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From: szabo@netcom.com (Nick Szabo)
Subject: Genetic landscapes
Message-ID: <szaboD0FuCq.Kvy@netcom.com>
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Date: Wed, 7 Dec 1994 11:24:25 GMT
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Xref: glinda.oz.cs.cmu.edu comp.ai.alife:1438 comp.ai.genetic:4500

A common implication of the terms "landscape" and "search space"
is that the space defined by the search operator is Euclidean, but 
several interesting classes of search spaces in machine learning 
are non-Euclidean.  In a non-Euclidean search space the
concept of "distance", and thus "gradients" and
"hill-climbing", cannot be defined in an easy to visualize
and analyze Euclidean manner.  In practice this means they do not
get defined, or that Euclidean visualizations and distance
measures are misapplied.

(This issue is orthogonal to the dynamic environment or
co-evolving fitness function issue, which also makes the
space non-Eucliean and difficult to impossible to 
visualize or analyze).

In genetic algorithms the space is defined by the genetic
operator.  For example, mutation gives us a well-defined,
easy to visualize Euclidean space, with each point one
bit apart.  Not so for crossover.  Nobody has formalized the space
for crossover; even worse few have seen the need for formalizing
it.  The "NK landscapes" do not count -- they are mutation landscapes,
and to apply them to genetic algorithms with crossover
leads only to silly visualizations like "teleporting across
the fitness space".

I heavily doubt that the space defined by genetic crossover is
Euclidean, and conclude that crossover makes GAs fundamentally different 
from hill-climbing techniques.

Some of the work done on randomly generated data structures (Flajolet
et. al.) might be well applied to determine crossover probability
distributions for GA and GP, define a distance measure that corresponds
to the distribution, and thus define crossover space.  Can anyone point 
to work along these lines?


-- 
Nick Szabo				szabo@netcom.com
