of Simple Genetic Algorithm

• Create a population of random genes
• For a specified number of generations
• Apply the fitness function to each member of the
  population
• Biasing toward the more fit individuals, create a
  pool of parents
• While there are not a number of children equal to
  the original population size
• Randomly select two parents and create two
  children either using cloning or crossover
• Apply mutation
• Completely replace the current generation with
  the children

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Basic Parameters for a Genetic Algorithm

• Population size
• Gene Length
• Number of crossover points
• Probability of using crossover vs. cloning
• Number of generations
• Scaling Factors
• More Complex/ Powerful applicatoin specific
  modifications

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Schema

• A Schema is a similarity template describing a
  subset of strings with similarities at certain
  string positions
• Consists of series of 1s, 0s, and s (for
  dont care)
• Examples
• 1
• Matches (100, 101, 110, 111)
• 10
• Matches (01000, 01001, 01010, 01011,
• 11000, 11001, 11011, 11011)
• The plural of the word schema is schemata.

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Counting Schemata

• For a binary schema of length k, there are 3k
  possible schemata
• Each position may contain 0, 1, or
• A binary string of length k can have membership
  in 2k different schemata
• Each position can contain the actual string
  digit, or
• Example
• 101 has membership in schemata (101, 10, 11,
  1, 01, 0, 1, )
• Schemata per populations
• Range from 2k (all strings are the same) to n2k
  (all strings have different schemata)

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Schema Terms

• Schema order o(H) is the number of fixed values
• Number of 1s and 0s
• 010 has an order of 3
• 0111 has an order of 4
• Schema defining length d(H) is the distance
  between first and last fixed value
• Number of positions where crossover can disrupt
  the schema
• 010 has a defining length of 4
• 0111 has defining length of 5
• 111 has a defining length of 4
• 00010 has a defining length of 7

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Effect of Reproduction on Expected Number of
Schemata in Population

• A particular schema grows as the ratio of average
  fitness of the schema to average fitness of the
  population
• m examples of a particular schema H at time t is
  m(H, t)

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Reproduction of Fit Schemata

• Suppose a particular schema H remains above
  average an amount c.

Starting at t0 and assuming a stationary value
of c, we obtain
Reproduction allocates exponentially increasing
and decreasing numbers of schemata to future
generations!
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Schema Disruption Due to Crossover

• Ps Probability of surviving crossover
• Pd Probability of being destroyed by crossover
• Pc Probability of crossover versus cloning
• l is the length of the string
• d(H) is the defining length of the schema

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Schema Disruption Due to Mutation

• Pm Probability of mutating each bit
• o(H) is the schema order (number of fixed values)
• Probability a particular schema is disrupted by
  mutation is
• (1-pm)o(H)
• For very small values of pm, we can approximate
  this to 1 o(H)pm

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Fundamental Theorem of Genetic Algorithms
Short, low-order, above-average schemata receive
exponentially increasing trials in subsequent
generations.
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Position of Bits on the Gene is Important

• Allow this to evolve as well
• Dynamically change the location of each bit of
  the gene
• Inversion
• Select subsequences of the gene and invert the
  data and the interpretation
• Has no effect on the genetic content, just the
  way the information is stored.
• Requires additional meta-information to be stored
  that describes the location of each bit.

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Inversion Example
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0
0 1 2 3 4 9 8 7 6 5 10 11 12 13 14 15
0 1 1 1 0 1 1 0 1 0 0 1 1 0 0 0
Randomly select two points in the sequence for
inversion. This example, uses 5 and 9. The bit
numbers and the data are inverted. Now bits 4
and 9 are adjacent and less likely to be
disrupted with crossover.
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Inversion Complicates Crossover
4 5 9 0 11 12 1 13 14 10 15 2 3 6 7 8
0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0
0 1 2 3 4 9 8 7 6 5 10 11 12 13 14 15
1 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1

• How do we combine these two genes with different
  bit orders?
• Insist that parents have same organization (not
  very good)
• Discard if crossover yields duplicate bit
  numbers
• Reorder one parent, chosen at random, to match
  the other
• Reorder the less fit of the two parents to match
  the other

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When Might GA provide a Good Solution?

• Highly multimodal functions
• Discrete or discontinuous functions
• High-dimensional functions, including many
  combinatorial ones
• Nonlinear dependence on parameters (interactions
  among parameters) epistasis makes it hard for
  others
• Often used for approximate solutions to
  NP-complete combinatorial problems

From Introduction to Genetic Algorithms Tutorial,
Erik D. Goodman, Gecco 2005
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Flavors of Evolutionary Computing

• Genetic Algorithms (GA)
• Evolutionary Programming (EP)
• Classifier Systems
• Evolving Neural Networks
• Genetic Programming (GP)
• Artificial Life (AL)

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Acknowledgement

• Schema theory equations from David E. Goldbergs,
  Genetic Algorithms in Search, Optimization and
  Machine Learning.