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EC Theory

EC Theory. Questions. What laws described the macroscopic behavior of GAs? How do low-level operators give rise to this behavior? On what types of problems are GAs likely to perform well or poorly? What does it mean for a GA to “perform well”? When will a GA outperform other search methods?.

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EC Theory

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  1. EC Theory

  2. Questions • What laws described the macroscopic behavior of GAs? • How do low-level operators give rise to this behavior? • On what types of problems are GAs likely to perform well or poorly? • What does it mean for a GA to “perform well”? • When will a GA outperform other search methods?

  3. Holland’s Schema Theorem • “an adaptive system must persistently identify, test, and incorporate structural properties hypothesized to give better performance in some environment.” • Schemas are formalizations of these structural properties

  4. Holland’s Analysis Showed… • While explicitly calculating the fitness of a population, a GA implicitly estimates the average fitness of a much larger number of schemas – “implicit parallelism” • Those schemas whose fitness estimates remain above average receive an exponentially increasing number of trials • Selection increasingly focuses the search on subsets of the search space with above average fitness

  5. Schemata • schema • a template • the new gene alphabet  {0,1,*} • allows exploration of similarities among chromosomes • represents all matching strings

  6. Schemata example: the schema (* 1 1 1 1 0 0 1 0 0) matches two strings: (0 1 1 1 1 0 0 1 0 0) and (1 1 1 1 1 0 0 1 0 0) Q. which strings does this schema match? (0 1 1 * 1 0 1 1 * *)

  7. Schemata • matches 2r strings • r: no. of (*) in schema • a string of length l matched by 2l schemata • for strings of length l  total 3l schemata • population size n: between 2l and n*2l schemata may be represented

  8. Schemata • about O(n3) (Holland’s estimate) schemata processed successfully  implicit parallelism • even though at each generation computation proportional to size of the population (O(n)) is performed • one of the main reasons for success of GAs

  9. Schema Properties • order of schema S: o(S) • no. of fixed (non-*) positions • defining length of schema S: d(S) • distance between first and last fixed string positions • fitness of schema S at time t: f(S,t) • the average fitness of all strings in population matched by S

  10. Schema Properties Examples: S1: (* * * * 0 * *) o(S1)=1 d(S1)=0 S2: (* 1 * * 0 * *) o(S2)=2 d(S2)=3 S3: (0 1 * 0 0 * 1) o(S2)=5 d(S2)=6

  11. Schema Processing let m(S,t) denote the expected no. of individuals matched by schema S at time t, what is m(S,t+1)? Assume: • genotype length: l • proportionate parent selection • one-point crossover • bitwise mutation • generational

  12. Schema Processing(Selection)

  13. Schema Processing(Crossover) If crossover site chosen uniformly at random, schema destroyed with probability: Survival probability of schema S:

  14. Schema Processing(Crossover) • example: • Consider • A=0111000 matched by • S1=*1****0 • S2=***10** • Assume crossover site=2 • What happens to both schemata?

  15. Schema Processing(Mutation) • for a schema to survive mutation with pm • all specified positions must survive • a single position survives with probability (1-pm) • schema survives with probability (1-pm)o(S) which is approximately (1-pmo(S) ) Survival probability of schema S:

  16. Schema Processing let m(S,t) denote the expected number of individuals belonging to schema S at time t

  17. The Schema Theorem • building blocks receive increasing trials in subsequent generations of a GA • building block • short (small d) • low-order (small o) • above average schemata (high f(S))

  18. The Building Block Hypothesis states that combining • short, low-order, above average schemata yields • high order schemata that also demonstrate • above average fitness

  19. The Building Block Hypothesis • the fundamental theorem of GAs. • shows in essence how GAs explore similarities • theoretical arguments against the schema theorem • research continues

  20. Modelling and Analyzing EAs • schema analysis and linkage analysis • Markov chain analysis • each population is a state in a Markov chain • transition matrices based on pc , pm and selection

  21. Modelling and Analyzing EAs (Markov chain analysis) • nth generation will certainly contain global optimum if • parent selection prob. for all individuals >0 • survival selection prob. for all individuals >0 • elitist survival selection • prob. of creating any solution by variation operators is >0 • a GA with pm>0 and elitism will always converge to global optimum

  22. Modelling and Analyzing EAs • dynamical systems approach • assumes infinite population • statistical mechanics approach • reductionist approaches • isolating different parts of system to examine seperately • neglect interaction effects but provides good insights

  23. Modelling and Analyzing EAs • No Free Lunch Theorems • if averaged over space of all possible problems, all non-revisiting,black-box algorithms exhibit the same performance • debated however basically accepted • lessons: • if a new algorithm performs best for a specific class of problems, it will perform poorly at some others • NFL can be circumvented through using problem specific knowledge

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