Genetic Algorithms - Program to Analysis and Mechanization | CS 591, Study notes of Programming Languages

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CS 591

Topics in Complex Adaptive Systems

Spring Semester, 2006

Genetic Algorithms

Stephanie Forrest

FEC 355E

http://cs.unm.edu/~forrest/cas-class-06.html [email protected] 505-277-

Genetic Algorithms

• Principles of natural selection applied to computation:
  • Variation
  • Selection
  • Inheritance
• Evolution in a computer:
  • Individuals (genotypes) stored in computer’s memory
  • Evaluation of individuals (artificial selection)
  • Differential reproduction through copying and deletion
  • Variation introduced by analogy with mutation and crossover
• Simple algorithm captures much of the richness seen in naturally evolving populations.

Example Performance Curve

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(^020406080100120140160180200220240260280300320340360380400420440460480500520535) Generation

Fitness

mean fitness max fitness

Multi-parameter Function Optimization

0 0 1 1 1 1 Bit string (Gray coded) 0 0 1 1 0 1 Base 2

1 5 Base 10

F(x, y) = yx^2 - x^4

Degray

F (0001111) = F (1,5) = 5 • 12 -1^4 = 4

Decimal Binary Gray code 0 000 000 1 001 001 2 010 011 3 011 010 4 100 110 5 101 111 6 110 101 7 111 100

Genetic Programming

• Evolve populations of computer programs:
  • Typically use the language Lisp.
  • Select a set of primitive functions for each problem.
  • Represent program as a syntax tree.
• Function approximation vs. function optimization.
• Crossover operator:
  • Exchange subtrees between individual program trees.
  • Schema Theorem?
• Many applications:
  • Optimal control (e.g., the pole balancing problem)
  • Circuit design
  • Symbolic regression (data fitting)

Genetic Programming

Expression X^2 = 3xy = y^2 ⇓ LISP (= (* x x) (3 x y) ( y y)) ⇓

x

x x

3 y

y y

Where did these ideas originate?

• Genetic algorithms (Holland, 1962)
  • Original idea was to create an algorithm that captured the richness of natural adaptive systems.
  • Emphasized the adaptive properties of entire populations and the importance of recombination mechanisms such as crossover.
  • Application to function optimization introduced by DeJong (1975).
• Evolutionstrategie (Rechenberg, 1965)
  • Emphasized the importance of selection and mutation, as
  • Mechanisms for solving difficult real-valued optimization problems.
• Evolutionary programming (Fogel et al., 1966)
  • Emphasis on evolving finite state machines.
• Genetic programming (Koza, 1992)

==> “Evolutionary Computation”

References

• J. H. Holland Adaptation in Natural and Artificial Systems. Univ. of Michigan Press (1975). Second Edition published by MIT Press (1992).
• D. E. Goldberg Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley (1989).
• M. Mitchell An Introduction to Genetic Algorithms. MIT Press (1996).
• S. Forrest “Genetic Algorithms: Principles of natural selection applied to computation.” Science 261:872-878 (1993).
• J. Koza Genetic Programming. MIT Press (1992).

Genetic Algorithms and Search

• High-dimensional search spaces:
  • All binary strings of length l.
  • All possible strategies for playing the game of chess.
  • All possible tours in the Travelling Salesman Problem.
• Genetic algorithms use biased sampling to search high-dimensional spaces: - Independent sampling. - Selection biases search towards high-fitness regions. - Crossover combines partial solutions from different strings.
• Partial solution formalized as “schema.”

1*** ***** 0 ****** 0***

***** 1 ******

Schemas

• Schemas capture important regularities in the search space:
• Implicit Parallelism: 1 individual samples many schemas simultaneously.
• Schema Theorem: Reproduction and crossover guarantee exponentially increasing samples of the observed best schemas.
• Order of a schema O(s) = number of defined bits. Defining length of a schema D(s) = distance between outermost bits.

Royal Road Schemas

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Fitness

schema 1 schema 2 schema 3

Study Question

• Given a population, consisting of
  • N individuals,
  • Each individual is L bits long,
• How many schemas are sampled by the population (in one generation)?
• Hint:
  • Minimum value is: 2L
  • Maximum value is Nx2L

Questions

• When will genetic algorithms work well and when they will not?
  • Not appropriate for problems where it is important to find the exact global optimum.
  • GA domains are typically those about which we have little analytical knowledge (complex, noisy, dynamic, poorly specified, etc.).
  • Would like a mathematical characterization that is predictive.
• What makes a problem hard for genetic algorithms?
  • Deception, multi-modality, conflicting schema information, noise, ...
• What makes a problem easy for genetic algorithms?
• What distinguishes genetic algorithms from other optimization methods, such as hill climbing?
• What does it mean for a genetic algorithm to perform well?

Building Blocks Hypothesis

(Holland, 1975)

1. GA initially detects biases in low-order schemas: GA obtains good estimates of schema average fitness by sampling strings.
2. Over time, information from low-order schemas is combined through crossover, and
3. GA detects biases in high-order schemas,
4. Eventually converging on the most fit region of the space.

Implies that crossover is central to the success of the GA.

• Claim: GA allocates samples to schemas in a near optimal way:
  • K-armed bandit argument.