General Lower Bounds for the Running Time of Evolutionary Algorithms

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1 Geeral Lower Bouds for the Ruig Time of Evolutioary Algorithms Dirk Sudholt Iteratioal Computer Sciece Istitute, Berkeley, CA 94704, USA Abstract. We preset a ew method for provig lower bouds i evolutioary computatio based o fitess-level argumets ad a additioal coditio o trasitio probabilities betwee fitess levels. The method yields exact or ear-exact lower bouds for LO, OeMax, ad all fuctios with a uique optimum. All lower bouds hold for every evolutioary algorithm that oly uses stadard mutatio as variatio operator. Itroductio Rigorous ruig time aalysis has emerged as a importat ad active area i evolutioary computatio. Results obtaied by mathematical argumets help to judge the performace of evolutioary algorithms EAs) o iterestig problems ad they ca be used to comparedifferet algorithms i a rigorousway. Ruig time aalyses have bee performed for may pseudo-boolea fuctios[] as well as for may problems from combiatorial optimizatio [2]. We cotribute to this developmet with a ew method for provig lower bouds for the ruig time of stochastic search algorithms. This method is applied to a very broad class of evolutioary algorithms for pseudo-boolea optimizatio. The resultig bouds are ot oly tight i a asymptotic sese. They cotai best possible leadig costats whe compared to upper bouds for the well-kow +) EA see page 3 for a defiitio). We also make a effort towards statig bouds with precise costats for all ivolved terms, without resortig to asymptotic otatio. 2 Previous Work There is a log history of results o pseudo-boolea optimizatio, icludig lower bouds. Already Droste, Jase, ad Wegeer [3] preseted a lower boud of Ω) for the +) EA o every -bit pseudo-boolea fuctio with uique global optimum. The costat factor precedig the -term is /2 e /2 ) Wegeer [] metios a lower boud ε) l c where ε > 0 is a arbitrarily small costat ad the costat c > 0 depeds o ε. Doerr, Fouz, ad Witt [4] preseted a lower boud o))el for the +) EA o the fuctio OeMaxx) := i= x i. This result was exteded by Doerr, Johase, ad Wize [5]. They proved that the same boud holds for the +) EA o every fuctio with a uique global optimum.

2 The fuctio LeadigOes,shortly LOx) := i= i j= x i, is aotherpopular test fuctio that couts the umber of leadig oes i the bit strig. Droste, Jase, ad Wegeer [3] showed that the ruig time of the +) EA o LO is at least c 2 with probability 2 Ω), for some costat c > 0. Black-box complexity of search algorithms as itroduced by Droste, Jase, ad Wegeer [6] is aother method for provig lower bouds. These bouds hold for all algorithms i a black-box settig where oly the class of fuctios to be optimized is kow, but the precise istace is hidde from the algorithm. Their results imply that every black-box algorithm eeds at least Ω/log ) fuctio evaluatios to optimize OeMax ad LO or, to be more precise, straightforward geeralizatios to fuctio classes). Very recetly Lehre ad Witt [7] preseted a more restricted black-box model. If oly uary operators are used that are ubiased w. r. t. bit values ad bit positios, every black-box algorithm eeds Ω log ) fuctio evaluatios for every fuctio with a uique global optimum. Recetly, drift aalysis has received a lot of attetio [8 0]. Assume a oegative potetial fuctio such that the optimum is reached oly if the potetial is 0. If the expected decrease drift ) of the potetial i oe geeratio is bouded from below, a upper boud o the expected optimizatio time follows. Coversely, a upper boud o the drift implies lower bouds o the expected optimizatio time. If there is a drift poitig away from the optimum o a part of the potetial s domai the expoetial lower bouds ca be show [8]. I this workwe showthat alsoamoredirect approachis sufficiet for provig good lower bouds. We itroduce the ew lower-boud method i Sectio 4, followed by applicatios for LO i Sectio 5 ad OeMax i Sectio 6. Sectio 7 trasfers the last result to all fuctios with a uique optimum. 3 Prelimiaries The presetatio i this work is for maximizatio problems, but it ca be easily adapted for miimizatio. The techique for provig lower bouds will be applied to a very geeral class of evolutioary algorithms. It cotais all EAs that geerate N idividuals uiformly at radom ad afterwards oly use stadard mutatios to geerate offsprig see Algorithm ). The optimizatio time is give by the time idex t that couts the umber of fuctio evaluatios. The paret selectio mechaism is very geeral as ay mechaism based Algorithm Scheme of a mutatio-based EA : create idividuals x,...,x {0,} uiformly at radom. 2: let t :=. 3: loop 4: select a paret x {x,...,x t} accordig to t ad fx ),...,fx t). 5: create x t+ by copyig x ad flippig each bit idepedetly with prob. /. 6: let t := t+. 7: ed loop

3 o the time idex t ad fitess values of previous search poits may be used. Ay mechaism for maagig a populatio fits i this framework. This icludes paret populatios ad offsprig populatios with arbitrary selectio strategies ad eve parallel evolutioary algorithms with spatial structures ad migratio. The +) EA is a well-kow special case with populatio size =. It maitais a sigle idividual x ad i every iteratio it creates x by stadard mutatio of x ad replaces x by x if fx ) fx) for maximizatio problems). We deote by +) EA a geeralizatio of the +) EA that is iitialized with a best idividual out of idividuals geerated uiformly at radom. We review the fitess-level method, also kow as the method of f-based partitios []. It yields upper bouds for EAs whose best fitess value i the populatio ever decreases. We call these algorithms elitist EAs. The optimizatio time is the umber of fuctio evaluatios util a global optimum is foud. Theorem Fitess-level method for provig upper bouds). For two sets A,B {0,} ad fitess fuctio f let A < f B if fa) < fb) for all a A ad all b B. Cosider a partitio of the search space ito o-empty sets A,...,A m such that A < f A 2 < f < f A m ad A m oly cotais global optima. For a mutatio-based EA A we say that A is i A i or o level i if the best idividual created so far is i A i. Cosider some elitist EA A ad let s i be a lower boud o the probability of creatig a ew offsprig i A i+ A m, provided A is i A i. The the expected optimizatio time of A o f without the cost of iitializatio) is bouded by m i= m PA starts i A i ) j=i s i m The caoical partitio is the oe i which A i cotais exactly all search poits with fitess i. It is kow that for LO the method applied to the caoical partitio yields a upper boud of i=0 e = e2 for the +) EA sice the probability of fidig a improvemet is lower bouded by the probability of flippig the first bit with value 0. This probability is at least / /) /e).for OeMaxweget a upper boud of i=0 e/ i) = e i= /i el+o) for the +) EA sice o level i there are i -bit mutatios that flip a 0-bit to ad hece improve the fitess. 4 Lower Bouds with Fitess-Levels The best lower bouds with fitess-level argumets kow so far were preseted by Wegeer i [], assumig fitess levels A,...,A m for some fitess fuctio f: Lemma. Let u i be a upper boud o the probability of a EA A creatig a ew offsprig i A i+ A m, provided A is i A i where A is i A i is defied as i Theorem ). The the expected optimizatio time of A o f is at least m i= PA starts i A i ) u i. i= s i.

4 The resultig lower bouds are very weak sice we oly look at the time it takes to leave the iitial fitess level ad the pessimistically assume that the optimum is foud. Makig a additioal assumptio about the trasitio probabilities betwee fitess levels allows for much better lower bouds. I the followig theorem γ i,j reflects the coditioal probability of jumpig from level i to level j, give that the algorithm leaves level i. Theorem 2. Cosider a partitio of the search space ito o-empty sets A,...,A m such that oly A m cotais global optima. For a mutatio-based EA A we agai say that A is i A i or o level i if the best idividual created so far is i A i. Let the probability of traversig from level i to level j i oe step be at most u i γ i,j ad m j=i+ γ i,j =. Assume that for all j > i ad some 0 < χ it holds m γ i,j χ γ i,k. ) k=j The the expected optimizatio time of A o f is at least m i= m i= PA starts i A i ) +χ u i m PA starts i A i ) χ j=i m u j=i+ j 2) u j. 3) If the same fitess levels are used ad s i = u i for all levels the 3) matches the upper boud from Theorem up to a factor of χ. Proof of Theorem 2. The secod boud immediately follows from the first oe sice 0 χ. Let E i be the miimum expected remaiig optimizatio time, where the miimum is take for all possible histories x,...,x t of previous search poits with x,...,x t A A i. By defiitio E E 2 E m = 0 as the coditios o the histories are subsequetly relaxed. By the law of total expectatio the ucoditioal expected optimizatio time is at least m i= PA starts i A i) E i, hece we oly eed to boud E i. The probability of leavig level i is at most u i ad the waitig time for this evet is at least /u i. I case level i is left, the remaiig time depeds o the ew level. We have E i u i + m j=i+ γ i,j E j. Assume for a iductio that for all j > i it holds E j u j +χ m k=j+ u k. The E i is at least u i + m j=i+ γ i,j +χ u j m u k k=j+ = + u i m u j=i+ j γ i,j +χ j k=i+ γ i,k )

5 where the equality holds sice o the left-had side every term /u k i the ier sum appears for all summads i +,...,j i the outer sum, weighted by γ i,k χ. Usig the precoditios o the γ-values, the last term equals m +χ u i j=i+ u j γ m i,j χ + k=j γ i,k u i +χ m j=i+ u j. Notethat i orderto apply the theorem,we olyhaveto fid a upper boud u i γ i,j o the probability of jumpig from level i to level j. I particular, we ca allow ourselves some slack i the defiitio of u i, which ca make it much easier to prove the desired coditio. Also ote that the theorem does ot require the sets A i to form fitess levels. The oly requiremet is that all global optima are cotaied i A m. Furthermore, global optima ca be replaced by ay other optimizatio goal such as fidig high-fitess idividuals. 5 A Lower Boud for LeadigOes Our first applicatio is for LO as here the γ-values ca be estimated i a very atural way. Theorem 3. Let X be a radom variable that describes the maximum LOvalue amog idividuals created idepedetly ad uiformly at radom. For every 2 the expected optimizatio time of every mutatio-based EA o LO is at least PX = i) i=0 e 2 ) i + 2 j=i+ ) j 4) ) Proof. Cosider the caoical partitio ad assume that the algorithm is o level i <. This implies that i the best idividual created so far the first i + bits are predetermied. I additio, i all idividuals created so far the bits at positios i+2,..., have ot cotributed to the fitess yet. These bits have bee iitialized uiformly at radom ad they have bee subjected to radom mutatios. It is easy to see that this agai results i uiform radom bits. More precisely, the probability that a specific bit j with j i + 2 i a specific idividual has a specific bit value 0 or is exactly /2 see the proof of Theorem 7 i Droste, Jase, ad Wegeer [3]). Cosider a idividual x that has bee selected as paret amog the created idividuals.letlox) = j i.weboudtheprobabilityofcreatigaoffsprig with k leadig oes for some i + k. Oe ecessary coditio is that the first j leadig oes do ot flip, which happes with probability /) j. The bit at positio j + is 0, hece it must be flipped. All bits at positios j + 2,...,i + must obtai the value i the offsprig. This probability is

6 determied by the umber of oes amog these bits. But clearly /) i j is a lower boud o this probability sice this reflects the best-case sceario that all these bits are i the paret. Sice 2 the probability of flippig a bit is ot larger tha the probability of ot flippig it.) The last ecessary coditio is to create exactly k i oes amog at positios i+2,...,. By the precedig argumets o the radomess of these bits, the probability of creatig exactly k i oes is 2 k+i := γ i,k if k < ad 2 k+i+ := γ i,k if k =. Puttig everythig together, we have that ) i γ i,k is a upper boud o the probability of jumpig to level k. Checkig the coditio o the γ-values, k=i+ γ i,k = k=i+ 2 k+i + 2 +i+ = ad for all i < j coditio ) holds sice γ i,k = 2 k i +2 i = 2 j+i+ = 2γ i,j. k=j k=j Settig χ = /2, the precoditios for Theorem 2 are fulfilled. Usig u i := /) i /, this proves the boud PX = i) ) i + ) j 2 i=0 j=i+ ad hece 4). The secod boud 5) follows by simple calculatios ad the followig case distictios. If 20 the claimed boud is egative, hece we ca assume 2. Observe that the boud 4) is ever larger tha e 2, eve for the special case X = 0. If e 2 the the probability that the optimum is ot foud durig the first e 2 idividuals created durig iitializatio is at most e 2 2 / for 3. This proves the claimed lower boud. If e 2 the PX > 4) e = / for 3. Pessimistically assumig that X = 4 i case X 4 ad estimatig the coditioal expected optimizatio time by 0 i case X > 4 results i the boud ) ) ) j 2 2 ) j=0 = ) ) ) 2 ) ) e ) 2 2 e 2 e ) 2 2 j=4log )+ ) 2 ) j

7 for 4. e Note that a term Θ) is, i geeral, ecessary sice with, say, = a EA will start with a average of Θ) leadig oes i the best search poit. For the +) EA u i γ i,j is the exact probability of jumpig from fitess level i to level j > i. Also recall that all coditios ) o the γ i,j -values hold with equality. Goig through the proof of Theorem 2 agai, we fid that i this special settig all estimatios are, i fact, equalities. This shows that the first lower boud o LO is exact for the +) EA. This proves that amog all mutatio-based EAs the +) EA is a optimal algorithm for the fuctio LO. For = we get the followig. Corollary. The expected optimizatio time of the +) EA o LO is exactly 2 i ) i + 2 i=0 j=i+ ) j = ) The factor precedig 2 coverges to e )/2 from below. To the author s kowledge this is the first time the leadig costat for the +) EA o LO has bee stated explicitly. This result also shows that additioal iformatio about trasitio probabilities betwee fitess levels is also useful for provig better upper bouds. 6 A Lower Boud for OeMax Theorem 4. The expected optimizatio time of every mutatio-based EA o OeMax is at least el 2log 6. Proof. Assume that 34 as otherwise the claim is trivial. If 2el the the probability that the first 2e l search poits geerated durig iitializatio fid the optimum is at most 2el 2 /2, which establishes the lower boud el. I the followig we assume 2el. Let l = /. Cosider the followig partitio A l,...,a. Let A i = {x x = i} for i > l ad A l cotai all remaiig search poits. With probability at least 2e i=0 i ) 2 /) for 34 the iitial populatio oly cotais idividuals o the first fitess level. Cosider a situatio where the algorithm is o fitess level i, i.e., the best-so-far search poit has had up to i oes. The probability of reachig ay specific higher fitess level j > i is maximal if a idividual with i oes is selected as paret. See Lemma i [5] for a formal proof.) For j > i let p i,j be the probability of the evet that mutatig a idividual with i oes results i a offsprig that cotais j oes. A ecessary coditio

8 for this evet is that i this mutatio either j i 0-bits flip to ad o -bit flips to 0 or that at least j i+ 0-bits flip to. This yields ) i p i,j j i j i ) j i i ) j+i i + j i+ ) ) j+i + i. ) j i+ For i l defie u i := i ) ) + i ) j i i ad γ i,j := where the prime idicates that these will ot be the fial variables used i the applicatio of Theorem 2. Observe that u i γ i,j = i ) ) + i ) j i i ) j i i = ) j+i + i ) ) j i ) j i i ) ) j+i + i p i,j. Sice Theorem 4 requires the γ i,j -variables to sum up to, we cosider the followig ormalized variables: u i := u i j=i+ γ i,j ad γ γ i,j i,j :=. As j=i+ γ i,j u i γ i,j = u i γ i,j p i,j, the coditios o the trasitio probabilities are fulfilled. The coditio γ i,j χ k=j γ i,j is equivalet to γ i,j χ k=j γ i,j ad γ i,k = k=j k=j ) k i i Usig i /, we have i ) j i k=0 ) k i = γ i,j i. i / = / 2. Hece, choosig χ := 2/ we obtai k=j γ i,k γ i,j )/i ) γ i,j /χ as required. Now that all coditios are verified, we proceed by estimatig the variables u i. Boudig the sum of the γ i,j -values as before, j=i+ γ i,j j=0 ) j i i 2.

9 Hece u i i i e i e ) ) + i + ) + e i) ) + 3 i e i e 5. Applyig Theorem 2 ad recallig that the algorithm is iitialized o the first fitess level with probability at least yields the upper boud ) 2 ) e 5 ) i i=l 8 ) l e i. i= Sice r i= /i lr for ay r R+, the boud is at least 8 ) el/) = 8 ) el) llog )) el el) 8e l) > el 2log 6. We remark that the lower boud does ot hold for all mutatio operators. The biased mutatio operator i [] leads to a boud Θ) for the +) EA o OeMax. 7 Geeralizatio to all Fuctios with Uique Optimum Usig argumets by Doerr, Johase, ad Wize [5], we ow trasfer the lower boud for OeMax to all fuctios with a uique global optimum. This yields a more precise result tha the Ω) boud by Lehre ad Witt [7]. I[5]theauthorsprovedthattheexpectedoptimizatiotimeofthe+) EA ooemaxisotlargerthatheexpectedoptimizatiotime ofthe+) EAo ay other fuctio with uique global optimum. Their proof exteds to arbitrary mutatio-based EAs i a straightforward way. Theorem 5. The expected umber of fuctio evaluatios for every mutatiobased EA A o every fuctio f with a uique global optimum is at least el 2log 6. Proof. For some a {0,} deote by f a the fuctio fx a) where deote the bit-wise exclusive or. Observe that this trasformatio does ot chage the

10 behavior of a mutatio-based EA i ay way, i.e., all mutatio-based EAs have the same rutime distributio o f a as o f. Hece, we do ot lose geerality if we trasform the fuctio f i such a way that is the global optimum. Let E f A deote the expected optimizatio time of A o f ad assume that the algorithm has already created search poits x,...,x t. Let E f A i) be the miimum expected remaiig optimizatio time for A give that A has oly created idividuals o the first i fitess levels so far, formally x,...,x t A 0 A i with A 0,...,A the caoical partitio for OeMax. Observethatbydefiitio,sicethecoditiosox,...,x t aresubsequetly restricted, E f A ) Ef A ) Ef A 0). Further defie a more specific ad slightly modified quatity for the +) EA : let ẼOeMax +) EA i) be defied like E OeMax +) EA i), but with the additioal coditio that the history x,...,x t cotais at least oe searchpoit i A i. Sice we have oly added a costrait, ẼOeMax +) EA i) E OeMax +) EA i). Followig Doerr, Johase, ad Wize [5], we ow prove iductively that for all i it holds E f A i) ẼOeMax +) EA i). Clearly E f A ) ẼOeMax +) EA ) = 0. Assume E f A j) ẼOeMax +) EA j) for all j > i. Let x be the ext offsprig costructed by A. If the best OeMax-valuesee so faris at most i ad x = k > i the the expected remaiig optimizatio time is at best E f A k) or larger). If the ew offsprig has a smaller umber of oes, the remaiig expected optimizatio time is still bouded below by E f A i). Thus, usig the assumptio of our iductio, E f A i) + + k=i+ k=i+ P x = k) E f A k)+p x i) E f A i) P x = k) ẼOeMax +) EA k)+p x i) E f A i). The best distributio for x is obtaied whe a paret z with exactly i oes is selected. Note that the probability of selectig such a z might be 0, i which cases the real boud is eve larger.) Let Z be the radom umber of oes whe mutatig z, the E f A i) + k=i+ O oe had this is equivalet to PZ = k) ẼOeMax +) EA k)+pz i) E f A i). + E f A i) k=i+pz = k) ẼOeMax +) EA k). 6) PZ i) O the other had for the +) EA o OeMax we have Ẽ+) OeMax EA i) = + PZ = k) ẼOeMax +) EA k)+pz i) ẼOeMax +) EA i), k=i+

11 which is equivalet to + Ẽ+) OeMax k=i+pz = k) ẼOeMax +) EA k) EA i) =. 7) PZ i) Takig6)ad7)togetheryieldsE f A i) ẼOeMax +) EA i).moreover,ẽoemax +) EA i) E+) OeMax EA i). As A ad +) EA are iitialized i the same way, they share the same distributio for the iitial fitess level. We coclude E f A EOeMax +) EA ad the boud follows from Theorem 4 applied to +) EA. As a side result, the proof has also show that the +) EA is a optimal algorithm for OeMax. 8 Coclusios Usig a adaptatio of the fitess-level method, we have preseted geeral lower bouds for the ruig time of mutatio-based evolutioary algorithms i pseudo-boolea optimizatio. The bouds for LO ad OeMax are exact or exactuptolower-ordertermswhecomparedtoupperboudsforthe+) EA. This is a rare occasio of results that are both very geeral ad very precise at the same time. I additio, we have prove that amog all mutatio-based EAs the +) EA for proper ) is a optimal algorithm for LO ad OeMax, with respect to the umber of fuctio evaluatios. The method itself is ot restricted to the ivestigated settig. It is ready to be applied to other search spaces ad further stochastic search algorithms. Ackowledgmet. The author was supported by a postdoctoral fellowship from the Germa Academic Exchage Service. Refereces. Wegeer, I.: Methods for the aalysis of evolutioary algorithms o pseudo- Boolea fuctios. I Sarker, R., Yao, X., Mohammadia, M., eds.: Evolutioary Optimizatio. Kluwer 2002) Oliveto, P.S., He, J., Yao, X.: Time complexity of evolutioary algorithms for combiatorial optimizatio: A decade of results. Iteratioal Joural of Automatio ad Computig 43) 2007) Droste, S., Jase, T., Wegeer, I.: O the aalysis of the +) evolutioary algorithm. Theoretical Computer Sciece ) Doerr, B., Fouz, M., Witt, C.: Quasiradom evolutioary algorithms. I: Geetic ad Evolutioary Computatio Coferece GECCO 0). 200) To appear. 5. Doerr, B., Johase, D., Wize, C.: Drift aalysis ad liear fuctios revisited. I: IEEE Cogress o Evolutioary Computatio CEC 0). 200) To appear.

12 6. Droste, S., Jase, T., Wegeer, I.: Upper ad lower bouds for radomized search heuristics i black-box optimizatio. Theory of Computig Systems 394) 2006) Lehre, P.K., Witt, C.: Black box search by ubiased variatio. I: Geetic ad Evolutioary Computatio Coferece GECCO 0). 200) To appear. 8. Oliveto, P.S., Witt, C.: Simplified drift aalysis for provig lower bouds i evolutioary computatio. I: Parallel Problem Solvig from Nature PPSN X). Volume 599 of LNCS., Spriger 2008) He, J., Yao, X.: A study of drift aalysis for estimatig computatio time of evolutioary algorithms. Natural Computig 3) 2004) Doerr, B., Johase, D., Wize, C.: Multiplicative drift aalysis. I: Geetic ad Evolutioary Computatio Coferece GECCO 0). 200) To appear.. Jase, T., Sudholt, D.: Aalysis of a asymmetric mutatio operator. Evolutioary Computatio 8) 200) 26