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Schema Theory and its Applications in Genetic Algorithms

This article discusses the concept of schema theory in Genetic Algorithms (GA) and its implications for selection, crossover, and mutation. It also explores the Building Blocks Hypothesis and the No Free Lunch Theorem in optimization algorithms. Additionally, it examines the relationship between function approximation and optimization problems, as well as the potential of neural networks in approximation problems.

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Schema Theory and its Applications in Genetic Algorithms

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  1. CAP6938Neuroevolution and Developmental EncodingEvolutionary Computation Theory Dr. Kenneth Stanley September 13, 2006

  2. Schema Theory (Holland 1975) • A building block is a set of genes with good values • Schemas are a formalization of buildings blocks • Schemas are bit strings with *’s (wildcards) • 1****0 is all 6-bit strings surrounded by 1 and 0 • Order 2: 2 defined bits • A schema defines a hyperplane • Example: 1** 1

  3. Schema Fitness • GA implicitly evaluates fitness for all its schemas • Average fitness of a schema is average fitness of all possible instances of it • A GA behaves as if it were really storing these averages

  4. Schema Theorem on Selection • Idea: Calculate approximate dynamics of increase and decreases of schema instances • Instances of H at time t: • Observed avg. fitness of H at time t: • Goal: Calculate • Using the fact that number of offspring is proportional to fitness: • Thus, increases or decreases in instances depends on schema average fitness

  5. Schema Theorem with Crossoverand Mutation • Question is what the probability is that schema H will survive a crossover or mutation • Let d(H) be H’s defining length • Probability that schema H will survive crossover: • Equation shows it’s higher for shorter schemas • Probability of surviving mutation:

  6. Total Schema Theorem • The expected number of instances of schema H taking into account selection, crossover, and mutation: • Meaning: Low-order schemas whose average fitness remains above the mean will increase exponentially. • Reason: Increase of non-disrupted schema proportional to

  7. Building Blocks Hypothesis(Goldberg 1989) • Crossover combines good schema into equally good or better higher-order schema • That is, crossover (or mutation) is not just destructive: It is a power behind the GA

  8. Questioning the BBH • Why would separately discovered building blocks be compatible? • What about speciation? • Hybridization is rare in nature • Gradual elaboration is safer • Schema Theorem and BBH assume fixed length genomes

  9. No Free Lunch Theorem (Wolpert and Macready 1996) • An attack on GAs and “black box” optimization • Across all possible problems, no optimization method is better than any other • “Elevated performance over one class of problems is exactly paid for in performance over another class.” • Implication: Your method is not the best • Or is it?

  10. Hill Climbing vs. Hill Descending • Isn’t hill climbing better overall? No

  11. Very Bad News • “If an algorithm performs better than random search on some class of problems then it must perform worse than random search on the remaining problems.” • “One should be weary of trying to generalize [previously obtained] results to other problems.” • “If the practitioner has knowledge of problem characteristics but does not incorporate them into the optimization algorithm…there are no formal assurances that the algorithm chosen will be at all effective.”

  12. Hope is not Lost • An algorithm can be better over a class of problems if it exploits a common property of that class • What is the class of problems known as the “real world?” • Characterizing a class has become important

  13. Function Approximation is a Subclass of Optimization • A function approximator can be estimated • Estimation means described in fewer dimensions (parameters) than the final solution • That is not true of optimization in general • f(x) can be as simple or as complex as we want • There may be a limit on the # of bits in f(x), i.e. the size of the memory, but we can use them however we want

  14. Exploiting Approximation • How can the structure of approximation problems be exploited? • Start with simple approximations • Complexify them gradually • Information about a function can be elaborated • Information is not accumulated in general optimization • Neural networks are approximators • Real world problems are often approximation problems

  15. Next Week: Neuroevolution (NE) • Combining EC with neural networks • Fixed-topology NE and TWEANNs • The Competing Conventions Problem Genetic Algorithms and Neural Networks by Darrell Whitley (1995) Evolving Artificial Neural Networks by Xin Yao (1999)Genetic set recombination and its application to neural network topology optimisation by Radcliffe, N. J. (1993). (Skim from section 4 on, except for 9.2)

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