Dr. Kenneth Stanley September 13, 2006 CAP6938 Neuroevolution and Developmental Encoding Evolutionary Computation Theory Dr. Kenneth Stanley September 13, 2006

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

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

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

Schema Theorem with Crossover and 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:

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

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

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

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?

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

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.”

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

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

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

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)