Sakanashi, h.; kakazu, y. (1994): co evolving genetic algorithm with filtered evaluation function


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Sakanashi, h.; kakazu, y. (1994): co evolving genetic algorithm with filtered evaluation function

  1. 1. t994 IEEE Symposium on Emerging Technologies & Factory Automa1ion Co-Evolving Genetic Algorithm with Filtered Evaluation Function Hidenoti SAKANASHI and Yukinoti KAKAZU Faculty of Engineering, Hokkaido University. North- 13. West-8, Kita-ku , Sapporo, 060, Japan. E-mail : {sakana. kakazu }@ Abscracr - As a function optimizer or a search procedure. Genetic search procedure applicable to a wide variety of problems, it is felt that such remarkable weak points should be overcome . Hence , much research into GA-hard problems has been presented, and many extended GAs have been proposed. Some of them focus on maintaining adequate diversity of the strings in the population, using sharing [4]. incest pre vention [3 ], crowding [10], introducing ecological frameworks [2], and so on. Almost all of them aim at solving the specific types of hard problem, and seem to be over-sensitive to parameter senings. Algorithms (GAs) are very powerful and have many advantages. Fundamental research concerning the internal behavior of GAs has highlighted their limitations as regards search performances for what are called GA-hard problems. The reason for these difficulties seems to be that GAs generate insufficient strategies for the convergence of populations . To overcome this problem, an extended GA. which we name the filtering-GA, that adopts the concept of co-evolution, is proposed. It has two GAs , and the y influence each other through their evaluation process. 3. CO-EVOLUTION I. INTRODUCTION In a static environment, agents, which have certain mechanisms for improving their performance and can change its own structure, will acquire stable structure and can show better performance. Then, whether that perfonnance is optimal or not for given environment, they wiU never change stable structure. Such structure is one of local optima. On the other hand, in the dynamicaUy vary ing environment, they may keep changing their struc ture to improve performance and avoid sticking into local optima. If the environment changes lo suppress their perfonnance impro vement, moreover, it is diffi cu lt for the agents to improve their performance, and it is expected that the structure acquired in such en vironment is more adaptive and robust. Here, consider the environmen l as problem, and agents as solutions or strategies for obtaining them. It is reported that such system can solve [6] and is called co-evolution system. The co-evolutionary system has competitive re lationship between its components. In above case, competitors have different roles, e .g. , problem and solutions , but it is not only a type of coevolutionary system. Tournament evaluation in genetic programming is adopted to reduce the computational load [!] . There also is a system using two types of agents with different aims, planning and scheduling, for solving the problem of manufacturing [8]. In this study , there are one environment and two agents . The environment is the objective function, and the agents are GAs. Two GAs influence evaluation processes of opponents, but they aren't competitors. One change the appearance of evaluation function of opponent, and the population of the opponent is used in evaluation of its population. Through this cyclic re lationship between two GAs, the system can avoid local optima without interfering optimization of the objective fW>Ction. As a function optimizer or a search procedure , GAs are very powerful, and can perform robust search through multi-point search using the population. So they have been applied to many fields {4]. Fundamental research concerning their internal beh.-ior, howe ver. has re v e aled the lim itations as regards search performance. This kind of problems singled out by suc h research are known as GA -hard pro blems. Whether the y are GA-hard or not, they may be optimized if there is some prior know ledge, but this si tuation never o ccurs as it is not necessary to use GAs for problems whose characteristics are already known. The reason for difficu lties seems to be that, for the convergence of po pulations, GAs generate insufficient strategies. To overcome this problem, in this paper, we proposed an e. tended GA whi ch adopts the concept of co-evolution. It has a mechanism for controlling the convergence of populations . From the functiona1 point of view , the system consists of three components: the observation of convergence. its force , and the repulsion against it. To realize these functions, the system proposed in this paper uses two GAs and a filter which stores the information showing the trend of convergence . Two GAs influence each other through their evaluation process. 2. GENETIC ALGORITHMS In the simp le GA as a multi-point search procedure . the search points of the objective function F, xi, are represented in the strings of characters, S,, and are stored in the population , wh ich is represented as [[S, ]]" in this paper. Here, supposing that the strings are composed of the binary digit characters [0,1)', N is the population size. Every string is mapped into the search space of the objective function by the decoding function, D. They are tested in the objective function and receive values using the evaluation function, G. and G(S,) = F(D{s,)) = F(x,) (! ) On the basis of those values, the genetic operators act on the population and produce the population of the ne xt generation. Through repeated reproduction of the population, strings can receive a larger value from the e valuation func tion. and the average of those values will increase with each population. The notion of the schema is important for analyzing the behavior of GAs. It consists of the trinary characters [0.1.# 1. The character # is called the don't care mark and matches both 0 and I. That is to say, the schema is a set of perfectly matching strings. In contrast to suings in the population that indicate points in the search space. the schemata specify regions thereof. Clearly, schema with a large order are likely to be e ffected by mutation. and schema with a long defining length wiU tend to be influenced by crossover. This fact is the essence of the schema theorem [7]. GA-hard problems !iteraUy mean types of problem which simple GA can't easily sol ve. Since GAs are known as a robust 0-7603-2t14-6/941$4.00 © 1994 IEEE. 4. FlLTIRING GENETIC ALGORITHM 4.1 ConcepiUal D yTifl!1Jics of F i ltering-GA GA-hard problems can be thought of as having a landscape with many slatic or dynamic local peaks, and we now assume that the objective is to maJtimize such a multimodal function . Without prior knowledge for solving the problems to which they are being applied. it is difficult for any search procedure to completely avoid the local peaks. To suppress mistaken convergence on the local peaks, the extended GA proposed in this paper, the filtering -GA, repeats its searches many times. In this way, it is possible to repeat the fast evolution of the population, and to discover many local optima. Furthennore, if the system can restart from an unexplored region, the search is expected to be more effective, and such search will find the most promising regions sequentially until the search space is completely explored or the system is terminated by an external control. To re liably locate many peaks in the landscape of a multimodal objective Function in one computation, the filtering-GA changes the effect of the objective function on the dynamics of the GA. If the evaluation value around a local optimal area already 454 ~-· - .. -- · . - -- ----
  2. 2. discovered decreases, the population will move toward another optimaJ area. In order _to realize ~ i s mec~s~ autonomously, we propose the followmg operatmns: ftltenng, covering, and extractl.on. Whether the original string S is or isn't io the region represented by the schema H , the new string S"" .. generated by the covering operation is in this region. If the expected evaluation value of the schema is large, it is quite possible that new strings generated by tllis tnapping will get high evaluation values. 4.1 .I Filtering Operation 4 .1.3 Schema Extraction The first operation of filtering is described here. The filter consists of two sequences of length I, which correspond to the length of the strings, and consist of real numbers, v" and v'. The jth element of the sequence V' is represented hy v; Using t.his filter. the The extracted schema is represented by H"' . The values of all its p.ositions are defined by the following equations: 0, if z<l - P.., , (7) 1, ifz>P,u• { #, otherwise. h = 't string S, is evaluated by the modified evaluation function, as follows: G••• (S.)=G(S,)- ±•:•. ,., ., _L:J s'J N (2 ) (8) •- where s, is the value {0,1) of the jth position of string S,. As seen whe re Pu, is the system parameter named the in this equation, if v; is large. the strings which have values h (0 or 1) in lhejth position will receive a small evaluation value. Then, if v;, in which j and h correspond to the local optima. has a large value, the population will move away from these local optima and IS expected to find another local optima. After it has converged on one local optima, the population can be led away to another local optima efficiently if the change of the filter is controlled properly. Because the system doesn 't know what the local and global optima are, it is impossible for the filter values to change when one of the local optima is found. Therefore. the degree of convergence ts used for deterrmmng the filter values. because the population tends to converge on pealcs in the landscape of a problem whether is calculated as: they are optimal or not. Namely, all threshold, 0.5 ~ Puo ~ 1, and z is the mean allele value at position j . Eq. 7 is named the extract function. v; = v ; *t,{ 4.2 Schematic Structure of Filtering-GA The filtering-GA consists of two GAs and a filter as shown in Fig. 2. OneGA is named GA,•. This is almost the same as a simple GA except in its evaluation of strings. Systems which consist of several GAs have already been proposed, such as the multi-level GA [Grefenstette86] and so on. The two GAs in the filtering-GA malce no hierarchy, and are not equal. They interact with each other and evolve like a co-evolution system. The strings of GAm are represented as S,w and the size of the population [[S,"Jt· is N,. Its suings are evaluated through t.he filtering operation. The other GA is named GAsch• and the members of its population are represented in the form of schemata {0.1.#) . Every schema S,"" in the v; G~S,) xcomp(h,sJ}. 0, if h >' S· , h comp(h ~s - ) = { ~' v 1• ot erwtse. extraction (3 ) population [[S,"" J(· of GAsch is evaluated using the covering (4) operation. The population size of [[S,""]]"· is given by N,. The filter is generated from GAsch and affects the evaluation of GA 50• That is, is the observed fllness of allele h at position j in the current population. lf there are many strings whi c h ha,~ e value h in the jth pos ition and their evaluation values are large. ': w ill be large. The filter generated by these equations lowers the evaluation values of strings which have value h in the jth position, in its evaluation of the next generation by eq. 2 . As a result of this procedure, the population moves away from the region where it has converged (Fig. 1). t~~~ "i"'jy,.t ~t ~ Fig. 2. Schematic diagram of filtering-GA. 4.3 Interaction among Two GAs and Objectivt Function Fig. 1. Image of filtering operation. 4 .3 .I Evaluation Functions 4 .1.2 Covering Opera tion The evaluation function of GAm• G,v, is given as the next equation. and every soing S,"" of GAstr is evaluated a.s follows : Th~ second ?peration of the filterin.g -GA is the covering operation. Th1.s operation forces the populauon to converge in a particular G,,(sw ) =G(S")-C, reg10n [9] . Consider a schema Has a template of strings which have windows at the position of the don't care marks #s. Then the new string s~· is generated by putting this template on one string S. This operation is formalized as follows: s- = cover(H.S), (5) ( ) _ _ 1 , if hi "#, 5 6 ' - hi, otherwtse. ±vf. (9) i•O where G is given in eq. 1. s,~ is the value of the jth position of S,'", v; is the jth value in V' of the filter, C, is the system variable, and 0 s C, s 2 . Because the filter stores the tendency of convergence, the population ofGA.a moves away from the region of convergence. At each generation. the system records the maximum and the average values calculated as G. If the best G value discovered by GAm in a certain generation exceeds the {s 155
  3. 3. average of all the G values obtained after P, generations ago. then it would seem that GAsu is currently searching the most promising region. Then, to weaken the influence of the filter on GA,~. C, is reduced 2 x 1/ P, . The filter never affects GA,., when C, = 0. If that condition is not satisfied. the search concerning the region which GAm is currently searching seems to finish or that region is not the most promising region. Then, tO stn:ngthen the influence of the filter on GA,u. and to cause the population of GA 5u to move away from that region, C, is increased l/ P,.. It is expected that the influence of the filter will reject the convergence of GAsu when C, =I, causing the power to avoid convergence when C, > I; and when C, = 2. such avoidance power will be negatively coequal to the convergence power of GA 50 with C, = 0. If it becomes C, < 0 or C, > 2 , C, is resettled at 0 or 2. P,. is named the history number and is a parameter of the system. Every schema of GAsch is evaluated by the following equation: C",(S;''•) = m;"'{c(cover(S,"" .s;•))}. easily to be optimized. On the other hand, if a poor arrangement (type- I) is used, the defining length of each building block is long (type-2) , and then GAs can not easily converge to the optimal solution. In both type, its global optima is 300. Discontinuous Function is defined as follows : kzK D,(S.), F(S,J= j D 112 IK s;) ' <( · , where (10) 4.3 .2 Generation of Filter The procedure for the generation of the filter is based on eq. 3 and is modified as follows : t: ••I ' I_,,i=0. ,., (4-0·8 ) Otherwise, I K =300 and D,(S)= L :. , s,. It is deceptive function. To optimize it, the schema of (0 ... 0# ... 11), in which all positions of left half have Os. must be achieved. Its global optimum is 1.2 x K at (0 ... 0 1. .. I) and its local optimum is K at (II... I ) . The standard parameters used in the simulations are shown in Table t. These parameters, for both the simple GA and the filtering-GA, aren't adjusted for each problem. So, these two GAs may be capable of exhibiting a higher performance than the results shown here. Fig. 3-a and 3-b repon the performances of the simple GA and the filtering-GA on type-! and type-2 of 30-Bit, Order-Three Deceptive Problem. These results were obtained by averaging over 5 independent runs. Although the population of the simple GA completely converged onto a local optimum in both type, the filtering-GA can find their optimal solutions. Fig. 4 reports the performances of the simple GA and the filtering-GA on Discontinuous Function. Although the population of the simple GA completely converged onto a local optimum at 240, the filtering-GA failed optimization only once in 5 trials. This failure was caused by the effect of the scheduling mechanism introduced at eq. 9. The accurate acquisition of those schemata is expected to be realized, if C, is changed in an adaptive way. In both function, the filtering-GA discovered the local optima as fast as the simple GA in the e"periments. Moreover, premature 'on vergence never occurred, because it is able 10 avoid it through using the filtering operation. Thus, the filtering-GA can fast search local optima, using large selection pressure, without convergence of the system. This is the equation for obtaining the expected maximum evaluation values of the schemata. The evaluation values obtained in this way imply the possibility of evolution for the population in those regions. Obviously, a large number of sampling points will ensure accurate estimation. Therefore, all the strings of GAm arc utilized in the evaluation of one schema of GAsch using the covering operation. _, • 1 ~{a(s,"") v,=N"- o(s.~')xcoml'h,s,,"")} , if kE{O.I} . (II) where o(H) is the order of the schema H. Based on the definition of the schema, the don't care mark,#, matches both 0 and 1. but it matches neither 0 nor I in the function comp(h,s) . The position of #s in each schema never 3ffects the filter, because it has no connection with the convergence of the search. The population of GA..:h evolves and changes according to eq. 10, and the filter generated from it varies, in each generation. This means that the evaluation function of GAm given m eq. 9 changes at each generation. Under the evaluation function, which changes drastically, the population doesn't evolve properly. Hence, the average of all the filters generated after P,. generations ago is used in eq. 9 . P, is the history number introduced at eq. 9. 6. CONCLUSION This paper proposed an extended genetic algorithm , named the fi ltering genetic algorithm (the filtering-GA ), as a multimodal function optimizer. The filtering-GA is intended to solve GA-hard multimodal functions, and to optimize GA-easy problems quickly. It consists of two GAs and a filter. Two GAs influences each other thorough their evaluation processes, and form a co-evolutionary system. They suppress the convergence of oppenents, and avoid sticking into local optima. The role of the filter is to smoothen the interaction between the two GAs . Using the se three parts. the filtering-GA can search widely throughout the search spaces of problems, and discover many local optima sequentially. Its conceptual and detailed definitions were given, and its functional framework was discussed. In order to examine the search performance of the filtering-GA, some experiments were carried out, then it exhibited a higher performance and faster search than the simple GA. 4.3 .3 Extraction nf Schema To avoid system convergence, we introduce the third interaction between the two GAs, named tl1e extraction. This operation is used to estimate where GA,u is exploring in the search space, and attracts the attention of GAscll to this region. In practice, the schema which indicates such a region is extracted from the population of GA, •. and is joined to the population of GAsch· To determine the evaluation value Of the extracted Schema, CK>(Ho') , it is Settled as max, G(S;'). Then the old schema. which has the lowest fitness value in the population of GAsch. is replaced by this new extracted schema H•'. REFERENCE {I] P. J. Angeline and J. B. Pollack, "Compttitive Environments Evolve Better Solutions for Complex Tasks," Proceedings of The Fifth lnternationLJI Confertnce on Gtnetic Algorithms, pp. 264-270, Morgan Kaufmann (1993 ). [2] Y. Davidor, T. Yamada and R. Nakano. "The ECOlogical Frameworl:: ll: Improving GA Performance At Vinually Zero Cost: Proceedings of The Fifth fnternational Conference on Generic Algorithms, pp. 171-176, Morgan Kaufmann ( 1993). [3] L, J. Eshelman, "The CHC Adaptive Search Algorithm : How to Have Safe Search When Engaging in Nontraditional Genetic Recombination," Foundations of Gtntti c Algorithms, pp. 265-283. Morgan Kaufmann (1991). 5. COMPUTER SIMULATION This section examines the search performance of the filtering·GA. The objective functions used here are 30-Bit, Order-Three Deceptive Problem (5] and Discontinuous FWiction {II]. 30-Bit, Order-Three Deceptive Problem is created as the sum of I 0 copies of the 3-bit function, where each copy is associated with the next three bits. If all bits associated with a particular subfunction are as close as possible, the function will 456
  4. 4. (4] D. E. Goldberg and J. Richardson, "Genetic Algorithms with Sharing for Multimodal Function Optimization, " Proce.dings of The Second International Conference on Gen<tic Algorithms, pp. 41-49, Lawrence Etlbaum Associates (1987). [4} D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning , p.412, (1989). [ ~} D. E. Goldberg, B. Korb and K. Deb, "Messy Genetic Algorithms: Motivation, Analysis, and First Results. " Complex Systems, 3, pp. 493-530, Complex Systems (1989). [6} W. D. Hillis, "Co-Evolving Parasites Improve Simulated Evolution as An Optimization Procedure ." Emergent Computation , pp. 228-234, The MIT Press ( 991 ). [7] J. H. Holland, Adaptation in Natural and Artificial Systems , University of Michigan Press ( 1975). (8] P. Husbands and F. Mill, "Simulated Co-Evolution as The Mechanism for Emergent Planning and Scheduling," Proceedings of The Fourth International Conferen ce on Genetic Algorithms , pp. 264-270, Morgan Kaufmann (199 ). [9} Y. Kakazu, H. Sakanashi and K. Suzuki , "Adapti ve Search Q) " OJ > c: .Q 1ij ::> iii > Q) 300 290 280 270 v; Q) .0 260 0 Strategy for Genetic AJgorithm s with Additional Genetic 400 600 generation (a) Type- I . 800 1000 0 Q) :::J iii > c: 200 200 400 600 generation (b) Type-2. BOO I 000 300 290 0 -~ Algorithms," Parallel Problem Solving from Nature , 2, pp. 31-320, Nonh-Holland ( 992). [10] S. W. Mahfoud, "Crowding and preselection revi sited, " Para/It/ Problem Solving from Nazure, 2, pp . 27-36, NonhHolland ( 1992). [II} K. Suzuki, H. Sakanashi andY . Kakazu, "Iterative Schema Extracting Operation for Genetic Algorithms ," Pro c. of :::J 280 iii > Q) 270 1n 2l 260 Australian and N ew Zealand C o nfe rence o n Inte ll igent lnformlltion Systems. p. 51 7-520 (1993). Fig. 3. Search performance of simple GA and filtering-GA on 30Bit, Order-Three Deceptive Problem. Table 1. Parameter settings used in simulations. Filtering-GA Simple GA 1000 population size 30 string length maximum generation 1000 on elilist strategy uniform crossover type 0.8 crossover rate 0.03 mutation rate history number extraction threshold GAstr GAsch 30 30 1000 on uniform 0.6 0. 03 30 Q) ::> 'iii 340 > 320 .Q 300 :::J 280 c: 30 000 off two-point 0. 8 0.05 ;;; iii > Q) 1ii Q) 260 240 .0 10 0.8 0 200 400 600 generation BOO 1 000 Fig. 4. Search performance of simple GA and filtering-GA on Discontinuous Function. 457