An Introduction to Genetic Algorithms - Boente

Results of the Formalization
How can this formalization help us to better understand or predict the GA's behavior? Vose and Liepins,
viewing G as a dynamical system, formulated a geometric picture of the GA's behavior and then used it to
prove some behavioral properties. The geometric picture is that the set of all possible vectors form a
surface S on which G acts to move from point to point. The initial point is , and iterating G from this point
forms a trajectory on S. In analyzing the dynamics of G, the first things to determine are the fixed points of G
on S—i.e., the set of such that . In other words, we want to know what points
have the property that, once the GA arrives there, it will not move away.
This general problem was not solved by Vose and Liepins (1991); instead they solved the separate problems
of finding the fixed points of F and and analyzing their properties. It is not difficult to show that the fixed
points of F (selection alone) are the populations that have completely converged to strings of equal fitness.
Vose and Liepins proved that only one class of these fixed points is stable: the set of fixed points
corresponding to the maximally fit strings in the search space. In other words, if the population converges to a
state that does not consist entirely of maximally fit strings, a small change in the fitness distribution of the
population might result in movement away from that fixed point. However, if the population is maximally fit,
then under any sufficiently small change in the fitness distribution, the GA will always return to that fixed
point.
Vose and Liepins then showed that working alone on has only one fixed point: the vector
consisting of equal probabilities for all strings in the search space. Likewise, working on has one fixed
point: all strings present in equal proportions. This means that, in the limit of an infinite population, crossover
and mutation working in the absence of selection will eventually produce maximally "mixed" populations
with equal occurrences of all strings.
Vose and Liepins left open the more difficult problem of putting F and M together to understand the
interacting effects of crossover and mutation. However, they conjectured that the formalism could shed light
on the "punctuated equilibria" behavior commonly seen in genetic algorithms—relatively long periods of no
improvement punctuated by quick rises in fitness. The intuition is that such punctuated equilibria arise from
the combination of the "focusing" properties of F and the "diffusing" properties of . The periods of
"stasis" correspond to periods spent near one of the unstable fixed points, and the periods of rapid
improvement correspond to periods spent moving (under the diffusing force of recombination) from the
vicinity of one fixed point to another. (Note that even though a fixed point is unstable, a dynamical system can
stay in its vicinity for some time.) These effects have yet to be rigorously quantified under this or any other
model. (The notion of focusing and mixing forces working together in a GA is discussed in less technical
detail in chapter 6 of Holland 1975.)
Vose and Liepins's formalization is an excellent first step toward a more rigorous understanding of and more
rigorous predictions about simple GAs. (For further work on Vose and Liepins's model see Whitley 1993a.)
However, one major drawback is that the formalization and its results assume an infinite population—that is,
they are phrased in terms of expectations. Assuming an infinite population is an idealization that simplifies
analysis; however, the behavior of finite populations can be very different as a result of sampling error.
A Finite−Population Model
Chapter 4: Theoretical Foundations of Genetic Algorithms
The infinite−population case involves deterministic transitions from to and thus from to
—in an infinite population there are no sampling errors. In contrast, modeling a finite population
requires taking account of the stochastic effects of sampling.
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