Average Convergence Rate of Evolutionary Algorithms

Transcription

1 1 Average Covergece Rate of Evolutioary Algorithms Ju He ad Guagmig Li Abstract arxiv: v3 [cs.ne] 2 Ju 215 I evolutioary optimizatio, it is importat to uderstad how fast evolutioary algorithms coverge to the optimum per geeratio, or their covergece rates. This paper proposes a ew measure of the covergece rate, called the average covergece rate. It is a ormalized geometric mea of the reductio ratio of the fitess differece per geeratio. The calculatio of the average covergece rate is very simple ad it is applicable for most evolutioary algorithms o both cotiuous ad discrete optimizatio. A theoretical study of the average covergece rate is coducted for discrete optimizatio. Lower bouds o the average covergece rate are derived. The it of the average covergece rate is aalyzed ad the the asymptotic average covergece rate is proposed. Idex Terms evolutioary algorithms, evolutioary optimizatio, covergece rate, Markov chai, matrix aalysis I. INTRODUCTION Evolutioary algorithms (EAs) belog to iterative methods. As iterative methods, a fudametal questio is their covergece rates: how fast does a EA coverge to the optimum per geeratio? Accordig to [1], existig results o the covergece rate of geetic algorithms ca be classified ito two categories. The first category is related to the eigevalues of the trasitio matrix associated with a EA. A lower boud of covergece rate is derived i [2] for simple geetic algorithms by aalyzig eigevalues of the trasitio matrix. The the work is exteded i [3] ad it is foud that the covergece rate is determied by the secod largest eigevalue of the trasitio matrix. The other category is based o Doebli s coditio. The upper boud o the covergece rate is derived usig Deobli s coditio i [4]. As to cotiuous optimizatio, the local covergece rate of EAs o the sphere fuctio, quadratic covex fuctios ad covex objective fuctios are discussed i [5] [7]. The research of the covergece rate covers various types of EAs such as isotropic algorithms [8], gee expressio programmig [9], multiobjective optimizatio EAs [1]. The relatioship betwee the covergece rate ad populatio size is ivestigated i [11], [12]. The covergece rate i previous studies [1] [4] is based o Markov chai theory. Suppose that a EA is modeled by a fiite Markov chai with a trasitio matrix P, i which a state is a populatio [13]. Let p t be the probability distributio of the tth geeratio populatio o a populatio space, π a ivariat probability distributio of P. The p t is called coverget to π if t p t π = where is a orm; ad the covergece rate refers to the order of how fast p t coverges to π [4]. The goal is to obtai a boud ǫ(t) such that p t π ǫ(t). But to obtai a closed form of ǫ(t) ofte is difficult i both theory ad practice. The curret paper aims to seek a covergece rate satisfyig two requiremets: it is easy to calculate the covergece rate i practice while it is possible to make a rigorous aalysis i theory. Ispired from covetioal iterative methods [14], a ew measure of the covergece rate, called the average covergece rate, is preseted. The paper is orgaized as follows: Sectio II defies the average covergece rate. Sectio III establishes lower bouds o the average covergece rate. Sectio IV discusses the coectios betwee the average covergece rate ad other performace measures. Sectio V itroduces a alterative defiitio of the average covergece rate if the optimal fitess value is ukow. Sectio VI cocludes the paper. II. DEFINITION AND CALCULATION Cosider the problem of miimizig (or maximizig) a fuctio f(x). A EA for solvig the problem is regarded as a iterative procedure (Algorithm 1): iitially costruct a populatio of solutios Φ ; the geerate a sequece of populatios Φ 1, Φ 2, Φ 3 ad so o. This procedure is repeated util a stoppig criterio is satisfied. A archive is used for recordig the best foud solutio. The fitess of populatio Φ t is defied by the best fitess value amog its idividuals, deoted by f(φ t ). Sice f(φ t ) is a radom variable, we cosider its mea value f t := E[f(Φ t )]. Let f opt deote the optimal fitess. The fitess differece This work was supported by EPSRC uder Grat No. EP/I989/1 (He), Natioal Sciece Foudatio of Guagdog Provice uder Grat No. S ad Shezhe Scietific Research Project uder Grat No. JCYJ (Li). Ju He is with Departmet of Computer Sciece, Aberystwyth Uiversity, Aberystwyth SY23 3DB, U.K. jqh@aber.ac.uk. Guagmig Li is with Shezhe Istitute of Iformatio Techology, Chia. ligm@sziit.com.c.

2 2 Algorithm 1 A EA with a archive 1: iitialize a populatio of solutios Φ ad set t ; 2: a archive records the best solutio i Φ ; 3: while the archive does t iclude a optimal solutio do 4: geerate a ew populatio of solutios Φ t+1 ; 5: update the archive if a better solutio is geerated; 6: t t+1; 7: ed while betwee f opt ad f t is. The covergece rate for oe geeratio is 1. (1) Sice 1, calculatig the above ratio is ustable i practice. Thus a ew average covergece rate for EAs is proposed i the curret paper. Defiitio 1: Give a iitial populatio Φ, the average (geometric) covergece rate of a EA for t geeratios is ( f opt f 1 R(t Φ ) := 1 ( 1 ) 1 ). (2) If f = f opt, let R(t Φ ) = 1. For the sake of simplicity, R(t) is short for R(t Φ ). The rate represets a ormalized geometric mea of the reductio ratio of the fitess differece per geeratio. The larger the covergece rate, the faster the covergece. The rate takes the maximal value of 1 at f t = f opt. Ispired from covetioal iterative methods [14, Defiitio 3.1], the average (logarithmic) covergece rate is defied as follows: R (t) := 1 t log. (3) Formula (3) is ot adopted sice its value is + at f t = f opt. But i most cases, average geometric ad logarithmic covergece rates are almost the same. Let α t := f opt f t / f opt f t 1. Usually α t 1 ad (α 1 α t ) 1, the log(α 1 α t ) (1 (α 1 α t ) ) 1. I practice, the average covergece rate is calculated as follows: give f(x) with f opt kow i advace, 1) Ru a EA for T times (T 1). 2) The calculate the mea fitess value f t as follows, 1 T ( f(φ [1] t )+ +f(φ [T] t ) ), (4) where f(φ [k] t ) deotes the fitess f(φ t ) at the kth ru. The law of large umbers guaratees (4) approximatig to the mea fitess value f t = E(f(Φ t )) whe T teds towards +. 3) Fially, calculate R(t) accordig to formula (2). The calculatio is applicable for most EAs o both cotiuous ad discrete optimizatio. We take a example to illustrate the average covergece rate. Cosider the problem of miimizig Ackley s fuctio: f(x) = 2exp{.2[ (x i +e) 2 /] 1 2 } exp[ cos(2πx i +2πe)/]+2+e, (5) i=1 where x i [ 32 e,32 e],i = 1,,. The optimum is ( e, e, ) ad f opt =. We compare the Multi-grid EA (MEA) [15] with the Fast Evolutioary Programmig (FEP) [16] uder the same experimet settig (where is 3 ad populatio size is 1). Ru the two EAs for 15 geeratios ad 1 times. Calculate f t accordig to (4) ad the R(t) accordig to (2). Fig. 1 illustrates the covergece rates of MEA ad FEP. The average covergece rate is differet from the progress rate such as f t f opt or logarithmic rate log f t f opt used i [17]. The progress rate measures the fitess chage; but the covergece rate measures the rate of the fitess chage. We demostrate this differece by a example. Let g(x) = 1f(x). I terms of f t f opt, the progress rate o g(x) is 1 times that o f(x). I terms of log f t f opt, the progress rate o g(x) is 1+2/log f t f opt times that o f(x). However, the average covergece rate is the same o both f(x) ad g(x). i=1

3 MEA FEP , 1,5 t Fig. 1. A compariso of the average covergece rates R(t) of MEA ad FEP o Ackley s fuctio. III. ANALYSIS FOR DISCRETE OPTIMIZATION Lookig at Fig. 1 agai, two questios may be raised: what is the lower boud or upper boud o R(t)? Does R(t) coverge or ot? For discrete optimizatio, a theoretical aswer is provided to these questios i this sectio. For cotiuous optimizatio, its aalysis is left for future research. I the rest of the paper, we aalyze EAs for discrete optimizatio ad assume that their geetic operators do ot chage with time. Such a EA ca be modeled by a homogeeous Markov chai [13] with trasitio probabilities Pr(X,Y) := Pr(Φ t+1 = Y Φ t = X),X,Y S, where populatios X,Y deote states of Φ t ad S deotes the set of populatios (called the populatio space). Let P deote the trasitio matrix with etries Pr(X, Y). A populatio is called optimal if it icludes a optimal solutio; otherwise called o-optimal. Let S opt deote the set of optimal populatios, ad S o = S \S opt. Because of the stoppig criterio, the optimal set is always absorbig, Trasitio matrix P ca be split ito four parts: Pr(Φ t+1 S o Φ t S opt ) =. (6) P = ( S opt S o ) S opt A O S o B Q (7) where A is a submatrix represetig probability trasitios amog optimal states; O a submatrix for probability trasitios from optimal states to o-optimal oes, of which all etries take the value of zero; B a submatrix deotig probability trasitios from o-optimal states to optimal oes; ad Q a submatrix for probability trasitios amog o-optimal states. Sice Φ t is a radom variable, we ivestigate the probability distributio of Φ t istead of Φ t itself. Let q t (X) deote the probability of Φ t at a o-optimal state X, q t (X) := Pr(Φ t = X). Let vector (X 1,X 2, ) represet all o-optimal states ad vector q T t deote the probability distributio of Φ t i the o-optimal set, where q t := (q t (X 1 ),q t (X 2 ), ) T. Here otatio q is a colum vector ad q T the row colum with the traspose operatio. For the iitial probability distributio, q where = (,, ) T. Oly whe the iitial populatio is chose from the optimal set, q =. Cosider probability trasitios amog o-optimal states oly, which ca be represeted by matrix iteratio q T t = q T t 1Q = q T Q t. (8) Defiitio 2: A EA is called coverget if q t = for ay q or Q t = O. It is equivalet to sayig that the probability of fidig a optimal solutio is 1 as t teds towards +. The mea fitess value f t is give as follows: f t := E[f(Φ t )] = X Sf(X)Pr(Φ t = X). (9) The it follows = X S o (f(x) f opt )q t (X). (1) Let vector f := (f(x 1 ),f(x 2 ), ) T deote the fitess values of all o-optimal populatios (X 1,X 2, ). The (1) ca be rewritte i a vector form where deotes the vector product ad 1 = (1,1, ) T. = q T t (f opt1 f), (11)

4 4 For a vector v, deote v T := v T (f opt 1 f). (12) Sice v = iff v = ; av = a v ad v 1 +v 2 v 1 + v 2, thus v is a vector orm. For a matrix M, let M be the iduced matrix orm, give by { v T } M M = sup v T : v. (13) Usig the above Markov chai model, we are able to estimate lower bouds o the average covergece rate. Theorem 1: Let Q be the trasitio submatrix associated with a coverget EA. For ay q, 1) The average covergece rate for t iteratios is lower-bouded by 2) The it of the average covergece rate for t geeratios is lower-bouded by R(t) 1 Q t. (14) R(t) 1 ρ(q), (15) where ρ(q) is the spectral radius (i.e., the supremum amog the absolute values of all eigevalues of Q). 3) Uder radom iitializatio (that is, Pr(Φ = X) > for ay X S o or q > ), it holds R(t) = 1 ρ(q). (16) 4) Uder particular iitializatio (that is, set 1 q = v/ v where v is a eigevector correspodig to the eigevalue ρ(q) with v but v. The existece of such a v is give i the proof), it holds for all t 1, Proof: 1) From (8): q T t = q T Qt, we have Hece R(t) = 1 ρ(q). (17) = qt t q T = qt Q t q T qt Q t q T = Q t. (18) 1 which proves the first coclusio. 2) Accordig to Gelfad s spectral radius formula [18, p.619], we get 1 Q t, (19) Qt = ρ(q). (2) The secod coclusio follows by combiig (2) with (14). 3) Sice Q, accordig to the extesio of Perro-Frobeius theorems to o-egative matrices [18, pp. 67], ρ(q) is a eigevalue of Q. There exists a eigevector v correspodig to ρ(q) such that v but v. I particular, ρ(q)v T = v T Q. (21) Let max(v) deote the maximum value of the etries i vector v. Due to radom iitializatio, q >. Let mi(q ) deote the miimum value of the etries i vector q. Set From (21), we get u = mi(q ) v. (22) max(v) ρ(q)u T = u T Q. (23) Thus vector u is a eigevector of ρ(q). Let w = q u. The from (22), we kow w. Sice q = u+w, w ad Q, we deduce that 1 For vector v = (v 1,v 2, ), deote v := i v i. q T t = qt Qt = (u+w) T Q t u T Q t = ρ(q) t u T. (24)

5 5 It follows that = q T t (f opt1 f) ρ(q) t u T (f opt 1 f) q T (f opt1 f). (25) ρ(q) u T (f opt 1 f) q T (f opt1 f). (26) Sice both u T (f opt 1 f) ad q T (f opt 1 f) are idepedet of t, we let t + ad get u T (f opt 1 f) q T (f = 1, (27) opt1 f) the we get R(t) = 1 ρ(q). (28) 1 ρ(q). (29) The third coclusio follows by combiig (29) with (15). 4) Set q = v/ i v i where v is give i Step 3. The q is a eigevector correspodig to the eigevalue ρ(q) such that ρ(q)q T = qt Q. From (8): qt t = q T t 1Q, we get Thus we have for ay t 1 = qt t (f opt 1 f) = ρ(q)t q T (f opt 1 f) q T (f. opt1 f) = ρ(q), (3) the R(t) = 1 ρ(q) which gives the fourth coclusio. The above theorem provides lower bouds o the average covergece rate. Furthermore, it reveals that R(t) coverges to 1 ρ(q) uder radom iitializatio ad R(t) = 1 ρ(q) for ay t 1 uder particular iitializatio. Similar to covetioal iterative methods [14, pp. 73], we call 1 ρ(q) the asymptotic average covergece rate of a EA, deoted by R. Accordig to (16), its value ca be approximately calculated as follows: uder radom iitializatio, R(t) approximates to 1 ρ(q) if t is sufficietly large. Note that this defiitio is differet from aother asymptotic covergece rate, give by log ρ(q) i [19]. I most cases, the two rates are almost the same sice usually ρ(q) 1 ad the logρ(q) (1 ρ(q)) 1. Sice 1 ρ(q) is idepedet of t ad iitializatio, hece usig asymptotic average covergece rate is coveiet for comparig two EAs, for example, to aalyze mixed strategy EAs [19]. IV. CONNECTIONS The average covergece rate is differet from other performace measures of EAs: the expected hittig time is the total umber of geeratios for obtaiig a optimal solutio [13]; ad fixed budget aalysis focuses o the performace of EAs withi fixed budget computatio [2]. However, there are some iterestig coectios betwee them. There exists a lik betwee the asymptotic average covergece rate ad the hittig time. Let m(x) be the expected umber of geeratios for a coverget EA to hit S opt whe startig from state X (called the expected hittig time). Deote m := (m(x 1 ),m(x 2 ), ) T where (X 1,X 2, ) represet all o-optimal states. Theorem 2: Let Q be the trasitio submatrix associated with a coverget EA. The 1/R is ot more tha m := max{m(x);x S o }. Proof: Accordig to the fudametal matrix theorem [21, Theorem 11.5], m = (I Q) 1 1, where I is the uit matrix. The m = (I Q) 1 1 = (I Q) 1 ρ((i Q) 1 ) = (1 ρ(q)) 1, (31) where the last equality takes use of a fact: (1 ρ(q)) 1 is a eigevalue ad spectral radius of (I Q) 1. The above theorem shows that 1/R is a lower boud o the expected hittig time. Followig Theorem 1, a straightforward coectio ca be established betwee the spectral radius ρ(q) ad the progress rate. Corollary 1: Let Q be the trasitio submatrix associated with a coverget EA.

6 6 1) Uder radom iitializatio (that is q > ), it holds ρ(q) t = 1. (32) 2) Uder particular iitializatio (that is, set q = v/ v where v is a eigevector correspodig to the eigevalue ρ(q) with v but v ), it holds for all t 1, ρ(q) t = 1. (33) The expoetial decay, ρ(q) t, provides a theoretical predictio for the tred of. The corollary cofirms that the spectral radius ρ(q) plays a importat role o the covergece rate [3]. We explai the theoretical results by a simple example. Cosider a (1+1) EA for maximizig the OeMax fuctio x where x = (s 1,,s ) {,1,}. Algorithm 2 A (1+1) elitits EA Oebit Mutatio: choose a bit of Φ t (oe idividual) uiformly at radom ad flip it. Let Ψ t deote the child. Elitist Selectio: if f(ψ t ) > f(φ t ), the let Φ t+1 Ψ t ; otherwise Φ t+1 Φ t. Deote subset S k := {x : x = k} where k =,,. Trasitio probabilities satisfy that Pr(Φ t+1 S k 1 Φ t S k ) = k ad Pr(Φ t+1 S k Φ t S k ) = 1 k. Writig them i matrix P (where submatrix Q i the bold fot): (34) The spectral radius ρ(q) = 1 1 ad the asymptotic average covergece rate R = 1. Notice that 1/R (= ) is less tha the expected hittig time (= ( )). I the OeMax fuctio, set = 1, ad the ρ(q) =.9 ad R =.1. Choose Φ uiformly at radom, ru the (1+1) EA for 5 geeratios ad 2 times, ad the calculate f t accordig to (4) ad R(t) accordig to formula (2). Sice Φ is chose uiformly at radom, f 5. Fig. 2 demostrates that R(t) approximates.1(= R ). Fig. 3 shows that the theoretical expoetial decay, ρ(q) t, ad the computatioal progress rate,, coicide perfectly..2 R(t) t Fig. 2. R(t) approximates.1 for the (1+1) EA o the OeMax fuctio with = 1. V. ALTERNATIVE RATE So far the calculatio of the average covergece rate eeds the iformatio about f opt. However this requiremet is very strog. Here we itroduce a alterative average covergece rate without kowig f opt, which is give as below, R (t) := 1 f t+δt f t f t f t δt, (35) where δt is a appropriate time iterval. Its value relies o a EA ad a problem.

7 7 6 4 ρ(q) t t Fig. 3. A compariso of the theoretical predictio ρ(q) t ad the computatioal result f opt f t for the (1+1) EA o the OeMax fuctio with = 1, f = 5, f opt = 1 ad ρ(q) =.9. For the (1+1) EA o the OeMax fuctio with = 1, we set δt = 1. Choose Φ uiformly at radom, ru the (1+1) EA for 6 geeratios ad 2 times, ad the calculate f t accordig to (4) ad R (t) accordig to formula (35). Due to δt = 1, R (t) has o value for t < 1 ad t > 5 accordig to formula (35). Fig. 4 demostrates that R (t) approximates.1(= 1 ρ(q)). But the calculatio of R (t) is ot as stable as that of R(t) i practice..2 R (t) t Fig. 4. R (t) approximates.1 for the (1+1) EA o the OeMax fuctio with = 1. The above average covergece rate coverges to 1 ρ(q) but uder stroger coditios tha that i Theorem 1. Theorem 3: Let Q be the trasitio submatrix associated with a coverget EA. 1) Uder particular iitializatio (that is, set q = v/ v where v is a eigevector correspodig to the eigevalue ρ(q) with v but v ), it holds for all t 1, R (t) = 1 ρ(q). (36) 2) Uder radom iitializatio (that is q > ), choose a appropriate δt such that g := (I Q δt )(f opt 1 f) > for a maximizatio problem (or g < for a miimizatio problem) 2. If Q is positive 3, the it holds Proof: From (8): q T t = q T t 1Q ad (11), we get R (t) = 1 ρ(q). (37) f t+δt f t = f t+δt f opt + = q T t (f opt1 f) q T t+δt (f opt1 f) = q T t (f opt 1 f) q T t Q δt (f opt 1 f) = q T t g. (38) 1) Sice q is a eigevector correspodig to the eigevalue ρ(q) such that ρ(q)q T = q T Q. From (38) ad (8): q T t = q T t 1Q, we get f t+δt f t q T t f t f t δt = g q T t δt g = q T Qt g q T Qt δt g = ρ(q) t ρ(q) t δt qt g q T g = ρ(q). (39) 2 It is always true for a large time iterval δt sice δ (I Q δt ) = I ad f opt1 f > for a maximizatio problem (or f opt1 f < for a miimizatio problem). 3 The coditio of positive Q could be relaxed to o-egative Q if takig a similar argumet to the extesio of Perro-Frobeius theorems to o-egative matrices [18, pp. 67].

8 8 The R (t) = 1 ρ(q) which gives the first coclusio. 2) Without loss of the geerality, cosider g >. Sice let f t+δt f t = qt t g f t f t δt q T t δt g = qt t δt Qδt g, (4) g q T t δt [q T t δt λ t = mi Qδt ] i i [q T t δt ], λ t = max i i where [v] i represets the ith etry i vector v. Accordig to Collatz formula [22] [23, Theorem 2], Hece for ay [g] i >, it holds [q T t δt Qδt ] i [q T t δt ], (41) i λ t = λ t = ρ(q δt ). (42) mi [q T t δt Qδt ] i [g] i i [q T t δt ] = ρ(q δt ), i[g] max [q T t δt Qδt ] i [g] i i i [q T t δt ] = ρ(q δt ). (43) i[g] i Usig mi{ a1 b 1, a2 b 2 } a1+a2 b 1+b 2 max{ a1 b 1, a2 Equivaletly The b 2 }, we get i [qt t δt Qδt ] i [g] i i [qt t δt ] i[g] i = ρ(q δt ). (44) q T t δt Qδt g q T t δt g = ρ(qδt ). (45) f t+δt f t f t f t δt = ρ(q δt ) = ρ(q). (46) Fially it comes to the secod coclusio. The theorem shows that the average covergece rate R (t) plays the same role as R(t). But the calculatio of R (t) is ot as stable as that of R(t) i practice. VI. CONCLUSIONS This paper proposes a ew covergece rate of EAs, called the average (geometric) covergece rate. The rate represets a ormalized geometric mea of the reductio ratio of the fitess differece per geeratio. The calculatio of the average covergece rate is simple ad easy to implemet o most EAs i practice. Sice the rate is ormalized, it is coveiet to compare differet EAs o optimizatio problems. For discrete optimizatio, lower bouds o the average covergece rate of EAs have bee established. It is prove that uder radom iitializatio, the average covergece rate R(t) for t geeratios coverges to a it, called the asymptotic average covergece rate; ad uder particular iitializatio, R(t) equals to the asymptotic average covergece rate for ay t 1. The aalysis of EAs for cotiuous optimizatio is differet from that for discrete optimizatio. I cotiuous optimizatio, a EA is modeled by a Markov chai o a geeral state space, rather tha a fiite Markov chai. So a differet theoretical aalysis is eeded, rather tha matrix aalysis used i the curret paper. This topic is left for future research. REFERENCES [1] L. Mig, Y. Wag, ad Y.-M. Cheug, O covergece rate of a class of geetic algorithms, i Proceedigs of 26 World Automatio Cogress. IEEE, 26, pp [2] J. Suzuki, A Markov chai aalysis o simple geetic algorithms, IEEE Trasactios o Systems, Ma ad Cyberetics, vol. 25, o. 4, pp , [3] F. Schmitt ad F. Rothlauf, O the importace of the secod largest eigevalue o the covergece rate of geetic algorithms, i Proceedigs of 21 Geetic ad Evolutioary Computatio Coferece, H. Beyer, E. Catu-Paz, D. Goldberg, Parmee, L. Spector, ad D. Whitley, Eds. Morga Kaufma Publishers, 21, pp [4] J. He ad L. Kag, O the covergece rate of geetic algorithms, Theoretical Computer Sciece, vol. 229, o. 1-2, pp , [5] G. Rudolph, Local covergece rates of simple evolutioary algorithms with Cauchy mutatios, IEEE Trasactios o Evolutioary Computatio, vol. 1, o. 4, pp , [6], Covergece rates of evolutioary algorithms for a class of covex objective fuctios, Cotrol ad Cyberetics, vol. 26, pp , 1997.

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