Evolutionary Computation

Transcription

1 Topic 9 Evolutionry Computtion Introduction, or cn evolution e intelligent? Simultion of nturl evolution Genetic lgorithms Evolution strtegies Genetic progrmming Summry Cn evolution e intelligent? Intelligence cn e defined s the cpility of system to dpt its ehviour to ever-chnging environment. According to Aln Turing, the form or ppernce of system is irrelevnt to its intelligence. Evolutionry computtion simultes evolution on computer. The result of such simultion is series of optimistion lgorithms, usully sed on simple set of rules. Optimistion itertively improves the qulity of solutions until n optiml, or t lest fesile, solution is found. The ehviour of n individul orgnism is n inductive inference out some yet unknown spects of its environment. If, over successive genertions, the orgnism survives, we cn sy tht this orgnism is cple of lerning to predict chnges in its environment. The evolutionry pproch is sed on computtionl models of nturl selection nd genetics. We cll them evolutionry computtion,, n umrell term tht comines genetic lgorithms, evolution strtegies nd genetic progrmming. Simultion of nturl evolution On 1 July 1858, Chrles Drwin presented his theory of evolution efore the Linnen Society of London. This dy mrks the eginning of revolution in iology. Drwin s clssicl theory of evolution,, together with Weismnn s theory of nturl selection nd Mendel s concept of genetics,, now represent the neo-drwinin prdigm. Neo-Drwinism is sed on processes of reproduction, muttion, competition nd selection. The power to reproduce ppers to e n essentil property of life. The power to mutte is lso gurnteed in ny living orgnism tht reproduces itself in continuously chnging environment. Processes of competition nd selection normlly tke plce in the nturl world, where expnding popultions of different species re limited y finite spce. Evolution cn e seen s process leding to the mintennce of popultion s ility to survive nd reproduce in specific environment. This ility is clled evolutionry fitness. Evolutionry fitness cn lso e viewed s mesure of the orgnism s ility to nticipte chnges in its environment. The fitness, or the quntittive mesure of the ility to predict environmentl chnges nd respond dequtely, cn e considered s the qulity tht is optimised in nturl life.

2 How is popultion with incresing fitness generted? Let us consider popultion of rits. Some rits re fster thn others, nd we my sy tht these rits possess superior fitness, ecuse they hve greter chnce of voiding foxes, surviving nd then reeding. If two prents hve superior fitness, there is good chnce tht comintion of their genes will produce n offspring with even higher fitness. Over time the entire popultion of rits ecomes fster to meet their environmentl chllenges in the fce of foxes. Simultion of nturl evolution All methods of evolutionry computtion simulte nturl evolution y creting popultion of individuls, evluting their fitness, generting new popultion through genetic opertions, nd repeting this process numer of times. We will strt with Genetic Algorithms (GAs)) s most of the other evolutionry lgorithms cn e viewed s vritions of genetic lgorithms. Genetic Algorithms In the erly 197s, John Hollnd introduced the concept of genetic lgorithms. His im ws to mke computers do wht nture does. Hollnd ws concerned with lgorithms tht mnipulte strings of inry digits. Ech rtificil chromosome consists of numer of genes, nd ech gene is represented y or 1: Nture hs n ility to dpt nd lern without eing told wht to do. In other words, nture finds good chromosomes lindly. GAs do the sme. Two mechnisms link GA to the prolem it is solving: encoding nd evlution. The GA uses mesure of fitness of individul chromosomes to crry out reproduction. As reproduction tkes plce, the crossover opertor exchnges prts of two single chromosomes, nd the muttion opertor chnges the gene vlue in some rndomly chosen loction of the chromosome. Bsic genetic lgorithms Step 3 Step 1: 1 Represent the prolem vrile domin s chromosome of fixed length, choose the size of chromosome popultion N,, the crossover proility p c nd the muttion proility p m. Step 2: 2 Define fitness function to mesure the performnce, or fitness, of n individul chromosome in the prolem domin. The fitness function estlishes the sis for selecting chromosomes tht will e mted during reproduction. Step 3: Rndomly generte n initil popultion of chromosomes of size N: x 1, x 2,..., x N Step 4: 4 Clculte the fitness of ech individul chromosome: f (x 1 ), f (x 2 ),..., f (x N ) Step 5: 5 Select pir of chromosomes for mting from the current popultion. Prent chromosomes re selected with proility relted to their fitness.

3 Step 6: 6 Crete pir of offspring chromosomes y pplying the genetic opertors crossover nd muttion. Step 7: 7 Plce the creted offspring chromosomes in the new popultion. Step 8: 8 Repet Step 5 until the size of the new chromosome popultion ecomes equl to the size of the initil popultion, N. Step 9: 9 Replce the initil (prent) chromosome popultion with the new (offspring) popultion. Step 1: Go to Step 4, nd repet the process until the termintion criterion is stisfied. Genetic lgorithms GA represents n itertive process. Ech itertion is clled genertion.. A typicl numer of genertions for simple GA cn rnge from 5 to over 5. The entire set of genertions is clled run. Becuse GAs use stochstic serch method, the fitness of popultion my remin stle for numer of genertions efore superior chromosome ppers. A common prctice is to terminte GA fter specified numer of genertions nd then exmine the est chromosomes in the popultion. If no stisfctory solution is found, the GA is restrted. Genetic lgorithms: cse study A simple exmple will help us to understnd how GA works. Let us find the mximum vlue of the function (15x x 2 ) where prmeter x vries etween nd 15. For simplicity, we my ssume tht x tkes only integer vlues. Thus, chromosomes cn e uilt with only four genes: Integer Binry code Integer Binry code Integer Binry code Suppose tht the size of the chromosome popultion N is 6, the crossover proility p c equls.7, nd the muttion proility p m equls.1. The fitness function in our exmple is defined y f(x) ) = 15 x x 2 The fitness function nd chromosome loctions Chromosome lel f(x) Chromosome string Decoded integer Chromosome fitness Fitness rtio, % X X X X X X x () Chromosome initil loctions x () Chromosome finl loctions. In nturl selection, only the fittest species cn survive, reed, nd therey pss their genes on to the next genertion. GAs use similr pproch, ut unlike nture, the size of the chromosome popultion remins unchnged from one genertion to the next. The lst column in Tle shows the rtio of the individul chromosome s fitness to the popultion s totl fitness. This rtio determines the chromosome s chnce of eing selected for mting. The chromosome s verge fitness improves from one genertion to the next.

4 Roulette wheel selection The most commonly used chromosome selection techniques is the roulette wheel selection X1: 16.5% X2: 2.2% X3: 6.4% X4: 6.4% X5: 25.3% X6: 24.8% Crossover opertor In our exmple, we hve n initil popultion of 6 chromosomes. Thus, to estlish the sme popultion in the next genertion, the roulette wheel would e spun six times. Once pir of prent chromosomes is selected, the crossover opertor is pplied. First, the crossover opertor rndomly chooses crossover point where two prent chromosomes rek, nd then exchnges the chromosome prts fter tht point. As result, two new offspring re creted. If pir of chromosomes does not cross over, then the chromosome cloning tkes plce, nd the offspring re creted s exct copies of ech prent. Crossover X6 i X2 i X1 i X5 i X2 i X5 i Muttion opertor Muttion represents chnge in the gene. Muttion is ckground opertor. Its role is to provide gurntee tht the serch lgorithm is not trpped on locl optimum. The muttion opertor flips rndomly selected gene in chromosome. The muttion proility is quite smll in nture, nd is kept low for GAs,, typiclly in the rnge etween.1 nd.1. X6' i 1 X2' i 1 1 X1' i X5' i X2 i X5 i Muttion X1" i 1 X2" i

5 X1 i The genetic lgorithm cycle Genertion i 1 1 f = 36 X2 i 1 f = 44 X3 i 1 f = 14 X4 i f = 14 X5 i f = 56 X6 i 1 1 f = 54 X1 i1 Genertion (i 1) 1 f = 56 X2 i1 1 1 f = 5 X3 i f = 44 X4 i1 1 f = 44 X5 i1 1 1 f = 54 X6 i f = 56 Crossover X6 i 1 1 X2 i X1 i X5 i 1 X2 i X5 i X6' i 1 X2' i 1 1 X1' i X5' i X2 i X5 i Muttion X1" i 1 X2" i Genetic lgorithms: cse study Suppose it is desired to find the mximum of the pek function of two vriles: f ( x, y) = (1 x) e x ( y 1) 3 3 x y ( x x y ) e where prmeters x nd y vry etween 3 3 nd 3. The first step is to represent the prolem vriles s chromosome prmeters x nd y s conctented inry string: x y We lso choose the size of the chromosome popultion, for instnce 6, nd rndomly generte n initil popultion. The next step is to clculte the fitness of ech chromosome. This is done in two stges. First, chromosome, tht is string of 16 its, is prtitioned into two 8-it 8 strings: nd Then these strings re converted from inry (se 2) to deciml (se 1): ( 111) 2 = = (138) 1 nd ( 11111) 2 = = (59) 1 Now the rnge of integers tht cn e hndled y 8-its, tht is the rnge from to (2 8 1), is mpped to the ctul rnge of prmeters x nd y, tht is the rnge from 3 3 to 3: 6 = To otin the ctul vlues of x nd y,, we multiply their deciml vlues y nd sutrct 3 from the results: x = ( 138) = nd y = ( 59) = Using decoded vlues of x nd y s inputs in the mthemticl function, the GA clcultes the fitness of ech chromosome. To find the mximum of the pek function, we will use crossover with the proility equl to.7 nd muttion with the proility equl to.1. As we mentioned erlier, common prctice in GAs is to specify the numer of genertions. Suppose the desired numer of genertions is 1. Tht is, the GA will crete 1 genertions of 6 chromosomes efore stopping. Chromosome loctions on the surfce of the pek function: initil popultion

6 Chromosome loctions on the surfce of the pek function: first genertion Chromosome loctions on the surfce of the pek function: locl mximum Chromosome loctions on the surfce of the pek function: glol mximum Performnce grphs for 1 genertions of 6 chromosomes: locl mximum pc =.7, pm =.1.7 Performnce grphs for 1 genertions of 6 chromosomes: glol mximum pc =.7, pm = Performnce grphs for 2 genertions of 6 chromosomes pc =.7, pm = F i t n e s s F i t n e s s F i t n e s s Best Averge.4 Best Averge.4 Best Averge G e n e r t i o n s G e n e r t i o n s G e n e r t i o n s

7 Steps in the GA development 1. Specify the prolem, define constrints nd optimum criteri; 2. Represent the prolem domin s chromosome; 3. Define fitness function to evlute the chromosome performnce; 4. Construct the genetic opertors; 5. Run the GA nd tune its prmeters. Evolution Strtegies Another pproch to simulting nturl evolution ws proposed in Germny in the erly 196s. Unlike genetic lgorithms, this pproch clled n evolution strtegy ws designed to solve technicl optimistion prolems. In 1963 two students of the Technicl University of Berlin, Ingo Rechenerg nd Hns-Pul Schwefel,, were working on the serch for the optiml shpes of odies in flow. They decided to try rndom chnges in the prmeters defining the shpe following the exmple of nturl muttion. As result, the evolution strtegy ws orn. Evolution strtegies were developed s n lterntive to the engineer s intuition. Unlike GAs,, evolution strtegies use only muttion opertor. Bsic evolution strtegies In its simplest form, termed s (11) 1)-evolution strtegy, one prent genertes one offspring per genertion y pplying normlly distriuted muttion. The (11) 1)-evolution strtegy cn e implemented s follows: Step 1: 1 Choose the numer of prmeters N to represent the prolem, nd then determine fesile rnge for ech prmeter: { x1 min, x1mx}, { x2 min, x2mx},..., { x Nmin, xnmx} Define stndrd devition for ech prmeter nd the function to e optimised. Step 2: 2 Rndomly select n initil vlue for ech prmeter from the respective fesile rnge. The set of these prmeters will constitute the initil popultion of prent prmeters: x 1, x 2,..., x N Step 3: 3 Clculte the solution ssocited with the prent prmeters: X = f (x 1, x 2,..., x N ) Step 4: 4 Crete new (offspring) prmeter y dding normlly distriuted rndom vrile with men zero nd pre-selected devition δ to ech prent prmeter: xi = xi (, δ ) i = 1, 2,..., N Normlly distriuted muttions with men zero reflect the nturl process of evolution (smller chnges occur more frequently thn lrger ones). Step 5: 5 Clculte the solution ssocited with the offspring prmeters: ( x x ) X = f 1, 2,..., x N

8 Step 6: 6 Compre the solution ssocited with the offspring prmeters with the one ssocited with the prent prmeters. If the solution for the offspring is etter thn tht for the prents, replce the prent popultion with the offspring popultion. Otherwise, keep the prent prmeters. Step 7: 7 Go to Step 4, nd repet the process until stisfctory solution is reched, or specified numer of genertions is considered. An evolution strtegy reflects the nture of chromosome. A single gene my simultneously ffect severl chrcteristics of the living orgnism. On the other hnd, single chrcteristic of n individul my e determined y the simultneous interctions of severl genes. The nturl selection cts on collection of genes, not on single gene in isoltion. Genetic progrmming One of the centrl prolems in computer science is how to mke computers solve prolems without eing explicitly progrmmed to do so. Genetic progrmming offers solution through the evolution of computer progrms y methods of nturl selection. In fct, genetic progrmming is n extension of the conventionl genetic lgorithm, ut the gol of genetic progrmming is not just to evolve it- string representtion of some prolem ut the computer code tht solves the prolem. Genetic progrmming is recent development in the re of evolutionry computtion. It ws gretly stimulted in the 199s y John Koz. According to Koz,, genetic progrmming serches the spce of possile computer lgorithms for progrm tht is highly fit for solving the prolem t hnd. Any computer progrm is sequence of opertions (functions) pplied to vlues (rguments), ut different progrmming lnguges my include different types of sttements nd opertions, nd hve different syntctic restrictions. Since genetic progrmming mnipultes progrms y pplying genetic opertors, progrmming lnguge should permit computer progrm to e mnipulted s dt nd the newly creted dt to e executed s progrm. For these resons, LISP ws chosen s the min lnguge for genetic progrmming. LISP structure LISP hs highly symol-oriented oriented structure. Its sic dt structures re toms nd lists.. An tom is the smllest indivisile element of the LISP syntx. The numer 21,, the symol X nd the string This is string re exmples of LISP toms. A list is n oject composed of toms ndor other lists. LISP lists re written s n ordered collection of items inside pir of prentheses.

9 How do we pply genetic progrmming to prolem? Before pplying genetic progrmming to prolem, we must ccomplish five preprtory steps: 1. Determine the set of terminls. 2. Select the set of primitive functions. 3. Define the fitness function. 4. Decide on the prmeters for controlling the run. 5. Choose the method for designting result of the run. The Pythgoren Theorem helps us to illustrte these preprtory steps nd demonstrte the potentil of genetic progrmming. The theorem sys tht the hypotenuse, c,, of right tringle with short sides nd is given y 2 2 c = The im of genetic progrmming is to discover progrm tht mtches this function. To mesure the performnce of the s-yet yet- undiscovered computer progrm, we will use numer of different fitness cses.. The fitness cses for the Pythgoren Theorem re represented y the smples of right tringles in Tle. These fitness cses re chosen t rndom over rnge of vlues of vriles nd. Side Side Hypotenuse c Side Side Hypotenuse c Step 1: 1 Determine the set of terminls. The terminls correspond to the inputs of the computer progrm to e discovered. Our progrm tkes two inputs, nd. Step 2: 2 Select the set of primitive functions. The functions cn e presented y stndrd rithmetic opertions, stndrd progrmming opertions, stndrd mthemticl functions, logicl functions or domin-specific functions. Our progrm will use four stndrd rithmetic opertions,,, nd,, nd one mthemticl function sqrt. Step 3: 3 Define the fitness function. A fitness function evlutes how well prticulr computer progrm cn solve the prolem. For our prolem, the fitness of the computer progrm cn e mesured y the error etween the ctul result produced y the progrm nd the correct result given y the fitness cse. Typiclly, the error is not mesured over just one fitness cse, ut insted clculted s sum of the solute errors over numer of fitness cses. The closer this sum is to zero, the etter the computer progrm. Step 4: 4 Decide on the prmeters for controlling the run. For controlling run, genetic progrmming uses the sme primry prmeters s those used for GAs.. They include the popultion size nd the mximum numer of genertions to e run. Step 5: 5 Choose the method for designting result of the run. It is common prctice in genetic progrmming to designte the est-so so-fr generted progrm s the result of run.

10 Once these five steps re complete, run cn e mde. The run of genetic progrmming strts with rndom genertion of n initil popultion of computer progrms. Ech progrm is composed of functions,,, nd sqrt,, nd terminls nd. In the initil popultion, ll computer progrms usully hve poor fitness, ut some individuls re more fit thn others. Just s fitter chromosome is more likely to e selected for reproduction, so fitter computer progrm is more likely to survive y copying itself into the next genertion. Crossover in genetic progrmming: Two prentl S-expressionsS sqrt sqrt sqrt ( ( (sqrt ( ( ) ( ))) ) ( )) ( ( (sqrt ( ( ) )) ) (sqrt ( ))) Crossover in genetic progrmming: Two offspring S-expressionsS sqrt sqrt sqrt ( ( (sqrt ( ( ) ( ))) ) (sqrt ( ( ) ))) ( (( ) ) (sqrt ( ))) Muttion in genetic progrmming A muttion opertor cn rndomly chnge ny function or ny terminl in the LISP S-expression. S Under muttion, function cn only e replced y function nd terminl cn only e replced y terminl. Muttion in genetic progrmming: Originl S-expressionsS sqrt sqrt sqrt Muttion in genetic progrmming: Mutted S-expressionsS sqrt sqrt sqrt ( ( (sqrt ( ( ) ( ))) ) ( )) ( ( (sqrt ( ( ) )) ) (sqrt ( ))) ( ( (sqrt ( ( ) ( ))) ) ( )) ( ( (sqrt ( ( ) )) ) (sqrt ( )))

11 In summry, genetic progrmming cretes computer progrms y executing the following steps: Step 1: 1 Assign the mximum numer of genertions to e run nd proilities for cloning, crossover nd muttion. Note tht the sum of the proility of cloning, the proility of crossover nd the proility of muttion must e equl to one. Step 2: 2 Generte n initil popultion of computer progrms of size N y comining rndomly selected functions nd terminls. Step 3: 3 Execute ech computer progrm in the popultion nd clculte its fitness with n pproprite fitness function. Designte the est- so-fr individul s the result of the run. Step 4: 4 With the ssigned proilities, select genetic opertor to perform cloning, crossover or muttion. Step 5: 5 If the cloning opertor is chosen, select one computer progrm from the current popultion of progrms nd copy it into new popultion. If the crossover opertor is chosen, select pir of computer progrms from the current popultion, crete pir of offspring progrms nd plce them into the new popultion. If the muttion opertor is chosen, select one computer progrm from the current popultion, perform muttion nd plce the mutnt into the new popultion. Step 6: 6 Repet Step 4 until the size of the new popultion of computer progrms ecomes equl to the size of the initil popultion, N. Step 7: 7 Replce the current (prent) popultion with the new (offspring) popultion. Step 8: 8 Go to Step 3 nd repet the process until the termintion criterion is stisfied. Fitness history of the est S-expressionS F i t n e s s, % G e n e r t i o n s sqrt B e s t o f g e n e r t i o n Wht re the min dvntges of genetic progrmming compred to genetic lgorithms? Genetic progrmming pplies the sme evolutionry pproch. However, genetic progrmming is no longer reeding it strings tht represent coded solutions ut complete computer progrms tht solve prticulr prolem. The fundmentl difficulty of GAs lies in the prolem representtion, tht is, in the fixed-length coding. A poor representtion limits the power of GA, nd even worse, my led to flse solution.

12 A fixed-length coding is rther rtificil. As it cnnot provide dynmic vriility in length, such coding often cuses considerle redundncy nd reduces the efficiency of genetic serch. In contrst, genetic progrmming uses high-level uilding locks of vrile length. Their size nd complexity cn chnge during reeding. Genetic progrmming works well in lrge numer of different cses nd hs mny potentil pplictions.