1. Dr. Kenneth Stanley September 13, 2006
CAP6938 Neuroevolution and Developmental Encoding Evolutionary 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 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:

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)