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Genetic algorithms

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Genetic algorithms. Gentle introduction. Jim Cohoon and Kimberly Hanks ... Melanie Mitchell, An Introduction to Genetic Algorithms. Evolutionary terminology ... – PowerPoint PPT presentation

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Title: Genetic algorithms


1
Genetic algorithms
  • Gentle introduction
  • Jim Cohoon and Kimberly Hanks

2
Genetic Algorithms in a slide
  • Premise
  • Evolution worked once (it produced us!), it might
    work again
  • Basics
  • Pool of solutions
  • Mate existing solutions to produce new solutions
  • Mutate current solutions for long-term diversity
  • Cull population

3
Originator
  • John Holland
  • Seminal work
  • Adaptation in Natural and Artificial Systems
    introduced main GA concepts, 1975

4
Introduction
  • Computing pioneers (especially in AI) looked to
    natural systems as guiding metaphors
  • Evolutionary computation
  • Any biologically-motivated computing activity
    simulating natural evolution
  • Genetic Algorithms are one form of this activity
  • Original goals
  • Formal study of the phenomenon of adaptation
  • John Holland
  • An optimization tool for engineering problems

5
Main idea
  • Take a population of candidate solutions to a
    given problem
  • Use operators inspired by the mechanisms of
    natural genetic variation
  • Apply selective pressure toward certain
    properties
  • Evolve a more fit solution

6
Why evolution as a metaphor
  • Ability to efficiently guide a search through a
    large solution space
  • Ability to adapt solutions to changing
    environments
  • Emergent behavior is the goal
  • The hoped-for emergent behavior is the design of
    high-quality solutions to difficult problems and
    the ability to adapt these solutions in the face
    of a changing environment
  • Melanie Mitchell, An Introduction to Genetic
    Algorithms

7
Evolutionary terminology
  • Abstractions imported from biology
  • Chromosomes, Genes, Alleles
  • Fitness, Selection
  • Crossover, Mutation

8
GA terminology
  • In the spirit but not the letter of biology
  • GA chromosomes are strings of genes
  • Each gene has a number of alleles i.e., settings
  • Each chromosome is an encoding of a solution to a
    problem
  • A population of such chromosomes is operated on
    by a GA

9
Encoding
  • A data structure for representing candidate
    solutions
  • Often takes the form of a bit string
  • Usually has internal structure i.e., different
    parts of the string represent different aspects
    of the solution)

10
Crossover
  • Mimics biological recombination
  • Ssome portion of genetic material is swapped
    between chromosomes
  • Typically the swapping produces an offspring
  • Mechanism for the dissemination of building
    blocks (schemas)

11
Mutation
  • Selects a random locus gene location with
    some probability and alters the allele at that
    locus
  • The intuitive mechanism for the preservation of
    variety in the population

12
Fitness
  • A measure of the goodness of the organism
  • Expressed as the probability that the organism
    will live another cycle (generation)
  • Basis for the natural selection simulation
  • Organisms are selected to mate with probabilities
    proportional to their fitness
  • Probabilistically better solutions have a better
    chance of conferring their building blocks to the
    next generation (cycle)

13
A Simple GA
  • Generate initial population
  • do
  • Calculate the fitness of each member
  • // simulate another generation
  • do
  • Select parents from current population
  • Perform crossover add offspring to the new
    population
  • while new population is not full
  • Merge new population into the current
    population
  • Mutate current population
  • while not converged

14
How do GAs work
  • The structure of a GA is relatively simple to
    comprehend, but the dynamic behavior is complex
  • Holland has done significant work on the
    theoretical foundations of Gas
  • GAs work by discovering, emphasizing, and
    recombining good building blocks of solutions
    in a highly parallel fashion.
  • Melanie Mitchell, paraphrasing John Holland
  • Using formalism
  • Notion of a building block is formalized as a
    schema
  • Schemas are propagated or destroyed according to
    the laws of probability

15
Schema
  • A template, much like a regular expression,
    describing a set of strings
  • The set of strings represented by a given schema
    characterizes a set of candidate solutions
    sharing a property
  • This property is the encoded equivalent of a
    building block

16
Example
  • 0 or 1 represents a fixed bit
  • Asterisk represents a dont care
  • 1100 is the set of all solutions encoded in 8
    bits, beginning with two ones and ending with two
    zeros
  • Solutions in this set all share the same variants
    of the properties encoded at these loci

17
Schema qualifiers
  • Length
  • The inclusive distance between the two bits in a
    schema which are furthest apart (the defining
    length of the previous example is 8)
  • Order
  • The number of fixed bits in a schema (the order
    of the previous example is 4)

18
Not just sum of the parts
  • GAs explicitly evaluate and operate on whole
    solutions
  • GAs implicitly evaluate and operate on building
    blocks
  • Existing schemas may be destroyed or weakened by
    crossover
  • New schemas may be spliced together from existing
    schema
  • Crossover includes no notion of a schema only
    of the chromosomes

19
Why do they work
  • Schemas can be destroyed or conserved
  • So how are good schemas propagated through
    generations?
  • Conserved good schemas confer higher fitness
    on the offspring inheriting them
  • Fitter offspring are probabilistically more
    likely to be chosen to reproduce

20
Approximating schema dynamics
  • Let H be a schema with at least one instance
    present in the population at time t
  • Let m(H, t) be the number of instances of H at
    time t
  • Let x be an instance of H and f(x) be its fitness
  • The expected number of offspring of x is
    f(x)/f(pop) (by fitness proportionate selection)
  • To know E(m(H, t 1)) (the expected number of
    instances of schema H at the next time unit), sum
    f(x)/f(pop) for all x in H
  • GA never explicitly calculates the average
    fitness of a schema, but schema proliferation
    depends on its value

21
Approximating schema dynamics
  • Approximation can be refined by taking into
    account the operators
  • Schemas of long defining length are less likely
    to survive crossover
  • Offspring are less likely to be instances of such
    schemas
  • Schemas of higher order are less likely to
    survive mutation
  • Effects can be used to bound the approximate
    rates at which schemas proliferate

22
Implications
  • Instances of short, low-order schemas whose
    average fitness tends to stay above the mean will
    increase exponentially
  • Changing the semantics of the operators can
    change the selective pressures toward different
    types of schemas

23
Theoretical Foundations
  • Empirical observation
  • GAs can work
  • Goal
  • Learn how to best use the tool
  • Strategy
  • Understand the dynamics of the model
  • Develop performance metrics in order to quantify
    success

24
Theoretical Foundations
  • Issues surrounding the dynamics of the model
  • What laws characterize the macroscopic behavior
    of GAs?
  • How do microscopic events give rise to this
    macroscopic behavior?

25
Theoretical Foundation
  • Hollands motivation
  • Construct a theoretical framework for adaptive
    systems as seen in nature
  • Apply this framework to the design of artificial
    adaptive systems
  • Issues in performance evaluation
  • According to what criteria should GAs be
    evaluated?
  • What does it mean for a GA to do well or poorly?
  • Under what conditions is a GA an appropriate
    solution strategy for a problem?

26
Theoretical Foundation
  • Hollands observations
  • An adaptive system must persistently identify,
    test, and incorporate structural properties
    hypothesized to give better performance in some
    environment
  • Adaptation is impossible in a sufficiently random
    environment

27
Theoretical Foundation
  • Hollands intuition
  • A GA is capable of modeling the necessary tasks
    in an adaptive system
  • It does so through a combination of explicit
    computation and implicit estimation of state
    combined with incremental change of state in
    directions motivated by these calculations

28
Theoretical Foundation
  • Hollands assertion
  • The identify and test requirement is satisfied
    by the calculation of the fitnesses of various
    schemas
  • The incorporate requirement is satisfied by
    implication of the Schema Theorem

29
Theoretical Foundation
  • How does a GA identify and test properties?
  • A schema is the formalization of a property
  • A GA explicitly calculates fitnesses of
    individuals and thereby schemas in the population
  • It implicitly estimates fitnesses of hypothetical
    individuals sharing known schemas
  • In this way it efficiently manages information
    regarding the entire search space

30
Theoretical Foundation
  • How does a GA incorporate observed good
    properties into the population?
  • Implication of the Schema Theorem
  • Short, low-order, higher than average fitness
    schemas will receive exponentially increasing
    numbers of samples over time

31
Theoretical Foundation
  • Lemmas to the Schema Theorem
  • Selection focuses the search
  • Crossover combines good schemas
  • Mutation is the insurance policy

32
Theoretical Foundation
  • Hollands characterization
  • Adaptation in natural systems is framed by a
    tension between exploration and exploitation
  • Any move toward the testing of previously unseen
    schemas or of those with instances of low fitness
    takes away from the wholesale incorporation of
    known high fitness schemas
  • But without exploration, schemas of even higher
    fitness can not be discovered

33
Theoretical Foundation
  • Goal of Hollands first offering
  • The original GA was proposed as an adaptive
    plan for accomplishing a proper balance between
    exploration and exploitation

34
Theoretical Foundation
  • GA does in fact model this
  • Given certain assumptions, the balance is
    achieved
  • A key assumption is that the observed and actual
    fitnesses of schemas are correlated
  • This assumption creates a stumbling block to
    which we will return

35
Traveling Salesperson Problem
36
Initial Population for TSP
(5,3,4,6,2)
(2,4,6,3,5)
(4,3,6,5,2)
(2,3,4,6,5)
(4,3,6,2,5)
(3,4,5,2,6)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
37
Select Parents
(5,3,4,6,2)
(2,4,6,3,5)
(4,3,6,5,2)
(2,3,4,6,5)
(4,3,6,2,5)
(3,4,5,2,6)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
Try to pick the better ones.
38
Create Off-Spring 1 point
(5,3,4,6,2)
(2,4,6,3,5)
(4,3,6,5,2)
(2,3,4,6,5)
(4,3,6,2,5)
(3,4,5,2,6)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
(3,4,5,6,2)
39
Create More Offspring
(5,3,4,6,2)
(2,4,6,3,5)
(4,3,6,5,2)
(2,3,4,6,5)
(4,3,6,2,5)
(3,4,5,2,6)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
(3,4,5,6,2)
(5,4,2,6,3)
40
Mutate
(5,3,4,6,2)
(2,4,6,3,5)
(4,3,6,5,2)
(2,3,4,6,5)
(4,3,6,2,5)
(3,4,5,2,6)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
(3,4,5,6,2)
(5,4,2,6,3)
41
Mutate
(5,3,4,6,2)
(2,4,6,3,5)
(4,3,6,5,2)
(2,3,4,6,5)
(2,3,6,4,5)
(3,4,5,2,6)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
(3,4,5,6,2)
(5,4,2,6,3)
42
Eliminate
(5,3,4,6,2)
(2,4,6,3,5)
(4,3,6,5,2)
(2,3,4,6,5)
(2,3,6,4,5)
(3,4,5,2,6)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
(3,4,5,6,2)
(5,4,2,6,3)
Tend to kill off the worst ones.
43
Integrate
(5,4,2,6,3)
(5,3,4,6,2)
(2,4,6,3,5)
(2,3,6,4,5)
(3,4,5,2,6)
(3,4,5,6,2)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
44
Restart
(5,3,4,6,2)
(2,4,6,3,5)
(5,4,2,6,3)
(2,3,6,4,5)
(3,4,5,2,6)
(3,4,5,6,2)
(3,5,4,6,2)
(4,5,3,6,2)
(5,4,2,3,6)
(4,6,3,2,5)
(3,4,2,6,5)
(3,6,5,1,4)
45
Genetic Algorithms
  • Facts
  • Very robust but slow
  • Can make simulated annealing seem fast
  • In the limit, optimal

46
Other GA-TSP Possibilities
  • Ordinal Representation
  • Partially-Mapped Crossover
  • Edge Recombination Crossover
  • Problem
  • Operators are not sufficiently exploiting the
    proper building blocks used to create new
    solutions.

47
Genetic Algorithms
  • Some ideas
  • Parallelism
  • Punctuated equilibria
  • Jump starting
  • Problem-specific information
  • Synthesize with simulated annealing
  • Perturbation operator

48
Heuristic H
Length(MST) lt Length(T)
Let T be the optimal tour.
49
Heuristic H
Tour T
Tour T
50
Perturbation of points
51
Perturbation of a Point
52
Mutation Operator
Points are perturbed in a normal distribution
centered around the original location and a
standard deviation which is a function of the
original interpoint distances.
53
Crossover Operator
54
Characteristics of Operators
  • Perturbed points tend to stay close to original
    locations, hence distances remain reasonable.
  • Small shifts in point position can have an effect
    on the MST, hence see many different solutions.

55
Improvement Over EC
Average Improvement 32.1
56
Improvement Over H
Average Improvement 15.1
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