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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); } 6

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