On the Virtues of Parameterized Uniform Crossover

Transcription

1 On the Vrtues of Parameterzed Unform Crossover Wllam M. Spears Naval Research aboratory Washngton, D.C USA Kenneth A. De Jong George Mason Unversty Farfax, VA USA Abstract Tradtonally, genetc algorthms have reled upon and 2-pont crossover operators. Many recent emprcal studes, however, have shown the benefts of hgher numbers of crossover ponts. Some of the most ntrgung recent work has focused on unform crossover, whch nvolves on the average /2 crossover ponts for strngs of length. Theoretcal results suggest that, from the vew of hyperplane samplng dsrupton, unform crossover has few redeemng features. However, a growng body of expermental evdence suggests otherwse. In ths paper, we attempt to reconcle these opposng vews of unform crossover and present a framework for understandng ts vrtues. Introducton One of the unque aspects of the work nvolvng genetc algorthms (GAs) s the mportant role that recombnaton plays. In most GAs, recombnaton s mplemented by means of a crossover operator whch operates on pars of ndvduals (parents) to produce new offsprng by exchangng segments from the parents genetc materal. Tradtonally, the number of crossover ponts (whch determnes how many segments are exchanged) has been fxed at a very low constant value of or 2. Support for ths decson came from early work of both a theoretcal and emprcal nature [Holland, 975; DeJong, 975]. However, there contnue to be ndcatons that there are stuatons n whch havng a hgher number of crossover ponts s benefcal [Syswerda, 989; Eschelman, 989]. Perhaps the most surprsng result (from a tradtonal perspectve) s the effectveness on some problems of unform crossover, an operator whch produces on the average /2 crossngs on strngs of length [Syswerda, 989]. Recent work by [Spears and De Jong, 990] has extended the theoretcal analyss of n-pont and unform crossover wth respect to dsrupton of samplng dstrbutons. However, they ponted out that dsrupton analyss alone s not suffcent n general to predct and/or select optmal forms of crossover. In partcular, they have shown that the populaton sze must also be taken nto account [DeJong and Spears, 990]. Ths paper extends that work by lookng at the propertes of a parameterzed unform crossover operator and by consderng two other aspects of crossover operators, namely, ther recombnaton potental and ther exploratory power. In ths context, a surprsngly postve vew of unform crossover emerges. 2 Dsrupton Analyss Holland provded the ntal formal analyss of the behavor of GAs by showng how they allocate trals n a near optmal way to competng low order hyperplanes f the dsruptve effects of the genetc operators used s not too severe [Holland, 975]. Snce mutaton s typcally run at a very low rate, t s generally gnored as a sgnfcant source of dsrupton. However, crossover s usually appled at a very hgh rate. So, consderable attenton has been gven to estmatng P d, the probablty that a partcular applcaton of crossover wll be dsruptve. Holland s ntal analyss of the samplng dsrupton of -pont crossover [Holland, 975] has been extended to n-pont and unform crossover [DeJong, 975; Spears and DeJong, 990]. These results are n the form of estmates of the lkelhood that the samplng of a kth order hyperplane (H k ) wll be dsrupted by a partcular form of crossover. It turns out to be easer mathematcally to estmate the complement of dsrupton: the lkelhood of a sample survvng crossover (whch we denote as P s ). As one mght expect, the results are a functon of both the order k of the hyperplane and ts defnng length (see the Appendx and [Spears and DeJong, 990] for more precse detals). We provde n Fgure a graphcal summary of a typcal nstance of these results for the case of 3rd order hyperplanes. The non-horzontal curves represent the survval

2 of 3rd order hyperplanes under n-pont crossover (n =...6). The horzontal lne represents the probablty of survval under unform crossover. Fgure hghlghts two mportant ponts. Frst, f we nterpret the area above a partcular curve as a measure of the cumulatve dsrupton potental of ts assocated crossover operator, then these curves suggest that 2-pont crossover s the least dsruptve of the n-pont crossover famly, and less dsruptve than unform crossover. Fnally, unlke n-pont crossover, unform crossover dsrupts all hyperplanes of order k wth equal probablty, regardless of how long or short ther defnng lengths are. 3 A Postve Vew of Crossover Dsrupton A recurrng theme n Holland s work s the mportance of a proper balance between exploraton and explotaton when adaptvely searchng an unknown space for hgh performance solutons [Holland, 975]. The dsrupton analyss of the prevous secton mplctly assumes that dsrupton of the samplng dstrbutons s a bad thng and to be avoded (e.g., a hgh dsrupton may stress exploraton at the expense of explotaton). However, ths s not always the case. There are mportant stuatons n whch mnmzng dsrupton hnders the adaptve search process by overemphaszng explotaton at the expense of needed exploraton. One of the clearest examples of ths s when the populaton sze s too small to provde the necessary samplng accuracy for complex search spaces [DeJong and Spears, 990]. To llustrate ths we have selected a 30 bt problem wth 6 peaks from [DeJong and Spears, 990]. The measure of performance s smply the best ndvdual found by the genetc algorthm. Ths s plotted every 00 evaluatons. Snce we are maxmzng, hgher curves represent better performance. Fgures 2 and 3 llustrate the effect of populaton sze on GA performance. Notce how unform crossover domnates 2-pont crossover on the 6-Peak problem wth a small populaton, but just the opposte s true wth a large populaton. One concluson of these results mght be that we should mantan a portfolo of crossover operators and study the effects of varous combnatons. We have been examnng another approach: achevng a better balance of exploraton and explotaton usng only unform crossover. We are ntrgued by ths possblty for two reasons: ts smplcty (only one crossover form) and ts potental for ncreased robustness because the dsruptve effect of unform crossover s not nfluenced by hyperplane defnng length. unform Evaluatons * 00 Fgure 2: 6-Peak (30 bts) - Populaton 20 P s pt unform / 2 Defnng ength pt Evaluatons * 00 unform Fgure. Survval of 3rd Order Hyperplanes Fgure 3: 6-Peak (30 bts) - Populaton 000 2

3 4 A Closer ook at Unform Crossover It s clear that the level of dsrupton provded by unform crossover s too hgh n many cases (e.g., when large populatons are used). Ths standard form of unform crossover swaps two parents alleles wth a probablty of. Suppose, however, that we parameterze unform crossover, where P 0 denotes the probablty of swappng. We can now consder the effect of decreasng P 0. Fgure 4 llustrates ths for 3rd order hyperplanes. Notce how the dsrupton of unform crossover can be controlled by lowerng P 0, wthout affectng the property that the dsrupton has no defnng length bas. In partcular, note that by smply lowerng P 0 to., unform crossover s less dsruptve (overall) than 2-pont crossover and has no defnng length bas! Ths suggests a much more postve vew of the potental of unform crossover, namely, an unbased recombnaton operator whose dsrupton potental can be easly controlled by a sngle parameter P 0. To test ths hypothess, we have run a number of experments n whch P 0 vared. As expected, we can ncrease and decrease performance on a gven problem wth a fxed populaton sze smply by varyng P 0. Fgure 5 llustrates ths on the 6-Peak problem. Note that n ths partcular case, a value of P 0 = 0.2 produced the best results. Referrng back to Fgures 3 and 4, we can now see why. For the 6-Peak problem, a populaton sze of 000 has suffcent samplng capacty to requre only the dsrupton level provded by 2-pont crossover. Unform crossover wth P 0 = 0.2 provdes approxmately the same level of dsrupton but wthout the length bas. P s pt.5 unform / 2 Defnng ength.0 unform. unform.2 unform pt Fgure 4. Survval of 3rd Order Hyperplanes Note that we do not need to consder the possblty of ncreasng P 0, due to the symmetry of unform crossover..2 unform.0 unform.5 unform Evaluatons * 00 Fgure 5: 6-Peak (30 bts) - Populaton 000 Is ths lack of length bas really mportant? Intutvely, t should help overcome representaton problems n whch mportant hyperplanes happen to have defnng lengths whch are adversely affected by the partcular n-pont crossover operator n use. Syswerda llustrated ths clearly wth hs "sparse -max" problem n whch 270 fake bts were appended to a 30-bt problem [Syswerda, 989]. One can show smlar results wth almost any problem. Fgure 6 llustrates ths on our 6-Peak problem appended wth 270 fake bts and the same evaluaton functon. Notce that, n comparson to the orgnal 30-bt problem shown n Fgure 5, the performance of 2-pont crossover s worse, whle the performance of unform crossover (P 0 =.2) remans essentally unchanged. How do we explan the drop n performance of 2-pont crossover? In ths case, the 30 mportant bts are all.2 unform Evaluatons * 00 Fgure 6: 6-Peak (300 bts) - Populaton 000 3

4 wthn a dstance of /0 of each other (where s the length of the strng). If we examne Fgure 4, we note that 2-pont crossover s less dsruptve wthn that range (0 to /0) of defnng lengths. In other words, the addton of 270 addtonal bts effectvely decreases the dsrupton of the mportant hyperplanes under 2-pont crossover. Ths effect s most obvous towards the end of the runs (see Fgure 6), where dsrupton s ncreasngly useful (due to the ncreasng homogenety of the populaton). Unform crossover s not nfluenced by the added 270 bts, snce t s nsenstve to defnng length. In summary, we see two mportant vrtues of unform crossover. The frst s the ease wth whch the dsruptve effect of unform crossover can be controlled by varyng P 0. Ths s useful n achevng the proper balance between exploraton and explotaton. The second vrtue s that the dsruptve potental of unform crossover does not depend on the defnng length of hyperplanes. Ths allows unform crossover to perform equally well, regardless of the dstrbuton of mportant alleles. 5 Recombnaton Potental Another possble vrtue of unform crossover that has been dscussed n the lterature s ts recombnaton potental. In comparng unform, and 2-pont crossover, Syswerda felt that unform crossover ganed sgnfcant advantage from ts ablty to combne small buldng blocks nto larger ones [Syswerda, 989]. He defned recombnaton potental as the ablty of crossover to create hgher order hyperplanes when the parents contan the necessary lower order hyperplanes. He provded an analyss showng unform crossover (P 0 =.5) to have a hgher recombnaton potental than and 2-pont crossover. Syswerda ponted out that recombnaton can be consdered to be a specalzed form of survval, n whch two lower order hyperplanes survve onto the same strng, resultng n a hgher order hyperplane. Ths observaton allowed Syswerda to construct a recombnaton analyss from hs survval analyss. However, snce hs survval analyss was lmted to and 2-pont crossover, and to unform crossover wth a P 0 of.5, hs recombnaton analyss was smlarly lmted. Ths motvated us to create a new recombnaton analyss n a smlar ven, snce our survval analyss ncludes all of n-pont crossover and a parameterzed unform crossover. In [Spears and DeJong, 990], we developed a survval analyss for n-pont crossover and a parameterzed (P 0 ) unform crossover. Detals of ths analyss, and our recombnaton analyss, are presented n the Appendx. Fgure 7 llustrates the relatonshps of the crossover operators n terms of ther recombnaton potental (we denote P r as the probablty of recombnaton). Note specfcally that there s evdence to support the clam that unform crossover (P 0 =.5) has a hgher P r.5 unform / 2 Defnng ength. unform.0 unform Fgure 7: 3rd Order Hyperplane Recombnaton pt recombnaton potental than the other crossover operators. However, t s even more nterestng to note that these relatonshps are qualtatvely dentcal to those shown n Fgure 4. In other words, f one operator s better than another for survval, t s worse for recombnaton (and vce versa). Ths observaton appears to hold for all k, and suggests very strongly that the recombnaton analyss tells us nothng new about crossover. 6 Exploraton Power It has also been ponted out that dsrupton does not necessarly mean useful exploraton. Crossover dsrupton smply mples that a hyperplane sample has been modfed by crossover n some way so as to no longer be a member of that hyperplane, wthout any ndcaton as to the possble forms that change mght take. The potental number of ways n whch a crossover operator can effect a change has been called ts exploratory power. It has been ponted out that unform crossover has the addtonal property that t has more exploratory power than n-pont crossover [Eschelman, 989]. To see that ths s true, consder the extreme case n whch one parent s a strng of all 0s and the other all s. Clearly unform crossover can produce offsprng anywhere n the space whle and 2-pont crossover are restrcted to rather small subsets. In general, unform crossover s much more lkely to dstrbute ts dsruptve trals n an unbased manner over larger portons of the space. The dffculty comes n analyzng whether ths exploratory power s a vrtue. If we thnk of explotaton as the based component of the adaptve search process, t makes sense to balance ths wth unbased exploraton. Clearly, ths exploratory power can help n the early generatons, partcularly wth smaller populaton szes, to make sure the whole space s well sampled. At the same 4

5 tme, some of ths exploratory power can be acheved over several generatons va repeated applcatons of and 2-pont crossover. Unfortunately, our current analyss tools do not allow us to make comparsons of propertes whch span generatons and are strongly affected by selecton. Hopefully we wll develop such tools and resolve questons of ths type n the near future. 7 Conclusons and Further Work The extensons to the analyss of n-pont and unform crossover presented n ths paper open up an nterestng and postve vew of the usefulness of unform crossover. There appear to be three potentally mportant vrtues of unform crossover. Frst, the dsrupton of hyperplane samplng under unform crossover does not depend on the defnng length of the hyperplanes. Ths reduces the possblty of representaton effects, snce there s no defnng length bas. Second, the dsrupton potental s easly controlled va a sngle parameter P 0. Ths suggests the need for only one crossover form (unform crossover), whch s adapted to dfferent stuatons by adjustng P 0. Fnally, when a dsrupton does occur, unform crossover results n a mnmally based exploraton of the space beng searched. The frst two vrtues have been confrmed both theoretcally and expermentally. At the same tme, t should be emphaszed that the emprcal studes presented are lmted to a carefully controlled expermental settng. The authors are currently workng on expandng these experments and on developng an exploraton theory for recombnaton operators. Our goal s to understand these nteractons well enough so that GAs can be desgned to be self-selectng wth respect to such decsons as optmal populaton sze and level of dsrupton. Acknowledgements We would lke to thank Dana Gordon for pontng out flaws n our prelmnary recombnaton analyss. References De Jong, Kenneth A. (975). An Analyss of the Behavor of a Class of Genetc Adaptve Systems, Doctoral Thess, Department of Computer and Communcaton Scences, Unversty of Mchgan, Ann Arbor. De Jong, K. A. & Spears, W. (990). An Analyss of of the Interactng Roles of Populaton Sze and Crossover n Genetc Algorthms, Proceedngs of the Frst Int l Conf. on Parallel Problem Solvng from Nature, Dortmund, Germany, October 990. Eschelman,., Caruana, R. & Schaffer, D. (989). Bases n the Crossover andscape, Proc. 3rd Int l Conference on Genetc Algorthms, Morgan Kaufman Publshng. Holland, John H. (975). Adaptaton n Natural and Artfcal Systems, The Unversty of Mchgan Press. Spears, W. & De Jong, K. A. (990). An Analyss of Mult-pont Crossover, Proceedngs of the Foundatons of Genetc Algorthms Workshop, Indana, July 990. Syswerda, Glbert. (989). Unform Crossover n Genetc Algorthms, Proc. 3rd Int l Conference on Genetc Algorthms, Morgan Kaufman Publshng. Appendx Summary of the Survval Analyss For n-pont crossover, P s s expressed n the order dependent form (P k,s ): P 2,s ( n,, ) = and Σ n n = 0 P k,s ( n,,,..., k ) = Σ n n = 0 n C s n P k,s (,,..., k ) Note that the survval of a kth order hyperplane under n- pont crossover s recursvely defned n terms of the survval of lower order hyperplanes. refers to the length of the ndvduals. The... k refer to the defnng lengths between the defnng postons of the kth order hyperplane. The effect of the recurson and summaton s to consder every possble placement of n crossover ponts wthn the kth order hyperplane. The correcton factor C s computes the probablty that the hyperplane wll survve, based on that placement of crossover ponts. Suppose that crossover results n x of the k defnng postons beng exchanged. Then the hyperplane wll survve f: ) the parents match on all x postons beng exchanged, or 2) f they match on all k x postons not beng exchanged, or 3) they match on all k defnng postons. Hence, the general form of the correcton s: C s = P eq x + P eq k x P eq k where P eq s the probablty of two parents sharng an allele at each locus, and the P k eq reflects an overlap wthn the 3 possbltes (and hence must be subtracted). 5

6 As an example, consder Fgure 8. The two parents are denoted by P and P2. In ths fgure, we represent the survval of a 4th order hyperplane. The hyperplane defnng postons are depcted wth crcles. Snce of the defnng postons wll be exchanged (under the 2-pont crossover shown), the probablty of survval s: P: P2: C s = P eq + P eq 3 P eq 4 Fgure 8: 4th Order Hyperplane Survval of the k defnng postons survvng n the same ndvdual (.e., x s a subset of the m + n defnng postons), then recombnaton wll occur f: ) the parents match on all of the x postons, or 2) f they match on all k x postons, or 3) they match on all k defnng postons. Hence, the general form of the recombnaton correcton C r s: C r = P eq x + P eq k x P eq k Note the smlarty n descrpton wth the survval correcton factor C s (the only dfference s n how x s defned). In other words, gven a kth order hyperplane, and two hyperplanes of order n and m, P k,r s smply P k,s wth the correcton factor redefned as above. As an example, consder Fgure 9. In ths fgure, we represent the recombnaton of 2 2nd order hyperplanes. One hyperplane s depcted wth crcles, and the other wth rectangles. Snce 3 of the defnng postons wll survve onto the same ndvdual (under the 2-pont crossover shown), the probablty of survval s: C r = P eq 3 + P eq P eq 4 For parameterzed unform crossover, P s s also expressed n an order dependent form (P k,s ): P k,s (H k ) = P: Σ k!" k#$ (P 0 ) ( P 0 ) k (P k eq + P eq P k eq ) = 0 where P 0 s the probablty of swappng two parents alleles at each locus. A graphcal representaton of these equatons has been shown prevously n Fgure 4. Recombnaton Analyss for N-Pont Crossover In our defnton of survval, t s possble for a hyperplane to survve n ether chld. Recombnaton can be consdered a restrcted form of survval, n whch two lower order hyperplanes survve to form a hgher order hyperplane. The dfference s that the two lower order hyperplanes (each of whch exsts n one parent) must survve n the same ndvdual, n order for recombnaton to occur. In the remanng dscusson we wll consder the creaton of a kth order hyperplane from two hyperplanes of order m and n. We wll restrct the stuaton such that the two lower order hyperplanes are non-overlappng, and k = m + n. Each lower order hyperplane s n a dfferent parent. We denote the probablty that the kth order hyperplane wll be recombned from the two hyperplanes as P k,r. An analyss of recombnaton under n-pont crossover s smple f one consders the correcton factor C s defned earler for the survval analyss. Recall that recombnaton wll occur f both lower order hyperplanes survve n the same ndvdual. If an n-pont crossover results n x P2: Fgure 9: 2nd Order Hyperplane Recombnaton Recombnaton Analyss for Unform Crossover The analyss of recombnaton under unform crossover also nvolves the analyss of the orgnal survval equaton. Note that, due to the ndependence of the operator (each allele s swapped wth probablty P 0 ), the survval equaton can be dvded nto three parts. The frst part expresses the probablty that a hyperplane wll survve n the orgnal strng: P k,s,org (H k ) = Σ k %& k'( (P 0 ) ( P 0 ) k k (P eq ) = 0 The second part expresses the probablty that a hyperplane wll survve n the other strng: P k,s,other (H k ) = Σ k )* k+, (P 0 ) ( P 0 ) k (P eq ) = 0 The fnal part expresses the probablty that a hyperplane wll exst n both strngs: 6

7 P k,s,both (H k ) = Σ k -. k/0 (P 0 ) ( P 0 ) k (P k k eq ) = P eq = 0 Then: P k,s (H k ) = P k,s,org (H k ) + P k,s,other (H k ) P k,s,both (H k ) Note, however, that ths formulaton allows us to express recombnaton under unform crossover. Agan, assumng the recombnaton of two non-overlappng hyperplanes of order n and m nto a hyperplane of order k: P k,r (H k ) = P m,s,org (H m ) P n,s,other (H n ) + P m,s,other (H m ) P n,s,org (H n ) P m,s,both (H m ) P n,s,both (H n ) Ths equaton reflects the decomposton of recombnaton nto two ndependent survval events. The frst term s the probablty that H m wll survve on the orgnal strng, whle H n swtches (.e., both hyperplanes survve on one parent). The second term s the probablty that both hyperplanes survve on the other parent. The thrd term reflects the jont probablty that both hyperplanes survve on both strngs, and must be subtracted. Fnally, t s nterestng to note that the last term s equvalent to P m n eq P eq = P k eq. 7