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Algorithm and selection Pressure

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					            The GENITOR Algorithm and Selection Pressure:
        Why Rank-Based Allocation of Reproductive Trials is Best
                                               Darrell Whitley
                                        Computer Science Department
                                          Colorado State University
                                           Fort Collins, CO 80523.
                                          whitley@cs.colostate.edu

                       Abstract                                  be consistent with Holland's fundamental theorem of
                                                                 genetic algorithms: the schema theorem. The schema
This paper reports work done over the past three                 theorem predicts changes in the sampling rate for a hy-
years using rank-based allocation of reproductive trials.        perplane from generation to generation + 1. This
                                                                                                    t                t
New evidence and arguments are presented which                   change satis es the inequality
suggest that allocating reproductive trials according to
rank is superior to tness proportionate reproduction.                ( + 1)        1?       ( ) (1 ? ( ))
                                                                                              l h
                                                                                                                      ( )
Ranking can not only be used to slow search speed,               P h; t                  Pc
                                                                                             ?1
                                                                                              L
                                                                                                           P h; t     F R P h; t

but also to increase search speed when appropriate.
Furthermore, the use of ranking provides a degree                    where represents a particular hyperplane and
                                                                            h

of control over selective pressure that is not possible             ( ) indicates the fraction of the population that are
                                                                 P h; t

with tness proportionate reproduction. The use of                members of a hyperplane at time .     h        t  indicates
                                                                                                                    FR

rank-based allocation of reproductive trials is discussed        the average tness of the members of relative to the
                                                                                                                h

in the context of 1) Holland's schema theorem, 2)                population average (the \Fitness Ratio" of hyperplane
DeJong's standard test suite, and 3) a set of neural             h .) The de ning length of the schema, ( ), includes
                                                                                                                    l h

net optimization problems that are larger than the               only the signi cant part of the de ning schema for .        h     L

problems in the standard test suite. The GENITOR                 is the length of the binary string or \genotype." is        Pc

algorithm is also discussed; this algorithm is speci cally       the probability that crossover will be applied (Holland
designed to allocate reproductive trials according to            1975).
rank.                                                                The schema theorem calculates the number of rep-
                                                                 resentations that a particular schema has in the pop-
Background and Motivation                                        ulation during one generation and calculates the aver-
                                                                 age tness of the individuals that possess the schema in
   In 1985 James Baker reported experiments where re-            question; this value, which will be referred to as , is     Sf

productive trials were allocated according to the rank           compared to the average tness of the population, ,               Pf

of the individual strings in the population rather than          to determine the tness ratio (ie:          =
                                                                                                           FR        ).
                                                                                                                    Sf =Pf

by individual tness relative to the population average.              One of the beauties of genetic search is that there is
His results were fairly successful, but seem not to have         no need to ever directly manipulate schemata. In actual
revolutionized the way people think about genetic algo-          implementation the number of trials that any individual
rithms. There are several reasons why his results have           receives during any generation is the individual tness
not received as much attention as they perhaps deserve.          If   divided by the average tness of all the individuals in
First, Baker used ranking to slow down convergence:              the population . Thus is actually a partial evalua-
                                                                                    Pf            If

this not only resulted in more accurate optimization,            tion of the 2 ? 1 schemata that make up an individual
                                                                                L


but also slower optimization. The experiments reported           genotype.
here indicate that it is also possible to use ranking to             The problem with ranking is that it doesn't seem to t
achieve faster search on easier optimization problems            with the schema theorem{but in fact there is no reason
and on more di cult optimizations problems, where a              that needs to be calculated as a direct function of .
                                                                      FR                                                          If
reduced but constant selective pressure can yield steady         Instead of using   If =P subf to calculate the tness ratio,
improvement resulting in more e cient optimization.              FR     =       ( ) can be substituted without changing
                                                                          Rank If
   A second reason that ranking has not received more            the intent of the schema theorem. In some ways it is
attention may be the standard test suite itself. The             actually more consistent with the schema theorem be-
test problems are all rather small and relatively easy.          cause it removes the need for other \parameters" that
It is quite common for researchers to try to solve these         are used to indirectly control selective pressure that are
problems using as few recombinations as possible{or to           not part of the schema theorem.
compare optimization after some xed number of re-                    This raises another issue. It can be argued that there
combinations. These measures can be deceptive. The               are only two primary factors (and perhaps only two fac-
GENITOR algorithm shows dramatic increases in per-               tors) in genetic search: population diversity and selec-
formance on larger problems such as the neural net ex-           tive pressure. These two factors are inversely related.
periments reported here.                                         Increasing selective pressure results in a faster loss of
   A third reason that ranking has probably not received         population diversity. Maintaining population diversity
more attention is that it does not at rst appear to              o sets the e ect of increasing selective pressure. In some

                                                             1
sense this is just another variation on the idea of explo-         be \super genotypes" that have an unusually high t-
ration versus exploitation that has been discussed by              ness ratio and thus dominate the search process; Baker
Holland and others. Many of the various parameters                 seems largely concerned with slowing down searches
that are used to \tune" genetic search are really indi-            that progress too fast because of \super individuals."
rect means of a ecting selective pressure and population           But ranking completely solves the scaling problem.
diversity. As selective pressure is increased, the search          Consider the following \lock and tumbler" problems
focuses on the top individuals in the population, but              (Ackley 1987; Davis 1988, personal communication).
because of this \exploitation" genetic diversity is lost.          Starting at the rst bit in a genotype, let C be the num-
Reducing the selective pressure (or using a larger popu-           ber of \1's" that occur in some prede ned sequence. For
lation) increases \exploration" because more genotypes             example, if the sequence is 0,1,2,...L, then C would be
and thus more schemata are involved in the search.                 the number of consecutive 1's that occurs as a pre x
   The research reported in this paper is based on the             of the binary encoding. Thus, the string 1111101011...1
idea that selective pressure and population diversity              has C = 5. Notice that 1's which are not part of the ini-
should be controlled as directly as possible. The                  tial pre x do not contribute to the value of C, only those
better one can understand and control the relationship             that occur before the rst 0 in the sequence. (The \lock"
of population diversity and selective pressure to the              sequence need not be 0,1,2,...L, but any sequence; e.g.
parameters used to \tune" a genetic algorithm, the                 5,20,1,...etc.) The C representing the bit count could be
more insight the researcher will gain toward improving             directly used as an evaluation function, but suppose this
genetic search. Selective pressure can be simply and               did not produce an adequate selection pressure. An-
directly controlled by allocating reproductive trials              other variation on this would be to use an evaluation
according to rank.                                                 function where the \evaluation" is C*10, or 50 in this
                                                                   example. Yet another possibility is to let the evaluation
Scaling Problems and Ranking                                       function be C! or 5! in this example. In this case \super
                                                                   individuals" would be an enormous problem because of
   Fitness proportionate reproduction can sometimes                the relative di erence in value between individuals with
lead to problems when conditions arise where the search            di erent evaluations. For a genetic algorithm using t-
is likely to (1) stagnate because the search lacks selective       ness proportionate reproduction these evaluation func-
pressure, or (2) prematurely converge because selection            tions create very di erent scaling problems. The task we
has caused the search to narrow too quickly. The most              are trying to solve, however, is really the same in every
usual cause of this is referred to as the \scaling prob-           case. To a rank-based algorithm such as G       E N I T OR
lem." Suppose the genetic algorithm is applied to a                these problems are all identical and there would be no
maximization problem where the performance values of               di erence in performance no matter which evaluation
genotypes in the population range from 100 to 1100 with            function is used.
an average of 550 and the tness ratio is calculated using             The use of various \scaling windows" is just one
individual tness over the population average. Initially,           of several parameters aimed at controlling selective
the selective pressure toward the top ranked genotype is           pressure or population diversity. Grefenstette (1986)
1100 550 or 2. Suppose this is a su cient selective pres-
     =                                                             discusses the following parameters: 1) population size,
sure to keep the search moving forward but so much as              2) crossover rate, 3) mutation rate, 4) generation gap,
to cause premature convergence. However, later in the              5) scaling window, and 6) selection strategy. All of
search the range may be 1000 to 1200 with an average               these in some way a ect the selective pressure. In
of 1100. Now the selective pressure is only 1200 1100 or
                                                  =                developing the G  E N I T OR   algorithm, there has been
1.09, which may not be adequate and the search stag-               an explicit e ort to remove as many parameters from
nates. (These values should not be taken as being pre-             the algorithm as possible. The G   E N I T OR   algorithm
cise de nitions of \appropriate selective pressure;" they          allows population diversity and selective pressure to be
are simply used to indicate that selective pressure can            directly controlled.
  uctuate.) To x this, we could subtract 1000 from all
the values in the above example, creating an e ective              Ranking, Fitness and Schema Theory
range of 0 to 200, an average of 100, and a selective
pressure of 2.0 again toward the top ranked genotype.                 The most serious objection to ranking is that it vio-
Maintaining adequate selective pressure as a population            lates the schema theorem. It might be argued that the
becomes more homogeneous is di cult because there                  average of the rank of the genotypes that sample a par-
is less variation in tness. John Grefenstette's Gene-              ticular hyperplane does not correspond to the rank of
sis implementation package and user's guide (Grefen-               the hyperplane's average tness. But looking at ranking
stette 1984) de nes various \scaling window" options               in this way fails to reveal what ranking actually does in
that work in this general fashion. But this represents a           the search space. Ranking acts as a function transfor-
deviation from a strict interpretation of the fundamen-            mation that assigns a new tness value to a genotype
tal theorem of genetic algorithms. Ranking is in some              based on its performance relative to other genotypes;
ways more consistent with the schema theorem because               in other words, rank translates into a assigned tness
it makes it unnecessary to introduce additional param-             value. Further, it assigns tness values so as to con-
eters that are not accounted for by the schema theorem             sistently \distance" the corners in the hypercube that
in order to control selective pressure. Selective pres-            are currently being sampled in terms of their relative
sure remains consistent throughout the search; scaling               tness{thus acting as a kind of smoothing function. No-
problems do not occur and tness di erences between                 tice that because of the smoothing e ect that ranking
genotypes can be exploited, regardless of the magnitude            has, the hyperplane with the \best average raw tness"
of those di erences.                                               may not be the same as the \best average assigned t-
   Baker used ranking in an e ort to stop premature                ness," but this is exactly the kind of e ect one wishes to
convergence. One cause of premature convergence may                achieve with ranking. It reduces the e ect of exagerated

                                                               2
di erences in tness.                                              are selected usingFR   to bias selection toward high per-
   If \disruptions" are ignored (and disruptions are the          formance genotypes. However, the theorem is indepen-
same whether one uses ranking or tness proprotionate              dent of how that tness ratio is derived{it can either be
reproduction) the bottom line is this: ranking will in-           calculated in the usual way or calculated as a function
crease the representation of schemata that have above             of the genotype's rank in the population. A derivation
average (mean) \assigned tness" and reduce represen-              and in-depth discussion of this formula is presented in
tation of schemata that have below average \assigned              (Whitley 1988b).
  tness." The implicit parallelism is the same; the ac-              GENITOR only produces one new genotype at a
tual tness values and thus the resulting representation           time, so inserting a single new individual is relatively
of schemata will be di erent, but information about               simple. Furthermore the insertion automatically ranks
2 ? 1 schemata is still gained with every string eval-
 L
                                                                  the individual relative to the existing pool{no further
uation. When the ranking function is linear, the mean             measure of relative tness is needed.
\assigned" tness will correspond to the median rank
in the population . When a nonlinear function is used,            Duplicates and Selective Pressure
the mean tness will be shifted toward the top of the
                                                          One way to increase a genetic algorithm's ability
population. It is true that if a nonlinear function is used
to assign tness the genetic algorithm can display hill-to accumulate high performance schemata is to apply
climbing tendencies. However, controlling the selectivecrossover probabilistically, which is one idea behind the
pressure controls the behavior of the algorithm.           (the crossover rate) in the standard schema theorem.
                                                                  Pc
                                                       In GENITOR this could be done by making a copy of
   A second objection is that ranking discards or ignores
information about the search space as revealed by the  the parent and introducing it as the new o spring. If            Pc
evaluation function. This objection again fails to see is introduced into the GENITOR formula, it does in fact
ranking as a function transformation that intentionallytheoretically increase the algorithm's ability to retain
discards information about the magnitude of tness      information about long schemata. However, this ap-
and uses relative magnitude instead. In many real      proach supplies no new information to the search since
applications the evaluation function is likely to be a it does not sample a new genotype and its corresponding
                                                       schemata.
heuristic that simply indicates which strings are better
than others. In most cases it may not be realistic to     Probabilistic crossover ( ) allows some genotypes to
                                                                                             Pc
use the value generated by the evaluation function to  remain intact. This allocates more trials to high per-
judge relative di erences in tness. Ranking (or even   formance genotypes since they are the most likely to
an approximate ranking) may the best one can expect    produce duplicates during the reproduction/selection
from an evaluation function. There is simply no reason phrase. The result is more copies of what appear to
to believe that most evaluation functions yield an     be high performance schemata, but this process fails to
\exact" measure of tness. And if they do not yield an  test these schemata (i.e. hyperplanes) in a new con-
\exact" measure of tness, then why insist that \exact  text until these duplicates undergo recombination. But
  tness" be used?                                      while this creates additional selective pressure toward
                                                       schemata contained in the duplicate genotype, it does so
The GENITOR Project                                    in the worst possible way: every duplicate means a loss
                                                       of genetic diversity in the population. The GENITOR
   GENITOR is an acronym for GENetic                   algorithm requires no such safeguard to guarantee the
ImplemenTOR, a genetic search algorithm that           survival of the better genotypes in the population.
di ers in two major ways from the standard genetic        The genetic algorithm community often cites the need
algorithms. One major di erence is the explicit use    to balance exploitation and exploration. However, this
of ranking. The second di erence is that GENITOR       highlights a deeper con ict in genetic search{population
abandons the generational approach and reproduces      diversity versus selective pressure. Increasing the se-
                                                       lective pressure tends to reduce diversity and increase
new genotypes on an individual basis. It does so in such
a way that parents and o spring can and typically do   search speed. Decreasing the selective pressure helps to
co-exist. Theoretical analyses suggests that GENITOR   maintain diversity, resulting in a slower, although more
may be less biased against schemata with long de ning  robust search. This observation is not novel to biological
lengths than the standard genetic algorithm. Because   researchers. However, this observation does add legiti-
GENITOR uses a one-at-time reproduction scheme, it     macy to our emphasis on selective pressure and popula-
also appears to achieve faster feedback relative to thetion size as the critical variables in a genetic algorithm.
rate at which new points of the search space are being Fitness proportionate reproduction does not automati-
sampled.                                               cally translate into \appropriate" selective pressure. In
   G E N I T OR                                        fact, as was the case with the \lock and tumbler" de-
                's ability to sample high performance hy-
perplanes is captured in the following formula:        scribed in Section 2, the \exact" value returned by the
                                                       evaluation function can vary depending on how the eval-
                                          ( ))         uation function is implemented. As another example,
        ( + 1)
       P h; t           ( )( ( ) ?
                      P h; t   P h; t
                                            P h; t
                                                       one can always square the value returned by the eval-
                                            Pz F R
                                                       uation function. For feed-forward neural networks the
                                                       error at the individual nodes is typically squared and
 +   ( )
     P h; t
                 1 ? (1 ? ( ) ) 2     1 1+ ( )   l h   then summed to generate the error term. Squaring error
              FR          P h; t   FR
                                               ?1      at output nodes can even cause genotypes to be ranked
                                                       di erently: e.g. 1 + 5 3 + 4 but 1 + 25 9 + 16
       Pz                                        L
                                                                                         <        ;            >         :
                                                                                             \exact" tness
where is the population size and ( z( ) ) is used to The point is that the notion of anat rst appear. value
                                        P h;t

                                                       is more arbitrary than it would
estimate the probability that some member of is re- culating tness as a function of rank is a simple Cal-
        Pz
                                        P   FR


placed.     FR
                                                     h
            is the tness ratio. Note that both parents                                                         and

                                                              3
        11                                                                   GRAPH OF BEST PERFORMANCE
        10                                  FUNCTION 5                                     std genetic algorthm (GA)
        9
        8                                                                                  std GA using ranking
        7                                                                                  GENITOR using fitness
        6
        5                                                                                  GENITOR using rank
        4
        3
        2
        1
t            0         200        400       600        800            1000

Figure 1: This graph shows best performance on function f5 from DeJong's standard test suite. The tests were
averaged over 100 experiments. The horizonal axis indicates the total number of genotypes evaluated: 100 in the
initial population, plus 900 generated by crossover. Performance is given on the vertical axis.

e ective way to obtain a greater degree of control over           recombinations, this will focus more search toward
selective pressure.                                               those parts of the search space that are least agreed
   In the empirical tests reported in this paper the \re-         upon in terms of their representation in the population.
duced surrogate" 2-point recombination operator devel-
oped by Booker (1987) was used. The importance of this            Empirical Results: The Standard Test
operator is stressed here for two reasons. First, it can
actually reduce the bias toward high ranking genotypes
                                                                  Suite
by preventing duplicates; this will help prevent them                The GENITOR program has been run on ve func-
from prematurely dominating the search. This is also              tions that are used as standard test functions for genetic
one of the goals of GENITOR; by removing as a vari-
                                              Pc                  algorithms.
able there will be fewer duplicates and more information             The standard genetic algorithm package used for com-
about hyperplanes in the population. This will not only           parative purposes is the Genesis program (developed
reduce premature convergence, it will also mean that              and graciously provided by John Grefenstette of the
more schemata will be tested in a larger number of con-           Naval Research Laboratory.) In the original implemen-
texts. Selective pressure is most fairly distributed in the       tation of GENITOR the program checked for and elim-
population when there are no duplicates. For example,             inated duplicates; that is not necessary in the current
duplicates that are ranked 5th and 6th in the popula-             implementation.
tion can end up with more selective advantage than the               We found duplicates could be controlled by reduc-
top ranked individual in the population. Preventing du-           ing the bias or selective pressure (i.e. by regulating
plicates removes an unwanted source of selective bias in          the number of reproductive trials allocated to the top
the GENITOR algorithm.                                            ranked genotypes). The reduced surrogate crossover
   Second, there is a subtle heuristic at work in the             operator also helped. Again, both algorithms used
operator which we had been working on independently               Booker's 2-point reduced surrogate crossover.
from Booker. By looking at the reduced surrogates,                   Typically researchers use four standard ways to
one is also heuristically reducing the search space. In           measure performance (DeJong 1975): online, o ine,
other words, the two parents are in agreement about               average, and best. Results for all 4 measures and
the values of certain bit positions. This agreement has           function descriptions have been presented elsewhere
the a ect of reducing that part of the search space               (Whitley 1988a). Also, since GENITOR varies in two
which is currently of interest, especially if one looks at        ways from a standard genetic algorithm, to accurately
the amount of agreement that exists across several re-            evaluate the di erence between the two algorithms
combinations. The reduced surrogate operator re ects              all four variations were tested: 1) generational re-
that agreement. This insures that o spring are selected           production using tness (Genesis), 2) generational
from a part of the search space about which the two               reproduction using ranking, 3) one-at-a-time reproduc-
parents are not in agreement. In particular o spring              tion using tness, and 4) one-at-a-time reproduction
generated in this way lie along a minimal edge path               using ranking (GENITOR). One thing the results
between the two parents in a hypercube corresponding              demonstrated is that sometimes search can be im-
to the problem encoding. Consider the following two               proved by controlling the selection bias. This is evident
strings: 0010000000 and 0000010100. By removing                   in those cases where the two ranked approaches did
all bits that match between the two strings we obtain             much better than the tness proportional approaches.
{1{0-0{ and {0{1-1{. Given that one crossover occurs              An example of this is given in Figure 1 for function
between these \end bits," the o spring will always be             5 (f5); function 5 tests the algorithms' ability to
di erent from the parent reduced surrogates. Crossing             achieve a global search. Considering all ve functions,
the reduced surrogates also means that a select part of           GENITOR performed best on f1, f3 and f4. No clear
the search space has been focused on. Over numerous               winner emerged on f2. On f5, as shown in Figure

                                                              4
1, the rank based generational approach did as well             error of 0.1 was achieved by increasing population size
as (or slightly better than) GENITOR. GENITOR                   to 1000, but to obtain these results required 50,000 re-
competed well by all measures; the standard genetic             combinations. Increasing the population with Genesis
algorithm never did much better than GENITOR, but               did not seem to help, at least not within a reasonable
GENITOR often did much better than the standard                 number of recombinations.
genetic algorithm.                                                 The adder problem involves adding 2 2-bit numbers.
                                                                This version of the adder problem had 2 2-bit inputs,
Empirical Results: Three Neural Net-                            4 hidden nodes and 3 output nodes. The layers were
works                                                           fully connected and encoded using 280 bits, 8 bits per
                                                                connection. For this problem results are averaged over
   The standard genetic algorithm has been tuned on             fewer runs because of the long execution times involved.
these same test functions for over a decade (DeJong             Initial results compared GENITOR and Genesis on the
1975). The optimization problems posed by neural net-           adder problem with a population size of 2000 after
works provide a new and more challenging test of ge-            100,000 recombinations. Here GENITOR did notically
netic algorithms. The genetic algorithms were used to           better than Genesis, but only reduced error to 2.48 (av-
optimize the connection weights in three neural prob-           eraged over 5 runs). Genesis had only reduced error
lems. These are 1) the exclusive-or (Xor) problem, 2)           to 5.8 after 100,000 recombinations (averaged over 13
a 424-encoder, and 3) an addition problem (Rumelhart            runs). The t-test indicates that both the o ine and best
1986). The following discussion assumes some famil-             performance are signi cantly di erent at the .01 level.
iarity with back propagation and feed-forward neural            Online performance was not signi cantly di erent. The
networks (Whitley 1988c).                                       Genesis code was run again and allowed to continue un-
   To optimize neural net connection weights, the prob-         til it achieved similar average best performance (2.48)
lem must rst be encoded as a binary string. It is as-           with a population size of 2000; It required 1,250,000
sumed that (1) nodes and arc connections are already            recombinations to reduce average error to 2.48{more
known to exist (i.e. the net is already con gured) and          than 5 times the number of recombinations required by
(2) each arc weight can take on a prede ned nite range          GENITOR (averaged in this case over 8 runs). With a
of settings. The genetic algorithm generates a binary           population of 5000 the GENITOR program reduced the
string representing arc weights in the net. The network         average error to 0.5 after 500,000 recombinations (av-
is then run in a feed-forward fashion for each training         eraged over 5 runs). Larger populations have not been
pattern just as if one were going to use back propaga-          attempted with Genesis because of the long execution
tion. The sum of the squared error in then accumulated;         time required.
this value represents the \performance" of the binary              These are not excellent results. We have subse-
genotype.                                                       quently achieved much better neural net optimization
   Eight bits were typically used to represent each con-        using increased selective pressure coupled with a
nection weight; on the 424 problem 4 bits were used.            special mutation operator to sustain diversity. This
These eight bits were interpreted as a signed magnitude         combination resulted in optimization with greater
number ranging between -127 and +127 with 0 occur-              accuracy (errors of 101 0 and below) and produced
ring twice (-0 and +0).                                         these results up to 10 times faster using a very small
   The Xor problem appears to be similar in di culty            population of 50 genotypes (Whitley 1989).
and size to the standard test problems. It is encoded us-
ing 72 bits and can be solved after 500 to 1500 recombi-        Creating Selective Pressure
nations. Both GENITOR and Genesis solved the prob-
lem easily. Using a population of 200 and using 1500               First, it should be noted that while more complex
recombinations the average error was 0.00001 per run            mechanisms for selective pressure could possibly be
using GENITOR. The average error for Genesis under              de ned, a single variable is adequate. Over the past
the same conditions was 0.006. A t-test comparing the           three year the GENITOR project has experimented
results for online and best performance indicates these         with several modes of selective pressure, but we have
were signi cantly di erent at the .01 level. O ine was          found that the following mechanism works as well as
not signi cantly di erent, largely because Genesis had          any we have tried and has the advantage of being
better early performance, but GENITOR had much bet-             simple. .ti 3n For a bias up to and including 2.0, a
ter later performance. On very small populations (i.e.          linear function is used to allocate reproductive trials.
a population size of 50) GENITOR did not perform as             (See Appendix One.) In the neural net experiments
well as Genesis.                                                reported here a selective bias of 1.5 was found to give
   On the other hand, the performance of GENITOR                the best overall results. Selective pressures above 1.5
improved as population size was increased. Population           lead to premature convergence, while values below
size did not have as much e ect on the performance of           this failed to drive the search hard enough to achieve
Genesis.                                                        similar results. A bias of 1.5 implies that the top
   The 424-encoder was implemented using 4 bits per             ranked individual in the population is 1.5 times more
link; the total length of the encoding was 88 bits, but         likely to reproduce (on one reproductive cycle) than
the solution space was much more di cult to search.             the median individual in the population. For selective
Initial results were averaged over 50 runs using a popu-        pressures greater that 2.0 a nonlinear allocation of
lation of 200. A t-test indicates that o ine, online and        trials is used. For example, a selective pressure of 5.0
best performance are all signi cantly di erent at the .01       implied that 5% of all reproductive opportunities go
level. Here GENITOR did noticeably better, but the to-          to the top ranked position in the populations. 5% of
tal average error is not really satisfactory. However, it       the remaining 95% of the reproductive opportunities
was found that better solutions could be generated us-          are given the second ranked position, etc. Any residual
ing larger population sizes for GENITOR. An average             opportunities are evenly distributed.

                                                            5
                                                                                    REFERENCES
Conclusions
                                                                     Ackley 1987] Ackley, D. A Connectionist Machine
   Several major conclusions concerning selective pres-           for Genetic Hillclimbing (Kluwer Academic Publishers,
sure are suggested.                                               1987).
   (1) The value returned by an evaluation function
should not be considered an \exact" measure of tness.                Baker 1985] Baker J. Adaptive selection methods for
The exact value returned can vary greatly depending               genetic algorithms, in: John Grefenstette (Ed.), Proc.
on how the function was implemented. There is sim-                of an International Conf. on Genetic Algorithms and
ply no reason to believe that the value returned by an            Their Applications. (L. Erlbaum, 1988, original pro-
evaluation function produces appropriate selective pres-          ceedings 1985) 101-111.
sure, and problems with \super individuals," stagnating              Booker 1987] Booker, L. Improving search in genetic
searches, and premature convergence provide ample ev-
idence that it does not.                                          algorithms, in: Lawrence Davis (Ed.), Genetic Algo-
   (2) Allocating reproductive trials according to rank           rithms and Simulated Annealing. (Morgan Kau mann,
prevents scaling problems, since ranking automatically            1987) 61-73.
introduces a uniform scaling across the population.                  DeJong 1975] DeJong, K. Analysis of the Behavior
   (3) Allocating reproductive trials according to rank           of a Class of Genetic Adaptive Systems. Ph.D. Disser-
provides a means for directly controlling selective pres-
sure. Other parameters such as those used with the                tation, University of Michigan.
Genesis package a ect selective pressure indirectly and              Grefenstette 1984] J. Grefenstette A user's guide to
imprecisely.                                                      GENESIS, Tech. Report CS-84-11, Computer Science
   (4) Ranking, coupled with one-at-a-time reproduction           Dept., Vanderbilt Univ., Nashville, TN.
gives the search greater focus. Once the GENITOR al-                 Grefenstette 1986] Grefenstette, J., Optimization of
gorithm nds a good genotype it stays in the population            Control Parameters for Genetic Algorithms. IEEE
until displaced by a better string. This means that the
algorithm is less prone to \wander" in the search space           Transactions on Systems, Man, and Cybernetics, SMC-
and will maintain an emphasis on the best schemata                16 (1) (1986): 122-128.
found so far. This also removes the need for \Probab-                Holland 1975] Holland, J. Adaptation in Natural and
listic Crossover."                                                Arti cial Systems. (Univ. of Michigan Press, Ann Ar-
   (5) Allocating reproductive trials according to rank           bor, 1975).
can be used to speed up genetic search. We are currently
further testing the use of specialized mutation and in-              Rumelhart 1986] Rumelhart, D., and McClelland, J.,
creased selective pressure to produce faster yet more             Parallel Distributed Processing: Explorations in the Mi-
accurate optimization. Selective mutation can help sus-           crostructure of Cognition, Vol I (MIT Press, Cambridge,
tain genetic diversity, which in turn allows a higher level       MA 1986).
of exploitation to be achieved via higher selective pres-            Whitley 1988a] Whitley D., and Kauth J.
sure (Whitley 1989).                                              GENITOR: a di erent genetic algorithm, in Proc. of
             ACKNOWLEDGMENTS                                      the Rocky Mountain Conf. on Arti cial Intelligence,
                                                                  Denver, CO (1988) 118-130.
This research was supported in part by a grant from                  Whitley 1988b] Whitley D., and Kauth J. Sampling
the Colorado Institute of Arti cial Intelligence (CIAI).          Long Schemata in Genetic Algorithms. Tech. Report
CIAI is sponsored in part by the Colorado Advanced
Technology Institute (CATI), an agency of the State               CS-88-105, Computer Science Dept., Colorado State
of Colorado. CATI promotes advanced technology                    Univ.
education and research at universities in Colorado for               Whitley 1988c] Whitley D. Applying Genetic Algo-
the purpose of economic development.                              rithms to Neural Network Learning. Proc. 7th Conf.
                                                                  for the Study of Arti cial Intelligence and Simulated
              APPENDIX ONE:                                       Behavior, Sussex, England. Pitman Publishing.
    A Linear Function for Selective Pressure                         Whitley 1989] Whitley, D., and Hanson, T. Opti-
                                                                  mizing Neural Networks Using Faster, More Accurate
   For selective pressure between 1.0 (random) and 2.0            Genetic Search. 1989 Genetic Algorithm Conference.
the following linear algorithm is used. The function              Morgan Kaufmann, Publishers. In Proc. International
\random()" returns a random fraction between 0 and                Joint Conference on Arti cial Intelligence.
1.
linear()
{ float bias = 1.5;     int index;    double sqrt;

  index = POPULATION_SIZE *
    (bias -
      sqrt (bias * bias - 4.0 (bias -1) * random()))
    / 2.0 / (bias -1);

return (index); }


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