The Gene Expression Messy Genetic Algorithm For. Hillol Kargupta & Kevin Buescher. Computational Science Methods division

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1 The Gene Expression Messy Genetic Alorithm For Financial Applications Hillol Karupta & Kevin Buescher Computational Science Methods division Los Alamos National Laboratory Los Alamos, NM, USA. Abstract This paper introduces the ene expression messy enetic alorithm (GEMGA) a new eneration of messy GAs that may nd many applications in nancial enineerin. Unlike other existin blackbox optimization alorithms, GEMGA directly searches for relations amon the members of the search space. The GEMGA is an O(jj k (` + k)) sample complexity alorithm for the class of order-k delineable problems (Karupta, 1995) (problems that can be solved by considerin no hiher than order-k relations) in sequence representation of lenth ` and alphabet set. The GEMGA is desined based on alternate perspective of natural evolution proposed by the SEARCH framework (Karupta, 1995) that emphasizes the role of ene expression. This paper also presents the test results for lare multimodal problems and identies possible applications to nancial enineerin. 1 Introduction Durin the last few years, nancial enineerin has emered as a major eld where the state of the art technoloies can be tested on hihly demandin rounds. Blackbox optimization alorithms (BBO) such as enetic alorithms, simulated annealin, and their numerous cousins have found many applications in system identication, optimization, and prediction problems in this eld. However, apart from the results of individual applications and the claim of superiority by each of these alorithms, there hardly exist any fundamental understandin and common round for desinin and comparin BBO alorithms. The SEARCH (Search Envisioned As Relation and Class Hierarchizin) framework introduced elsewhere (Karupta, 1995) oered an alternate perspective of This author can be reached at, P.O. Box 1663, XCM, Mail Stop F645, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. hillollanl.ov blackbox optimization (BBO) in terms of relations, classes and partial orderin. SEARCH is primarily motivated by the observation that searchin for optimal solution in a BBO is essentially an inductive process and in absence of any relation amon the members of the search space, induction is no better than enumeration (Watanabe, 1969). SEARCH decomposed BBO into three spaces: (1) relation, (2) class, and (3) sample spaces. SEARCH also identied the importance of searchin for appropriate relations in BBO. No BBO alorithm can eciently solve a reasonably eneral class of problems unless it searches for relations. Karupta (1995) also showed that the class of order-k delineable problems can be solved in SEARCH with sample complexity polynomial in problem size, desired quality and reliability of the solution. Despite the existence of so many BBO alorithms, it is quite unfortunate that there hardly exist any BBO alorithm that realizes the importance for detectin appropriate relations. Neither do they realize the critical decomposition of decision makin that a BBO alorithm should do in the relation and class spaces. Clearly, we need alorithms that are desined in a more constructive manner and tell us what class of problems they can solve eciently. This paper takes a step toward that direction. I introduce a new class of BBO alorithm called the ene expression messy GA (GEMGA), that directly search for relations followin the lessons from the SEARCH framework. GEMGA is an O(jj k (` + k)) sample complexity alorithm for order-k delineable problems in sequence representation of lenth ` and alphabet set. Section 2 describes dierent aspects of GEMGA. Section 3 presents the test results for lare multimodal, order-k delineable problems. Finally, Section 4 concludes this paper with a discussion on possible immediate applications of GEMGA in nancial enineerin. 1

2 2 The GEMGA In this section I present the Gene Expression Messy GA (GEMGA) an O(jj k (` + k)) sample complexity alorithm for order-k delineable problems in sequence representation of lenth ` and alphabet set. Desin of GEMGA is based on the alternate perspective of evolution, developed by SEARCH that emphasize the computational role of ene expression. Section 2.1 discusses the representation in GEMGA. Section 2.2 explains the population sizin in GEMGA. This is followed by Section 2.3 that describes the main operators, transcription, selection, and recombination. Section 2.4 presents of the overall mechanisms. 2.1 Representation GEMGA uses a sequence representation. Each sequence will be called a chromosome. Every member of this sequence is called a ene. A ene is a data structure, containin the locus, value, and weiht. The locus determines the position of the member in the sequence. The locus does not necessarily have to be the same as the physical position of the ene in the chromosome. For example, the ene with locus i, may not be at the i-th position of the chromosome. When the chromosome is evaluated, however the ene with locus i ets the i-th slot. This positional independence in codin was introduced elsewhere (Goldber, Korb, & Deb, 1989) to enforce the proper consideration for all relations dened by the representation. GEMGA does not depend on the particular sequence of codin. For a iven ` bit representation, the enes can be placed in arbitrary sequence. A ene also contain the value, which determines the value of the ene, which could be any member of the alphabet set,. The relation space is explicitly evaluated usin the weihts associated with each member. Weihts take a positive real number except at the initial stae. All weihts are initialized to -1.. No two members with the same locus are allowed in the sequence. In other words, unlike the oriinal messy GA (Goldber, Korb, & Deb, 1989) no under or overspeciction are allowed. A population in GEMGA is a collection of such chromosomes. 2.2 Population sizin GEMGA requires at least one instance of the optimal order-k class in the population. For a sequence representation with alphabet, a randomly enerated population of size k is expected to contain one instance of an optimal order-k class. The population size in GEMGA is therefore, n = c k, where c is a constant. When the sinal from the relation space is clear, a small value for c should be sucient. However, if the relation comparison statistic produces a noisy sinal, this constant should statistically take care the samplin noise from the classes dened by any order-k relation. Since GEMGA uses sequence representation, the relation space contains total 2` relations. However, GEMGA processes only those relations with order bounded by a constant, k. In practice, the order of delineability (Karupta, 1995) is often unknown. Therefore, the choice of of population size in turn determines what order of relations will be processed. For a population size n, the order of relations processed by GEMGA is, k = lo(n=c)=lojj. If the problem is order-k delineable (Karupta, 1995) with respect to the chosen representation and class comparison statistics then GEMGA will solve the problem otherwise not. In that case a hiher population size should be used to consider hiher order relations. 2.3 Operators GEMGA has four primary operators, namely: (1) transcription, (2) class selection, (3) strin selection, and (4) recombination. Each of them is described in the followin Transcription As mentioned before, the weiht space in GEMGA chromosomes is used to process relations. The transcription operator detects the appropriate order-k relations. Comparin relations require a relation comparison statistics. GEMGA does not process the relations in a centralized lobal fashion; instead it evaluates relations locally in a distributed manner. Every chromosome tries to determine whether or not it has an instance of a ood class belonin to some relation. In GEMGA, the quality of a relation is determined by the quality of its ood classes distributed over the population. Aain, no centralized processin of relations is performed. The transcription operator is a deterministic one. It considers one ene at a time. The value of the ene is randomly ipped to note the chane in tness. For a minimization problem, if that chane cause a improves the tness (i.e. tness decreases) then the oriinal instance of the ene certainly do not belon to the instance of the best class of a relation, since tness can be further improved. Transcription sets the correspondin weiht of the ene to zero. On the other hand if the tness worsens (i.e. tness increases) then the oriinal ene may belon to a ood class; at least that observation does not say it otherwise. The correspondin weiht of the ene is set to the absolute value of the chane in tness. Finally, the value of that ene is set to the oriinal value and the tness of the chromosome is set to the oriinal tness. In other words, ultimately transcription does not chane anythin in a chromosome except 2

3 // pick is the currently considered ene Transcription(CHROMOSOME chrom, int pick) f double phi, delta; int dummy; double dwt; dwt = chrom[pick].weiht(); if(dwt >. OR dwt == -1.) f phi = chrom.fitness(); dummy = chrom[pick].value(); // Chane the value randomly chrom[pick].perturbvalue(); // Compute new tness chrom[pick].evaluatefitness(); // Compute the chane in tness delta = chrom[pick].fitness() - phi; // For minimization problem if(delta <.) delta =.; // Set the weiht if(dwt < delta OR delta ==.) chrom[pick].setweiht(delta); // Set the value to the oriinal value chrom[pick].setvalue(dummy); // Set the oriinal tness chrom[pick].setfitness(phi); Fiure 1: Transcription operator for minimization problem. For maximization problem, if delta< absolute value of delta is taken and otherwise delta is set to. the weihts. For a maximization problem the conditions for the weiht chane are just reversed. The same process is continued deterministically for all the ` enes in every chromosome of the population. Fiure 1 shows the pseudo-code for the transcription operator. For enes with hiher cardinality alphabet set () this process is repeated for some constant C < jj times. The followin section describes the two kinds of selection operators used in GEMGA, which correspond to the selective pressures in protein and DNA spaces of natural evolution Selection Once the relations are identied, selection operator is applied to make more instances of better classes. GEMGA uses two kinds of selections (1) class selection and (2) strin selection. Each of them is described in the followin: ClassSelection(chrom1, chrom2) CHROMOSOME chrom1, chrom2; f int i; for(i=; i<problem lenth; i++) f if(chrom1[i].weiht() > chrom2[i].weiht() ) chrom2[i] = chrom1[i]; else if(chrom2[i].weiht() > chrom1[i].weiht() ) chrom1[i] = chrom2[i]; Fiure 2: Class selection operator in GEMGA. A consistent codin (where chrom1[i] and chrom2[i] has common locus) is used in place of messy codin for the sake of illustration. Class Selection: The class selection operator is responsible for selectin individual classes from the chromosomes. Better classes detected by the transcription operator are explicitly chosen and iven more copies at the expense of bad classes in other chromosomes. Fiure 2 describes the operator. Two chromosomes are randomly picked; the weihts of the enes are compared and the ene with hiher weiht overwrites the correspondin ene in other chromosome with lower weiht. Strin Selection: This selection operator ives more copies of the chromosomes. A standard binary tournament selection operator (Goldber, Korb, & Deb, 1989) is used. Binary tournament selection randomly picks up two chromosomes from the population, compares their objective function values, and ives one additional copy of the winner to the population at the expense of the looser chromosome. The followin section describes the recombination operator in GEMGA Recombination Fiure 3 shows the mechanism of the recombination operator in GEMGA. It randomly picks up two chromosomes from the population and considers all the enes in the chromosomes for possible swappin. It randomly marks one amon them. If the weiht of a ene from the marked chromosome is reater than that of the correspondin ene from the other chromosome then it swaps the enes. The followin section describes the overall mechanism of the alorithm. 3

4 Recombination(chrom1, chrom2, pc) CHROMOSOME chrom1, chrom2; double pc; f int i; GENE dummy; for(i=; i<problem lenth; i++) f if(chrom1[i].weiht() >= chrom2[i].weiht() AND Random() < pc) f dummy = chrom1[i]; chrom1[i] = chrom2[i]; chrom2[i] = dummy; Fiure 3: Recombination operator in GEMGA. A consistent codin (where chrom1[i] and chrom2[i] has common locus) is used in place of messy codin for the sake of illustration. Random() enerates a random number in between and 1; pc is a number between and The alorithm GEMGA has two distinct phases: (1) primordial stae and (2) juxtapositional stae. The primordial stae simply applies transcription operator for ` enerations, deterministically considerin every ene in each eneration. Durin this stae the population of chromosomes remains unchaned, except that the weihts of the enes chane. This is followed by the juxtapositional stae, in which the selection and recombination operators are applied iteratively. Fiure 4 shows the overall alorithm. The lenth of the juxtapositional stae can be rouhly estimated as follows. If t be the total number of enerations in juxtapositional stae, then for binary tournament selection, every chromosome of the population will convere to same instance of classes when 2 t = n, i.e. t = lo n= lo 2. Substitutin n = cjj k lo c+k lo jj, we et,t =. A lo 2 constant factor of t is recommended for actual practice. Clearly the number of enerations in juxtapositional stae is O(k). Let us now compute the overall sample complexity of GEMGA. Since the population size is O(jj k ) and the primordial stae continues for C` = O(`) enerations, the overall sample complexity, SC = O(jj k (` + k)) GEMGA is a direct realization of the lessons from the SEARCH framework. Followin SEARCH, it can be reconized that the sample complexity is also a funcvoid GEMGA() f POPULATION Pop; int i, j, k, C, k max; // Initialize the population at random Initialize(Pop); i = ; // Primordial stae While(i < C) f // C is a constant j = ; Repeat f // Identify better relations Transcription(Pop, j); // Increment eneration counter j = j + 1; Until(j == Problem lenth) i = i + 1; k = ; // Juxtapositional stae Repeat f // Select better strins Selection(Pop); // Select better classes ClassSelection(Pop); // Produce osprin Recombination(Pop); Evaluate(Pop); // Evaluate tness // Increment eneration counter k = k + 1; // k max is of O(lo(Problem lenth)) Until ( k > k max ) Fiure 4: Pseudo-code of GEMGA. The constant C < jj, where jj is the cardinality of the alphabet set. tion of the desired quality of the solution and the reliability of the process. However, the implementation of GEMGA throuh distributed local evaluation of relations and classes outweihs the satisfaction of quantifyin the success probability that is straiht forward in case of centralized comparison (as it was in SEARCH) from the practical perspective. Therefore, the reader must realize the dependence of the sample complexity on the desired accuracy of the solution and reliability, implicit in the above aruments. The followin section presents the test results. 3 Test Results Desinin a test set up requires careful consideration. An ideal set up should contain problems with dier- 4

5 ent dimensions of problem diculty, such as multimodality, bounded inappropriateness of relation space (BIRS), problem size, scalin, noise. The GEMGA has been tested aainst problems with all of these dimensions of diculties (Karupta, 1996). However, because of limited space, in this section, we present the performance of GEMGA for problems with only massive multimodality and controlled amount of BIRS. For all functions the averae number of function evaluations per success (AFPS) is measured. For every function we choose the desired solution value (DSV) a priori. We say the alorithm was successfull if it reaches the DSV. Deceptive trap functions (Ackley, 1987) are used as basic buildin blocks for desinin this test suite. A trap function can be dened as follows: f(x) = ` if u = ` = `? 1? u otherwise; where u is the number of 1-s in the strin x and ` is the lenth of the sequence used for representin the variable x. Goldber, Deb, and Clark (1992) showed that such deceptive problems can be used to desin problems of bounded diculty. In a trap function de- ned over a sequence of lenth ` the order of delineability is ` with respect to the class averae comparison statistics. Althouh GEMGA does not work usin the class averae comparison statistic (i.e. when classes are compared with respect to the distribution means) this ives us a simple way to capture the main essence. When multiple number of such functions are concatenated with each other a problem dened over a sequence of lenth ` with order-` delineability can be desined. Since the order of delineability directly controls the BIRS, such concatenated functions can be eectively used for desinin problems with BIRS by controllin the `. Such functions have only `=` proper relations amon the?`` order-5 relations that must be detected in order to nd the lobal solution. Therefore, searchin for the appropriate relations is not a trivial job in this class of problems. Apart from BIRS, such functions also oer multimodality. If we carefully observe, we shall note that a trap function has two peaks. One of them corresponds to the strin with all 1-s and the other is the strin with all -s. If we desin a problem by concatenatin m such functions, it will have a total of 2 m local optima and amon them only one will be the lobally optimal solution.clearly this class of problems are massively multimodal and has bounded inappropriateness of the relation space, dened by the representation. For testin the GEMGA, a test function is constructed by concatenatin multiple numbers of trap functions, each with ` = 5. Therefore the order of de- Averae Function Evaluations Per Success (AFPS) Recombination probability. Gene exchane probability (pc). Class selection probability 1. Strin selection probability. Table 1: GEMGA parameters for TF1. AFPS Problem size Fiure 5: Growth of the number of function evaluations with problem size. The population size for all problem sizes was 3. All the results are averae of ve independent runs. lineability is ve. As we increase the number of functions, in other words the overall problem lenth `, the deree of BIRS remains constant, but the deree of multimodality increases exponentially. For ` = 2, the overall function contains 4 subfunctions; therefore, an order-5 bounded 2-bit problem has 2 4 local optima, and amon them, only one is lobally optimal. The GEMGA is tested aainst order-5 deceptive problems of dierent sizes. Table 1 shows the GEMGA parameters used for all of them. Fiure 5 shows the averae number of function evaluations from ve independent runs needed to nd the DSV for dierent problem sizes. For all problems, the DSV is set to the lobally optimal solution, which is equal to problem size, `. The population size is chosen as described earlier in this paper. The chosen population size for all the problems was 3. The sample complexity clearly rows linearly and the population size is constant. Fiure 6 and 7 show the radual detection of the relations durin the primordial and juxtapositional staes for a 3-bit order-5 deceptive problem. Each ure represent the relation space of the whole population at a certain eneration. The x-axis denotes the weihts in the enes, ordered on the basis of the locus of the ene. In other words the values alon the x-axis correspond to the actual value of the locus of a ene in 5

6 a chromosome. The y-axis corresponds to the dierent members in the population. The z-axis, perpendicular to the pae denotes the real valued weihts of the correspondin ene in the correspondin chromosome. Since the test function is comprised of order-5 trap functions, for any particular ene in a chromosome, there are only 4 other enes that are related with it. The complete relation space has a cardinality of 2 3. Amon? 3 5 order-5 relations there are only 6 relations that correctly correspond to the actual dependencies dened by the problem. GEMGA needs to detect the relations that relate enes with loci ranin from to 4 toether, from 5 to 9 toether and so on. Fiure 6 show that these relations are radually detected in dierent chromosomes that contain ood classes from those relations. Finally, at the end of primordial stae (Fiure 7 (top)) all the relations are detected. Fiure 7(middle), (bottom) shows the processin of classes durin juxtapositional stae. More instances of ood classes are produced by selection and they are exchaned amon dirent strins to create hiher order relations that nally lead to the optimal solution. The followin section concludes this paper. 4 Possible Financial Applications The GEMGA awaits many possible applications in - nancial enineerin. The main strenths of GEMGA compared to the existin BBS alorithms comes from the careful attention paid to the dierent decision makin aspects of BBS, particularly for detectin appropriate relations. Since the GEMGA adequately search for relations dened by the representation, it is likely to be applicable for wider class of problems. Moreover, the GEMGA precisely states the class of BBO problems that it can solve in polynomial sample complexity. Therefore, application of the GEMGA for critical nancial application is very promisin. Some of the immediate possible applications could be in the followin areas: (1) nonlinear system identication, (2) classication problems, and (3) prediction and forcastin. Another strenth of GEMGA is that it is inherently suitable for parallel processin, more than the simple GA. GEMGA is a linear sample complexity alorithm and the linear part is contributed by the primordial stae. Since primordial stae works only by transcription operator, which is unary. Every strin can in fact be iven to an independent processor. Such hih deree of parallelism makes it suitable for run time nancial applications. The followin section concludes this paper. 5 Conclusion This paper presents a brief introduction of GEMGA and the test results for lare problems with milions of local optima and bounded inappropriateness of the representation. GEMGA eliminates many problems of the previous versions of messy GAs. The main improvements are (1) explicit processin of relations and classes, (2) eliminatin the need for a template solution, (3) reducin the population size from O(2 k`) to O(2 k ) for order-k delineable problems in sequence representation of lenth ` and (4) reducin the runnin time by a lare factor. Experimental results clearly showed that GEMGA can detect appropriate relations eciently. GEMGA has been tested for dierent class of problems. Detailed description of those experiments can be found elsewhere (Karupta, 1996). 6 Acknowledement The early staes of this work was supported by AF- SOR Grant F and the Illinois enetic Alorithm Laboratory. The followin staes of the desin and experimentation have been performed at Los Alamos national Laboratory under the auspices of the US. Department of Enery. References Ackley, D. H. (1987). A connectionist machine for enetic hill climbin. Boston: Kluwer Academic. Goldber, D. E., Deb, K., & Clark, J. H. (1992). Genetic alorithms, noise, and the sizin of populations. Complex Systems, 6, 333{362. Goldber, D. E., Korb, B., & Deb, K. (1989). Messy enetic alorithms: Motivation, analysis, and rst results. Complex Systems, 3 (5), 493{53. (Also TCGA Report 893). Karupta, H. (1995, October). SEARCH, Polynomial Complexity, and The Fast Messy Genetic Alorithm. Doctoral dissertation, Department of Computer Science, University of Illinois at Urbana-Champain, Urbana, IL 6181, USA. Also available as IlliGAL Report 958. Karupta, H. (1996). Search, evolution, and the ene expression messy enetic alorithm. Los Alamos National Laboratory Report LA-UR Watanabe, S. (1969). Knowin and uessin - A formal and quantative study. New York: John Wiley & Sons, Inc. 6

7 Fiure 6: The relation space durin primordial eneation 1 (top), 1 (middle), and 2 (bottom) Fiure 7: The relation space at the end of primordial eneation (top), juxtapositional enerations 1 (middle) and 4 (bottom). 7