New Crossover Scheme for Parallel Distributed Genetic Algorithms Parallel and Distributed Computing and Systems 2000 (November 6-9, 2000 Las Vegas, Nevada) New Crossover Scheme for Parallel Distributed Genetic Algorithms Mitusnori MIKI, Tomoyuki HIROYASU, Jun-ichi YOSHIDA, Ikki OHMUKAI Thank you chairperson for introducing me. Now I talk about our study "New Crossover scheme for parallel distributed genetic algorithms". Department of Knowledge Engineering and Computer Science Doshisha University Kyoto, Japan

Background Applications of parallel processing Many studies on parallelization of numerical analysis. Few studies on parallel optimization. Parallel optimization is promising. Parallel processing of mathematical programming Parallelization of analysis is popular. New approach：parallelization of sensitivity analysis Parallelization of heuristic search methods Simulated Annealing(SA)，Genetic Algorithms(GA)， Cellular Automata(CA)，Neural Network(NN) Parallelization is needed to reduce high computational costs. Studies on such parallelization is not yet enough. In the area of applications of parallel processing, the parallelization of numerical analysis has been well investigated. However, the studies on parallelization of optimization have not been well performed. We think that studies on parallel optimization can be considered to be promising. There are two types of studies on parallel optimization. One is the parallelization of mathematical programming. In this area, the parallelization of analysis is very popular, but this is not a parallel optimization. Another direction of parallel optimization is the parallelization of heuristic search methods. Heuristic search methods include simulated annealing, genetic algorithms, and so on. These methods need high computational power, so the parallelization of these methods is very important. But, the studies on such parallelization is not yet enough.

Genetic Algorithms (1) f (x1, x2) Genetic Algorithms(GAs) (3, 5 ) 1 1 Optimization method based on the mechanism of natural selection and natural genetics. Candidate solution encode decode (3, 5 ) Individuals: search points -Each individual has design variables. -Design variables are encode from real value to binary (chromosome). I,d like to explain the genetic algorithms, briefly. Genetic algorithms are optimization method based on the mechanism of natural selection and natural genetics. In GAs, there are many search points, called individuals. Each individual has design variables. This is the case of two design variables. For example, this point has (3,5). These design variables are encode from real value to binary. This binary string called chromosome. The new searching points are generated by genetic operators such as selection, crossover, mutation, in each generation. After some generations, we may get a good solution in result. One of the disadvantages of GAs is the high calculation cost, GAs need a lot of iterations and it takes much time. One of the solution of ttis problem is performing GAs on parallel computers. Individual : 1 1 1 x1 x2 chromosome

Genetic Algorithms (2) Disadvantage The new searching points are generated by genetic operators. - Selection, Crossover, Mutation many generations Iteration Evaluation Selection Crossover Mutation Initialization End(find solution) Genetic operator Get a good solution in result. Disadvantage I,d like to explain the genetic algorithms, briefly. Genetic algorithms are optimization method based on the mechanism of natural selection and natural genetics. In GAs, there are many search points, called individuals. Each individual has design variables. This is the case of two design variables. For example, this point has (3,5). These design variables are encode from real value to binary. This binary string called chromosome. The new searching points are generated by genetic operators such as selection, crossover, mutation, in each generation. After some generations, we may get a good solution in result. One of the disadvantages of GAs is the high calculation cost, GAs need a lot of iterations and it takes much time. One of the solution of ttis problem is performing GAs on parallel computers. High calculation cost - GAs needs a lot of iteration Performing GAs on parallel computers

Parallel Distributed Genetic Algorithms Parallel Distributed GA (PDGA) Single Population GA (SPGA) subpopulation individual Population migration A population is divided into subpopulations. Each subpopulation assigned to each processor of a parallel computer. Canonical GA is performed in each subpopulations. Migration: Exchange of individuals among subpopulations. PDGAs can show better performance than SPGAs. Here let me explain the PDGAs. This figure shows conventional GAs. Many individuals are in a single population. We call this GA "Single Population GA", SPGA. This figure illustrates Parallel Distributed GA, PDGA. In PDGAs, a large population is divided into several smaller subpopulations, and each processors of a parallel computer are assigned to each subpopulations. Conventional GAs are executed in each subpopulation, separately. After certain number of generations, some individuals are exchanged among subpopulations which is called migration. In PDGAs, the variety of the individuals in the whole population is increased by the migration. Because of this large variety, PDGAs can show better performance than SPGAs.

Crossover operator Crossover parameters Produce new individuals by recombining a parent's chromosome - Direct information exchange - Main search operator Crossover parameters The performance of GAs depends on the choice of parameters The subject of our study is about the crossover operator in PDGAs. The crossover is a genetic operator that produces new individuals by recombining parent's chromosomes. The crossover is employed to perform direct information exchange between individuals in a population. Therefore, the performance of GAs depends on the type crossover operator and good choice of crossover rate. There are many researches about the crossover operator and the crossover rate in SPGAs. However, in PDGAs, appropriate crossover operator has not been studied yet. /* Some commonly used crossover rates vary between 0.45 and 0.95. However, the best setting of the crossover rate for SPGAs depends on the choice of other aspects of the overall algorithm, or the problem to be solved. Several crossover operators have been proposed, such as 1-point crossover, N-point crossover, Uniform crossover, and so on. And the role of these crossover operators has been investigated. */ Crossover operator 1-point , n-point, uniform, etc… Crossover rate Optimal crossover rate depends on the problems (commonly used : 0.45 – 0.9)

The growth of solutions in PDGAs Search good schemata in each subpopulations. Native Immigrant Recombine the schemata between the immigrant and the native. crossover The search mechanism of SPGAs and PDGAs are diffrerent. In PDGAs, small subpopulations tend to converge rapidly. For example, an individual in subpopulation B has a chromosome like this. This red bit indicates the good schemata found in subpopulation B. An individual in subpopulation A has a chromosome like this.The bit compositions of the chromosomes in subpopulations A and B are different. However, migration exchange individuals. Furthermore, different good schemata are growing in each subpopulation, and these schemata are mixed by the crossover after migration. Hence, optimal solution can be found by PDGAs, quickly. Here, we introduce two terms about migration. “a native”, which is an individual in the destination subpopulation. And, a “hybrid”, which is an individual generated from a native and a immigrant. From this viewpoint, the role of crossover in PDGAs can be considered as follows. (1)The first is to raise good schemata in each subpopulation To raise good schemata, it is necessary not to disrupt parent’s schema. Therefore the PDGAs with 1X and 2X shows better performance than that with UX, as shown in the previous result. (2)The second is to recombine the schemata between the immigrants and the natives. In previous experimental results, the crossover rate 1.0 shows a good performance in PDGAs. 1.0 is the setting to maximize the number of hybrids. That is, it is important to recombine good schema grown in each subpopulation by crossover after migration. Thus, the role of crossover operator is different before and after migration. Hybrid Optimal solution Hybrid individuals play Important role

The growth of solutions in PDGAs Small subpopulations tend to converge rapidly. Hybrid Native Immigrant crossover Migration exchange individuals The search mechanism of SPGAs and PDGAs are diffrerent. In PDGAs, small subpopulations tend to converge rapidly. For example, an individual in subpopulation B has a chromosome like this. This red bit indicates the good schemata found in subpopulation B. An individual in subpopulation A has a chromosome like this.The bit compositions of the chromosomes in subpopulations A and B are different. However, migration exchange individuals. Furthermore, different good schemata are growing in each subpopulation, and these schemata are mixed by the crossover after migration. Hence, optimal solution can be found by PDGAs, quickly. Here, we introduce two terms about migration. “a native”, which is an individual in the destination subpopulation. And, a “hybrid”, which is an individual generated from a native and a immigrant. From this viewpoint, the role of crossover in PDGAs can be considered as follows. (1)The first is to raise good schemata in each subpopulation To raise good schemata, it is necessary not to disrupt parent’s schema. Therefore the PDGAs with 1X and 2X shows better performance than that with UX, as shown in the previous result. (2)The second is to recombine the schemata between the immigrants and the natives. In previous experimental results, the crossover rate 1.0 shows a good performance in PDGAs. 1.0 is the setting to maximize the number of hybrids. That is, it is important to recombine good schema grown in each subpopulation by crossover after migration. Thus, the role of crossover operator is different before and after migration. Different good schemata are mixed by the crossover after migration. Optimal solution can be found by PDGAs, quickly. Optimal solution

The role of Crossover in PDGAs(1) The role of crossover in PDGAs can be considered as follows. 1 Raise good schemata in each subpopulation. Native Immigrant Recombine the schemata between the immigrant and the native. 2 The search mechanism of SPGAs and PDGAs are diffrerent. In PDGAs, small subpopulations tend to converge rapidly. For example, an individual in subpopulation B has a chromosome like this. This red bit indicates the good schemata found in subpopulation B. An individual in subpopulation A has a chromosome like this.The bit compositions of the chromosomes in subpopulations A and B are different. However, migration exchange individuals. Furthermore, different good schemata are growing in each subpopulation, and these schemata are mixed by the crossover after migration. Hence, optimal solution can be found by PDGAs, quickly. Here, we introduce two terms about migration. “a native”, which is an individual in the destination subpopulation. And, a “hybrid”, which is an individual generated from a native and a immigrant. From this viewpoint, the role of crossover in PDGAs can be considered as follows. (1)The first is to raise good schemata in each subpopulation To raise good schemata, it is necessary not to disrupt parent’s schema. Therefore the PDGAs with 1X and 2X shows better performance than that with UX, as shown in the previous result. (2)The second is to recombine the schemata between the immigrants and the natives. In previous experimental results, the crossover rate 1.0 shows a good performance in PDGAs. 1.0 is the setting to maximize the number of hybrids. That is, it is important to recombine good schema grown in each subpopulation by crossover after migration. Thus, the role of crossover operator is different before and after migration. Hybrid

The role of crossover in PDGAs (2) Before migration: To raise good schemata in each subpopulation Best combinatorial crossover (BCX) To search the best combination of the schemata of parent individuals After migration: To recombine the schemata between the immigrants and the natives Before migration, the primary role of crossover is to raise good schemata in each subpopulation. After migration, the primary role of crossover is to recombine the schemata between the immigrants and the natives. Here we proposed new crossover operator that enhance these two roles. First, we propose best combinatorial crossover which searches the best combination of the schemata of parent individuals. Second, we propose hybridization crossover which increase the number of hybrid individuals. Hybridization crossover (HX) To increase the number of hybrid individuals

Best Combinatorial Crossover To search the best combination of the schemata of parent individuals Parents All possible children (candidate children) are generated from parents using 1X. - Not generate children with the same chromosome - Parent individuals are included in the candidate children Candidate children Evaluate candidate children. Two of the best children are selected. The purpose of BCX is to search the best combination of the schemata of parent individuals. This figure illustrates the behavior of the BCX. In BCX, all possible children are generated from the parents using the 1X. As show in this figure, parent individuals are included in the candidate children. #In this phase, generating all children is redundant. #そこで実際には，重複する個体は生成しないようにしています． After evaluating these children, the two of the best children are selected and survive as the members of the next generation. Therefore, BCX can grow good schemata in a deterministic manner. Grow good schemata in a deterministic manner.

Test functions Rastrigin Schwefel Griewank Ridge n The performance of GAs is examined with these standard test functions, the Rastrigin, Schwefel, Griewank, and Ridge functions. All functions are 10 dimensional, while their 2-dimensional shapes are shown in these figures. The Rastrigin and Schwefel functions have no epistasis between design variables. On the other hand, The Griewank and Ridge function have epistasis. Therefore, it is difficult to find optimal solution by GAs.

Performance of GAs with BCX (Fitness-Eval) Griewank Ridge To demonstrate the effectiveness of BCX, we compare the performances of BCX and 1X with four functions, Rastrigin, Griewank, Schwefel, and Ridge. this figure shows the histories of the best fitness values for the Griewank and Ridge functions. This(X) axis shows the number of evaluations and this (Y) axis the fitness value. In the former stage of the search, the variety of individuals is large. So, a lot of the candidate children are generated in an BCX operation. Therefore PDGAs with BCX require huge number of evaluation calculation The fitness value of the PDGA with 1X increases faster than that of BCX. However, in the latter stage of the search, the number of candidate children are decrease because the variety of individuals are small. BCX shows better performance than 1X and find optimal solution quickly. As show in these figures, BCX has high search ability. (Population size = 400, Crossover rate = 1.0, Mutation rate = 1/L) BCX : find optimum for all functions 1X : find optimum for only Rastrigin

Hybridization Crossover Hybridization rate: the percentage of the hybrid individuals in each subpopulation. Next , we propose new crossover scheme for PDGA, called Hybridization Crossover (HX) The purpose of HX is to mate the immigrants only with the natives in each subpopulation, as shown in this figure. Here, we define a new parameter called the hybridization rate. The hybridization rate represents the percentage of the hybrid individuals in each subpopulation. In PDGAs with conventional crossover, the maximum value of the hybridization rate is 0.5, when crossover rate is 1.0 and migration rate is 0.5, which is the maximum value for these parameters. That is, the hybrid individuals that plays important role in search are generated for only 50% of the subpopulation size. But, using PDGAs with the HX, the hybridization rate can be increased beyond 0.5. The maximum value of the hybridization rate becomes 1.0 with HX, and the hybridization rate can be varied 0% to 100%. 0.5 1.0 Mate the immigrants only with the natives in each subpopulations. ※Crossover rate 1.0, Migration rate 0.5

Performance of HX Effect of hybridization rate is remarkable in the former stage of the search. The former stage: the difference in the solutions in each subpopulation is large. Hybrid individuals play an important role for the global search. To investigate the effect of the hybridization rate in PDGAs, the experiments for the 4 functions in performed. The hybridization crossover scheme and 1X are used in this experiment. The crossover rate is 1.0. This figure shows the history of the best fitness for the Ridge function. It shows that the effect of hybridization rate is remarkable in the former stage of the search, but it is not remarkable in the latter stage. In the former stage, the difference in the solutions in each subpopulation is large, and the hybrid individuals play an important role for the global search. But in the latter stage, the difference becomes very small, and the hybrid individuals are very similar to the natives or immigrants. Thus the effect of HX is decreased in this stage. Crossover rate = 1.0, Mutation rate = 1/L

Conclusion In PDGA, the role of crossover is changed before and after migration. We proposed new crossover scheme that enhance these two roles Best combinatorial crossover that grows the good schemata in a deterministic manner. Hybridization Crossover that recombine the schemata between immigrants and natives. Now I’d like to conclude as follows. We investigate the appropriate crossover operator and crossover rate for PDGAs. From the results of these experiments, we proposed two new crossover scheme for PDGAs. The PDGA with proposed schemes showed good performance for test functions. The PDGA with proposed schemes showed good performance.

Genetic Algorithms f (x1, x2) Genetic Algorithms (3, 5 ) 1 1 1 x1 x2 Candidate solution Optimization method based on the mechanism of natural selection and natural genetics. encode decode (3, 5 ) Iteration Evaluation Selection Crossover Mutation Initialization End(find solution) Genetic operator I,d like to explain the genetic algorithms, briefly. Genetic algorithms are optimization method based on the mechanism of natural selection and natural genetics. In GAs, there are many search points, called individuals. Each individual has design variables. This is the case of two design variables. For example, this point has (3,5). These design variables are encode from real value to binary. This binary string called chromosome. The new searching points are generated by genetic operators such as selection, crossover, mutation, in each generation. After some generations, we may get a good solution in result. One of the disadvantages of GAs is the high calculation cost, GAs need a lot of iterations and it takes much time. One of the solution of ttis problem is performing GAs on parallel computers. Individual : 1 1 1 x1 x2 chromosome

BCX with HX Generations BCX BCX+HX BCX BCX+HX Population Size

Appendix

Experiment (1) Compare the effect of three crossover operators Crossover rate Mutation rate 1X, 2X, UX 0.1, 0.3, 0.6, 1.0 0 /L, 0.01 /L, 0.1 /L, 1/L (2) Examine the optimal crossover rate operators To investigate the optimal crossover operator and the crossover rate for PDGAs, we performed two experiments. The effect of crossover operators are examined by using three types of crossover, 1X, 2X, and UX. And the effect of crossover rate is examined by using these combination of the crossover rates of 0.1, 0.3, 0.6, 1.0, and the mutation rates of 1/L(one-over-L), 0.1, 0.01, 0/L, where L is the length of the chromosome. Other parameters are shown in this table.(Next Page) L: length of chromosome

Experimental result (summary) Crossover rate Crossover operator PDGA SPGA Performance Performance These figures show the summary of experimental result ． This (left) figure shows the effect of crossover rate in PDGA and SPGA. This (X) axis indicates the crossover rate, so the maximum value is 1.0. This axis shows the performance of the GAs. In SPGA, optimal crossover rate exists. However it depends on various factors: such as mutation rate, population size, objective function, and so on. On the other hand, optimal crossover rate in PDGA is always 1.0. This figure shows the effect of crossover operator in PDGA. The performance of 1X and 2X are better than that of UX. 1X and 2X tend to preserve the schemata of the parent individuals. However, UX tends to disrupt schemata and cannot exploit the information of the previous generation. In PDGA, UX disrupts the schemata of the immigrants after the migration, and the effect of migration is decreased. So, the necessary function of the crossover operator appropriate for the PDGAs. From this experimental result, we think the search mechanism of PDGA is as follows. 1.0 1X 2X UX Crossover rate Crossover operator Prevent and recombine parent’s schemata Crossover rate 1.0

Rastrigin Schwefel Griewank Ridge Parameters Population size Crossover rate Mutation rate Max generation Num. subpopulations Migration interval Migration rate Test functions 400 --- 2000 generations 8 5 generations 0.5 Rastrigin　Schwefel　Griewank　Ridge

Find good solution quickly Performance of PDGAs SPGA – population size 400 Premature convergence Cannot find optimal solution PDGA – subpopulation size 50 num. subpopulation 8 - local search in each subpopulation - migration maintain the variety of whole population Find good solution quickly

Effect of crossover rate (SPGA) SPGA(Rastrigin) Population size 160 800 Mutation rate Optimal crossover rate depends on population size and mutation rate

Effect of crossover rate (PDGA) PDGA(Rastrigin) Population size 160 800 Mutation rate Optimal crossover rate is 1.0

Effect of crossover rate Optimal crossover rate (population size 400, mutation rate 1/L) SPGA PDGA Rastrigin 0.3 1.0 Griewank 0.3 1.0 Schwefel 0.6 1.0 SPGA: Optimal crossover rate depends on population size and mutation rate PDGA: Optimal crossover rate is always 1.0

Effect of crossover operator GAs with 1X and 2X performs better than with UX

Performance of GAs with BCX (table) Rastrigin 0.0 147,040 247,286 Schwefel -0.127 800,000 236,272 Griewank 794,261 Ridge -0.468 764,499 BCX find global optimum on the all functions

Reduction of evaluate calculations in BCX Disadvantage of BCX - A huge number of evaluations are required. - Some candidate have same chromosome. 1 ２ ３ Make crossover operations only between different genes with respect to the parent genes. 1 ２ ３ 961927 In BCX, a huge number of evaluations are required to find two of the best children. Therefore, reduction of the number of evaluations is important. If parents have similar gene in their chromosomes, many children with the same chromosome are produced by BCX. The effective way to achieve this is to make the crossover operations only between different genes with respect to the parent chromosome. This can be done using flag string, where the unique chromosomes from two parents are represented and no redundant candidate children are generated. There are no child with the same chromosome. And the number of evaluation is decreased by using this scheme, shown in this figure. Candidate children 242719

Empirical studies Crossover rate Optimal setting of crossover parameters for SPGAs Crossover rate De Jong (1975) 0.6 Grefenstette (1986) 0.95 Back (1996) 0.75 – 0.95 - Optimal crossover rate depends on the problems - Higher crossover rate cause premature convergence Crossover operator There are many researches about crossover operator and crossover rate in SPGAs. Some commonly used crossover rates vary between 0.45 and 0.95. However, the best setting of the crossover rate for SPGAs depends on the choice of other aspects of the overall algorithm, or the problem to be solved. Several crossover operators are proposed and investigated the role of these operators. For example, 1-point crossover, N-point crossover, Uniform crossover, and so on. However, the best setting of crossover operator and crossover rate in PDGAs has not been determined. So, we investigates the choice of crossover operator and, the setting of crossover rate for PDGAs. Holland (1975) 1-point crossover Syswerda (1989) uniform crossover De Jong (1992) Formal analysis of the role of multipoint crossover in genetic algorithms

A weak point of canonical crossover 1 2 Crossover point is decided at random - Effectiveness of a crossover operation depend on stochastic factor :3 :2 - 改良よりも改悪の方が多い（wu 97） :5 1 :0 Crossover operator which raise good schemata in each subpopulation :3 2 :2 In canonical crossover operator, crossover point is decided at random. thus the effectiveness of a crossover operation depends on the stochastic factor. So, we proposed new crossover operator, called Best Combinatorial Crossover(BCX) to raise good schemata in each subpopulation. Best Combinatorial Crossover (BCX) Building brock

Crossover operator 1-point Crossover: 1X Uniform Crossover: UX

The role of crossover in PDGAs Before migration: To raise good schemata in each subpopulation Best combinatorial crossover (BCX) To maximize the exploitation of the schemata of parent individuals After migration: To recombine the schemata between the immigrants and the natives 先に述べた最適な交叉率の実験で．．．crossover rate of 1.0 shows a good performance in PDGAs. This result indicate that, it is important not only to grow schemata in each subpopulation, but also to recombine them by crossover after migration in PDGAs. The effective recombination is provided by the hybridization between natives and immigrants. Therefore, we think that the performance of PDGAs can be increased by increasing the number of hybrids. so, we propose Hybridization Crossover (HX), to increase the number of hybrid individuals. Hybridization crossover (HX) To increase the number of hybrid individuals

Hybridization rate PDGA PDGA + Hybridization crossover 0.5 0.5 0.3 0.5 Hybridization rate: the percentage of the hybrid individuals in each subpopulation PDGA PDGA + Hybridization crossover Here, we define a new parameter called hybridization rate. Hybridization rate shows the percentage of the hybrid individuals in each subpopulation. In PDGAs with normal crossover, the maximum value of the hybridization rate is 0.5, when crossover rate is 1.0 and migration rate is 0.5. That is, the hybrid individuals that plays important role in search are generated for only 50% of the subpopulation size. But, PDGAs with HX can increase the hybridization rate beyond 0.5. The maximum value of the hybridization rate becomes 1.0 with HX, and the hybridization rate can be varied 0% to 100%. Migration rate 0.5 0.5 0.3 Hybridization rate 0.5 1.0 0.6

Hybridization Crossover Recombining good schemata by crossover after migration is important The effective recombination is provided by the hybridization between natives and immigrants The performance of PDGAs can be increased by increasing the number of hybrids

Experimental result (outline) Crossover rate Crossover operator PDGA Prevent schema １Ｘ SPGA Performance exploitation ２Ｘ Disrupt schema UＸ These graphs show the outline of experimental result ． This (right ) graph shows the effect of crossover rate in PDGA and SPGA. This (X) axis indicate crossover rate, so the maximum value is 1.0. This axis shows the performance of GAs. In SPGA, optimal crossover rate exists. However it depends on various factors: for example mutation rate, population size, object function, and so on. On the other hand, optimal crossover rate in PDGA is always 1.0. This graph shows the nature of crossover operators. 1X and 2X tend to prevent the schemata of parents, and UX tends to disrupt that. 1X and 2X perform better than UX.1X and 2X tend to preserve the schemata of the parent individuals. However,UX tends to disrupt schemata and cannot exploit the information of the previous generation. PDGA,UX disrupts the schemata of the migrants, and the effect of migration is decreased. So , the condition of appropriate crossover operator for PDGA is to prevent and recombine the schemata of parent. このような違いは，SPGAとPDGAの解探索メカニズムが異なるために生じています． 1.0 exploration Crossover rate Prevent and recombine parent’s schemata Crossover rate 1.0