An Introduction to Genetic Algorithms - Boente

Deceiving a Genetic Algorithm
Chapter 4: Theoretical Foundations of Genetic Algorithms
The theory of schemas has been used by some in the GA community to propose an answer to "What makes a
problem hard for a GA?" As described above, the view is that competition among schemas roughly proceeds
from low−order schema partitions at early times to higher−order schema partitions at later times. Bethke
(1980) reasoned that it will be hard for a GA to find the optimum of a fitness function if low−order partitions
contain misleading information about higher−order partitions. The following extreme example illustrates this.
Call a schema H a "winner" if its static average fitness is highest in its partition. Suppose that any schema
whose defined bits are all ones is a winner except for the length L schema 1111···1, and let 0000···0 be a
winner. In principle, it should be hard for a GA to find 0000···0, since every lower−order partition gives
misleading information about where the optimum is likely to be found. Such a fitness function is termed "fully
deceptive." (The term "deceptive" was introduced by Goldberg (1987).) Fitness functions with lesser amounts
of deception are also possible (i.e., some partitions give correct information about the location of the
optimum). Bethke used "Walsh Transforms"—similar to Fourier transforms—to design fitness functions with
various degrees of deception. For reviews of this work, see Goldberg 1989b,c and Forrest and Mitchell 1993a.
Subsequent to Bethke's work, Goldberg and his colleagues carried out a number of theoretical studies of
deception in fitness functions, and deception has become a central focus of theoretical work on GAs. (See,
e.g., Das and Whitley 1991; Deb and Goldberg 1993; Goldberg 1989c; Liepins and Vose 1990, 1991; Whitley
1991.)
It should be noted that the study of GA deception is generally concerned with function optimization; the
deception in a fitness function is assumed to make it difficult to find a global optimum. In view of the
widespread use of GAs as function optimizers, deception can be an important factor for GA practitioners to
understand. However, if one takes the view that GAs are not function optimizers but rather "satisficers" or
payoff maximizers, then deception may be less of a concern. As De Jong (1993, p. 15) puts it, the GA (as
formulated by Holland) "is attempting to maximize cumulative payoff from arbitrary landscapes, deceptive or
otherwise. In general, this is achieved by not investing too much effort in finding cleverly hidden peaks (the
risk/reward ratio is too high)."
Limitations of "Static" Schema Analysis
A number of recent papers have questioned the relevance of schema analysis to the understanding of real GAs
(e.g., Grefenstette 1993; Mason 1993; Peck and Dhawan 1993). Here I will focus on Grefenstette's critique of
the "Static Building Block Hypothesis."
The following qualitative formulation of the Schema Theorem and the Building Block Hypothesis should now
be familiar to the reader: The simple GA increases the number of instances of low−order,
short−defininglength, high−observed−fitness schemas via the multi−armed−bandit strategy, and these
schemas serve as building blocks that are combined, via crossover, into candidate solutions with increasingly
higher order and higher observed fitness. The rationale for this strategy is based on the assumption that the
observed and static fitnesses of schemas are correlated; some potential problems with this assumption have
been pointed out in the previous sections.
Grefenstette (1993, p. 78) claims that much work on GA theory has assumed a stronger version that he calls
the "Static Building Block Hypothesis" (SBBH): "Given any low−order, short−defining−length hyperplane
[i.e., schema] partition, a GA is expected to converge to the hyperplane [in that partition] with the best static
average fitness (the 'expected winner')." This is stronger than the original formulation, since it states that the
GA will converge on the actual winners of each short, low−order partition competition rather than on the
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