Running Time Analysis of the (1+1)-EA for OneMax and LeadingOnes under Bit-wise Noise

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1 Ruig Time Aalysis of the +-EA for OeMax ad LeadigOes uder Bit-wise Noise Chao Qia Uiversity of Sciece ad Techology of Chia Hefei 3007, Chia Wu Jiag Uiversity of Sciece ad Techology of Chia Hefei 3007, Chia ABSTRACT Previous ruig time aalyses of evolutioary algorithms EAs i oisy eviromets ofte studied the oe-bit oise model, which flips a radomly chose bit of a solutio before evaluatio. I this paper, we study a atural extesio of oe-bit oise, the bit-wise oise model, which idepedetly flips each bit of a solutio with some probability. We aalyze the ruig time of the +-EA solvig OeMax ad LeadigOes uder bit-wise oise for the first time, ad derive the rages of the oise level for polyomial ad superpolyomial ruig time bouds. The aalysis o LeadigOes uder bit-wise oise ca be easily trasferred to oebit oise, ad improves the previously kow results. CCS CONCEPTS Theory of computatio Theory of radomized search heuristics; KEYWORDS Noisy optimizatio, evolutioary algorithms, ruig time aalysis, computatioal complexity ACM Referece format: Chao Qia, Chao Bia, Wu Jiag, ad Ke Tag. 07. Ruig Time Aalysis of the +-EA for OeMax ad LeadigOes uder Bitwise Noise. I Proceedigs of GECCO 7, Berli, Germay, July 5-9, 07, 8 pages. DOI: INTRODUCTION Evolutioary algorithms EAs have bee widely applied to solve real-world optimizatio tasks, where the exact objective i.e., fitess evaluatio of cadidate solutios is ofte Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. Copyrights for compoets of this work owed by others tha ACM must be hoored. Abstractig with credit is permitted. To copy otherwise, or republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. Request permissios from permissios@acm.org. GECCO 7, Berli, Germay 07 ACM /7/07... $5.00 DOI: Chao Bia Uiversity of Sciece ad Techology of Chia Hefei 3007, Chia biacht@mail.ustc.edu.c Ke Tag Uiversity of Sciece ad Techology of Chia Hefei 3007, Chia ketag@ustc.edu.c impossible, while we ca obtai oly a oisy oe [3, 3]. However, previous theoretical aalyses of EAs maily focused o oise-free optimizatio, where the fitess evaluatio is exact. For ruig time aalysis, a leadig theoretical aspect [, 4], oly a few pieces of work o oisy evolutioary optimizatio have bee reported. Droste [6] first aalyzed the +-EA o the OeMax problem i the presece of oe-bit oise ad showed the maximal oise probability p = log / allowig a polyomial ruig time, where is the problem size. Gieße ad Kötzig [] recetly studied the LeadigOes problem, ad proved that the expected ruig time is polyomial if p /6e ad expoetial if p = /. For iefficiet optimizatio of the +-EA uder high oise levels, some implicit mechaisms of EAs were proved to be robust to oise. I [], it was show that the µ+- EA with a small populatio of size Θlog ca solve Oe- Max i polyomial time eve if the probability of oe-bit oise reaches. The robustess of populatios to oise was also proved i the settig of o-elitist EAs [4, 7]. However, Friedrich et al. [8] showed the limitatio of populatios by provig that the µ+-ea eeds super-polyomial time for solvig OeMax uder additive Gaussia oise N 0, σ with σ 3. This difficulty ca be overcome by the compact geetic algorithm cga [8] ad a simple At Coloy Optimizatio ACO algorithm [9], both of which fid the optimal solutio i polyomial time with a high probability. The ability of explicit oise hadlig strategies was also theoretically studied. Qia et al. [9] proved that the threshold selectio strategy is robust to oise: the expected ruig time of the +-EA usig threshold selectio o OeMax uder oe-bit oise is always polyomial regardless of the oise level. For the +-EA solvig OeMax ad LeadigOes uder oe-bit or additive Gaussia oise, the resamplig strategy was show able to reduce the ruig time from expoetial to polyomial i high oise levels [8]. Akimoto et al. [] also proved that resamplig with a large sample size ca make optimizatio uder additive ubiased oise behave as optimizatio i a oise-free eviromet. The iterplay betwee resamplig ad implicit oise-hadlig mechaisms has bee statistically studied i [0]. The studies metioed above maily cosidered the oebit oise model, which flips a radom bit of a solutio before

2 GECCO 7, July 5-9, 07, Berli, Germay C. Qia et. al. Table : For the ruig time of the +-EA solvig OeMax ad LeadigOes uder prior oise models, the rages of oise parameters for a polyomial upper boud ad a super-polyomial lower boud are show below. +-EA bit-wise oise p, bit-wise oise, q oe-bit oise OeMax Olog /, ωlog / Olog /, ωlog / [] Olog /, ωlog / [6] LeadigOes Olog /, ωlog / Olog / 3, ωlog / /6e, = / []; Olog /, ωlog / evaluatio with probability p. However, the oise model, which ca chage several bits of a solutio simultaeously, may be more realistic ad eeds to be studied, as metioed i the first oisy theoretical work [6]. I this paper, we study the bit-wise oise model, which is characterized by a pair p, q of parameters. It happes with probability p, ad idepedetly flips each bit of a solutio with probability q before evaluatio. We aalyze the ruig time of the +-EA solvig OeMax ad LeadigOes uder bit-wise oise with two specific parameter settigs p, ad, q. The rages of p ad q for a polyomial upper boud ad a super-polyomial lower boud are derived, as show i the middle two colums of Table. For the +-EA o LeadigOes, we also trasfer the ruig time bouds from bit-wise oise p, to oe-bit oise by usig the same proof procedure. As show i the bottom right of Table, our results improve the previously kow oes []. The rest of this paper is orgaized as follows. Sectio itroduces some prelimiaries. The ruig time aalysis o OeMax ad LeadigOes is preseted i Sectios 3 ad 4, respectively. Sectio 5 cocludes the paper. PRELIMINARIES I this sectio, we first itroduce the optimizatio problems, evolutioary algorithms ad oise models studied i this paper, respectively, ad the preset the aalysis tools that we use throughout this paper.. OeMax ad LeadigOes I this paper, we use two well-kow pseudo-boolea fuctios OeMax ad LeadigOes. The OeMax problem as preseted i Defiitio. aims to maximize the umber of -bits of a solutio. The LeadigOes problem as preseted i Defiitio. aims to maximize the umber of cosecutive -bits coutig from the left of a solutio. Their optimal solutio is... briefly deoted as. It has bee show that the expected ruig time of the +-EA o OeMax ad LeadigOes is Θ log ad Θ, respectively [7]. Defiitio. OeMax. The OeMax Problem of size is to fid a bits biary strig x such that x = arg max x {0,} fx = i= xi. Defiitio. LeadigOes. The LeadigOes Problem of size is to fid a bits biary strig x such that x = arg max x {0,} fx = i. i= j= xj. Bit-wise Noise There are maily two kids of oise models: prior ad posterior [, 3]. The prior oise comes from the variatio o a solutio, while the posterior oise comes from the variatio o the fitess of a solutio. Previous theoretical aalyses ofte focused o a specific prior oise model, oe-bit oise. As preseted i Defiitio.3, it flips a radom bit of a solutio before evaluatio with probability p. However, i may realistic applicatios, oise ca chage several bits of a solutio simultaeously rather tha oly oe bit. We thus cosider the bit-wise oise model. As preseted i Defiitio.4, it happes with probability p, ad idepedetly flips each bit of a solutio with probability q before evaluatio. To the best of our kowledge, oly bit-wise oise with p = ad q [0, ] has bee recetly studied. Gieße ad Kötzig [] proved that for the +-EA o OeMax, the expected ruig time is polyomial if q = Olog / ad super-polyomial if q = ωlog /. I this paper, we will study two specific bit-wise oise models: p [0, ] q = ad p = q [0, ], which are briefly deoted as bit-wise oise p, ad bit-wise oise, q, respectively. Defiitio.3 Oe-bit Noise. Give a parameter p [0, ], let f x ad fx deote the oisy ad true fitess of a biary solutio x {0, }, respectively, the { f fx with probability p, x = fx with probability p, where x is geerated by flippig a uiformly radomly chose bit of x. Defiitio.4 Bit-wise Noise. Give parameters p, q [0, ], let f x ad fx deote the oisy ad true fitess of a biary solutio x {0, }, respectively, the { f fx with probability p, x = fx with probability p, where x is geerated by idepedetly flippig each bit of x with probability q..3 +-EA The +-EA as described i Algorithm is studied i this paper. For oisy optimizatio, oly a oisy fitess value f x istead of the exact oe fx ca be accessed, ad thus step 4 of Algorithm chages to be if f x f x. Note that the reevaluatio strategy is used as i [5, 6, ]. That is, besides evaluatig f x, f x will be reevaluated i each iteratio of the +-EA. The ruig time is usually defied as the umber of fitess evaluatios eeded to fid

3 Ruig Time Aalysis of the +-EA uder Bit-wise Noise GECCO 7, July 5-9, 07, Berli, Germay a optimal solutio w.r.t. the true fitess fuctio f for the first time [, 6, ]. ALGORITHM +-EA. Give a fuctio f over {0,} to be maximized, it cosists of the followig steps:. x := uiformly radomly selected from {0, }.. Repeat util the termiatio coditio is met 3. x :=flip each bit of x idepedetly with prob. /. 4. if fx fx 5. x := x..4 Aalysis Tools The process of the +-EA solvig OeMax or LeadigOes ca be directly modeled as a Markov chai {ξ t} + t=0. We oly eed to take the solutio space {0, } as the chai s state space i.e., ξ t X = {0, }, ad take the optimal solutio as the chai s optimal state i.e., X = { }. Give a Markov chai {ξ t} + t=0 ad ξˆt = x, we defie its first hittig time FHT as τ = mi{t ξˆt+t X, t 0}. The mathematical expectatio of τ, E[[τ ξˆt = x]] = + i=0 ip τ = i, is called the expected first hittig time EFHT startig from ξˆt = x. If ξ 0 is draw from a distributio π 0, E[[τ ξ 0 π 0]] = x X π0xe[[τ ξ0 = x]] is called the EFHT of the Markov chai over the iitial distributio π 0. Thus, the expected ruig time of the +-EA startig from ξ 0 π 0 is equal to + E[[τ ξ 0 π 0]], where the term correspods to evaluatig the iitial solutio, ad the factor correspods to evaluatig the offsprig solutio x ad reevaluatig the paret solutio x i each iteratio. Note that we cosider the expected ruig time of the +-EA startig from a uiform iitial distributio i this paper. I the followig, we give three drift theorems that will be used to derive the EFHT of Markov chais i the paper. LEMMA.5 ADDITIVE DRIFT []. Give a Markov chai {ξ t} + t=0 ad a distace fuctio V x, if for ay t 0 ad ay ξ t with V ξ t > 0, there exists a real umber c > 0 such that E[[V ξ t V ξ t+ ξ t]] c, the the EFHT satisfies that E[[τ ξ 0]] V ξ 0/c. LEMMA.6 SIMPLIFIED DRIFT [5, 6]. Let X t, t 0, be real-valued radom variables describig a stochastic process. Suppose there exists a iterval [a, b] R, two costats δ, ɛ > 0 ad, possibly depedig o l := b a, a fuctio rl satisfyig rl = ol/ logl such that for all t 0 the followig two coditios hold:. E[[X t X t+ a < X t < b]] ɛ,. P X t+ X t j X t > a rl + δ j for j N 0. The there is a costat c > 0 such that for T := mi{t 0 : X t a X 0 b} it holds P T cl/rl = Ωl/rl. LEMMA.7 SIMPLIFIED DRIFT WITH SELF-LOOPS [0]. Let X t, t 0, be real-valued radom variables describig a stochastic process. Suppose there exists a iterval [a, b] R, two costats δ, ɛ > 0 ad, possibly depedig o l := b a, a fuctio rl satisfyig rl = ol/ logl such that for all t 0 the followig two coditios hold:. a<i<b : E[[X t X t+ X t =i]] ɛ P X t+ i X t =i,. i > a, j N 0 : P X t+ X t j X t =i rl P Xt+ i Xt =i. +δ j The there is a costat c > 0 such that for T := mi{t 0 : X t a X 0 b} it holds P T cl/rl = Ωl/rl. 3 THE ONEMAX PROBLEM I this sectio, we aalyze the ruig time of the +- EA o OeMax uder bit-wise oise. Note that for bit-wise oise, q, it has bee proved that the maximal value of q allowig a polyomial ruig time is log /, as show i Theorem 3.. THEOREM 3.. [] For the +-EA o OeMax uder bitwise oise, q, the expected ruig time is polyomial if q = Olog / ad super-polyomial if q = ωlog /. For bit-wise oise p,, we prove i Theorems 3.4 ad 3.5 that the maximum value of p allowig a polyomial ruig time is log /. Istead of usig the origial drift theorems, we apply the upper ad lower bouds for the +-EA o stochastic OeMax i []. Let x k deote ay solutio with k umber of -bits, ad f x k deote its oisy objective value, which is a radom variable. Lemma 3. ituitively meas that if the probability of recogizig the true better solutio by oisy evaluatio is large, the ruig time ca be polyomially upper bouded. O the cotrary, Lemma 3.3 shows that if the probability of makig a right compariso is small, the ruig time ca be expoetially lower bouded. Both of them are proved by applyig stadard drift theorems, ad ca be used to simplify our aalysis. LEMMA 3.. [] Suppose there is a positive costat c /5 ad some < l / such that k < : P f x k < f x k+ l ; k < l : P f x k < f x k+ c k, the the +-EA optimizes f i expectatio i O log + Ol iteratios. LEMMA 3.3. [] Suppose there is some l /4 ad a costat c 6 such that l k < : P f x k < f x k+ c k, the the +-EA optimizes f i Ωl iteratios with a high probability. THEOREM 3.4. For the +-EA o OeMax uder bit-wise oise p,, the expected ruig time is polyomial if p = Olog /. PROOF. We prove it by usig Lemma 3.. For ay positive costat b, suppose that p b log /. We set the two parameters i Lemma 3. as c = mi{ b log, b} ad l =, ]. 5 c

4 GECCO 7, July 5-9, 07, Berli, Germay C. Qia et. al. For ay k <, f x k f x k+ implies that f x k k + or f x k+ k, either of which happes with probability at most p. By the uio boud, we get P f x k f x k+ p For ay k < l, we easily get b log = lc l. P f x k f x k+ lc < c k. By Lemma 3., we kow that the expected ruig time is O log + Ob log /c, i.e., polyomial. THEOREM 3.5. For the +-EA o OeMax uder bit-wise oise p,, the expected ruig time is super-polyomial if p = ωlog / ωlog / ad expoetial if p = Olog /. PROOF. We use Lemma 3.3 to prove it. Let c = 6. The case p = ωlog / ωlog / is first aalyzed. For ay positive costat b, let l = b log. For ay k l, we get P f x k f x k+ P f x k = k P f x k+ k. To make f x k = k, it is sufficiet that the oise does ot happe, i.e., P f x k = k p. To make f x k+ k, it is sufficiet to flip oe -bit ad keep other bits uchaged by oise, i.e., P f x k+ k p k+. Thus, P f x k f x k+ p p k + e Sice c k c l = ωlog /. = cb log, the coditio of Lemma 3.3 holds. Thus, the expected ruig time is Ωb log where b is ay costat, i.e., super-polyomial. For the case p = Olog /, let l =. We use aother lower boud p for P f x k = k, sice it is sufficiet that o bit flips by oise. Thus, we have P f x k f x k+ p p k + e = Ω. Sice c k c, the coditio of Lemma 3.3 holds. Thus, the expected ruig time is Ω, i.e., expoetial. 4 THE LEADINGONES PROBLEM I this sectio, we first aalyze the ruig time of the +- EA o LeadigOes uder bit-wise oise p, ad bit-wise oise, q, respectively. The, we trasfer the aalysis from bit-wise oise p, to oe-bit oise; the results are complemetary to the kow oes recetly derived i []. 4. Bit-wise Noise p, For bit-wise oise p,, we prove i the followig three theorems that the expected ruig time is polyomial if p = Olog / ad super-polyomial if p = ωlog /. Their proofs are accomplished by applyig additive drift aalysis, the simplified drift theorem with self-loops ad the simplified drift theorem, respectively. THEOREM 4.. For the +-EA o LeadigOes uder bitwise oise p,, the expected ruig time is polyomial if p = Olog /. PROOF. We use Lemma.5 to prove it. For ay positive costat b, suppose that p b log /. Let LOx deote the true umber of leadig -bits of a solutio x. We first costruct a distace fuctio V x as, for ay x with LOx = i, V x = + c + c i, where c = 4b log +. It is easy to verify that V x X = { } = 0 ad V x / X > 0. The, we ivestigate E[[V ξ t V ξ t+ ξ t = x]] for ay x with LOx < i.e., x / X. Assume that curretly LOx = i, where 0 i. We divide the drift ito two parts: positive E + ad egative E. That is, E[[V ξ t V ξ t+ ξ t = x]] = E + E. For the positive drift, we eed to cosider that the umber of leadig -bits is icreased. By mutatio, we have P LOx i + = i, sice it eeds to flip the i + -th bit which must be 0 of x ad keep the i leadig -bits uchaged. For ay x with LOx i +, f x < f x implies that f x i or f x i +. Note that, P f x i = p i, sice at least oe of the first i leadig -bits of x eeds to be flipped by oise; P f x i + = p i, 3 sice it eeds to flip the first 0-bit of x ad keep the leadig -bits uchaged by oise. By the uio boud, we get P f x f x = P f x < f x p i+ p i +, 4 where the last iequality is by p = Olog /. Furthermore, for ay x with V x i +, V x V x + c i+ + c i= c + c i. 5 By combiig Eqs., 4 ad 5, we have E + i c + c i c + c i, 6 where the last iequality is by i. e 3 For the egative drift, we eed to cosider that the umber of leadig -bits is decreased. By mutatio, we have P LOx i = i, 6 sice it eeds to flip at least oe leadig -bit of x. For ay x with LOx i where i, f x f x implies

5 Ruig Time Aalysis of the +-EA uder Bit-wise Noise GECCO 7, July 5-9, 07, Berli, Germay that f x i or f x i. Note that, P f x i p i, 7 sice for the first i bits of x, it eeds to flip the 0-bits whose umber is at least ad keep the -bits uchaged by oise; P f x i = p i, 8 sice at least oe leadig -bit of x eeds to be flipped by oise. By the uio boud, we get P f x f x p p i p i+.9 Furthermore, for ay x with LOx i, V x V x + c i. 0 By combiig Eqs. 6, 9 ad 0, we have E i p i + p + c i p e 3 + c i + c i. Thus, by subtractig E from E +, we have E[[V ξ t V ξ t+ ξ t = x]] + c i c 6 p 3 + c i 4b log + b log 6 3 6, where the secod iequality is by c = 4b log + ad p b log /. Note that V x + c e c = e 4b log + = e 4b. By Lemma.5, we get E[[τ ξ 0]] 6 e 4b = O 4b+, i.e., the expected ruig time is polyomial. THEOREM 4.. For the +-EA o LeadigOes uder bit-wise oise p,, if p = ωlog / o, the expected ruig time is super-polyomial. PROOF. We use Lemma.7 to prove it. Let X t = x 0 be the umber of 0-bits of the solutio x after t iteratios of the +-EA. Let c be ay positive costat. We cosider the iterval [0, c log ], i.e., the parameters a = 0 i.e., the global optimum ad b = c log i Lemma.7. The, we aalyze the drift E[[X t X t+ X t = i]] for i < c log. As i the proof of Theorem 4., we divide the drift ito two parts: positive E + ad egative E. That is, E[[X t X t+ X t = i]] = E + E. For the positive drift, we eed to cosider that the umber of 0-bits is decreased. For mutatio o x where x 0 = i, let X ad Y deote the umber of flipped 0-bits ad -bits, respectively. The, X Bi, ad Y B i,, where B, is the biomial distributio. Let P mutx, x deote the probability of geeratig x by mutatig x. To estimate a upper boud o E +, we assume that the offsprig solutio x with x 0 < i is always accepted. Thus, we have E + i P mutx, x i x 0 = k P X Y = k x : x 0 <i = i = i i k= k i j= j k= j=k k= P X = j P Y = j k k P X = j P Y = j k j= j P X = j = i. For the egative drift, we eed to cosider that the umber of 0-bits is icreased. We aalyze the i cases where oly oe -bit is flipped i.e., x 0 = i +, which happes with probability. Assume that LOx = k i. If the j-th where j k leadig -bit is flipped, the offsprig solutio x will be accepted i.e., f x f x if f x j ad f x j. Note that, P f x j = p+p j p j, where the equality is sice it eeds to keep the j leadig -bits of x uchaged, ad the last iequality is by p = o; P f x j = p j 3 = p j + j p e j pj 3, where the equality is sice at least oe of the first j leadig -bits of x eeds to be flipped by oise. Thus, we get P f x f x pj 6. 4 If oe of the i k o-leadig -bits is flipped, LOx = LOx = k. We ca use the same aalysis procedure as Eq. 4 i the proof of Theorem 4. to derive that P f x f x p k +, 5 where the secod iequality is by p = o. Combiig all the i cases, we get E k pj 6 + i k i+ i 6 j= pkk + + i k pk e 36 + i k. 6 By subtractig E from E +, we get E[[X t X t+ X t = i]] i pk 36 i k. 6 To ivestigate coditio of Lemma.7, we also eed to aalyze the probability P X t+ i X t = i. For X t+ i, it is ecessary that at least oe bit of x is flipped ad the offsprig x is accepted. We cosider two cases: at least oe of the k leadig -bits of x is flipped; the k leadig - bits of x are ot flipped ad at least oe of the last k bits is flipped. For case, the mutatio probability is k

6 GECCO 7, July 5-9, 07, Berli, Germay C. Qia et. al. ad the acceptace probability is at most p k+ by Eq. 9. For case, the mutatio probability is k k k ad the acceptace probability is at most. Thus, P X t+ i X t = i p + k. 7 Whe k < p, we have E[[X t X t+ X t = i]] i i k 6 k p/ 7c log k 6 4 p+ k 8, where the secod iequality is by k > p ad i < c log, the third iequality is by p = ωlog / ad the last is by k > p. Whe k p, we have E[[X t X t+ X t = i]] i pk 9 36 c log p 44 p 88 p + k, 576 where the secod iequality is by p = o ad i < c log, the third is by p = ωlog / ad the last is by k p. Combiig Eqs. 7, 8 ad 9, we get that coditio of Lemma.7 holds with ɛ =. 576 For coditio of Lemma.7, we eed to show P X t+ X t j X t = i rl P X +δ j t+ i X t = i for i. For P X t+ i X t = i, we aalyze the cases where oly oe bit is flipped. Usig the similar aalysis procedure as E, except that flippig ay bit rather tha oly -bit is cosidered here, we easily get P X t+ i X t = i pkk k 6. 0 For X t+ X t j, it is ecessary that at least j bits of x are flipped ad the offsprig solutio x is accepted. We cosider two cases: at least oe of the k leadig -bits is flipped; the k leadig -bits are ot flipped. For case, the mutatio j probability is at most k ad the acceptace probability is at most p k+ by Eq. 9. For case, the mutatio j probability is at most k k ad the acceptace j j probability is at most. Thus, we have P X t+ X t j X t = i k j p k + j + k k j j pkk+ 4 + k j pkk+ + k 44 j j By combiig Eq. 0 with Eq., we get that coditio of Lemma.7 holds with δ = ad rl = 44. Note that l = b a = c log. By Lemma.7, the expected ruig time is Ωc log, where c is ay positive costat. Thus, the expected ruig time is super-polyomial. THEOREM 4.3. For the +-EA o LeadigOes uder bitwise oise p,, the expected ruig time is expoetial if p = Ω. PROOF. We use Lemma.6 to prove it. Let X t = i be the umber of 0-bits of the solutio x after t iteratios of the +-EA. We cosider the iterval i [0, / ]. To aalyze the drift E[[X t X t+ X t = i]] = E + E, we use the same aalysis procedure as Theorem 4.. For the positive drift, we have E + i = o. For the egative drift, we re-aalyze Eqs. 4 ad 5. P f x j p j Thus, Eq. 4 becomes From Eqs. ad 3, we get that ad P f x j pj. 3 P f x f x p j 3 j. For Eq. 5, we eed to aalyze the acceptace probability for LOx = LOx = k. Sice it is sufficiet to keep the first k + bits of x ad x uchaged i oise, Eq. 5 becomes P f x f x p k+ p k+. 3 By applyig the above two iequalities to Eq. 6, we have k E p j j + + i k k =Ω, e 3 j= where the equality is by p = Ω. Thus, E + E = Ω. That is, coditio of Lemma.6 holds. Sice it is ecessary to flip at least j bits of x, we have P X t+ X t j X t j j j!, j which implies that coditio of Lemma.6 holds with δ = ad rl =. Note that l = /. Thus, by Lemma.6, the expected ruig time is expoetial. 4. Bit-wise Noise, q For bit-wise oise, q, we prove i the followig three theorems that the expected ruig time is polyomial if q = Olog / 3 ad super-polyomial if q = ωlog /. The proof idea is similar to that for bit-wise oise p,. The mai differece led by the chage of oise is the probability of acceptig the offsprig solutio, i.e., P f x f x. THEOREM 4.4. For the +-EA o LeadigOes uder bitwise oise, q, the expected ruig time is polyomial if q = Olog / 3. PROOF. The proof is very similar to that of Theorem 4.. The chage of oise oly affects the probability of acceptig the offsprig solutio i the aalysis. For ay positive costat b, suppose that q b log / 3. For the positive drift E +, we eed to re-aalyze P f x f x i.e., Eq. 4 i the proof of Theorem 4. for the paret x with LOx = i ad the offsprig x with LOx i +. By bit-wise oise, q, Eqs. ad 3 chage to P f x i = q i ; P f x i+= q i q. Thus, by the uio boud, Eq. 4 becomes P f x f x q i + q i q 4 = q i+ qi + /,

7 Ruig Time Aalysis of the +-EA uder Bit-wise Noise GECCO 7, July 5-9, 07, Berli, Germay where the last iequality is by q = Olog / 3. For the egative drift E, we eed to re-aalyze P f x f x i.e., Eq. 9 i the proof of Theorem 4. for the paret x with LOx = i where i ad the offsprig x with LOx i. By bit-wise oise, q, Eqs. 7 ad 8 chage to P f x i q q i, P f x i = q i. Thus, by the uio boud, Eq. 9 becomes P f x f x q q i + q i 5 = q i q i q q i+q, where the secod iequality is by q i i q ad q > 0 for q = Olog / 3. By applyig Eq. 4 ad Eq. 5 to E + ad E, respectively, Eq. chages to E[[V ξ t V ξ t+ ξ t = x]] + c i 4b log c i c 6 qi+ 3 b log That is, the coditio of Lemma.5 still holds with. Thus, 6 the expected ruig time is polyomial. THEOREM 4.5. For the +-EA o LeadigOes uder bit-wise oise, q, if q = ωlog / o/, the expected ruig time is super-polyomial. PROOF. We use the same aalysis process as Theorem 4.. The oly differece is the probability of acceptig the offsprig solutio due to the chage of oise. For the positive drift, we still have E + i, sice we assume that x is always accepted i the proof of Theorem 4.. For the egative drift, we eed to re-aalyze P f x f x for the paret solutio x with LOx = k ad the offsprig solutio x with LOx = j where j k +. For j k, to derive a lower boud o P f x f x, we cosider the j cases where f x = l ad f x l for 0 l j. Sice P f x = l = q l q ad P f x l = q l, Eq. 4 chages to j P f x f x q l q q l qj l=0 = qj + q q 6 j q j qj q qj, where the last iequality is by q j qj / sice q = o/. For j = k + i.e., LOx = LOx = k, we ca use the same aalysis as Eq. 4 to derive a lower boud /, sice the last iequality of Eq. 4 still holds with p = o/. Thus, Eq. 5 also holds here, i.e., P f x f x. 7 By applyig Eqs. 6 ad 7 to E, Eq. 6 chages to E qk + i k. 6 Thus, we have E[[X t X t+ X t = i]] = E + E i qk i k. 6 For the upper boud aalysis of P X t+ i X t = i i the proof of Theorem 4., we oly eed to replace the acceptace probability p k+ i the case of LOx < LOx with k + q i.e., Eq. 5. Thus, Eq. 7 chages to P X t+ i X t = i k + q + k q + k. To compare E[[X t X t+ X t = i]] with P X t+ i X t = i, we cosider two cases: k < q ad k q. By usig q = ωlog / ad applyig the same aalysis procedure as Eqs. 8 ad 9, we ca derive that coditio of Lemma.7 holds with ɛ =. 9 For the lower boud aalysis of P X t+ i X t = i, by applyig Eqs. 6 ad 7, Eq. 0 chages to qkk + P X t+ i X t = i + k 6. For the aalysis of X t+ X t j, by replacig the acceptace probability p k+ i the case of LOx < LOx with k + q, Eq. chages to P X t+ X t j X t = i qkk+ 4 + k j j qkk + + k j That is, coditio of Lemma.7 holds with δ =, rl = 48. Thus, the expected ruig time is super-polyomial. THEOREM 4.6. For the +-EA o LeadigOes uder bitwise oise, q, the expected ruig time is expoetial if q = Ω/. PROOF. We use Lemma.6 to prove it. Let X t = i be the umber of 0-bits of the solutio x after t iteratios of the +-EA. We cosider the iterval i [0, / ]. To aalyze the drift E[[X t X t+ X t = i]], we use the same aalysis procedure as the proof of Theorem 4.. We first cosider q = Ω/ o. We eed to aalyze the probability P f x f x, where the offsprig solutio x is geerated by flippig oly oe -bit of x. Let LOx = k. For the case where the j-th where j k leadig -bit is flipped, as the aalysis of Eq. 6, we get P f x f x If q j <, qj 4 qj q j qj q. ; otherwise, qj qj q qj. Thus, we have P f x f x mi{/4, qj/}. For the case that flips oe o-leadig -bit i.e., LOx = LOx = k, to derive a lower boud o P f x f x, we cosider f x = l ad f x l for 0 l k. Thus, k P f x f x q l q q l + q k+ q k qk l=0 + q k+ = + qk q,

8 GECCO 7, July 5-9, 07, Berli, Germay where the last iequality is by q = o. By applyig the above two iequalities to Eq. 6, we get k E { mi e 4, qj } + i k. j= If k, k j= mi{, qj } = Ω sice q = Ω/. If 4 k <, i k = Ω sice i. Thus, E = Ω. For q = Ω, we use the trivial lower boud q for the probability of acceptig the offsprig solutio x, sice it is sufficiet to flip the first leadig -bit of x by oise. The, E iq kq + i kq = = Ω. e e Thus, for q = Ω/, we have E[[X t X t+ X t = i]] = E + E i Ω = Ω. That is, coditio of Lemma.6 holds. Its coditio trivially holds with δ = ad rl =. Thus, the expected ruig time is expoetial. 4.3 Oe-bit Noise For the +-EA o LeadigOes uder oe-bit oise, it has bee kow that the ruig time is polyomial if p 6e ad expoetial if p = []. We exted this result by provig i Theorem 4.7 that the ruig time is polyomial if p = Olog / ad super-polyomial if p = ωlog /. The proof ca be accomplished as same as that of Theorems 4., 4. ad 4.3 for bit-wise oise p,. This is because although the probabilities P f x f x of acceptig the offsprig solutio are differet, their bouds used i the proofs for bit-wise oise p, still hold for oe-bit oise. THEOREM 4.7. For the +-EA o LeadigOes uder oe-bit oise, the expected ruig time is polyomial if p = Olog /, super-polyomial if p = ωlog / o ad expoetial if p = Ω. PROOF. We re-aalyze P f x f x for oe-bit oise, ad show that the bouds o P f x f x used i the proofs for bit-wise oise p, still hold for oe-bit oise. For the proof of Theorem 4., Eqs. ad 3 chage to P f x i = p i, P f x i + = p, ad thus Eq. 4 still holds; Eqs. 7 ad 8 chage to P f x i p, P f x i = p i, ad thus Eq. 9 still holds. For the proof of Theorem 4., Eqs. ad 3 chage to P f x j = p j, P f x j = p j, ad thus Eq. 4 still holds. For the proof of Theorem 4.3, Eq. still holds by the above two equalities; Eq. 3 still holds sice the probability of keepig the first k + bits of a solutio uchaged i oe-bit oise is p k+ p k+. 5 CONCLUSION C. Qia et. al. I this paper, we theoretically study the +-EA solvig OeMax ad LeadigOes uder bit-wise oise. We derive the rages of oise parameters for the ruig time beig polyomial ad super-polyomial, respectively. The previously kow rages for the +-EA solvig LeadigOes uder oe-bit oise are also improved. I the future, we shall improve the curretly derived bouds o LeadigOes, as they do ot cover the whole rage of oise parameters. ACKNOWLEDGMENTS We wat to thak the reviewers for their valuable commets. This work was supported by the NSFC , , the Fudametal Research Fuds for the Cetral Uiversities ad the CCF-Tecet Ope Research Fud. REFERENCES [] Y. Akimoto, S. Astete-Morales, ad O. Teytaud. 05. Aalysis of rutime of optimizatio algorithms for oisy fuctios over discrete codomais. Theoretical Computer Sciece , [] A. Auger ad B. Doerr. 0. Theory of Radomized Search Heuristics: Foudatios ad Recet Developmets. World Scietific, Sigapore. [3] L. Biachi, M. Dorigo, L. Gambardella, ad W. 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