Convergence to the Optimum

Transcription

1 An Adative Mutation Scheme in Genetic Algorithms or Fastening the Convergence to the Otimum Sima (Etaner) Uyar, Gulsen (Cebiroglu) Eryigit, Sanem Sariel Istanbul Technical University, Comuter Engineering Deartment Maslak TR Istanbul, Turkey. ABSTRACT Mutation rate arameter is considered to be one o the most sensitive o the arameters that a genetic algorithm works with. It has been shown that through using a mutation rate variation scheme that adats the mutation rate arameter during the run o the algorithm, the time to ind the otimum is decreased. In this study, a mutation rate adatation scheme, that adats the mutation rate searately or each gene location on the chromosome based on the eedback taken rom the success and ailure rates o the individuals in the current oulation, is roosed. Through tests using the onemax roblem, it is shown that the roosed mutation adatation scheme allows aster convergence than the other similar aroaches chosen or comarisons. The results are very romising and romote urther research. Keywords : adative mutation rate, arameter control, convergence rate, genetic algorithms 1. Introduction Genetic algorithms [3] (GA) belong to a class o biologically insired otimization aroaches that model the basic rinciles o classical Mendelian genetics and Darwinian theory o evolution. Due to their robust nature, genetic algorithms are used in a wide variety o alications. However one o the major drawbacks o working with genetic algorithms is that erormance largely deends on the aroriate setting o some arameters: namely oulation size, crossover and mutation rates. These arameters interact with each other, making it even harder to ind otimal settings. However mutation rate is considered to be one o the most sensitive o these arameters. It has been traditionally regarded as a background oerator that mainly works as an insurance olicy [10], rotecting alleles rom being lost rom the oulation but it has been shown that the mutation rate value largely aects the general behavior o the algorithm. There has been extensive work to investigate the exact nature o mutation [1], [4] [5], [8]. The techniques develoed to set these arameters are classiied by Eiben et al. [2] as: arameter tuning and arameter control. For arameter tuning, the arameter values are set in advance, beore the run and are ket constant during the whole execution o the algorithm. In arameter control techniques, arameters are initialized at the start o execution and are allowed to change during the run. Parameter control techniques are classiied mainly into three grous based on the tye o change they introduce: deterministic: the arameter value is udated according to some deterministic rule, adative: the arameter value is udated based on some eedback taken rom the oulation sel-adative: the arameter is evaluated and udated by the evolutionary algorithm itsel. In this study, an adative mutation rate strategy that uses eedback obtained rom the current oulation and increases or decreases the mutation rate accordingly or each locus on the chromosome is introduced. Even though using eedback rom the current state o the search seems to be a useul aroach, it has not been studied much within the scoe o canonical GAs [11]. This aroach is tested against reviously ublished methods or mutation rate control, on a simle one-max roblem. The results are seen to be romising and romote urther study.

2 The rest o this aer is organized as ollows: Section 2 introduces the roosed mutation rate adatation aroach. In section 3 the dierent tested methods are resented. In section 4, the exerimental setu and the results are given and discussed; Section 5 rovides a conclusion, based on the results while roviding ossible directions or uture work and concludes the aer. 2. Gene Based Adative Mutation GA In this aer, a Gene Based Adative Mutation () method is roosed. This aroach exeriments with varying mutation rate values during the run, using eedback rom the oulation. Dierent rom other known mutation adatation strategies, in each locus has its own mutation rate value. An adative aroach or adjusting mutation rates or the gene locations based on the eedback obtained by observing the relative success or ailure o the individuals in the oulation is used. In, there are two dierent mutation rates deined or each locus: a mutation rate value or those genes that have an allele value o "1" at that locus and another mutation rate value or those that have a "0". In the reroduction hase, the aroriate mutation rate is alied based on the gene allele value. Initially all mutation rates are set to a deault value in the seciied boundaries. During the GA run, the mutation robabilities and or all loci are udated at each generation using the eedback taken rom the relative success or ailures o those individuals having a "1" or "0" at that locus resectively. For a maximization roblem, the udate rule or the mutation rate subunits or one gene location can be seen in Eq.1. This udate rule is alied searately or each locus. + + = = + γ, γ, γ, + γ, ( S / P ) > 1 ( ) S / P 1 ( S / P ) > 1 ( ) S / P 1 The m i (i=0,1) value or a locus corresonds to the rate o mutation subunit that will be alied when the gene value is i in the corresonding gene. S is the average itness o the individuals with an allele "1" or the corresonding gene location. P is the average itness o the oulation. γ is the udate value or the mutation rates. (1) As a result o the udates at each generation, m i values are allowed to oscillate within the limits deined by lower and uer bounds. I an udate causes a mutation rate to exceed the uer limit, the corresonding mutation rate is set to the uer bound value and i it causes a mutation rate to go below the lower limit, the corresonding mutation rate is set to the lower bound value. Another arameter that uses is the initial mutation rate value. All arameters are determined emirically. As will be shown in the analysis o the exeriments, rovides raid convergence. For unimodal objective unctions, this raid convergence rovides a valuable reinement. However, the ast convergence eature may cause the rogram to get stuck at local otima, esecially or the multimodal objective unctions. This roblem is exlored in detail in [6] or sel-adative mutations, however the results can easily be extended to adative mutation schemes too. For the uroses o this aer, a unimodal unction is used or the tests. The roblems that may arise as a result o ast convergence are not within the scoe o this aer and will be exlored in detail in a uture study. 3. Aroaches Chosen or Comarisons The aim o the exeriments is to show that rovides aster rates o convergence, as well as exloring its convergence behavior comared to other similar arameter control aroaches ound in literature. As given in Section 1, arameter control aroaches are categorized based on the tye o change that is alied to the arameter. There are dierent ormulations and imlementations o each tye o arameter control ound in literature. A reresentative scheme, which is shown to give good erormance, is chosen rom each category and used or the comarisons Deterministic Aroach The deterministic mutation rate schedule rovides the mutation rate to decrease rom a value (generally, 0.5) to the otimum mutation rate (generally 1/L) without using any eedback rom the oulation. The deterministic mutation rate schedule imlementation roosed in [7] was reorted in [11] as having the most succeul results or hard combinatorial roblems. Based on this method, the timevarying mutation rate is calculated using the ormula given in Eq. 2. In this ormula, t is the current generation number and T is the maximum number o generations. In the original roosal or Eq. 2, the k value is chosen as 1.

3 t k L 2 = 2 + * t (2) T 1 = L x i i= 1 (5) 3.2. Sel-Adative Aroach In the sel-adative aroach, the arameters are encoded into the chromosomes and undergo mutation and recombination. The basic idea is that better arameter values lead to better individuals and these arameter values will survive in the oulation since they belong to the surviving individuals. Bäck et al. [3] reer to this aroach also as online learning. In their work, they roose a sel-adatation mechanism o a single mutation rate er individual. The mutation o this mutation rate value gives the new mutation rate through Eq. 3. In this equation, γ is the learning rate and controls the adatation seed. It is taken as 0.22 in [3] and also in this study. 1 1 ' = (1 +.ex( γ. N(0,1))) (3) 3.3. Individually Adative Aroach In this study, an individually adative GA method () [29] is chosen or the comarisons. In this method, the robabilities o crossover and mutation are adated deending on the itness values o the individuals. The adatation o the c and m allows the individuals having itness values o over-average to maintain their genetic material, while orcing the individuals with sub-average itness values to disrut. The mutation rate adatation rule is given in Eq. 4. In this equation, ƒ denotes the itness value o the individual, ƒ max denotes the best itness value o the current generation, and ƒ denotes the average itness value o the current generation. In [9], the constants k 2 and the k 4 are chosen as 0.5. m m = k ( = k 2 4 max 4. Exeriments ) /( < max The aroach is exected to reduce the number o generations (or itness evaluations) to locate an otimal individual. To investigate this eect, the one-max roblem, which is unimodal and easy or a simle GA, is used or the tests. The main aim o this roblem is to maximize the number o 1s in a binary reresented string o length L. The otimum or this unction is L. More ormally the itness unction can be deined as in Eq. 5 where x i reresents the ith character in the string. ), (4) Tests are erormed or our dierent string lengths: L=200, L=400, L=800 and L=1600 to exlore the eects o the length o the string on the number o generations required to irst locate the otimum. For all tests, the rogram imlementation or each chosen aroach is run 50 times. All arameter settings are determined emirically to rovide the best erormance or each aroach. Some settings are the same or all aroaches: number o generations: 1500 oulation size: 250 individuals arent selection: tournament selection with tournament sizes o two recombination: two-oint cross over with c =1.0 oulation dynamics: strictly generational Some extra arameters are used by the methods chosen or comarisons. The settings or these values are given in Table-1 where is used or the deterministic aroach, or the sel adative aroach, or the adative GA aroach and or the gene based adative mutation aroach roosed in this study. Table-1 Extra arameter settings k=1.2 (Eq. 2) initial mutation rate = 1/L lower mutation rate limit = k 2 =1/L (Eq. 4) initial mutation rate = 0.02 mutation rate lower limit = mutation rate uer limit = 0.2 mutation udate amount = The statistical calculations or the number o generations required to locate the otimum individuals are given in Table-2, where µ is the mean number o generations needed to locate the best individual, σ is the standard deviation o this value, CI is the 99% conidence interval calculated or the location o the mean. Since one-max is a unimodal unction, all aroaches excet or are able to ind the otimum or all string lengths. The lots o the number o generations needed to ind the otimum or all methods averaged over 50 runs are given in Fig. 1. As can be seen rom Fig. 1 and Table-2, reduces the number o stes required to ind the otimum solution. Based on the results in Table-2, seems to generate

4 romising results or all o the L values. and seem to generate very close results, which is to be exected since the advantage o using comes rom not having to ind otimal mutation rates beore the run. However in this study, is imlemented using otimal rates or each test roblem, causing and to erorm similarly. The drawback o deterministic aroach is the high generation number needed to locate the best itness value. However, when L value is increased, the results become accetable. Because the initial mutation rate value is 0.4 aroximately, which is unnecessarily high or small L values. Although the erorms well or the small values o L comared to,, and, when the L value increases, its erormance decreases. The reason o this is that when the L value increases the k 2 value in Eq. 4 becomes very small. Table-2 Statistical calculations or number o generations required to ind the best individual L=200 L=400 µ σ CI µ σ CI L=200 L=400 L=800 L=800 L=1600 µ σ CI µ σ CI * * * (*) is not able to ind the otimum in 1500 generations Fig.1. Best itness values observed through generations or all methods averaged over 50 runs L=1600

5 5. Conclusion and Future Work In this study, a mutation rate adatation aroach () or each gene location in a binary reresentation, based on the relative erormance o individuals is roosed. Because each gene location has a dierent arameter or controlling the rate o mutation at that location, the roosed aroach is more suited or roblems where the eistasis between genes is low to none. Since the mutation rate is adated based on the itness values o the best erorming individuals, it is exected that astens the convergence o the GA to the otimum. This has been conirmed through tests erormed on a simle one-max roblem. Since this roblem is a unimodal unction ast convergence to the otimum is not a roblem. However or multimodal itness landscaes, to escae rom local otima, extra convergence tests should be alied, and necessary recautions to restore diversity should be taken. Even though as a result o these reliminary tests, the overall erormance o seems to be very romising or the chosen tye o roblems, there is still more work to be done to be able to make healthy generalizations. First o all, the arameter settings or, such as the lower and uer bound values, initial mutation rate and the mutation udate values have been determined exerimentally. More exeriments need to be erormed to see the eects o these arameters on erormance more thoroughly. Secondly, the test roblem set can be extended to include dierent tyes o roblem domains. This also would allow the eects o dierent degrees o eistasis to be exlored. Thirdly, a convergence control mechanism should be added to. [5] Ochoa G., "Setting the Mutation Rate: Scoe and Limitations o the 1/L Heuristic", Proceedings o Genetic and Evolutionary Comutation Conerence, Morgan Kaumann (2002). [6] Rudolh G., "Sel-Adative Mutations May Lead to Premature Convergence", IEEE Transactions on Evolutionary Comutation, Vol. 5., No. 4, , IEEE (2001). [7] Smith J. E., Fogarty T. C., "Oerator and Parameter Adatation in Genetic Algorithms", Sot Comuting 1, , Sringer-Verlag (1997). [8] Sears W. M., "Crossover or Mutation", Proceedings o Foundations o Genetic Algorithms 2, Morgan Kaumann (1993). [9] Srinivas, M., Patnaik, L. M., "Adative Probabilities o Crossover and Mutation in Genetic Algorithms", IEEE Transactions on Systems, Man and Cybernetics, Vol. 24, No. 4., , IEEE (1994). [10] Stanhoe S.A., Daida J.M., "An Individually Variable Mutation-Rate Strategy", Proceedings o the Sixth Annual Conerence on Evolutionary Programming (1997). [11] Thierens D., "Adative Mutation Control Schemes in Genetic Algorithms", Proceedings o Congress on Evolutionary Comuting, IEEE (2002). Reerences [1] Bäck T., "Otimal Mutation Rates in Genetic Search", Proceedings o 5th International Conerence on Genetic Algorithms, Morgan Kaumann (1993). [2] Eiben A. E., Smith J. E., Introduction to Evolutionary Comuting, Sringer-Verlag, Berlin Heidelberg New York (2003). [3] Goldberg D. E., Genetic Algorithms in Search Otimization and Machine Learning, Addison Wesley (1989). [4] Hinterding R., Gielewski H., Peachey T. C., "The Nature o Mutation in Genetic Algorithms", Proceedings o the 6th International Conerence on Genetic Algorithms, , Morgan Kaumann (1995).