THEORY OF GENETIC ALGORITHMS WITH α-selection. André Neubauer

Transcription

1 THEORY OF GENETIC ALGORITHMS WITH α-selection André Neubauer Informaton Processng Systems Lab Münster Unversty of Appled Scences Stegerwaldstraße 39, D Stenfurt, Germany Emal: ABSTRACT Genetc algorthms are random heurstc search (RHS) algorthms for adaptve systems wth a wde range of applcatons n search, optmsaton, pattern recognton and machne learnng as well as sgnal processng. Despte ther wdespread use a general theory s stll lackng. A promsng approach s offered by the dynamcal system model whch descrbes the stochastc trajectory of a populaton under the dynamcs of a genetc algorthm wth the help of an underlyng determnstc heurstc functon and ts fxed ponts. However, even for the smple genetc algorthm (SGA) wth ftnessproportonal selecton, crossover and mutaton the determnaton of the populaton trajectory and the fxed ponts of the heurstc functon s unfeasble for practcal problem szes. In order to smplfy the mathematcal analyss α-selecton s ntroduced n ths paper. Based on ths strong selecton scheme t s possble to derve the dynamcal system model and the respectve fxed ponts n closed form. In addton to the theoretcal analyss expermental results are presented. Index Terms Genetc algorthm, α-selecton, random heurstc search, dynamcal system model. 1. INTRODUCTION Genetc algorthms mmc the basc mechansms of bologcal evoluton and molecular genetcs n smplfed form. These random heurstc search (RHS) algorthms are based on populatons of ndvdual solutons whch evolve accordng to selecton and genetc operators lke crossover and mutaton. Snce they were proposed by HOLLAND [1] genetc algorthms have been used n a wde range of applcatons, e.g. for search as well as numercal and combnatoral optmsaton problems []. Despte ther successful applcatons genetc algorthms stll lack a general and strngent theory whch gudes the practtoner n talorng the algorthm to a specfc applcaton doman [3]. The frst attempt to theoretcally descrbe the smple genetc algorthm (SGA) was the so-called schema theorem [1] whch gves a lower bound for the expected number of ndvduals representng partcular hyperplanes or schemata n the search space wthn the next generaton. Recently, the schema theory and ts mplcatons lke the so-called buldng block-hypothess have been crtcsed for lack of explanatory power [3]. As an alternatve, a dynamcal system model was ntroduced for the SGA by VOSE and hs co-workers [4, 5, 7, 8]. In ths model the SGA s stochastc dynamcs s modeled accordng to an underlyng determnstc heurstc functon whch descrbes the expected populaton trajectory of the SGA. Accordng to theoretcal and emprcal evdence ths populaton trajectory s attracted by the fxed ponts of the heurstc functon. The determnaton of these fxed ponts, however, becomes unfeasble for practcal problem szes. In order to smplfy the mathematcal analyss α- selecton s ntroduced n ths paper. The paper s organsed as follows. After a descrpton of the SGA over search spaces of bnary l-tuples n secton, the dynamcal system model s summarsed n secton 3. The underlyng heurstc functon s formulated for the SGA wth ftness-proportonal selecton, 1-pont or unform crossover and btwse mutaton. The genetc algorthm wth α-selecton s ntroduced n secton 4 based on the noton of the best ndvdual randomly matng wth all other ndvduals n the current populaton. The dynamcal system model of a genetc algorthm wth α-selecton s derved n secton 5. The fxed ponts of the correspondng heurstc functon are calculated analytcally and compared to smulaton results n secton 6 showng a close agreement between theory and experment. A bref summary and concluson s gven n secton 7.. SIMPLE GENETIC ALGORITHM In ths secton, the SGA wth ftness-proportonal selecton, 1-pont or unform crossover and btwse mutaton [7] s descrbed whch s used for the maxmsaton of a ftness functon f. Ths ftness functon f s defned over the search space Ω=Z l = {0, 1} l consstng of bnary l-tuples a = (a 0,a 1,...,a l 1 ),.e. f :Ω R..1. Algorthm The SGA works over populatons defned as multsets P (t) = (a 0 (t),a 1 (t),...,a r 1 (t)) consstng of r ndvdual bnary 137

2 l-tuples a k (t) =(a k,0 (t),a k,1 (t),...,a k,l 1 (t)) Ω wth ftness values f (a k (t)). For the creaton of offsprng ndvduals n each generaton t random genetc operators lke crossover χ Ω and mutaton μ Ω are appled to parental ndvduals whch are selected accordng to ther ftness values as shown n Fg. 1. The populaton P (0) s ntalsed approprately, e.g. by randomly choosng ndvduals n Ω. t := 0; whle of adaptaton true do for k =0to r 1 do select parental l-tuples b(t) and c(t); apply crossover χ Ω and mutaton μ Ω a k (t +1):=μ Ω (χ Ω (b(t),c(t))); evaluate ftness f (a k (t + 1)); t := t +1;.. Selecton Fg. 1. Smple genetc algorthm (SGA). For ftness-proportonal selecton each ndvdual a k (t) n the populaton P (t) s selected wth a probablty p k (t) whch s drectly proportonal to ts ftness f (a k (t)) 0. The ndvdual selecton probablty s then gven by.3. Crossover p k (t) = f (a k (t)) r 1 j=0 f (a j(t)). The crossover operator χ Ω : Ω Ω Ω takes two selected ndvdual l-tuples a = (a 0,a 1,...,a l 1 ) and b = (b 0,b 1,...,b l 1 ) and randomly generates an offsprng l-tuple c =(c 0,c 1,...,c l 1 ) accordng to c = χ Ω (a, b). For 1-pont crossover a poston 1 λ l 1 s chosen wth unform probablty and the two l-tuples (a 0,...,a λ 1, b λ,...,b l 1 ) and (b 0,...,b λ 1,a λ,...,a l 1 ) are generated one of whch s randomly chosen as offsprng c. The 1-pont crossover operator s appled wth crossover probablty χ. In case of unform crossover each bt of the offsprng c s chosen wth probablty χ from the parental l-tuple a and wth probablty 1 χ from the parental l-tuple b or vce versa..4. Mutaton In the SGA wth bnary l-tuples the btwse mutaton operator μ Ω :Ω Ω s defned by randomly flppng each bt of the ndvdual l-tuple a =(a 0,a 1,...,a l 1 ) wth small mutaton probablty μ. A typcal value s μ 1 l. 3. DYNAMICAL SYSTEM MODEL FOR THE SIMPLE GENETIC ALGORITHM In vew of the search space Ω of bnary l-tuples we dentfy each bnary l-tuple (a 0,a 1,...,a l 1 ) wth the nteger number a = a 0 l 1 + a 1 l a l 1 + a l 1 0. The respectve ftness s gven by f a = f(a). Ths leads to the search space Ω={0, 1,...,n 1} wth cardnalty Ω = n = l. Based on ths bnary number representaton we defne the btwse modulo- addton a b, btwse modulo- multplcaton a b and btwse bnary complement a, e.g. 3 5 = = 110 = 6, 3 5 = = 001 = 1 and 3=011 = 100 = 4 for l-tuples of length l =3. Vce versa, the nteger a Ω s also vewed as a column vector (a 0,a 1,...,a l 1 ) T so that we can defne the scalar product a T b = a 0 b 0 + a 1 b a l b l + a l 1 b l 1.In the followng we wll also make use of the all-one l-tuple 1 whch corresponds to the nteger n 1 = l 1. Furthermore, the ndcator functon [ = j] =δ,j s defned by [ = j] =1 f = j and 0 f j. In the dynamcal system model [7] the SGA s dynamcs s compactly formulated by defnng the populaton vector p = j P [j = (p 0,p 1,...,p n 1 ) T. Each component p = 1 r ] gves the proporton of the element Ω n the current populaton P. The populaton vector p s an element of the smplex Λ={p R n : p 0 Ω p =1}. For a populaton of sze r the number of possble populaton vectors s gven by ( ) n+r 1 r. In the lmt of nfnte populatons wth r the populaton vectors are dense n the smplex Λ. For smplcty we wll take the smplex Λ as the defnng regon of the populaton vector p. Therefore, the results obtaned are strctly vald only for large populatons wth r 1. The SGA s now descrbed as an nstance of RHS τ :Λ Λ accordng to p(t +1) = τ (p(t)) wth τ depng on the random selecton and genetc operators. As outlned n [7] τ can be equvalently represented by a sutable heurstc functon G : Λ Λ whch for a gven populaton vector p yelds the probablty dstrbuton G (p). Ths probablty dstrbuton G (p) = Pr{ndvdual s sampled from Ω} s sampled to generate the next populaton as llustrated n Fg.. For the SGA the heurstc functon G = M F s gven by the composton of the selecton operaton F and the mxng operaton M comprsng crossover χ Ω and mutaton μ Ω. The transton probabltes of the RHS τ are gven by the formula Pr{τ (p) =q} = r! Ω G (p) rq (rq )! The trajectory p, τ (p), τ (p),... approxmately follows the trajectory p, G (p), G (p),... of the determnstc dynamcal system defned by the heurstc functon G wth E{τ (p)} = G (p). Because of the mean quadratc devaton E{ τ (p) G(p) } = 1 r (1 G(p) ) the RHS τ behaves lke the determnstc dynamcal system model n the lmt of. 138

3 G p(0) τ p(1) τ p()... G (p(0)) G G (p(1)) sample n Ω Fg.. Genetc algorthm as RHS τ wth heurstc functon G. nfnte populatons wth r. Accordng to expermental evdence the heurstc functon G shows punctuated equlbra,.e. phases of relatve stablty nearby the fxed ponts ω = G (ω) of the heurstc functon G dsrupted by sudden transtons to another dynamcal equlbrum near another fxed pont. We call ths the fxed pont hypothess of genetc algorthms Selecton For ftness-proportonal selecton the correspondng heurstc functon F (p) = Pr{ndvdual selected} s gven by [7] f p F (p) j Ω f = f p j p j f T p wth ftness vector f =(f 0,f 1,...,f n 1 ) T and ndvdual ftness values f Crossover For the crossover operator χ Ω appled to the ndvdual l- tuples a and b wth crossover probablty χ we make use of a random crossover mask m Ω accordng to χ Ω (a, b) = a m m b or χ Ω (a, b) =a m m b each wth probablty 1. For 1-pont crossover the crossover mask m s randomly chosen from Ω accordng to the probablty dstrbuton [7] χ m = 1 χ, m =0 χ/(l 1), m = λ 1 wth 1 λ l 1 0, otherwse In case of unform crossover the probablty dstrbuton s gven by [7] { 1 χ + χ χ m = l, m =0 χ l., m > Mutaton The mutaton operator μ Ω can be descrbed wth the help of the random mutaton mask m Ω accordng to μ Ω (a) = a m. The mutaton mask m s randomly chosen from Ω accordng to the probablty dstrbuton [7] μ m = μ 1Tm (1 μ) l 1Tm Heurstc The mxng operaton M comprses crossover χ Ω and mutaton μ Ω. Wth the help of the probablty dstrbutons for crossover and mutaton masks we obtan Pr{c = μ Ω (χ Ω (a, b))} j Ω μ j Pr{χ Ω (a, b) =c j} j Ω μ j Ω χ + χ [a b = c j]. Ths yelds Pr{c = μ Ω (χ Ω (a, b))} = M a c,b c wth the help of the n n mxng matrx M,j μ v χu + χ u [ u u j = v]. The heurstc functon G follows accordng to G (p) = Pr{ndvdual s sampled from Ω} F (p) u F(p) v Pr{μ Ω (χ Ω (u, v)) = } F (p) u F(p) v M u,v F (p) u F(p) v M u,v. By defnng the mxng operaton M :Λ Λ accordng to M (p) p u p v M u,v the heurstc functon s gven by the composton G = M F. In order to determne the heurstc functon G the mxng matrx M has to be calculated. Ths can be done effcently wth the help of the WALSH transform [6]. For a matrx A the WALSH transform s  = W A W wth the orthogonal n n WALSH matrx W,j = n 1/ ( 1) Tj. Correspondngly, the WALSH transform of a vector x yelds x = W x. In Fg. 3 the WALSH matrx s llustrated for n = 6 =64. Wth the WALSH transform of the mutaton mask dstrbuton μ = n 1/ (1 μ) 1T the WALSH transform of the mxng matrx M s gven by M,j = n [ j =0] μ j k Ω j (χ k + χ k j ) wth Ω k = { Ω: k =0}. Although the mxng matrx M can be calculated effcently (for 1-pont crossover a closed-form expresson for the WALSH transform M s known [7]) the fxed pont analyss of the heurstc functon G becomes nfeasble even for moderate l due to ts depence on the populaton vector p va F (p) and the cardnalty n = l of the search space Ω. Therefore, n the next secton we ntroduce the genetc algorthm wth α-selecton whch yelds a smpler heurstc functon and even allows to derve a closedform soluton for the fxed ponts. 139

4 5.1. Heurstc The α-ndvdual b s selected as the frst parent for creaton of a new offsprng, whereas the second parent s chosen unformly at random from the current populaton accordng to the probablty dstrbuton Pr{ndvdual j s selected} = p j. The heurstc functon G s then gven by G (p) j Ω p j Pr{μ Ω (χ Ω (b, j)) = } j Ω p j M b,j. Wth the n n system matrx A,j = M b,j Fg. 3. Illustraton of the WALSH matrx W for n = 6 = GENETIC ALGORITHM WITH α-selection Let the current populaton be descrbed by the populaton vector p and let b = argmax {f : Ω p > 0} be the best ndvdual n the current populaton. Clearly, ths best ndvdual or α-ndvdual b deps on the populaton vector p (for ease of notaton we omt ths depence n the followng). In the genetc algorthm wth α-selecton the α- ndvdual b s mated wth ndvduals randomly chosen from the current populaton wth unform probablty. The genetc algorthm wth α-selecton s shown n Fg. 4. t := 0; whle of adaptaton true do select α-ndvdual b(t) as frst parent; for k =0to r 1 do select second parent c(t) randomly; apply crossover χ Ω and mutaton μ Ω a k (t +1):=μ Ω (χ Ω (b(t),c(t))); evaluate ftness f (a k (t + 1)); t := t +1; Fg. 4. Genetc algorthm wth α-selecton. 5. DYNAMICAL SYSTEM MODEL FOR THE GENETIC ALGORITHM WITH α-selection In the followng we determne the dynamcal system model for the genetc algorthm wth α-selecton by dervng the heurstc functon G (p) for a gven populaton vector p. we obtan the lnear system of equatons for the new populaton vector q = G (p) =A p. Wth the permutaton matrx (σ b ),j =[ j = b] and the twst (M ),j = M j, of the symmetrc mxng matrx M = M T the system matrx A can also be expressed as A = σ b M σ b. Due to the α-selecton scheme we gan a smpler heurstc functon G whch s completely descrbed by the α-ndvdual b and the mxng matrx M or system matrx A, respectvely. 5.. Fxed Ponts Because of the lnear relatonshp G (p) =A p n case of a gven α-ndvdual b the fxed ponts ω = G (ω) of the heurstc functon G are obtaned from the egenvectors of the system matrx A to egenvalue λ =1,.e. A ω = ω. To ths, we consder the WALSH transform of both sdes of the equaton q = A p accordng to q = W q = W A p = W A W W p =  p where we have made use of the orthogonalty of the WALSH matrx W. For an egenvector v wth egenvalue λ we obtan A v = λ v  v = λ v. The system matrx A and ts WALSH transform  have the same egenvalues wth egenvectors whch are also related by the WALSH transform. It s an easy but lengthy task to calculate the WALSH transform of the system matrx A,j = M b,j leadng to Â,j =( 1) bt ( j) M j,j. The system matrx A as well as ts WALSH transform  dep on the α-ndvdual b. For crossover χ Ω and mutaton μ Ω defned by masks t can be shown that the WALSH transform  s a lower trangular matrx wth egenvalues λ gven by the dagonal elements λ = Â, = M 0,. Ths yelds the egenvalues λ = (1 μ)1t k Ω (χ k + χ k ) 140

5 of the system matrx A. It s a remarkable fact that the egenvalues do not dep on the α-ndvdual b. Takng nto account Ω χ =1and μ< 1 t s evdent from ths expresson that the egenvalues are bounded by 0 λ 1. The system matrx A and ts WALSH transform  are therefore postve semdefnte. Because of λ 0 =1and 0 λ 1 μ for 1 n 1 there exsts a sngle egenvector ω whch s a fxed pont of the heurstc functon ω = G (ω) =A ω. Both the system matrx A and the fxed pont ω dep on the α-ndvdual b. Accordng to the fxed pont hypothess the populaton wll stay near ths fxed pont ω. If a new best or α- ndvdual b Ω s found the genetc algorthm wll converge to the new fxed pont thus showng punctuated equlbra,.e. phases of relatve stablty nearby a fxed pont dsrupted by sudden transtons to another fxed pont. The fxed pont ω can be determned explctly wth the help of the WALSH transform. Frst, we note that due to ω =  ω and the lower trangular matrx Â,j =( 1) bt ( j) M j,j we can recursvely calculate the WALSH transform of the fxed pont accordng to 1 1 ω = Â,j ω j 1 Â, j=0 for 1 n 1 startng wth ω 0 = n 1/ whch ensures that Ω ω =1. The fxed pont s then obtaned va the nverse WALSH transform ω = W ω. 6. EXPERIMENTAL RESULTS For the shfted and nverted ONEMAX problem wth ftness functon f = 1 T ( 0 ) wth 0 = 18 (correspondng to the number of 0 s n the bnary representaton of 0 Ω) Fg. 5 shows the theoretcal and estmated populaton vectors p n generaton t =5. Here, we have assumed a genetc algorthm wth α-selecton for l =8and n = 8 = 56 usng 1-pont crossover wth crossover probablty χ =1, btwse mutaton wth mutaton probablty μ = 1 l and populaton sze r = 100. The EUCLIDean dstance of the smulated populaton vector p and the fxed pont ω s There s a close match between the theoretcal predcton and the expermental result. 7. CONCLUSION The dynamcal system model represents the current state of the art of genetc algorthm theory [3, 7]. For practcal problem szes, however, the determnaton of the populaton trajectory and the fxed ponts of the underlyng heurstc functon G becomes unfeasble. In ths paper, α-selecton has been ntroduced n order to smplfy the mathematcal analyss and to enable the dervaton of the dynamcal system model and the fxed ponts of the heurstc functon n closed form. Expermental results show close agreement to the theoretcal p p Fg. 5. Populaton vector accordng to the dynamcal system model wth heurstc functon G (top) and ts estmate from a sngle run of the genetc algorthm wth α-selecton (bottom). predctons. The dynamcal system model for the genetc algorthm wth α-selecton deps on the α-ndvdual b. Ifn the next generaton ths ndvdual s lost or a better ndvdual s sampled from the search space Ω the heurstc functon G changes due to the depence of the system matrx A on the α-ndvdual b, thus showng punctuated equlbra. Snce α- selecton s a strong selecton scheme future research wll be orented towards genetc algorthms wth weaker selecton. 8. REFERENCES [1] HOLLAND, J.H.: Adaptaton n Natural and Artfcal Systems An Introductory Analyss wth Applcatons to Bology, Control, and Artfcal Intellgence. Cambrdge: Frst MIT Press Edton, 199 [] MITCHELL, M.: An Introducton to Genetc Algorthms. Cambrdge: MIT Press, 1996 [3] REEVES, C.R.; ROWE, J.E.: Genetc Algorthms Prncples and Perspectves, A Gude to GA Theory. Boston: Kluwer Academc Publshers, 003 [4] VOSE, M.D.; WRIGHT, A.H.: Smple Genetc Algorthms wth Lnear Ftness. In: Evolutonary Computaton. Vol., No. 4, pp , 1994 [5] VOSE, M.D.: Modelng Smple Genetc Algorthms. In: Evolutonary Computaton. Vol. 3, No. 4, pp , 1995 [6] VOSE, M.D.; WRIGHT, A.H.: The Smple Genetc Algorthm and the Walsh Transform Part I, Theory. In: Evolutonary Computaton. Vol. 6, No. 3, pp , 1998 [7] VOSE, M.D.: The Smple Genetc Algorthm Foundatons and Theory. Cambrdge: MIT Press, 1999 [8] VOSE, M.D.: Random Heurstc Search. In: Theoretcal Computer Scence. Vol. 9, No. 1-, pp ,