Benchmark Fitness Landscape Analysis

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1 Bechmark Fitess Ladscape Aalysis Galia Merkuryeva, Vitalijs Bolshakovs Departmet of Modellig ad Simulatio Riga Techical Uiversity Riga, Latvia Abstract Various techiques of fitess ladscape aalysis for the determiatio of optimisatio problem hardess for evolutioary algorithms are proposed i the literature. However, a few implemetatios of these techiques ad their applicatio i practice are described owadays. I the paper comparative statistical ad iformatio aalysis for bechmark fitess fuctios such as Sphere, Rastrigi, Rosebrock ad Ackley fuctios is performed. Both statistical ad iformatio measures for bechmark fitess ladscapes are estimated ad iterpreted i the optimisatio cotext. The sesitivity aalysis is performed to determie how the coditios of experimetal aalysis ad the oise factor will impact the target measures ad their estimatios. Keywords - fitess ladscape, fitess ladscape aalysis, iformatio measures, statistical measures, geetic algorithm I. INTRODUCTION Nowadays, evolutioary algorithms are ofte used to solve complex optimisatio problems. However, i some cases they may be ot eough computatioally efficiet. The hardess of a optimisatio problem ad the ability of the evolutioary algorithm to perform a efficiet search of the optimal solutio ca be determied by usig fitess ladscape aalysis [-4]. The problem is hard to solve with a evolutioary algorithm if its fitess ladscape has a large umber of structures, which disturb a search of the global solutio. The mai ladscape feature which iflueces the problem difficulty for a optimisatio algorithm is the ladscape ruggedess. To measure the degree of ruggedess of the problem search space, several statistical ad iformatio measures are proposed [5]. The paper presets practical implemetatio of the fitess ladscape aalysis techiques i order to estimate measures of fitess ladscapes for some bechmark fuctios with the kow structure. Also, the paper gives a aalysis o how radom oise i the fitess fuctio ca affect the measures of fitess ladscapes. The paper is orgaised as follows. Sectio II gives basic iformatio about the fitess ladscape aalysis, ad mai statistical ad iformatio measures of fitess ladscapes are described. Samples of bechmark fitess ladscapes are described i Sectio III. Sectio IV presets some results of a experimetal aalysis of statistical ad iformatio measures for bechmark fitess ladscapes. The effects of a radom oise o the estimates of these measures is aalysed i Sectio V. I Sectio VI, results of optimisatio experimets with a geetic algorithm (GA) for bechmark fitess fuctios are discussed i the cotext of their ladscapes aalysis. The last sectio presets the summary of performed research. This paper is a exteded versio of the paper preseted i the UkSIM Fourth Europea Modellig Symposium o Computer Modellig ad Simulatio EMS200 [6]. II. FITNESS LANDSCAPE ANALYSIS Fitess ladscape cosists of three mai compoets []: the set of geotypes, the fitess fuctio that evaluates the geotypes ad geetics operators that defie eighbourhood relatios betwee the geotype sets. A ladscape ca be iterpreted by a surface i a search space that defies the fitess for each potetial solutio. I this case, searchig a optimal solutio may be iterpreted as walkig o the fitess ladscape surface towards the highest hill, with overcomig other hills ad valleys. The structure of the fitess ladscape iflueces the performace of a evolutioary algorithm. There are several characteristics associated with the ladscape optima that defie the structure of fitess ladscapes [2] i terms of modality, epistasis, ruggedess, ad deceptiveess. The modality evaluates a umber of optima i a search space ad a optima desity. The epistasis refers to geotype fitess depedece o multiple gees iteractio. Both high modality ad epistasis lead to a more rugged fitess ladscape. Such rugged search spaces are harder to search compared to a smoother ladscape with low epistasis ad modality [3, 4]. A umber of differet techiques has bee developed for aalysis of fitess ladscapes by evaluatig their characteristics. These techiques ca be divided i two groups that provide statistical aalysis ad iformatio aalysis of fitess ladscapes. I statistical aalysis techiques, several correlatio metrics for characterisig the structure of the ladscape are proposed. The autocorrelatio fuctio is used to measure the ruggedess of the ladscape [7]. I case of a low autocorrelatio betwee two sets of fitess poits separated by some distace, these poits have dissimilar values, ad the ladscape is more rugged. Aother correlatio metric used i practice is the correlatio legth. It defies a distace beyod which two sets of fitess poits becomes ucorrelated [8]. The magitude of this legth idicates the 38 ISSN: x olie, prit

2 smoothess of the ladscape. A loger correlatio legth idicates smoother ladscape while a shorter would idicate a more rugged ladscape. Fitess ladscape aalysis is performed o a sequece of fitess values obtaied i a radom walk. The followig formulas are used for estimatio of the fitess ladscape statistical measures [4]. Autocorrelatio fuctio r(s) betwee sets of fitess poits separated with s solutios is calculated by: s f j j f f j s f r( s) 2 f j f j where f j is value of the sequece of fitess values {f j } j=. Correlatio legth τ is defied as: lr Iformatio aalysis is aimed to obtai more iformatio about the structure of the fitess ladscape, comparig to statistical aalysis. I particular, it allows estimatig the diversity of the local optima, modality of ladscape ad the degree of regularity of radom walks [3]. Iformatio aalysis is performed based o the sequece of fitess values obtaied by a radom walk o the ladscape. The cocept of etropy proposed i classical iformatio theory is used as a basic cocept to quatify the ruggedess of a ladscape. Four iformatio measures are proposed i literature [3]. The iformatio cotet ad partial iformatio cotet are two measures of the etropy or amout of fitess chage ecoutered durig the walk i the obtaied ladscape path. They idicate the ruggedess ad the modality of the ladscape path. The iformatio stability ε* characterizes the magitude of the ladscape path slope durig the walk. The desity-basi iformatio aalyses the variety of flat ad smooth sectios o the ladscape. Both statistical ad iformatio aalysis techiques suppose that reviewed fitess ladscapes are statistically isotropic [3]. The followig formulas are used for estimatio of fitess ladscape iformatio measures [3]. Iformatio cotet H(ε) is defied as follows; H P pq log P 6 pq pq where the parameter ε cotrols the sesitivity for measurig the etropy H(ε) ad is a real umber from the iterval [0, ], where is the maximum fitess differece of the sequece {f j } j=. Probabilities P [pq] represet frequecies of possible sub-blocks pq of elemets from the strig S(ε)= s, s 2,...,s of symbols s i [,0,]. Partial iformatio cotet M(ε) is defied by: M,0,0 where Φ(,0,0) couts the slopes of the optima, that are represeted by strig S(ε). For further explaatios we refer to [3]. Fially, desity-basi iformatio h(ε) is iterpreted by: P pp log 3 P pp h p,0, where probabilities P [pp] represet frequecies of the subblocks pp from the strig S(ε) [3]. III. BENCHMARK FITNESS LANDSCAPES A. Sample fitess ladscapes The followig fitess fuctios used for bechmarkig of geetic algorithms [9, 0, ad ] are selected for estimatig ad aalysig statistical ad iformatio measures of bechmark fitess ladscapes, i.e. Sphere, Rastrigi, Rosebrock, ad Ackley fuctios. All these fuctios ca be defied i the same search domai with a similar umber of variables ad ca be easily graphically iterpreted for two variables. The Sphere fuctio for the vector defied as follows [9]: f Sphere x x X x,..., x of variables is,..., 2 x i i It is a cotiuous, covex, quadratic ad uimodal fuctio with oe global miimum equal to f Sphere = 0 i the poit x i = 0, i =..., ad does ot have the local optima. The Sphere fuctio ad its cotour plot are show i Figure. Figure. Sphere fuctio with 2 variables ad its cotour plot. So called Rastrigi fuctio has more rugged ladscape ad i case of variables is defied by: 2 f Rastrigi x x 0 xi 0 cos2 xi,..., I case of the fuctio miimisatio, it also has oe global optimum i poit of x i 0, i... with the value f Rastrigi = 0. But this fuctio is highly multimodal havig the local optima produced by its cosie compoet (see Fig. 2). i 39 ISSN: x olie, prit

3 B. Search space ad eigbourhood structure For all four fuctios a umber of variables is equal to =2, ad a search domai is defied by: 5 x i 5, i, 2 Figure 2. Rastrigi fuctio with 2 variables ad its cotour plot. The Rosebrock fuctio of variables is calculated by: 2 2 x,..., x x x x 2 f 00 Rozebrock i i i i It is cotiuous, o-covex, quartic ad uimodal fuctio with the global miimum equal to 0 i the poit x i =, i =..., located iside a parabolic shaped flat valley (see Fig. 3). While the valley ca be easily foud, it could be difficult to coverge to the global miimum. Two types of solutio represetatio are used, i.e. realvalue ecodig ad biary represetatio. I the first case variables x ad x 2 are coded as real umbers with a resolutio factor of 0.0, e.g. -3,49 ad 0,84 correspodigly. Biary coded chromosomes have legth of 20, where first 0 bits code x, but others code x 2. Each variable is coded by a formula: z x 5 02,4 where z is a decimal umber coded with a biary strig of 0. Sample biary chromosome is show i Fig x = 3, x 2 = 0, Figure 5. Sample of biary chromosome. Figure 3. Rosebrock fuctio with 2 variables ad its cotour plot. Fially, the Ackley fuctio of variables is defied by: f 2 x,..., x 20 e 20 exp 0,2 xi exp cos2x i Like the Rastrigi fuctio, it has the local optima, but the slope to the optima is expoetial. The Ackley fuctio is highly multimodal ad rugged (see Fig. 4). i Figure 4. Ackley fuctio with two variables ad its cotoir plot. i Both statistical ad iformatio aalysis techiques use by a radom walk geerated fitess sequeces, where a mutatio operator defies the way, i which solutios i a search space are coected topologically. For real-value ecodig, a mutatio operator chages each variable i a chromosome by +0.0 or 0.0 with the probability equal to /3. If all variables x i stays uchaged, the mutatio operator is re-applied util at least oe variable i this chromosome chages its value. For biary represetatio, the bitflip operator is used that chage the value of a radomly selected bit to the opposite value. Thereby eight differet fitess ladscapes are aalysed i this paper, i.e. four differet bechmark fuctios ad two types of search space for each fuctio. IV. EXPERIMENTAL ANALYSIS A. Sigle experimets Express aalysis was performed for the Sphere, Rastrigi ad Rosebrock fitess ladscapes with variables coded by real umbers. First, by a radom walk the oly sigle path i a geotype space was geerated, ad correspodig fitess values for compared ladscapes i the pheotype space were determied (e.g., see Fig. 6 for Rastrigi fuctio). The autocorrelatio for differet fuctios ad differet lags was calculated by formula (). Obtaied correlograms are show i Figure ISSN: x olie, prit

4 TABLE I. INFORMATION MEASURES OF LANDSCAPE OF RASTRIGIN FUNCTION ε H(ε) M(ε) h(ε) Figure 6. Proceeded radom path i geotype space. From statistical aalysis, the Rastrigi fuctio has more rugged ladscape that ca be also see o the fuctio cotour plot. Its autocorrelatio values for lags ad 2 are smaller tha oes for the Sphere ad Rosebrock fuctios. Autocorrelatio fuctio for lag more tha 2 teds to zero ad does ot provide ay iformatio about its ladscape. Also, the correlatio legth defied by formula (2) is less for the Rastrigi fuctio ad equal to 2.35, while for two others fuctios the correlatio legth is 3.20 ad 3.30 correspodigly. Moreover, the Sphere ad Rosebrock fuctios have quite close autocorrelatio values (Fig. 7). B. Multiple experimets For more detailed aalysis of bechmark fitess ladscapes, a software prototype for calculatio both statistical ad iformatio measures was developed ad applied. To estimate structural measures of these ladscapes, multiple experimets were performed ad ivolved multiple radomly geerated paths with the startig poit uiformly distributed i the geotype space. I the first series of experimets, the effect of the path legth o the statistical ad iformatio measures for bechmark ladscapes was aalysed. For each legth of the path o the fitess ladscape 0 experimets were performed, ad measures were estimated by their average values. Autocorrelatio fuctios calculated for differet bechmark fuctios ad lags showed idetical results (Fig. 8). While correlatio measures show depedece o the legth of the path geerated by a radom walk, the behaviour of the iformatio cotet measures does ot demostrate this effect. Figure 7. Correlograms of Sphere, Rastrigi ad Rosebrock fuctios. The the sequeces of fitess values were trasformed to strigs of symbols S(ε) for differet ε, e.g. for the Sphere fuctio ad ε= 0.04, S(ε) was defied as Iformatio measures H(ε), M(ε) ad h(ε) for bechmark ladscapes were calculated accordig to formulas (3)-(5), e.g. iformatio measures obtaied for the ladscape of the Rastrigi fuctio are give i Table I. From iformatio aalysis, for Rastrigi fuctio with a more rugged ladscape measures of the iformatio cotet H(ε) ad the modality defied by the partial iformatio cotet M(ε) for differet ε are larger tha oes for Sphere ad Rosebrock fuctios. Fially, the iformatio stability for Rastrigi fuctio is low ad equal to ε* Figure 8. Depedece of autocorrelatio o legth of time series. I the secod series of experimets, the autocorrelatio for differet bechmark ladscapes ad lags was defied for two types of solutio represetatio. I this case, 20 sequeces of fitess values o the ladscapes were geerated by a radom walk with 200 poits i each. Correlograms obtaied for real-value ad biary coded bechmark ladscapes (Fig. 9) show the higher autocorrelatio for realvalue coded fitess ladscapes that make easier search process. But the differece betwee correlogramms for differet ladscapes with the same represetatio type is mior. 4 ISSN: x olie, prit

5 Figure 9. Correlograms for real-value ad biary coded fitess ladscapes. Figure 0. Sphere fuctio correlograms for differet legth values. Figure. Iformatio cotet of the Sphere fuctio obtaied i 0 experimets ad averaged over all experimets. Figure 2. Iformatio cotet of the Rosebrock fuctio. Figure 3. Iformatio cotet for biary coded Sphere fuctio Correlograms i Fig. 0 are obtaied for real-value coded Sphere ladscape ad show the higher autocorrelatio for the loger legth of the path i a radom work. Furthermore, the depedece o lag becomes less chaotic ad more liear for large legths, for example, whe L is equal to 200. I the third series of experimets, differet iformatio measures for all bechmark ladscapes ad differet ε values were estimated. The results show the higher iformatio cotet for the smaller values of calculatio ε with a explicit peak at ε=0.03 for the real-value coded Sphere fuctio (Fig.). However, at ε = 0 that provides takig ito accouts ay small movemets o the ladscape ad L=200, iformatio measures become idetical ad do t give ew iformatio about performace of the fitess ladscape. I this case, the iformatio cotet teds to (see Fig. ), the partial iformatio cotet to 0.5, ad the desity-basi iformatio to At the same time, smaller values of the iformatio cotet for the Rosebrock fuctio (Fig. 2) compared to the Sphere (Fig. ) idicate the higher degree of flatess with respect to rugged areas of the ladscape. Fig. 3 illustrates the depedece of iformatio cotet measure H(ε) o ε for differet legth L of the path obtaied by a radom walk o the Sphere fuctio. The curves obtaied showed o sigificat differeces for the five differet legths L. C. Local ladscape experimets Multiple experimets were also performed for three small areas o the Sphere bechmark ladscape located ear the global miimum, close to the local maximum ad betwee them. A startig poit i a radom walk i each local search was fixed ad defied by <0.05; 0.0>, <4.49; 4.82>, ad 42 ISSN: x olie, prit

6 < 3.7;.23>, correspodigly. The results obtaied for the Sphere fuctio show (Fig. 4) o sigificat sesitivity of the autocorrelatio to a startig poit locatio associated to the local structures with fitess mootoic chages. The magitude of fitess chage durig the walk betwee eighbourig poits has a big impact o the iformatio cotet measures. Therefore iformatio cotet (Fig. 5) shows high sesitivity to a parameter ε so that ε values eed to be carefully defied for differet local areas of the ladscape. V. NOISE AND MEASURES OF LANDSCAPES Additioal experimets were performed to defie the affects of a radom oise i fitess o statistical ad iformatio measures of the bechmark fitess ladscape. I this case, the fitess fuctio f* is described as follows: f* = f + ξ where f is the true fitess fuctio, ad ξ is a term that represets oise effects ot accouted i f that is treated as a statistical error ad assumed to be ormally distributed with mea zero ad variace σ 2. The results of experimets with 20 rus ad 200 poits i each ru show that both statistical ad iformatio measures are quite sesitive to oise. For the real-value coded Sphere fuctio with the rage of values [0; 50], a radom oise has a explicit impact o the degree of the autocorrelatio for σ 2 = Furthermore, with icrease of variace, comparig with the true fitess fuctio the autocorrelatio gets lower for shorter lags ad higher for loger lags. Additioally, a radom oise icreases the etropy of the true fitess ladscape structures. Graphs of the iformatio cotet (Fig. 7) show the etropy of these ladscape structures geerated by a radom oise for differet sesitivity values ε. VI. GA OPTIMISATION EXPERIMENTS To see the correlatio betwee the results of fitess ladscape aalysis ad hardess of a real problem for a evolutioary algorithm, a series of optimisatio experimets were performed with bechmark fitess fuctios. GA with oe poit crossover with rate 0.75 ad above described mutatio operator was used to estimate a cumulative probability of success (CPS) [0] for differet bechmark ladscapes. Multiple optimisatio experimets with differet umbers of geeratios were performed for both biary coded ad real-valued fitess ladscapes. I each series, 00 optimisatio experimets were performed with populatio of 500 chromosomes. The results of optimisatios experimets show that for biary coded Sphere, Rastrigi ad Ackley fuctios the global optimum is foud i about 50-60% rus of more tha 20 geeratios. The CPS does t become higher for a larger umber of geeratios. I case of the Rosebrock fuctio the global optimum is foud i 0-20% rus. Figure 4. Autocorrelatio aalysis of Sphere fuctio, for differet areas of ladscape. Figure 5. Iformatio cotet aalysis of Sphere fuctio, for differet areas of ladscape. Figure 6. Autocorrelatio of Sphere fuctio with radom oise. Figure 7. Iformatio cotet of Sphere fuctio with radom oise. 43 ISSN: x olie, prit

7 Figure 8. Solutios for 2 geeratios with real-valued chromosomes (Rosebrock fuctio). Figure 9. Solutios for 2 geeratios with biary chromosomes (Rosebrock fuctio). I geeral (except the Rosebrock fuctio), GA foud solutios o the real-value coded bechmark ladscape are better tha o the biary coded ladscape that was predicted withi statistical aalysis. As the autocorrelatio betwee eighbourhood fitess poits is higher (see Fig. 9), it is easier for GA to move to a poit with better fitess. Nevertheless, optimisatio experimets show that for applied GA the Rosebrock fuctio is harder tha three other bechmark fuctios. This could happe due to techiques of fitess ladscape aalysis do ot idetify the structures that make search with GA difficult. Also, o iformatio about crossover operator is used i fitess ladscape aalysis. To fid out the reasos of the GA performace, dyamics of populatios i differet geeratios was aalysed. Distributio of foud solutios i populatios i the fifth (crosses) ad sixteeth geeratios (poits) i the search space is illustrated i Figure 8 where solutios are ecoded as real-valued chromosomes. I the fifth geeratio all solutios are located o the both sides of the valley (see Fig. 3). But the global optimum <, > is o the outer edge of the cloud of solutios i the sixteeth geeratio. Here, the populatio teds to the cetre of the valley, ot to the global optimum. This tedecy was ot observed for the GA workig with biary chromosomes. For the biary ecoded fitess ladscape of Rosebrock fuctio, i the fifth geeratio solutios cover all valley area ad i the sixteeth geeratio populatio coverges to the global optimum. For other three fuctios GA coverge to the global optimum much better for both types of ecodig. I the fifth geeratio solutios are located ear local optimums, ad i the sixteeth geeratio almost all solutios are close to the global optimum. VII. CONCLUSIONS Fitess ladscape aalysis provides efficiet techiques for determiatio of the optimisatio problem hardess for evolutio algorithms. A umber of differet techiques ad measures (autocorrelatio, correlatio legth, iformatio cotet, iformatio stability, etc.) have bee developed for aalysig structural characteristics of fitess ladscapes that ifluece performace of a evolutioary algorithm. I particularly, these allow measurig ruggedess of the ladscape, estimatig its modality ad the diversity of the local optima, the degree of regularity of radom walks. The result of this aalysis is depedet o may parameters such as represetatio type of the solutios, legth of the part i the radom walk, sesitivity parameter that cotrol measurig itself, etc. Also, statistical ad iformatio measures of fitess ladscape aalysis are very sesitive to radom oise that icreases the etropy of the true fitess ladscape structures. Fitess ladscape aalysis for bechmark fuctios allows fidig out relatioships betwee the ladscape measures ad these parameters; ad betwee the results of fitess ladscape aalysis ad hardess of a real problem for a evolutioary algorithm by performig optimisatio experimets with bechmark fitess fuctios. I future research, a set of bechmark fuctios will be exteded, icludig fuctios that have more rugged fitess ladscape ad are complicated i structure with a variety of differet local structures. This will provide more detailed iformatio o the correlatio betwee the fitess ladscape aalysis measures ad the real hardess of evolutioary algorithm for these ladscapes. ACKNOWLEDGMENTS This work has bee supported by the Europea Social Fud withi the project Support for the implemetatio of doctoral studies at Riga Techical Uiversity. REFERENCES [] T. Joes, Evolutioary Algorithms, Fitess Ladscapes ad Search, Albuquerque: The Uiversity of New Mexico, p. [2] C. R. Reeves, J. E. Rowe, Geetic Algorithms - Priciples ad Perspectives. A Guide to GA Theory, Spriger, p. [3] V. K. Vassilev, T. C. Fogarty, J. F. Miller, Iformatio Characteristics ad the Structure of Ladscapes, Evolutioary Computatio, vol. 8, issue, Mar. 2000, pp ISSN: x olie, prit

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