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94 HYPOTHESIS SPACE SEARCH
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As illustrated above, GAS employ a randomized beam search method to seek a maximally fit hypothesis This search is quite different from that of other learning methods we have considered in this book To contrast the hypothesis space search of GAS with that of neural network BACKPROPAGATION, for example, the gradient descent search in BACKPROPAGATION smoothly from one hypothesis to a moves new hypothesis that is very similar In contrast, the GA search can move much more abruptly, replacing a parent hypothesis by an offspring that may be radically different from the parent Note the GA search is therefore less likely to fall into the same kind of local minima that can plague gradient descent methods One practical difficulty in some GA applications is the problem of crowding Crowding is a phenomenon in which some individual that is more highly fit than others in the population quickly reproduces, so that copies of this individual and very similar individuals take over a large fraction of the population The negative impact of crowding is that it reduces the diversity of the population, thereby slowing further progress by the GA Several strategies have been explored for reducing crowding One approach is to alter the selection function, using criteria such as tournament selection or rank selection in place of fitness proportionate roulette wheel selection A related strategy is "fitness sharing," in which the measured fitness of an individual is reduced by the presence of other, similar individuals in the population A third approach is to restrict the kinds of individuals allowed to recombine to form offspring For example, by allowing only the most similar individuals to recombine, we can encourage the formation of clusters of similar individuals, or multiple "subspecies" within the population A related approach is to spatially distribute individuals and allow only nearby individuals to recombine Many of these techniques are inspired by the analogy to biological evolution
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941 Population Evolution and the Schema Theorem
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It is interesting to ask whether one can mathematically characterize the evolution over time of the population within a GA The schema theorem of Holland (1975) provides one such characterization It is based on the concept of schemas, or patterns that describe sets of bit strings To be precise, a schema is any string composed of Os, Is, and *'s Each schema represents the set of bit strings containing the indicated 0s and Is, with each "*" interpreted as a "don't care" For example, the schema 0*10 represents the set of bit strings that includes exactly 0010 and 01 10 An individual bit string can be viewed as a representative of each of the different schemas that it matches For example, the bit string 0010 can be thought * of as a representative of 24 distinct schemas including 00**, O 10, ****, etc Similarly, a population of bit strings can be viewed in terms of the set of schemas that it represents and the number of individuals associated with each of these schema The schema theorem characterizes the evolution of the population within a GA in terms of the number of instances representing each schema Let m(s, t) denote the number of instances of schema s in the population at time t (ie, during the tth generation) The schema theorem describes the expected value of m(s, t 1) in terms of m(s, t) and other properties of the schema, population, and GA algorithm parameters The evolution of the population in the GA depends on the selection step, the recombination step, and the mutation step Let us start by considering just the effect of the selection step Let f (h) denote the fitness of the individual bit string h and f(t) denote the average fitness of all individuals in the population at time t Let n be the total number of individuals in the population Let h E s n p, indicate that the individual h is both a representative of schema s and a member of the population at time t Finally, let 2(s, t) denote the average fitness of instances of schema s in the population at time t We are interested in calculating the expected value of m(s, t + l), which we denote E[m(s,t I)] We can calculate E[m( s , t + I)] using the probability distribution for selection given in Equation (9 I), which can be restated using our current terminology as follows:
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Now if we select one member for the new population according to this probability distribution, then the probability that we will select a representative of schema s is
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The second step above follows from the fact that by definition,
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Equation (92) gives the probability that a single hypothesis selected by the G A will be an instance of schema s Therefore, the expected number of instances of s resulting from the n independent selection steps that create the entire new generation is just n times this probability
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Equation (93) states that the expected number of instances of schema s at generation t 1 is proportional to the average fitness i ( s , t ) of instances of this schema at time t , and inversely proportional to the average fitness f ( t ) of all members of the population at time t Thus, we can expect schemas with above average fitness to be represented with increasing frequency on successive generations If we view the G A as performing a virtual parallel search through the space of possible schemas at the same time it performs its explicit parallel search through the space of individuals, then Equation (93) indicates that more fit schemas will grow in influence over time While the above analysis considered only the selection step of the GA, the crossover and mutation steps must be considered as well The schema theorem considers only the possible negative influence of these genetic operators (eg, random mutation may decrease the number of representatives of s, independent of O(s,t ) ) , and considers only the case of single-point crossover The full schema theorem thus provides a lower bound on the expected frequency of schema s, as follows:
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Here, p, is the probability that the single-point crossover operator will be applied to an arbitrary individual, and p, is the probability that an arbitrary bit of an arbitrary individual will be mutated by the mutation operator o(s) is the number of defined bits in schema s, where 0 and 1 are defined bits, but * is not d(s) is the distance between the leftmost and rightmost defined bits in s Finally, 1 is the length of the individual bit strings in the population Notice the leftmost term in Equation (94) is identical to the term from Equation (93) and describes the effect of the selection step The middle term describes the effect of the single-point crossover operator-in particular, it describes the probability that an arbitrary individual representing s will still represent s following application of this crossover operator The rightmost term describes the probability that an arbitrary individual representing schema s will still represent schema s following application of the mutation operator Note that the effects of single-point crossover and mutation increase with the number of defined bits o(s) in the schema and with the distance d ( s ) between the defined bits Thus, the schema theorem can be roughly interpreted as stating that more fit schemas will tend to grow in influence, especially schemas
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containing a small number of defined bits (ie, containing a large number of *'s), and especially when these defined bits are near one another within the bit string The schema theorem is perhaps the most widely cited characterization of population evolution within a GA One way in which it is incomplete is that it fails to consider the (presumably) positive effects of crossover and mutation Numerous more recent theoretical analyses have been proposed, including analyses based on Markov chain models and on statistical mechanics models See, for example, Whitley and Vose (1995) and Mitchell (1996)
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