A Rigorous View On Neutrality

Transcription

1 A Rigorous View O Neutrality Bejami Doerr Michael Gewuch Nils Hebbighaus Frak Neuma Algorithms ad Complexity Group Max-Plack-Istitut für Iformatik Saarbrücke, Germay Departmet of Computer Sciece Christia-Albrechts-Uiversity of Kiel Kiel, Germay Abstract Motivated by eutrality observed i atural evolutio ofte redudat ecodigs are used i evolutioary algorithms. May experimetal studies have bee carried out o this topic. I this paper we preset a first rigorous rutime aalysis o the effect of usig eutrality. We cosider a simple model where a layer of costat fitess is distributed i the search space ad poit out situatios where the use of eutrality sigificatly ifluece the rutime of a evolutioary algorithm. I. INTRODUCTION Evolutioary algorithms have bee successfully applied to differet kids of optimizatio problems. Oe importat issue whe desigig such a algorithm is the ecodig of possible solutios. Ofte there is o direct represetatio but a special geotype-pheotype mappig which i may cases itroduces redudacy. The use of redudacy is ofte motivated by the eutrality which ca be foud i atural evolutio. It has bee cosidered i several experimetal studies whether redudacy ca sigificatly help to come up with better algorithms [, 4, 5, 8, 9]. The mai aim of our ivestigatios is to uderstad the effect of eutrality a little bit deeper from a theoretical poit of view. This is also the case for all rigorous aalyses carried out i the past. The first rigorous result o the rutime behavior of EAs has bee obtaied i [0] where it is prove that the wellkow (+) EA is able to optimize the fuctio ONEMAX withi a expected umber of O( log ) iteratios. The fuctio ONEMAX has also bee cosidered i other scearios as the startig poit for differet rutime aalyses [2, 3, 2]. After the first result for the (+) EA i [0] it has bee show i [6] that this algorithm ca solve some Log Path fuctios i expected polyomial time. Later o differet results have bee obtaied for the optimizatio of pseudo Boolea fuctios with differet properties (see e. g. [4, 8]). The (+) EA has also bee subject of differet ivestigatios o some of the best-kow combiatorial optimizatio problems [7,, 2] ad seems to be a stadard algorithms for ivestigatig the behavior of EAs with respect to their rutime behavior. Algorithms workig with a larger populatio size have bee cosidered i [7, 20]. We examie situatios where eutrality does (or does ot) help to speed up computatios compared with algorithms ot usig such a mechaism. I our ivestigatios we aalyze a simple model of eutrality already examied i [5, 6] by a experimetal study. Our aim is to carry out a rigorous aalysis of the rutime for the model ad fuctios ivestigated i these papers. The model of eutrality aalyzed i our ivestigatios uses a sigle layer of costat fitess. I the case of the fuctio ONEMAX usig such a eutrality mechaism may lead to a expoetial rutime while the optimizatio time is a small polyomial if o eutrality mechaism is used [4]. I [5, 6] it has bee stated that eutrality might be useful whe cosiderig a special class of deceptive fuctios. We prove that the optimizatio time of the (+) EA is expoetial for all possible fuctio values the layer ca attai. Afterwards, we preset a fuctio where eutrality is provably helpful. I particular we show that eutrality might tur a optimizatio time that is expoetial with probability close to ito a expected polyomial optimizatio time if the right eutrality layer is used. The aalysis for our fuctio cosiders the mixig time of the (+) EA. Usig mixig time argumets similar to [3] for the aalysis of evolutioary algorithms may be of idepedet iterest ad also helpful for aalyzig evolutioary algorithms i other situatios. The outlie of the paper is as follows. I Sectio II we itroduce the cosidered algorithm ad our model of eutrality. The fuctios cosidered i [5, 6] are aalyzed rigorously i Sectio III ad IV. Sectio V shows our example where usig eutrality is provably helpful. We fiish with some coclusios ad the statemet of some topics for future work. II. ALGORITHMS AND NEUTRALITY We cosider the optimizatio of pseudo Boolea fuctios f : {0, } R. Our aim is to ivestigate how eutrality may ifluece the search process whe dealig with fuctios of differet kids. We cosider the model ivestigated i [5, 6] where a costat fitess layer that is idetically distributed i the whole search space is used. This is doe by addig a extra eutrality bit which decides whether a search poit is o the eutrality layer are ot. Usig eutrality, we cosider the search space {0, } m, where m = +. The fuctio value of the fuctio f : {0, } m R with eutrality is give by the origial fuctio f if the eutrality bit x m is set to 0. I the case that x m =, the search poit is o the eutrality layer ad the fuctio value f layer is retured. Hece, we set /07$25.00 c 2007 IEEE 259

2 { f f(x,...,x (x) := ) : x m =0 f layer : x m =. We cosider the well-kow (+) EA workig o bit strigs of legth m ad flippig i each mutatio step each bit with probability /m. I additio the algorithm uses a elitism selectio method, where the offsprig replaces its paret if its fuctio value is at least as good as the value of the paret. Algorithm : (+) EA ) Choose a iitial solutio x {0, } m 2) Repeat Create x by flippig each bit of x with probability /m. If f (x ) f (x), set x := x. I the case that the origial fuctio f is cosidered the algorithm works with bit strigs of legth ad each bit is flipped with probability /. Aalyzig the (+) EA with respect to its rutime behavior, we are iterested i the umber of costructed solutios util a optimal oe has bee created for the first time. This is called the rutime or optimizatio time of the cosidered algorithm. Ofte, the expectatio of this value is cosidered ad called the expected optimizatio time. For coveiece we make some defiitios. For every bit strig x {0, } m we deote by x {0, } the strig obtaied from x by removig the eutrality bit. Furthermore, we defie x := i= x i ad x 0 := x. Note that x respectively x 0 are the umbers of s respectively 0s of x (igorig the eutrality bit). I the followig we proof some properties that are very useful for our aalyses of the effect of eutrality. Lemma provides the probability theoretical basis, whereas Lemma 2 is formulated more related to the (+) EA ad ca thus be directly applied i some of our proofs. We use the followig covetio: the s-fold compositio ϕ ϕ of a fuctio ϕ :Ω Ω, Ω some domai, is deoted by ϕ s. Furthermore, ϕ 0 should deote the idetical mappig o Ω. Lemma : Let Ω := {0, } ad x Ω be uiformly distributed. Let N Ω ad ψ, ϕ :Ω Ωbe uiformly distributed radom variables. Moreover, let X 0 be the evet that x N, ad for s let X s be the evet that (ψ ϕ s )(x) N. The ( t ) Prob X s (t +) s=0 for every t N Proof: We have ( t ) Prob X s Prob(x Ω \ N) s=0 +Prob( s t : X s ) t + Prob( X s ). s= Sice x, ψ, ad ϕ are uiformly distributed, it is immediate by trivial iductio that the radom variable (ψ ϕ s )(x) is uiformly distributed for s =,...,t. Thus, Prob( X s )= for all s t ad the statemet of the Lemma follows. Lemma 2: Let Ω := {0, } ad defie a special subset N := {y Ω:f(y) <f layer }. Let x be the iitial solutio of the ( + ) EA. Let us assume that x is eutral ad x is uiformly distributed i Ω. The all solutios x up to step t of the ( + ) EA (icludig the iitial solutio) stay eutral ad satisfy x N with probability P eu at least (t +) Ω\N. Proof: For coveiece we deote here the iitial solutio by x 0 ad the solutio after step s by x s. Recall that x 0 is eutral ad observe that x,...,x i stay i ay case eutral if x 0,..., x i N. Thus we ca lower boud the probability P eu by the probability Prob( 0 s t : x i N). To estimate this probability, let us defie the radom variables ψ, ϕ :Ω Ω, where ψ chages ay bit of a give strig i Ω idepedetly with probability /m, ad { ψ(y) with probability /m, ϕ(y) = y with probability /m. Both radom variables are uiformly distributed. We simulate the geeratio of a offsprig x of a solutio x of the (+) EA by x =(ψ( x),ν), where the eutral bit ν differs from the eutral bit of x with chace /m. Ifxis eutral ad x, x N, the the first bits of the ew solutio are geerated by ϕ( x i ). We claim that Prob( 0 s t : x i N) Prob( x 0 N, s t :(ψ ϕ s )( x 0 ) N). Ideed, let x 0 N ad (ψ ϕ s )( x 0 ) N for all s t. By iductio o s we show that x s N ad x s = ϕ s ( x 0 ). Obviously, x 0 = ϕ 0 ( x 0 ) N. Assume ow that x i N ad x i = ϕ i ( x 0 ) for 0 i s. It is easily see (e.g., by trivial iductio) that x s is ecessarily eutral. If x s deotes its offsprig, the x s = ψ( x s )=(ψ ϕ s )( x 0 ) N. Thus x s N ad x s = ϕ( x s )=ϕ s ( x 0 ). Hece our claim is verified. Due to Lemma the proof is complete. III. ANALYSIS FOR ONEMAX I [5, 6] experimets for the fuctio ONEMAX have bee carried out. ONEMAX is the simplest o-trivial fuctio that ca be cosidered ad has bee served as a startig poit for aalyzig a wide rage of evolutioary computatio methods [2 4, 0, 2]. It is well kow that the expected optimizatio time of the (+) EA without usig eutrality is O( log ) o each liear pseudo-boolea fuctio (icludig ONEMAX) [4]. The fuctio ONEMAX is defied as ONEMAX(x) = x i. i= IEEE Cogress o EvolutioaryComputatio(CEC 2007)

3 I the followig we will carry out a rutime aalysis of the (+) EA o its extesio to eutrality. The followig result shows that if the value of the eutrality layer is oly a little bit smaller tha the expected ONEMAX-value of the iitial solutio, eutrality does ot effect the magitude of the expected optimizatio time. Theorem 3: Choosig f layer = c, c</2 a costat, the expected optimizatio time of the (+) EA o ONEMAX is Θ( log ). Proof: The lower boud holds for ay pseudo-boolea fuctio with a sigle global optimum usig the ideas of the coupo collectors theorem [9]. The correspodig proof ca be foud i [4]. It remais to show the upper boud. If x m =holds for the iitial solutio x the solutio is o the eutrality layer. The expected time util a solutio z with z m =0is determied is O(m). Up to this momet, all solutios are accepted, sice they all have the same fuctio value f layer. Thus, for each idividual x the strig x is uiformly distributed i {0, }. Hece, usig stadard Cheroff bouds oe gets z >cwith probability e Ω().If such a z with z m =0ad z >f layer has bee produced o solutio with eutrality bit is accepted by the (+) EA afterwards. Therefore, the (+) EA behaves i this case as the (+) EA o the origial fuctio ONEMAX ad its expected optimizatio time is upper bouded by O( log ) due to fitess layer argumets give i [4]. It remais to reduce the case that the first solutio z with z m =0has ONEMAX-value at most c to the first oe. The probability for this evet is e Ω() as already stated. We show that i this case after a expected umber of O( 2 ) steps of the (+) EA a solutio y with ONEMAX(y) > f layer ad y m =0will be determied. Note that the cotributio of this O( 2 ) mutatio steps to the expected rutime is O( 2 )e Ω() = e Ω(). Let us cosider a solutio x with x c ad its offsprig z. The expected umber of s i z is x + x 0 x x +( 2c). Thus, i a expected umber of less tha 2c steps the (+) EA produces a solutio x with x > c. Sice the probability that the eutrality bit flips i this last step is m = Ω( ), after a expected umber of O( 2 ) (i this special situatio) a solutio z with z >cad z m =0is produced. Thus, we have reduced this special case to the ormal case cosidered above. Altogether, we have show that the (+) EA determies the optimal solutio i expected time Θ( log ). The ext result shows that if the value of the eutrality layer is a bit larger tha the expected ONEMAX-value of the iitial solutio, eutrality does effect the magitude of the expected optimizatio time i a couterproductive way. Theorem 4: Choosig f layer = c, /2 <c<acostat, the (+) EA does ot determie the optimum o ONEMAX i e (2c )2 2 steps with probability /2 e Ω(). I particular, the expected optimizatio time of the (+) EA is expoetial. Proof: With probability /2 the iitial solutio of the (+) EA is eutral ad we ca apply Lemma 2. We defie the set N := {y Ω:ONEMAX(y) <c}. Usig Cheroff bouds for a x chose uiformly at radom from {0, } we get =Prob(ONEMAX(x) c) e (2c )2 6. Applyig Lemma 2, i the first e (2c )2 2 steps of the (+) EA the curretly best solutio stays eutral ad i N with probability e Ω(). This proves the theorem. IV. TRAP FUNCTION I this sectio we study deceptive fuctios also cosidered from a experimetal poit of view i [5, 6]. Deceptive fuctios seem to be particularly hard to optimize by evolutioary algorithms as they mislead the search process. We ivestigate TRAP z (x) := { z (z x ) if x z z ( x z) if x >z. I the case z =, we are dealig with the fuctio TRAP for which a lower boud o the expected rutime has bee give i [4]. We are cosiderig the case where the slope-chage locatio z is give by z = λ for some /2 <λ<. Theorem 5: If /2 <λ< ad z := λ, the optimizatio time of the (+) EA o the fuctio TRAP z is lower bouded by /2 with probability e Ω(). I particular, the expected optimizatio time of the (+) EA is expoetial. Proof: Let x be the iitial idividual of the (+) EA. It satisfies x < λ+/2 2 with probability e Ω() usig Cheroff bouds. The oly steps accepted by the (+) EA are the followig. Either a offsprig with at least the umber of s as the curret solutio or a offsprig with more tha z = λ s is produced. I the former case the (+) EA is mislead i the directio of the local optimum 0. For the latter case, a mutatio step with at least d := 4 flippig bits is eeded. The probability of such a step is at most ( d ) d d! = Ω(). We cosider a phase of 3 mutatio steps. The probability that a idividual x with x >zis produced i this phase is clearly e Ω() by the last argumet. O the other had, i a expected umber of O( log ) steps the (+) EA has reached the search poit 0 sice the fuctio TRAP z behaves just like the fuctio ZEROMAX o the set {x {0, } x z}. Usig Markov s iequality, there is a costat c>0 such that with probability at least /2 the search poit 0 is reached i at most c log steps. I the phase of 3 steps we repeat this short phase of c log steps more tha times. Thus, the probability that the (+) EA has determied the strig 0 as curret solutio after 3 steps is at least e Ω(). Now the oly mutatio step accepted by the (+) EA is the step flippig all bits of the curret solutio 0 at oce. The probability for such a mutatio is. Thus, the (+) EA determies the optimum i the ext /2 steps with probability Ω(). This proves the Theorem. 2007IEEE Cogress o EvolutioaryComputatio(CEC 2007) 2593

4 We ow ivestigate the effect of eutrality for the behavior of the (+) EA o the fuctio TRAP z. Like i the last sectio we choose the eutrality level as f layer = c for a fixed costat 0 <c<. More precisely, we examie this effect for two differet rages of c depedig o the slopechage locatio z = λ. I the ext theorem we cosider the rage 0 <c<( ). If we choose c i this rage, the set N = {y Ω:TRAP z (y) <f layer } (with Ω={0, } ) satisfies N = e Ω(). This is the essetial property for the proof of the Theorem. Theorem 6: If 0 <c<( ), the optimizatio time of the (+) EA o the fuctio TRAP z is lower bouded by /2 with probability e Ω(). Proof: Set ε := ( ) c>0. We ow show that all x with x ( + ε) 2 satisfy Trap z( x) >f layer. For this ε aim we use the fact that + ε < εfor large eough. Let x {0, } with x ( + ε) 2. The TRAP z ( x) = z (z x ) +ε λ (λ 2 ) = ( ε ) =( ε ε ) > ( ε) = c. After a expected umber of at most em steps the (+) EA has determied a solutio x with x m =0. Util this momet all offsprigs were accepted as solutio by the (+) EA, sice they all share the same fuctio value f layer. Hece, x is uiformly distributed i {0, }. Usig Cheroff bouds, the probability that x ( + ε 2 ) 2 is e Ω(). The oly chace for the (+) EA to produce a accepted solutio y with y > x is a mutatio step flippig Θ() 0 bits. The probability for such a step is Ω() as already show i the proof of Theorem 5. Now we are i the situatio of the proof of Theorem 5, sice the (+) EA is ot affected by the eutrality heceforth (apart from switchig the eutrality bit from time to time). This implies that the (+) EA has ot determied the optimum i /2 mutatio steps with probability e Ω(). The performace of the (+) EA o the fuctio TRAP z with f layer = c ad ( ) <c< is the cotet of the ext theorem. N = {y Ω:TRAP z (y) <f layer } satisfies Ω\N = e Ω(),ifcis i this rage. This is the reaso, why the (+) EA behaves differet i this situatio. However, the optimizatio time of the (+) EA stays expoetial with probability expoetially close to. Theorem 7: If ( ) <c<, the optimizatio time of the (+) EA o the fuctio TRAP z is lower bouded by e α, α>0 a costat, with probability e Ω(). Proof: Set ε := c ( ) > 0. We prove that all x with ( ε) 2 x z satisfy TRAP z ( x) <f layer. For this ε aim we use the fact that + +ε < εfor large eough. Let x {0, } with ( ε) 2 x z. The TRAP z ( x) = z (z x ) ε λ (λ 2 ) = ( + ε ) =( + ε + +ε ) < ( + ε) = c. We already kow that x λ+/2 2 holds for the iitial solutio with probability e Ω().Ifx m =we ca apply Lemma 2. Sice N = {y Ω : TRAP z (y) < f layer } {y Ω:( ε) 2 x Ω\N z}, we get = e Ω() by Cheroff bouds. Hece, Lemma 2 assures that there is a costat α>0 such that after e α steps of the (+) EA the curret solutio is o the eutrality level ad thus ot optimal with probability e Ω(). We ow cosider the case where the eutrality bit of the iitial solutio is 0. The behavior is the same as the (+) EA o TRAP z as log as the eutrality bit is ot set to. Hece, the umber of oes ca ot decrease withi this phase with probability e Ω(). If a solutio x with TRAP z (x) >f layer has bee obtaied before the eutrality bit is set to oe, o solutio o the eutrality layer is accepted ad we are agai i the situatio of the (+) EA o TRAP z. It remais to cosider the case where the eutrality bit is set to oe before the (+) EA has obtaied a solutio x with TRAP z (x) >f layer. Let us deote the iitial idividual of the (+) EA by x ad the first solutio of the (+) EA with eutrality bit oe by y. With probability e Ω() we have x, y N. Moreover, sice with probability e Ω() o mutatio steps of the (+) EA that icrease the umber of oes of the curret solutio were accepted, the probability distributios of x ad y have the followig property. We ca get the probability distributio of y from the oe of x by shiftig weight from the right to the left. Let us cosider v, w, {0, } with v w < k for a k. The probability that the (+) EA produces i the ext t steps a solutio with at least k oes whe startig i v is clearly less or equal tha the success probability for the (+) EA startig i w. Usig these two argumets, the probability to reach the optimum i e α steps, α>0 a appropriate costat, startig from the elemet y is upper bouded by the probability to reach the optimum whe startig i x (the iitial solutio of the (+) EA). This reduces our cosideratios to the case that the iitial solutio x satisfies x m =ad completes the proof. V. NEUTRALITY HELPS The results of the previous sectio show that the use of the cosidered eutrality mechaism is ot able to reduce the rutime sigificatly for the class of deceptive fuctios cosidered i [5, 6]. The aim of this sectio is to poit out situatios where the use of eutrality ca sigificatly reduce the rutime, i. e. from expoetial to polyomial. We cosider IEEE Cogress o EvolutioaryComputatio(CEC 2007)

5 a deceptive fuctio with two local optimal search poits 0 ad. I additio there is a polyomial fractio of search poits i the search space that have a better fitess value tha the two local optima. I the case that such a search poit of fitess at least has bee obtaied the well-kow fuctio LeadigOes [4] directs the search to a sigle global optimum. W. l. o. g. we assume that is eve i the followig. We ivestigated the fuctio PEAK defied as PEAK(x) := 2 x : x < 2 x 2 : x > 2 + LeadigOes(x) : x = 2. as well as its extesio to eutrality PEAK. We defie by P = {x {0, } : i= x i = 2 } the set of search poits that are o the peak level. First, we ivestigate the (+) EA o the origial fuctio. As log as o search poit with fitess value at least has bee obtaied the search is mislead to oe of the local optima. This has the followig cosequece. Theorem 8: The optimizatio time of the (+) EA o PEAK is Ω(( 4 )/2 ) with probability o(). Proof: W. l. o. g. we assume the is eve. The umber of bit strigs whose umber of oes is exactly /2 is ( ) 2 /2 π /2 2 usig Stirlig s formula. Hece, the iitial solutio cosists of exactly /2 oes with probability ( ) 2 =Θ( /2 ), /2 ad the probability that the umber of oes lies i the iterval [/2 /4,/2+ /4 ] is O( /4 /2 )=O( /4 ). We cosider a phase of 3/2 steps uder the coditio that the iitial solutio has Hammig distace at least /4 to ay solutio with exactly /2 oes. The umber of flippig bits i a sigle mutatio step is asymptotically Poisso-distributed ad therefore the probability of havig a step with /4 flippig bits withi this phase is o(). This implies that the smallest Hammig distace to a solutio with exactly /2 oes does ot decrease withi this phase with probability o(). Let x be such a solutio whose fitess value is k. The probability of achievig a solutio with a higher fitess value is at least ( ) /2 k ( /) /2 k e as there are at least /2 k -bit flips which lead to a improvemet. This implies that a solutio with fitess value exactly /2 (oe of the search poits 0 ad ) has bee obtaied after a expected umber of O( log ) steps. The probability that such a solutio has ot bee obtaied withi a phase of 3/2 steps is o() usig Markov s iequality. Afterwards, the probability of producig from oe of the search poits 0 of a solutio with exactly /2 is upper bouded by ( ) ( ( ) ) /2 (/) /2 ( /) /2 = O. /2 4 All failure probabilities to do ot obtai the search poit 0 or are o(), which implies that the optimizatio time is Ω(( 4 )/2 ) with probability o(). While the optimizatio is expoetial with probability asymptotically close to, a polyomial optimizatio ca be prove whe eutrality is used. I this case the eutrality layer may prevet the algorithm from beig mislead ito the directio of a local optimum. Theorem 9: Let ε>0. Choosig f layer = c, 0 <c</2 a costat, the optimizatio time of the (+) EA o PEAK is O( 2 log ) with probability /2 ε. Proof: With probability /2 the iitial solutio satisfies x m =ad therefore PEAK (x) =f layer. The probability to produce a solutio z with PEAK(z) f layer is e Ω() which also holds for a polyomial umber of steps. The probability to obtai a solutio z with PEAK(z) is Θ( /2 ) ad the probability of flippig the bit z m i this step is /m if the failure has ot happeed. Usig Markov s iequality such a solutio has bee obtaied withi 2 steps with probability o(). It remais to cosider the time util a solutio z with exactly /2 leadig oes has bee obtaied. Note, that o solutio z with z m = is accepted if we have obtaied for the first time a solutio with exactly /2 oes. Let k be the umber of leadig oes i the curret solutio x. A improvemet ca be achieved by flippig the bit x k+ of x ad oe of the /2 k -bits ot cotributig to the leadig oes values. Hece, the probability for a improvemet is at least /2 k ( /) 2 /2 k e 2 The expected time util a LeadigOes-value of /2 has bee obtaied is O( 2 log ) by summig up the differet waitig times to achieve a improvemet. Hece, usig Markov s iequality O( 2 log ) steps are eough with probability ε. After havig chose x m = for the iitial solutio x all failure probabilities have bee bouded by o(), which proves the theorem. For the fitess layer values ivestigated i Theorem 9 there is oly a costat probability that the search is successful. However a failure with aother costat probability obtaiig oe of the local optima is still possible. This would imply a expected expoetial optimizatio time. We ca prevet such a iefficiet behavior by choosig a larger value of f layer which leads to a expected polyomial optimizatio time. Theorem 0: Choosig f layer = c, /2 < c < a costat, the expected optimizatio time of the (+) EA o PEAK is O( 2 log ). Proof: We cosider the followig three phases: (i) the phase util the first idividual with set eutrality bit is produced, (ii) the phase util a idividual o the peak level P is 2007IEEE Cogress o EvolutioaryComputatio(CEC 2007) 2595

6 determied ad the eutrality bit is uset, ad (iii) the phase of optimizig the leadig oes problem o the peak level. I phase (i) the eutrality bit has to be set to oe, which happes after a expected umber of O() mutatio steps. The mai problem i phase (ii) is, that we caot assume a uiform distributio of the curret idividual, sice i the first phase (util a idividual with x m =is produced) the (+) EA guides the curret idividual either to the strig 0 or to the strig. I the followig, we show that the curret idividual is almost uiformly distributed i {0, } after Θ( log ) mutatio steps by aalyzig the mixig time of the (+) EA. Let x deote the first idividual with x m = ad let y be the N s idividual with eutrality bit set after x is produced, where N log. For every bit of the y i we upper boud the differece Prob(y i = x i ) Prob(y i x i ). We have Prob(y i = x i ) Prob(y i x i ) N = = i=0 2 i N i=0 ( ) i ( ) N i N ( i=0 2 (i+) ) i ( ) N i ( ) i ( ) N i =( +( ))N ( 2 ) log e 2log =. 2 Usig Prob(y i = x i )+Prob(y i x i ) =, we obtai Prob(y i =) 2 ( ) ad Prob(y 2 i =0) 2 ( ), 2 respectively. Thus, we get for every z {0, } Prob(y = z) [ 2 ( )] 2 e 2. Hece, after log steps or more, idepedet of the startig vertex, the probability to be o a particular vertex is Θ(2 ). Therefore, the probability for a idividual to be o the peak level P is Θ( P 2 )=Θ( /2 ) after this umber of steps. The probability of turig off eutrality i a step that produces a solutio of P is /. Hece, a expected umber of O( 3/2 ) steps suffices to produce a solutio x P where x m =0 holds. The optimal solutio of P is obtaied afterwards withi O( 2 log ) steps as show i the proof of Theorem 9. VI. CONCLUSIONS The ifluece of eutrality has bee cosidered i several experimetal studies. We have carried out the first rigorous rutime aalysis of a evolutioary algorithms usig eutrality. The model of eutrality that has bee subject of this paper has already bee cosidered i [5, 6]. We have show that there is o sigificat advatage for usig eutrality for the fuctio ONEMAX ad the class of deceptive fuctios ivestigated i these papers (i. e. i the case of the proposed deceptive fuctios, the expected optimizatio time is expoetial regardless of the value of the fitess layer). I cotrast to this, we have poited out that eutrality may be helpful for deceptive fuctios whe the umber of search poits beig early optimal is at least a polyomial fractio of the search space. The fuctios cosidered i this paper are artificial oes which should make clear some effects eutrality might have o the rutime of evolutioary algorithms. For future work, it would be iterestig to see whether a positive effect of eutrality with respect to the rutime ca also be prove for some real-world problem (e. g. a particular combiatorial optimizatio problem). REFERENCES [] M. Collis. Fidig eedles i haystacks is harder with eutrality. I Proc. of GECCO 05, pages ACM Press, [2] S. Droste. Aalysis of the (+) ea for a dyamically bitwise chagig oemax. I Proc. of GECCO 03, volume 2724 of LNCS, pages Spriger, [3] S. Droste. Not all liear fuctios are equally difficult for the compact geetic algorithm. I Proc. of GECCO 05, pages ACM, [4] S. Droste, T. Jase, ad I. Wegeer. O the aalysis of the (+) evolutioary algorithm. Theor. Comput. Sci., 276:5 8, [5] E. Galvá-López ad R. Poli. A empirical ivestigatio of how ad why eutrality affects evolutioary search. I Proc. of GECCO 06, pages ACM Press, [6] E. Galvá-López ad R. Poli. Some steps towards uderstadig how eutrality affects evolutioary search. I Proc. of PPSN 06, volume 493 of LNCS, pages , [7] O. Giel ad I. Wegeer. Evolutioary algorithms ad the maximum matchig problem. I Proc. of STACS 03, volume 2607 of LNCS, pages Spriger, [8] T. Jase ad I. Wegeer. Evolutioary algorithms - how to cope with plateaus of costat fitess ad whe to reject strigs of the same fitess. IEEE Tras. Evolutioary Computatio, 5(6): , 200. [9] R. Motwai ad P. Raghava. Radomized Algorithms. Cambridge Uiversity Press, 995. [0] H. Mühlebei. How geetic algorithms really work: mutatio ad hillclimbig. I Proc. of PPSN 92, pages Elsevier, 992. [] F. Neuma ad I. Wegeer. Radomized local search, evolutioary algorithms, ad the miimum spaig tree problem. I Proc. of GECCO 04, volume 302 of LNCS, pages Spriger, [2] F. Neuma ad C. Witt. Rutime aalysis of a simple at coloy optimizatio algorithm. I Proc. of ISAAC 06, volume 4288 of LNCS, pages Spriger, [3] I. Pak ad V. H. Vu. O mixig of certai radom walks, cutoff pheomeo ad sharp threshold of radom matroid processes. Discrete Applied Mathematics, 0 (2-3):25 272, 200. [4] A. M. Raich ad J. Ghaboussi. Implicit represetatio i geetic algorithms usig redudacy. Evolutioary Computatio, 2(2): , IEEE Cogress o EvolutioaryComputatio(CEC 2007)

7 [5] F. Rothlauf. Populatio sizig for the redudat trivial votig mappig. I Proc. of GECCO 03, volume 2724 of LNCS, pages Spriger, [6] G. Rudolph. How mutatio ad selectio solve log path problems i polyomial expected time. Evolutioary Computatio, 4(2):95 205, 996. [7] T. Storch. O the choice of the populatio size. I Proc. of GECCO 04, volume 302 of LNCS, pages Spriger, [8] M. Toussait ad C. Igel. Neutrality ad self-adaptatio. Natural Computig, 2(2):7 32, [9] K. Weicker ad N. Weicker. Burde ad beefits of redudacy. I Proc. of FOGA 00, pages Morga Kaufma, 200. [20] C. Witt. A aalysis of the (μ+) ea o simple pseudoboolea fuctios. I Proc. of GECCO 04, volume 302 of LNCS, pages Spriger, [2] C. Witt. Worst-case ad average-case approximatios by simple radomized search heuristics. I Proc. of STACS 05, volume 3404 of LNCS, pages Spriger, IEEE Cogress o EvolutioaryComputatio(CEC 2007) 2597