An Introduction to Genetic Algorithms - Boente

function of the average error of the 20 networks over the chosen subset of 20 mappings—low average error
translated to high fitness. This fitness was then transformed to be a percentage, where a high percentage meant
high fitness.
Using this fitness measure, the GA was run on a population of 40 learning rules, with two−point crossover
and standard mutation. The crossover rate was 0.8 and the mutation rate was 0.01. Typically, over 1000
generations, the fitness of the best learning rules in the population rose from between 40% and 60% in the
initial generation (indicating no significant learning ability) to between 80% and 98%, with a mean (over
several runs) of about 92%. The fitness of the delta rule is around 98%, and on one out of a total of ten runs
the GA discovered this rule. On three of the ten runs, the GA discovered slight variations of this rule with
lower fitness.
These results show that, given a somewhat constrained representation, the GA was able to evolve a successful
learning rule for simple single−layer networks. The extent to which this method can find learning rules for
more complex networks (including networks with hidden units) remains an open question, but these results
are a first step in that direction. Chalmers suggested that it is unlikely that evolutionary methods will discover
learning methods that are more powerful than back−propagation, but he speculated that the GA might be a
powerful method for discovering learning rules for unsupervised learning paradigms (e.g., reinforcement
learning) or for new classes of network architectures (e.g., recurrent networks).
Chalmers also performed a study of the generality of the evolved learning rules. He tested each of the best
evolved rules on the ten mappings that had not been used in the fitness calculation for that rule (the "test set").
The mean fitness of the best rules on the original mappings was 92%, and Chalmers found that the mean
fitness of these rules on the test set was 91.9%. In short, the evolved rules were quite general.
Chalmers then looked at the question of how diverse the environment has to be to produce general rules. He
repeated the original experiment, varying the number of mappings in each original environment between 1
and 20. A rule's evolutionary fitness is the fitness obtained by testing a rule on its original environment. A
rule's test fitness is the fitness obtained by testing a rule on ten additional tasks not in the original
environment. Chalmers then measured these two quantities as a function of the number of tasks in the original
environment. The results are shown in figure 2.26. The two curves are the mean evolutionary fitness and the
mean test fitness for rules that were tested in an environment with the given number of tasks. This plot shows
that while the evolutionary fitness stays roughly constant for different numbers of environmental tasks, the
test fitness increases sharply with the number of tasks, leveling off somewhere between 10 and 20 tasks. The
conclusion is that the evolution of a general learning rule requires a diverse environment of tasks. (In this case
of simple single−layer networks, the necessary degree of diversity is fairly small.)
THOUGHT EXERCISES
1.
Chapter 2: Genetic Algorithms in Problem Solving
Using the function set {AND, OR, NOT} and the terminal set {s –1 , s 0 , s +1 }, construct a parse tree (or
Lisp expression) that encodes the r = 1 majority−rule CA, where s i denotes the state of the
neighborhood site i sites away from the central cell (with indicating distance to the left and +
indicating distance to the right). AND and OR each take two arguments, and NOT takes one
argument.
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