69

Population fixed-points for functions of uni ta t ion

J o n a t h a n E . R o w e A r t i f i c i a l Intelligence G r o u p

Dept . Compute r I n f o r m a t i o n Science De M o n t f o r t Univers i ty

.Mil ton Keynes M K 7 6HP Great B r i t a i n

j roweS-dmu.ac.uk

Abstract

T h e dynamica l systems model for the s imple genetic a l g o r i t h m due to N'ose [12] can be s impl i f ied to the ca,se of zero crossover, and to fitness funct ions t ha t d iv ide the search space in to re la t ively few equivalence classes. T h i s produces a low-dimens iona l system for which the fixed-point can be calculated; i t is the leading eigenvector o f the system. Th i s technique, applied elsewhere [11] to Royal Road funct ions, is adapted in this paper to apply to f tmctions of i m i t a t i o n . In f in i te popu la t i on fixed-points are calculated for some simple examples, i nc lud ing t r ap funct ions t ha t have previously been analysed in terms o f deception [2]. The effects of eigenvectors outside the popu la t ion space are explored, and finite popu la t ion behaviour is examined in this way. T w o surpris ing theorems are demonst ra ted for in f in i te populat ions; t ha t every popu la t ion d i s t r i b u t i o n is the fi.xed-point for some fitness funct ion; and tha t fitness functions exist for which evo lu t ion goes "backwards" towards the global m i n i m u m . Theore t ica l results are backed up w i t h exper iments using real, finite popu la t i on genetic a lgor i thms .

1 Introduction

The d y n a m i c a l systems model for the s imple genetic a l g o r i t h m (SG.-V) suggested earlier by Michae l Vose [12] represents popula t ions as vectors p= ( p o . . . . . Ps-1), where s is the size o f the search space ( tha t is, the number o f different chromosomes). In this vector, pj is the p r o p o r t i o n of the popu la t ion taken up by the chromosome corresponding to j . The set o f

Jonathan E. R o w e

vectors:

A = | ( : i - o , . . . , a - . - i ) : X ] , i j = > 0

[ i=° )

is Icnown as the simplex. Clearly, al l real populat ions correspond to points w i t h i n the s implex . However, i t is theore t ica l ly interest ing to also consider vectors outside the s implex (w^ith some Xj < 0) , but whose components s t i l l sum to 1. I n par t icu la r , i t w i l l be seen that fixed-points outside the s implex may afTect the behaviour inside the s implex .

T h e ac t ion of the SGA on a popu la t i on p is given by the heurist ic operator

g = M o T

(where o denotes operator compos i t ion ) . Each component [Qp)] is the p r o b a b i l i t y t ha t chromosome j is sampled in the next generat ion. I t is known tha t Qp is the expected next generat ion, and t h a t the actual t r ans i t ion rule f rom one generation to the next approaches Qp as the p o p u l a t i o n size tends to in f in i ty . For this reason, this is sometimes referred to as the in f in i te popu la t i on mode l .

T gives the effects of p ropor t iona l selection and is defined in terms o f a diagonal m a t r i x :

where f(k) is the fitness o f chromosome k. The selection operator is then:

where (f){p) is the average (mean) fitness of the popu la t ion p given by

Note tha t this def in i t ion is val id also for ;> outside the s implex.

T h e effects of one-point crossover and m u t a t i o n (at a given rate) are given by M (see [12] for detai ls) . For the case of zero cro.s,sover, this is s imply a m a t r i x M such tha t Mtj is the p r o b a b i l i t y t ha t chromosome j is mu ta t ed to chromosome i. Assuming a non-zero m u t a t i o n rate and a non-zero fitness func t ion , the fo l lowing facts are k n o w n about this case:

1. F ixed- jx i in t s o f C7 are eigenvectors of G = A / 5 , once they have been normalised so tha t their components sum to one.

2. Eigenvalues of G give the average fitness of the corresponding fixed-point popula t ions .

3. Exact ly one normalised eigenvector of G' is in the s implex A.

4. T h i s eigenvector corresponds to the (simple) m a x i m a l eigenvalue.

5. T h e sequence p,Qp,Q'-p, . . . converges for each p € .'\t is, Q is fnriif^ed).

Population Fixed-Points for Functions of Unitation 71

The first two points fol low f rom simi^le algebra. The other points come f rom the Perron-Frobenius theorem, which applies because G is posi t ive (see for example [7]) , and the fact t ha t X is convex.

In practice, i t may be diff icul t to calculate the eigensystem o f G, no t least because the matrices involved are extremely large. G is an s x s m a t r i x where s is the number o f chromosomes. Thus s = 2" , for b inary strings o f length n. T h e s i tua t ion can be made more t rac table i f one can define a set o f equivalence classes on the set of chromosomes which is respected by the fitness funct ion . T h i s is the approach taken by van Ni inwegen. Cru tchf ie ld and M i t c h e l l [ I I ] in analysing the dynamics o f a s imple genetic a l g o r i t h m on the cla.ss o f Royal Road fitness functions [4]. A n overview of thei r results can be found i n [10]. T h i s approach is feasible because the number of dis t inguishable fitness values t ha t can be taken on i n a Royal Road funct ion is very l i m i t e d (see [12] for a further theoret ical t reatment o f this idea) .

The m a i n focus of the s tudy in [ I I ] is the effect of finite popula t ions (for cer ta in m u t a t i o n rates) in p roduc ing metastable states. D u r i n g a typ ica l run , the p o p u l a t i o n w i l l tend t o l inger in certain n o n - o p t i m a l states. Th i s is due to the influence o f fixed-points outside the s implex . I f there is a fixed-point outside, but very close to the s implex, the force \Qp — p\ w i l l be very smal l close to that fixed-point, since Q is cont inuous. So a real p o p u l a t i o n , inside the s implex , w i l l have an expected next generation very close to itself, i f i t is near t h a t fixed-point. For a finite popu la t ion , there are only a l i m i t e d number o f viable poin ts w i t h i n the s implex , and so the popu la t ion may stay in the v i c i n i t y of tha t po in t u n t i l a freak m u t a t i o n sends i t a su i tab ly long distance away. '

T h i s paper applies the same technique to functions of u n i t a t i o n . These are functions of b i t s t r ings t h a t depend on ly on the number of ones in a s t r ing . The set of chromosomes can therefore be d iv ided up in to equivalence classes based on this u n i t a t i o n value. Once appropr ia te selection and m u t a t i o n matrices have been defined, the fixed-points o f some s imple fitness funct ions w i l l be s tudied. I t w i l l be seen tha t the in terplay between fitness func t ion and the m u t a t i o n rate can be c r i t i ca l in de t e rmin ing where in the search space the fixed-point w i l l l ie.

2 Statistical dynamics for functions of unitation

Suppose we have a func t ion o f u n i t a t i o n :

» : (0 , 1 } " -)?

where t((.f) depends only on the number of ones in x and is always non-negative. T h a t is, u is en t i re ly defined by a funct ion

/ : { 0 , . . . , n }

By considering on ly the equivalence classes of chromosomes w i t h equal numbers o f ones, we can represent an inf in i te popu la t ion by a vector

= ( P O . P l Pn]

' A n alternative suggestion for finding metastable states for finite populations is to perform all eigensystem calculations to a precision of 1/m, where m is the population size. Experiments wi th this approach have so far had only occasional sticcess.

Jonathan E. Rowe

where p t is t l ie p r o p o r t i o n of t l ie ] )opnla t io i i having exactly k ones. We now define the selection m a t r i x S to be a diagonal m a t r i x whose entries are

S k , , = .f(A')

The m u t a t i o n m a t r i x M, who.se entries give the probabi l i t ies tha t a s t r ing w i t h j ones becomes a s t r ing w i t h / ones, is given by

k = tl / = ( 1 ^ / V /

where q is the m u t a t i o n p robab i l i ty , and dj;y is the Kronecker del ta funct ion:

I \f X = y

The m a t r i x . 1 / is the same as the m a t r i x T in [11],

We now have MSp

C/v = • {fW

where Q is the heurist ic operator, adapted for our u n i t a t i o n equivalence clas.ses. Given any popu la t i on p, the expected next generation is Qp. and moreover this is the s amp l ing d i s t r i b u t i o n for the next generat ion. I f r is a fixed-point of C7 then

MSv

{f)iv)

and ,so Gi'^ MSv = {f){v)c •

which means v is an eigenvector of 6' = MS w i t h its average fitness as the corresponding eigenvalue.

,A.s ou t l ined above, i f there are no zero fitness values, the only (normalised) eigenvector in the s implex is the leading one, which demonstrates tha t there is a unique fi.xed-point for non-zero funct ions of u n i t a t i o n . I f the fitness funct ion can take on a zero value, then a p o p u l a t i o n compr i s ing copies of such a zero-valued s t r ing is also a fixed-point. However, any s t r ing w i t h zero fitne.ss w i l l not get selected for the next generation under p ropo r t i ona l selection, and so in practice we can ignore such cases.

We are now in a pos i t ion to consider some example functions of u n i t a t i o n and calculate the eigenvectors of G'. A l l calculat ions presented were computed using M a t h e m a t i c a ver­sion 2.2.3 r u n n i n g on Microsoft W i n d o w s 95. A number of exper imenta l runs o f a genetic a l g o r i t h m are also presented. Unless otherwise stated, a popu la t ion size o f 500 is used ( in i t i a l i sed at r a n d o m ) , w i t h experiments las t ing for 200 generations, and averaged over 10 runs. Er ro r bars o f one s tandard dev ia t ion are shown. For these runs i t should be noted tha t Stochastic f ln iversa l Sam])l ing (SI.'S) [1] was used, rather than p ropor t iona l selection. SUS ensures tha t , w i t h i n the l i m i t s of the given popu la t ion size, a popu la t ion as near to tha t exjiected for p ropor t iona l selection is generated. The above theory should s t i l l produce a good a p p r o x i m a t i o n of the fixed-jjoints for this selection scheme. However, i^opulations arc

proportion

Population Fixed-Points for Functions of Unitation 73

unitation 5 10 15 20

Figure 1: F ixed-poin t popu la t i on d i s t r i b u t i o n for onemax: n = 20, f/ = 0.05.

l ike ly to exh ib i t much less va r ia t ion f r o m one generation to the next t h a n w i t h p r o p o r t i o n a l selection. T h e advantage of this is t ha t i t should be seen more clearly when a f ixed-poin t has been reached. A further advantage is t ha t SUS is c o m p u t a t i o n a l l y more efficient t h a n p r o p o r t i o n a l selection, enabl ing more experiments to be conducted.

3 Simple examples of quasispecies

3 . 1 T h e o n e m a x p r o b l e m

T h e s tandard onemax prob lem is defined by /(A-) = k and so we have:

Sk.k = k

Figure 1 shows the f ixed-point d i s t r i b u t i o n calculated as the leading eigenvector o f G , for the case n = 20, g = 0.05. The corresponding eigenvalue (mean fitness) is 12.94. I t can be seen tha t th i s d i s t r i b u t i o n is clustered some distance f rom the global o p t i m u m . Such a p o p u l a t i o n is sometimes refered to as a quasispecies [3].

F igure 2 shows the results of 10 exper imenta l runs. T h e Euclidean distance f r o m the current p o p u l a t i o n vector to the f ixed-point (averaged over these runs) is shown for 200 generations, a long w i t h error bars o f one s tandard dev ia t ion . I n each run , the i n i t i a l p o p u l a t i o n was filled w i t h strings selected w i t h a u n i f o r m p r o b a b i l i t y d i s t r i b u t i o n . T h e average p o p u l a t i o n fitness t ra jec tory for the same 10 runs is also shown. I t can be seen tha t the eigenvalue of 12.94 provides a good est imate of the average popu la t i on fitness once the popula t ions have converged.

3 .2 T h o m p s o n ' s e x p e r i m e n t

I t has been suggested [9] t ha t genetic a lgor i thms may be good for finding robust solut ions to problems, as they tend to settle on areas o f the search space tha t are re la t ively flat w i t h respect to m u t a t i o n . T h i s suggestion by A d r i a n T h o m p s o n was i l lus t ra ted exper imen ta l ly w i t h a func t ion of u n i t a t i o n on five bi ts defined by:

/ (O) = 10

74 Jonathan E. Rowe

distance mean fitness

0.4 0.35

0.3 0.25

0.2 0.15

0.1 0.05 I

14 13 12 11 10

9 8

r ~ —

generation C 50 100 150 200 C ) 50 100 150 200

Figure 2: G A behaviour on the oi iemax prob lem n — '20, q = 0.0-5. Left: distance of popu la t ion f rom f ixed-point . R igh t : average popu la t i on fitness.

proportion distance

generation

Figure 3: Left: f ixed-point popu la t ion for Thompson ' s func t ion (q = 0.2). R i g h t : distance o f p o p u l a t i o n f r o m this f ixed-point .

/ ( I ) = 0

/ ( 2 ) = 0

/ ( 3 ) = 0

/ ( 4 ) = 5

/ ( 5 ) = 9

T h o m p s o n demonst ra ted tha t a popu la t ion s t a r t i ng ent i rely on the global o p t i m u m w i l l move across to the s u b o p t i m u m , which is broader ( though lower) and therefore more robust . The inf in i te p o p u l a t i o n fixed-point can be calculated using the eigenvector m e t h o d and is shown in Figure 3. T h i s provides an explanat ion for Thonrpson's exper imenta l result, wh ich indicated tha t the global o p t i m u m is unstable. A graph showing the distance f rom the current popu la t i on to the fixed-point over 200 generations (averaged over 10 runs) is also shown, . \ ga in , the i n i t i a l popula t ions are filled w i t h strings .selected w i t h a u n i f o r m p r o b a b i l i t y d i s t r i b u t i o n .

Population Fixed-Points for Functions of Unitation 75

Figure 5: T h e fixed-point popu la t i on for the t rap func t ion of F igure 4 is located near to the o p t i m u m for q = 0.02 (left) bu t is near the false o p t i m u m for q = O.Ob ( r i g h t ) .

4 Trap functions and deception

T r a p funct ions are piecewise l inear functions of u n i t a t i o n , w i t h a g loba l o p t i m u m occur ing w i t h n ones and a local o p t i m u m at 0 ones (see Figure 4 ) . The idea is t ha t , under cer ta in circumstances, a G A w i l l be "mis led" in to the false o p t i m u m . Indeed, the in f in i te fixed-p o i n t popu la t i on t y p i c a l l y resides near one or other of the o p t i m a . I f the m u t a t i o n rate is sma l l enough, then the popu la t i on can sit on the steeper g loba l o p t i m u m slope. For larger m u t a t i o n rates, however, this slope is too steep and the fi.xed-point now resides near the local o p t i m u m (see figure 5) .

These functions have been analysed in terms of the a m o u n t o f deception they con ta in [2]. T h e no t ion of deception has already been questioned £is a u.seful t o o l for analysing finite p o p u l a t i o n behaviour [5, 8] . We now see tha t the effect of deception i n a fitness func t ion is dependent on the operators t ha t are chosen. Decept ion analysis looks at a landscape independent of the operators. However, operators and landscape are in te r l inked : some m u t a t i o n rates mean tha t the area sur rounding the global o p t i m u m is unstable, even for funct ions t ha t are not fu l ly deceptive.

T h e graphs of distance f rom current popu la t ion to fixed-point are shown in Figure 6. For

76 Jonathan E. Rowe

distance distance

0.5 0.5

0.4 I 0.3 t

•

0.4 t

50 T00^~T50" "200 generation

0.1

0.3

0.2

0 N M W * W « » i » « 4 m m g e n e r a t i o n 50 100 150 200 0

Figure 6: Distance f rom current popu la t i on to f ixed-point for the t r ap funct ion w i t h q = 0.0.5 (left) and q = 0.02 ( r i gh t ) , h i i t i a l popula t ions are d rawn f rom u n i f o r m d i s t r i b u t i o n o f str ings (left) and un i fo rm d i s t r i b u t i o n of u n i t a t i o n class ( r i g h t ) .

the higher \a lue o f m u t a t i o n {q = 0.05). the popula t ion conxcrges to the fixed-point near the false o | ) t i m i m i . However, runs tha t use the lower n n i t a t i o n rate ((/ = 0.02) exh ib i t va r i ­able behaviour, depending on the way the i n i t i a l popula t ion is chosen. I f the popu la t ion is initiali.sed using a un i fo rm d i s t r i b u t i o n , in which different u n i t a t i o n values have an equal number o f expected representatives, then the evolved popu la t ion is close to the predicted fixed-point near the o p t i m u m for each r u n . However, i f the s tandard r a n d o m i n i t i a l pop­u l a t i on is used, in which different strings are equally l ikely, the evolved popu la t i on heads towards the false o p t i m u m in each run . T h i s indicates tha t we have a metastable state caused by a fixed-point jus t outside the s implex . Inspection of the second eigenvector ( tha t is, w i t h the second largest eigenvalue) indicates tha t i t is extremely close to the s implex ( i t has one negative component o f -.3 x 1 0 " ' ' ' after normal i s ing) . T h i s eigenvector is i l lus t ra ted in Figure 7. .K graph of distance f rom current popu la t ion to this (normal ised) eigenvector is also shown, and confirms that this is indeed act ing as an a l t r ac to r .

5 Evolution can go backwards

I t is usual ly taken for granted that an evolu t ionary process w i l l tend to favour ind iv idua l s w i t h higher fitness values, and indeed this assumpt ion is b u i l t i n to the mechanism of propor­t i ona l selection. T h i s describes the dynamic re la t ion o f one generation to the next . However, in the fixed-point l i m i t , things can be very different. I t is possible tha t the p r o p o r t i o n o f i nd iv idua l s in a popu la t ion , w i t h i n a given fitne-ss class, be inversely proportional to tha t fitness. To achieve this , we require tha t the generational operator G have the fo l lowing as an eigenvector (su i tably scaled):

k = 0 ,n. Using the def in i t ion o f G gives:

MSp = Xf

for some A. Rut the diagonal entries of .S' are s imply the fitne.ss values. T h a t is:

Sk.k = f(k-)

Population Fixed-Points for Functions of Unitation

proportion distance

unitation K#N'llWil>.'^ti»i 100 150 200

generation

Figure 7: Left : the second normal ised eigenvector for the t r ap func t ion o f Figure 4 is located very near to the s implex and so creates a metastable state in i ts v i c i n i t y [q = 0.02). R i g h t : distance f rom current p o p u l a t i o n to metastable state. The i n i t i a l popula t ions were chosen according to u n i f o r m d i s t r i b u t i o n of strings.

Therefore:

Ml =\p .

where 1 is the vector compr i s ing al l ones. F r o m this we can f ind the entries of p

and so (choosing A = 1):

T h i s func t ion is tabula ted i n Table 1 for ?7 = 20, q = 0.0-5, and i l lus t r a t ed in Figure 8, a long w i t h its f ixed-point .

I t should be noted tha t the reason this me thod works is t ha t we are deal ing w i t h equivalence classes o f ind iv idua l s . T w o ind iv idua l s are equivalent i f they .share the same u n i t a t i o n value. Because o f the way u n i t a t i o n is defined, these classes are of different sizes. I t is therefore possible to assign different fitness values t o these classes in such a way tha t the numbers of i nd iv idua l s i n a class, t imes the allocated fitness value, equals 1. I f the equivalence classes had equal sizes, then a l l the allocated values wou ld be equal and there wou ld be no surprises. T h i s phenomenon, then, is ent i rely due to the way u n i t a t i o n breaks up the search space in to cla.s,ses o f unequal size.

Exper imen t s show tha t for smal l popula t ions , the t ra jec tory is towards a metastable state, given by the t h i r d eigenvector, i l l u s t r a t ed in Figure 9, which is ex t remely close to the s implex . T h e second eigenvector is some distance f rom the s implex and is assumed to have no effect ( a l though see the discussion i n the next section). Since this metastable d i s t r i b u t i o n is close t o the i n i t i a l d i s t r i b u t i o n given the s tandard r a n d o m generation m e t h o d , experiments were r u n on this example using a u n i f o r m i n i t i a l popu la t ion (sampled so tha t each u n i t a t i o n value is equally l i k e l y ) . A popu la t i on of 50 was used. Larger popula t ions d r a w n i n this way are very close to the t rue fixed-point and so the meta-stable state is avoided. The graph of distance f rom current popu la t i on to the metastable state is shown i n Figure 10. Not ice t h a t

Jonathan E . Rovve

k 0 1 2 3 4 5 6 f (k) 2.64269 1.2.5.528 0.983518 0.91732 0.902926 0.900398 0.900044

k 7 8 9 10 11 12 13 f (k ) 0.900004 0.9 0.9 0.9 0.9 0.9 0.900004

k 14 15 16 17 18 19 20 f (k ) 0.900044 0.900398 0.902926 0.91732 0.983518 1.2.5528 2.64269

Table 1: A fitne.ss funct ion over 20 bits tha t has a f ixed-point p o p u l a t i o n in which the p r o p o r t i o n o f ind iv idua l s w i t h i n each fitness cla.ss is inverselv p ropo r t i ona l to t ha t fitness n = 20,f/ = 0.05.

fitness

3

2.5

2?

1.5 I \ 1

0.5

1 -

proportion

0.06 0.05 0.04 0.03 0.02 0.01

10 15 unitation unitation

20 10 15 20

Figure 8: Left: graph of the fitne.ss f tn ic t ion tabula ted in Table 1. R igh t : i ts fixed-point p o p u l a t i o n = 20, q = 0.05).

Population Fixed-Points for Functions of Unitation

proportion

0.2

s 0.175

0.15 0.125

0.1 0.075

0.05 0.025

unitation 0 5 10 15 20

Figure 9: T h e t h i r d eigenvector for the fitness func t ion shown i n Figure S.n = 20,g = 0.0.5

0 50 100 150 200

Figure 10: Left: distance f rom current popu la t ion to metastable state for the fitness func t ion .shown i n Figure 8. R igh t : The average p o p u l a t i o n fitness over the same runs (n = 20, q = 0.05). T h e i n i t i a l popula t ions were chosen according to a u n i f o r m d i s t r i b u t i o n o f u n i t a t i o n classes. A popu la t i on size of 50 was used.

this metastable state is centred a round the global m i n i m u m , a l though the object ive was to max imise the func t ion . P l o t t i n g the average popu la t ion fitness over t i m e reveals t h a t i t is decreasing as the system evoh'es towards this m i n i m u m . -

6 Evolution can go anywhere

I t is n a t u r a l to wonder wha t can be said i n general about the shape o f in f in i te popu la t i on f ixed-points . Are the bel l curves already seen typical? A r e there any l i m i t s on the kinds o f p o p u l a t i o n tha t can f o r m a f ixed-point? How w i l l the i n t r o d u c t i o n o f crossover affect the k inds of f ixed-points t ha t can be found? I n fact, i t turns ou t t ha t any p o p u l a t i o n d i s t r i b u t i o n can be a fixed-point for some fitness func t ion , even w i t h non-zero, one-point crossover, given cer ta in reasonable assumptions about crossover and m u t a t i o n rates.

T h e o r e m 1 Given a population p g A, there exists a fitness function such that p is a

"The fluctuation at generation 12.5 is due to a random drift in one of the runs.

distance mean fitness

0.5

80 Jonathan E. Rowe

Ji.ved-point of C, ~ .VI oj^. provided ( ) < ( / < 1/2 and tlie crossover rate is less than 1.^

Suppose we have a popu la t ion \ector p. We wish to find a fitness funct ion such tha t p is an f ixed-poin t of Q — M o T. I f we scale the fitness funct ion so tha t the average fitness of p is 1, the fixed-point equat ion becomes;

M o Sp = p

N o w note t ha t S — d i a g ( / ) . T h a t is, it is the diagonal m a t r i x whose entries are values o f / , considered as a vector. Therefore

p = \[ o d i a g ( / ) p = .V( o d i a g ( p ) /

and so

/ = d i a g ( / > ) - ' o . M - ' p (2)

T h u s the required fitne.ss func t ion / exists so long as M is inver t ib le . Th i s is known to be the case under the condi t ions stated above [13. ()].

For an example, we w i l l r e tu rn to the zero crossover case, and ask for a fitness funct ion for wdiich the fixed-point is the u n i f o r m d i s t r i b u t i o n across the space of u n i t a t i o n equivalence classes. T h a t is, we require pk = l / ( " - I - 1). Subs t i t u t i ng in to equat ion 2 (replacing M. w i t h the m u t a t i o n m a t r i x M) gives;

/ = - V / " ' l

and so

f{k)^Y.^i;} 1=0

A t y p i c a l such landscape is i l lus t ra ted in Figure 11, where the funct ion is defined over 10 bits and rj = 0.08. For popula t ions o f size .500, the behaviour of the genetic a l g o r i t h m is qui te interest ing. The popu la t i on seems to wander at r a n d o m around the search space, never se t t l ing down . A typ ica l r i i u over 2000 generations is shown in Figure 12.

I t w o u l d appear tha t the popu la t i on is a l ternately v i s i t i ng the fixed-point and a nearby metastable state. A n inspection of the first three eigenvectors reveals t ha t the second normal ised eigenvector is a long way f rom the s implex (components o f the order of 1 0 ' ° ) , whereas the t h i r d is (juite close. I t m igh t be thought tha t i t is the t h i r d tha t is creat ing the metastable state. However, experiments reveal that th is is not the case and, over many runs, popula t ions do not approach this state. The answ'er is to reconsidei- the second eigenvector, in the l ight of the fo l lowing two observations (which assume zero cro.ssover);

T h e o r e m 2 If a i>opulation vector p lits on the line joining two normalised eigenvectors of G. then so does Qp. Moreover, tip uUl be closer to the eigenvector with the greater eigenvalue.

Consider two eigenvectors and re. A po in t on the l ine connect ing them may be expres.sed as

p = Ac + (1 - \)w

"'The generalisation of tliis theorem to include crossover is due to one of the reviewers.

Population Fixed-Points for Functions of Unitation

fitness

unitation 2 4 6 r To

Figure 11: A landscape over 10 bits t ha t has a fixed-point popu la t ion tha t is u n i f o r m across the u n i t a t i o n search space for ? = 0.08

distance

500 TOOO T500 2000 generation

Figure 12: A typ ica l t ra jec tory (p lo t t ed as distance f rom the fixed-point) for the fitness func t ion shown in Figure 11, over 2000 generations.

Jonathan E. Rowe

for some scalar A. T h e n

where

MSp

imp) MSjXv + (1 - X)w)

{f){Xv + il - X)w)

X{f){v)v •+{1 - X)(f)(w)w

X{f}(v) + (1 - X){f)(w]

ftv + (1 - p)w

X{f)(v) + {1 - X){f}(w)

I f , say, > {f){w) then the popu la t ion w i l l move closer to v.

T h i s observat ion is of i tself interest ing. Given tha t there is on ly one eigenvector ac tua l ly i n the s implex, i t follows tha t the flow of p follows a s t ra ight l ine f r o m each of the other eigenvectors to the fixed-point.

T h e o r e m 3 Let v be the fixed-point in the simplex and w another normalised eigenvector. Let p G A be on the line connecting v and w. Then there ts a constant k such that

A g a i n , set p = Ay 4- (1 - X)w. Since p 6 A note tha t A > 0. T h e n some algebra produces:

( />(p) , p - p = A ( l - A ) M M z L i ^ ( , _ , „ )

S u b s t i t u t i n g

| I - A | =

and t a k i n g norms gives:

\p-v\

Choosing k t o be the m a x i m u m value of A tha t keeps p w i t h i n the s implex gives the result . Note t h a t i f v and lu are far apar t then A w i l l be close to 1.

T h e consequence of these two observations is tha t , i f the second eigenvalue is extremely close to the leading eigenvalue then, even i f the eigenvectors are far apart , the distance moved by a p o p u l a t i o n w i t h i n the s implex is smal l a l l along the l ine connect ing the two eigenvectors. R e t u r n i n g to our example, the second eigenvector is indeed far f rom the s implex, bu t the eigenvalue, 0.9997, is close to t ha t of the fixed-point (who.se eigenvalue is 1). T h i s indicates t h a t there is no t s i m p l y a metastable state bu t a metastable l ine. A l i t t l e geometry allows us to calculate the shortest distance f r o m a p o p u l a t i o n to this l ine. We now p l o t th i s distance over 2000 generations (averaged over 10 runs) . Figure 1.3 shows tha t popula t ions indeed head towards t ha t l ine and then stay more or less on i t .

distance

Population Fixed-Points for Functions of Unitation 83

0.25

0.2

0.15

0.1

0.05

m i i generation 0 500 1000 1500 2000

Figure 13: Distance f rom current popu la t i on t o metastable l ine for the fitness func t ion shown i n Figure 11 , over 2000 generations.

7 Conclusion

T h e d y n a m i c a l systems model o f the G A [12] can be s impl i f ied i n the case of zero crossover. B y considering functions which take on a l i m i t e d number o f values, the d imens iona l i t y o f the system can be reduced to the po in t where calculat ions are t ractable . T h i s m e t h o d , used i n [11] t o s tudy Roya l Road functions, has been adapted to apply to functions o f u n i t a t i o n . T h e inf in i te p o p u l a t i o n fixed-points o f these fitness functions, given a m u t a t i o n rate, are readi ly calculated for problems of modest size.

R has been shown theoret ical ly tha t any popu la t i on d i s t r i b u t i o n is po t en t i a l l y a fixed-point of some landscape (even w i t h crossover). I n par t icu la r , there exist non-flat fitness funct ions for which the fixed-point popu la t ion is d i s t r i bu t ed u n i f o r m l y across the entire search space o f u n i t a t i o n classes. We have also constructed fitness functions for which the final p o p u l a t i o n represents the number of ind iv idua l s w i t h a given u n i t a t i o n in inverse p r o p o r t i o n t o the fitness of t ha t u n i t a t i o n value. Exper iments conf i rm tha t evo lu t ion appears to go backwards on such landscapes.

T h e effect o f non-leading eigenvectors on finite p o p u l a t i o n behaviour has also been s tudied . I t has been shown tha t they can create metastable states as previously predic ted. I t has also been shown t h a t i f the second eigenvalue is close to the first, then a l ine t h r o u g h the s implex connect ing the eigenvectors w i l l be metastable, even i f the corresponding eigenvectors are far apar t . More generally, i t has been shown t h a t i f p is on a l ine connect ing any t w o eigenvectors, then so is Qp (assuming zero crossover). T h e force, \Qp — p\s p r o p o r t i o n a l to the difference between the corresponding eigenvalues.

8 Acknowledgements

I w o u l d l ike to acknowledge the many useful comments o f the reviewers o f th i s paper, and helpful conversations w i t h Michael Vose and Alden W r i g h t .

Jonathan E . Rowe

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