Genetic Algorithms

Li M. Fu

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Introduction

• The algorithms are based on simulated evolution.
• Basic elements include populations, generations, genetic operations, stochastic selection, fitness functions.
• Applications include learning, optimization, and genetic programming.

Genetic algorithms are algorithms for optimization and learning based on the mechanism of genetic evolution. The field of genetic algorithms was founded by John Holland (1975). Such algorithms have been applied to problems that are difficult and important, and are being applied to artificial intelligence systems.

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Topics

• Genetic algorithms
• Genetic programming
• Models of evolution

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A genetic algorithm consists of five components:

• A chromosomal representation of solutions (Chromosomes carry genetic material inside a cell nucleus.)
• A way to initialize the population of chromosomes
• An evaluation function to rate solutions
• Genetic operators to alter the composition of chromosomes during reproduction, e.g., crossover
• Parameter settings for the algorithm, e.g., population size and probabilities of applying genetic operators

Over time, the population evolves to contain better individuals and finally converges to globally optimal (or near optimal) results.

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An Analysis

Let m(H, t) denote the number of chromosomes of a particular configuration H at time t. Suppose during reproduction, a chromosome is copied according to its fitness, and the old population is replaced by the new population. Then, we may estimate the number at time t + 1 by (Goldberg 1989)

		    f(H)
m(H, t+1) = m(H, t)-----
		    E[f]

where f(H) is the fitness of H and E[f] is the average fitness of the entire population. Starting at t = 0 and assuming stationary fitness values, we obtain the equation

		     f(H)
m(H, t+1) = m(H, 0)(-----)t
		     E[f]

Thus, it is clear that the numbers of favorable (unfavorable) configurations will increase (decrease) exponentially.

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Schema Theory

• A schema is any string composed of 0s, 1s, and *'s.
• Better fit schemas are selected as evolution goes on.

• 	      f(s)              d(s)
  E[m(s, t+1)]=-----m(s, t)(1 - pc----)(1 - pm)o(s)
  	      E[f]              l-1
  
  pc: probability of crossover (single-point)
  pm: probability of mutation (single-point)
  l: length of a chromosome
  d(s): distance between the leftmost and rightmost defined bits in s
  o(s): number of defined bits
  

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A Simple Genetic Algorithm

• Initialize the population of chromosomes.
• If the termination criterion is met, exit; else repeat the following steps:
  • Select one or more parents (chromosomes) to reproduce children (chromosomes). Selection is stochastic, but parents with higher evaluations are favored. Children are produced by applying the genetic operators to the parents.
  • Children are evaluated, and promising ones are inserted into the population. The entire or a subset of population may be replaced.

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Example

A chromosome is represented as a fixed-length string of 0 and 1. Two strings undergo crossover at a random position so that all bits are swapped from that position through the end. Suppose the initial population consists of four strings:

No.  String  Fitness  
1  111000  90 
2  010101  240 
3  001011  150 
4  101110  320  

Now, suppose chromosomes 2 and 3 are selected for reproduction and crossover occurs at position 3 to yield two new strings:

No.  String  Fitness  
5  011011  470 
6  000101  80  

Assume that the population size is fixed. At this point, chromosome 5 can replace chromosome 1 to give rise to a new population. Iterations like this continue until no further improvement for the population is seen.

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Example

• The problem domain: Use GA to learn a rule set or disjunctive concepts.
• Representation:

  IF a1 = T and a2 = F Then c = T 
  
  IF a2 = T THEN c = F
  
  

• Genetic Operation:

  Parents:
      a1    a2   c        a1  a2  c
  h1: 1[0   01   1       11  1]0  0
  
  h2  0[1   1]1   0       10  01  0
  
  Children:
      a1   a2   c 
  h3: 11   10   0  
  
      a1   a2   c       a1  a2  c       a1  a2  c
  h4  00   01   1       11  11  0       10  01  0
  

• Fitness Function:

  Fitness (h) = (correct(h))2
  

• Extensions:

      a1   a2   c       a1  a2  c    Spe  Gen
      10   01   1       11  10  0     1    0
  

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Example

• The problem domain: Use GP to write a computer program for stacking blocks. concepts.
• Define Objects:

  CS (current stack)
  
  TB (top current block)
  
  NN (not necessary)
  

• Define Operators (Functions):

  (MS x) (move to stack)
  
  (MT x) (move to table)
  
  (EQ x y) (equal)
  
  (NOT x)
  
  (DU x y) (do x until y returns T)
  

• GP Parameters:

  Population size: 300
  Fitness: based on 166 training examples
  

• Results: 10 generations

  (EQ (DU (MT CS) (NOT CS)) (DU (MS NN) (NOT NN)))
  

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Evolution: Genetics and Learning

• Lamarckian hypothesis
• Baldwin effect

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Comments on Genetic Algorithms

A list of features has been identified which make a domain suitable for the application of genetic algorithms (Davis 1987): Domains should be multimodal; domains should not exhibit a high degree of epistasis (epistasis is the inhibition of one part of a solution by the action of another); domains should exhibit detectable and encodable regularities. The building block hypothesis has been suggested. Simple genetic algorithms depend on the recombination of building blocks to seek the best solutions. If the building blocks are misleading, the algorithms may still be clever enough to figure out (near) optimal solutions (by mutation, for example) but may take a longer time.

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Summary

• What are the genetic algorithms?
• Are they similar to neural networks? In what sense?
• Why are they interesting?
• What are their weaknesses?

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Web Material

Genetic Algorithms

Genetic Programming