corel ventura - gec02 po · e\Ábcd on_`Ä ¿og _½ !_ cdg` òodn+g ÁÂ ±yf]oyf¿zcd_`...

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EVOLUTION STRATEGIES Guenter Rudolph, chair

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Page 1: Corel Ventura - GEC02 PO · e\Ábcd On_`Ä ¿Og _½ !_ cdg` òodn+g Á ±Yf]OYf¿Zcd_` fnãÆ¡|lÜmÝ3È ±ÇøX q » nâo _`Ä ¿Og`_ÃÁØ cd _`o¶Ç ~ ob]Z K[ Y\Ä _½ +o jK

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Page 2: Corel Ventura - GEC02 PO · e\Ábcd On_`Ä ¿Og _½ !_ cdg` òodn+g Á ±Yf]OYf¿Zcd_` fnãÆ¡|lÜmÝ3È ±ÇøX q » nâo _`Ä ¿Og`_ÃÁØ cd _`o¶Ç ~ ob]Z K[ Y\Ä _½ +o jK
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EVOLUTION STRATEGIES 237

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An Analysis of the Role of O�spring Population Size in EAs

Thomas Jansen

Krasnow Institute

George Mason University

Fairfax, VA 22030

[email protected]

Kenneth De Jong

Krasnow Institute

George Mason University

Fairfax, VA 22030

[email protected]

Abstract

Evolutionary algorithms (EAs) are general

stochastic search heuristics often used to

solve complex optimization problems. Unfor-

tunately, EA theory is still somewhat weak

with respect to providing a deeper under-

standing of EAs and guidance for the practi-

tioner. In this paper we improve this situa-

tion by extending existing theory on the well-

known (1+1) EA to cover the (1 + �) EA,

an EA that maintains an o�spring popula-

tion of size �. Our goal is to understand how

the value of � a�ects expected optimization

time. We compare the (1 + �) EA with the

(1+1) EA and prove that on simple unimodal

functions no improvements are obtained for

� > 1. By contrast there are more complex

functions for which sensible values of � can

decrease the optimization time from expo-

nential to a polynomial of small degree with

overwhelming probability. These results shed

light on the role of � and provide some guide-

lines for the practitioner.

1 INTRODUCTION

Evolutionary algorithms (EAs) are a broad class of

stochastic search heuristics that include genetic algo-

rithms (Goldberg 1989), evolution strategies (Schwe-

fel 1995), and evolutionary programming (Fogel 1995).

While there are countless reports of successful appli-

cations, EA theory is often concerned with either only

isolated aspects of the algorithm or extremely simpli-

�ed versions. One example are investigations of the

so-called (1+1) EA, a very simple EA that uses a par-

ent population of size 1 in each generation to create a

single o�spring by bit-wise mutation and replaces the

parent by its o�spring if the �tness if the o�spring

is not inferior to that of its parent (see, for exam-

ple, Rudolph (1997), Garnier, Kallel, and Schoenauer

(1999), and Droste, Jansen, and Wegener (2002)).

It has been known for some time that a simple (1+1)

EA is often at least as eÆcient as much more so-

phisticated EAs (Juels and Wattenberg 1995; Mitchell

1995). This motivates the search for situations when

the (1+1) EA is outperformed by more sophisticated

EAs. For example, if we increase the parent population

size �, i.e., a (� + 1) EA and introduce multi-parent

reproduction, there are problems for which the use of

uniform or 1-point crossover can reduce the expected

optimization time from exponential to a polynomial of

small degree (Jansen and Wegener 2001).

In this paper we focus on the role of the o�spring pop-

ulation size � in (1 + �) EAs. In particular, we would

like to know when it makes sense to have � > 1, and

if so, what are reasonable values for �. Our intuition

suggests that, for simple unimodal functions, simple

hill-climbing optimization procedures such as a (1+1)

EA are eÆcient and not likely to be outperformed by

(1 + �) EAs. However, as the complexity of the func-

tions to be optimized increases, there should be bene-

�ts for having � > 1.

We formally address these issues by extending the

analysis of the (1+1) EA as described in Droste,

Jansen, and Wegener (2002). In particular, we are

able to characterize the slowdown due increasing �

on two well-known unimodal test functions: OneMax

and LeadingOnes. In addition, we exhibit a class

of functions for which increasing � in a quite sensible

manner reduces the expected optimization time from

en to n2, where n is the dimension of the search space.

The result of this analysis is an improved understand-

ing of the role of o�spring population size and a better

sense of how to choose � for practical applications.

In the next section, we give a formal description of

the (1 + �) EA and describe the two well-known test

EVOLUTION STRATEGIES238

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functions investigated. In Section 3 we present results

on the expected optimization time of the (1 + �) EA

for these test problems. In Section 4 we present an

example function that allows us to see when the use

of an o�spring population can reduce the optimiza-

tion time from exponential to a polynomial of small

degree. However, modifying this example function we

see that also the opposite behavior can be observed,

i. e., the use of a quite small population increases the

optimization from O(n2) to exponential. In Section 5,

we conclude and describe possible future research.

2 DEFINITIONS

The (1 + �) EA to be analyzed is an evolutionary al-

gorithm that maintains a parent population of a size

one to produce in each generation � o�spring indepen-

dently and replaces the parent by a best o�spring if

the �tness of this o�spring is not inferior to the �tness

of its parent. The internal representation is a �xed

length binary string of length n and o�spring variation

is the result of bit-wise mutation. For convenience we

assume the optimization goal is maximization. More

formally, we wish to maximize f : f0; 1gn ! R using:

Algorithm 1 ((1 + �) EA).

1. Choose x 2 f0; 1gn uniformly at random.

2. For i := 1 To � Do

Create yi by flipping each bit

in x independently with

probability 1=n.

3. m := max ff (y1) ; : : : ; f (y�)g.4. If m � f(x) Then

Replace x by one yi chosen

uniformly at random from

fyi j (1 � i � �) ^ (f (yi) = m)g.5. Continue at line 2.

Clearly, by setting � := 1, we get the well-known (1+1)

EA. Our goal is to analyze the e�ects that � > 1 has

on optimization performance. As usual in theoretical

analysis we measure the optimization time in terms of

the number of function evaluations needed before x be-

comes a globally optimal point for the �rst time. If G

is the number of generations, i. e., the number of times

lines 2{4 are executed, before this happens, we have

T = 1+� �G for the optimization time T . Note that T

is a fairer measure than the number of generations G.

However, on a parallel machine a speed-up of the fac-

tor � can be achieved. Since T obviously is a random

variable, we are mostly concerned with E (T ), the ex-

pected optimization time. Of course, other properties

of T can be of interest, too.

3 PERFORMANCE ON SIMPLEUNIMODAL FUNCTIONS

Theoretical analyses of evolutionary algorithms of-

ten begin with simply structured example problems.

The perhaps best known binary string test problem is

OneMax, where the function value is de�ned to be

the number of ones in the bitstring. Another slightly

less trivial example is LeadingOnes, where the func-

tion value is given by the number of consecutive ones

counting from left to right.

One way of analyzing algorithm performance on test

problems is via \growth curve analysis" in which closed

form expressions are sought that express the amount

of time required to solve a problem as a function of

the \size" of the problem (Corman, Leiserson, Rivest,

and Stein 2001). For example, it is known that the

expected optimization time E (T ) of the (1+1) EA on

OneMax is �(n logn), i.e., as the size of the binary

search space f0; 1gn increases, the increase in E (T ) is

well-approximated by n logn, and on LeadingOnes

the expected optimization time is ��n2�(Droste,

Jansen, and Wegener 2002).

Both of these functions are examples of unimodal func-

tions that can be optimized eÆciently by simple hill-

climbing algorithms such as a (1+1) EA. As a con-

sequence, intuitively, one shouldn't expect a (1 + �)

EA to outperform a (1 + 1) EA on such functions.

Rather, one might expect a decrease in performance

as we increase �. In this section we show this more

formally by extending the (1+1) EA growth curve

analysis to (1 + �) EAs. Interestingly, doing this for

LeadingOnes is far simpler than for OneMax.

3.1 PERFORMANCE ON LEADINGONES

Theorem 1.1. For the (1 + �) EA on the function

LeadingOnes : f0; 1gn ! R, E (T ) = ��n2 + n � ��

holds if � is bounded above by a polynomial in n.

Proof. We begin with the upper bound and distin-

guish two di�erent cases with respect to the number

of o�spring �. First, we assume that � � en holds.

Note, that we want to prove that the expected op-

timization time is ��n2�. We begin with the upper

bound. Obviously, it is suÆcient if the current func-

tion value is increased by at least 1 at least n times

in order to reach the unique global optimum 1n. Dur-

ing the optimization the probability to create an o�-

spring with larger function value is bounded below by

(1=n) � (1 � 1=n)n�1 � 1=(en), since it is always suf-

�cient to mutate exactly the left-most bit with value

0. The probability that in one generation no o�spring

EVOLUTION STRATEGIES 239

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has larger function value than x is therefore bounded

above by (1� 1=(en))�. Thus, with probability at

least 1�(1� 1=(en))� � 1�e��=(en) the current string

x is replaced by one of its o�spring with larger function

value. Since we have � � e � n, obviously �=(en) � 1

holds. For t 2 R with 0 � t � 1 we have 1�e�t � t=2.

Thus, the probability that in one generation at least

one o�spring improves upon its parent x is bounded

below by �=(2en) for � � en. Therefore, the expected

number of generations for an improvement of the func-

tion value is bounded above by 2en=�. We see, that

the expected number of generations the (1 + �) EA

needs to optimize LeadingOnes is bounded above by

2en2=�. So, the expected optimization time is at most

� � 2en2=� = O�n2�as claimed.

Now, assume that � > en holds. As we noticed above,

the probability that in one generation the (1 + �) EA

creates an o�spring with larger function value than its

parent is bounded below by 1 � e��=en. Since we as-

sume � > en, this is bounded below by 1� e�1. Thus,

the expected number of generations until we have n

such generations is bounded above by O(n) implying

O(� � n) as upper bound on E (T ).

In order to derive a lower bound, we remember that �

is bounded above by some polynomial. So, there exists

a constant k, such that � � nk holds. We see that

the probability of producing an o�spring where k + 2

pre-speci�ed bits are mutated is bounded above by

1=nk+2. Thus, the probability that such an individual

is not created among n � � o�spring is bounded below

by 1� �1=nk+2� � (n � �) > 1� 1=n. Now, we consider

only the �rst n=(4k + 8) = �(n) generations. There

at most nk+1=(4k + 8) o�spring are produced. Thus,

with probability at least 1 � 1=n no such individual

is created. Now, we work under the assumption that

this is the case in the observed run. With probability

1=2, at least n=2 bits in the initial string are 0. It

is known (Droste, Jansen, and Wegener 2002), that

the bits that are to the right of the leftmost bit with

value 0 are random and independent during the run.

Thus, with probability at least (1=2)�1=n, after n=12

generations the global optimum is not reached. This

implies (�n) as a lower bound on E (T ).

This theorem con�rms our intuition that a (1+�) EA

will not outperform a (1 + 1) EA on LeadingOnes

since no value of � > 1 will reduce the expected op-

timization time below �(n2). However, the more in-

teresting implication of this theorem is that, from a

growth curve analysis point of view, there is no signi�-

cant slowdown if � is anywhere in the range 1 � � � n,

since ��n2 + n � �� = �

�n2�for � = O(n).

This is due to the fact that, in each step of the

LeadingOnes function, the probability of creating an

o�spring that is better than its parent is �(1=n) (i.e.,

in ES terminology, the mutation operator has an al-

most constant success probability of �(1=n)). Thus,

producing up to n o�spring in each generation before

checking for improvements results in no substantial

waste of time.

3.2 PERFORMANCE ON ONEMAX

For OneMax things are less obvious. Although it is

fairly straightforward to show that a (1+�) EA cannot

do better than a (1 + 1) EA, it is considerably more

diÆcult to pin down precisely when increasing � be-

gins to signi�cantly a�ect performance. This is due to

the fact that the success probability of mutation for

OneMax varies over time rather than remaining al-

most constant as in LeadingOnes. As we approach

the global optimum of 1n, the probability of creating

an o�spring that is better than its parents approaches

�(1=n). Hence, as we saw with LeadingOnes, any

� � n is acceptable.

However, at the beginning, starting with a randomly

generated binary string, the probability of success is

with overwhelming probability larger than 1=c for a

constant c, much larger than 1=n. Hence, unless � is

a constant independent of n, there will be at least an

initial slowdown. Consequently, we see that there is

no simple intuitive answer as to when an increase in

� degrades performance. But it is possible to bracket

� with useful upper and lower bounds. We begin with

a theorem that establishes an upper bound on E (T )

and indirectly sets a lower bound for �:

Theorem 1.2. For the (1 + �) EA on the function

OneMax : f0; 1gn ! R, E (T ) = O(n logn + n�)

holds.

Proof. Assume that the current string x has Hamming

distance d from the global optimum, i. e., d = n �OneMax(x). The probability to create an o�spring y

withOneMax(y) > OneMax(x) is bounded below by

d=n(1� 1=n)n�1 � d=(en). Thus, the probability not

to increase the function value of x in one generation

is bounded above by (1� d=(en))� � e�d��=(en). So,

the probability to increase the function value of x in

one generation is bounded below by 1 � e�d��=(en) �1�1= (1 + d � �=(en)) = (d��)=(en+d��). Thus, E (T )is bounded above by

��nX

d=1

en+ d�

d�= �

n+

en

nXd=1

1

d

!= O(n�+n logn):

EVOLUTION STRATEGIES240

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In particular, if � is bounded above by logn, i.e.,

� = O(logn), then the expected optimization time

is bounded above by n logn, which is no worse than

a (1 + 1) EA on OneMax. Similarly, a useful lower

bound on E (T ) can be obtained that can be used to

establish an upper bound on �:

Theorem 1.3. For the (1 + �) EA on the function

OneMax : f0; 1gn ! R, E (T ) = (� � n= logn) holdsif � is bounded above by a polynomial in n.

Proof. The probability to create an o�spring y by

mutating x with OneMax(y) � OneMax(x) + d is

bounded above by�n

d

�� 1

nd� 1

d!<�ed

�dfor all x 2 f0; 1gn and all d 2 f1; 2, . . . , n �OneMax(x)g. Therefore, the probability to create

one such y within n � � independent tries is bounded

above by n���(e=d)d. We conclude that the probability

to create one such y within n �� independent tries withd 2 flog(n); : : : ; n�OneMax(x)g is bounded above

by n � � � (e= logn)logn = e�(log(n) log logn). Thus, the

probability that within n generations the Hamming

distance to the optimum is not decreased by at least

log(n) in one single generation is 1�2�(log(n) log logn).

Since after random initialization the Hamming dis-

tance to the global optimum is at least n=2 with prob-

ability 1=2, this implies ((n= logn) � �) as a lower

bound on E (T ) as claimed.

Hence, if we allow � to increase faster than log2 n,

we begin to see a signi�cant slowdown relative to the

n logn performance of a (1+1) EA onOneMax. Com-

bining this with the previous theorem we see that keep-

ing � < logn means we'll do no worse than a (1 + 1)

EA. This still leaves open the possibility that there

are values of � � log2 n that result in performance im-

provements over a (1+1) EA. However, as our intuition

suggests, this is not the case. We show this formally

by proving that, whenever � grows more slowly thanpn= logn (an upper bound much larger than log2 n),

E (T ) is bounded below by (n logn):

Theorem 1.4. For the (1 + �) EA on the function

OneMax : f0; 1gn ! R, E (T ) = (n logn) holds if

� = o (pn= logn).

Proof. It is easy to see that at some point of time

the Hamming distance between the current string

x and the global optimum is within fdpn=2e , . . . ,dpneg with probability very close to 1. We con-

sider a run of the (1 + �) EA after this point of time.

Then, the probability to create an o�spring y with

OneMax(y) � OneMax(x) + 3 is bounded above by�pn

3

��1=n3 < 1=n3=2. Thus, the probability to create at

least one such individual within � �n log n independent

tries is bounded above by 1 � �1� 1=n3=2���n log n

=

O (� � logn=pn) = o(1). Now, we continue under the

assumption that no such individual is created. Then,

the Hamming distance can only be decreased by 1 or

2 in each generation. Given that the current Ham-

ming distance between x and the global optimum is d,

the probability to create an o�spring that decreases

this Hamming distance is bounded above by 2d=n.

Thus, the probability to do so in one generation is

bounded above by 1 � (1� 2d=n)�< 4� � d=n. Thus,

the expected number of generations is bounded below

byPp

n

d=1 n=(4� � d) = ((n=�) � logn). This implies

E (T ) = (n logn).

The previous 3 theorems collectively tell us that we can

increase � up to logn without a signi�cant degradation

in performance, but not beyond log2 n, leaving the in-

terval between logn and log2 n unresolved. The �nal

two theorems of this section narrow that gap consid-

erably by increasing the lower bound slightly and de-

creasing the upper bound to within a constant factor

of the lower bound. We begin with the lower bound:

Theorem 1.5. For the (1 + �) EA on the function

OneMax : f0; 1gn ! R, E (T ) = �(n logn) holds if

� � (lnn) � (ln lnn)=(2 ln ln lnn).

Proof. The lower bound follows from Theorem 1.4.

For � = O(logn) the upper bound follows from Theo-

rem 1.2. So, we assume � � lnn from now. In partic-

ular, we de�ne := �= lnn and have � 1.

We divide a run of the (1 + �) EA into two disjoint

phases and count the number of function evaluations

separately for each phase. Let T1 denote this num-

ber for the �rst phase and let T2 denote this num-

ber for the second phase. The �rst phase starts af-

ter random initialization and continues as long as

OneMax(x) � n � n= ln lnn holds for the current

string x. The second phase starts immediately after

the end of the �rst phase and ends when the global

optimum is reached. Obviously, for the expected opti-

mization time E (T ) we have E (T ) = E (T1) + E (T2)

with these de�nitions.

In order to get an upper bound on E (T2), we recon-

sider the proof of Theorem 1.2. We see that E (T2)

is bounded above by � � Pn= ln lnn

d=1 (en + d � �)=(d ��) = � � ((n=ln lnn) + (en=�) �P

d=1 n= ln lnn1=d) =

O (� � (n= ln lnn) + n logn) = O(n logn). Here we

need � = O(log(n)= log logn).

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Now we consider the �rst phase. Since we have

OneMax(x) � n�n= ln lnn, the probability to create

an o�spring y by mutation of x with OneMax(y) �OneMax(x) + is bounded below by

�n= ln lnn

���1

n

��1� 1

n

�n�

��

n

� n � ln lnn�

� 1e= e�(1+ ln + ln ln lnn):

We conclude that the probability to create at least one

such individual y in one generation is bounded below

by 1��1� e�(1+ ln + ln ln lnn)�� � �=((e lnn)+�) �

1=(e + 1) since we have � (ln lnn)=(2 ln ln lnn) by

assumption. Obviously, after less than n= such gen-

erations the �rst phase ends. This implies O(��n= ) =O(n log n) as upper bound on E (T1).

We cannot prove that (lnn)(ln lnn)=(2 ln ln lnn) is a

sharp upper bound in the sense that any value for �

larger than this implies that the expected optimization

is worse than n logn. However, we can show that it is

true when � grows asymptotically faster than that.

Theorem 1.6. For the (1 + �) EA on the function

OneMax : f0; 1gn ! R, E (T ) = !(n logn) holds if

� = ! ((lnn)(ln lnn)= ln ln lnn).

Proof. We use a di�erent proof strategy in order to

derive this tighter lower bound. We derive an upper

bound on the expected decreasement in the Hamming

distance in one generation. Then we use this upper

bound in order to prove that it is not likely that the

Hamming distance is decreased by a certain amount

in a pre-de�ned number of generations.

In analogy to the proof of Theorem 1.4 it is easy to

see that at some point of time the Hamming distance

between the current string x and the global optimum

is within fdn=(2e)e ; : : : ; dn=eeg with probability very

close to 1. From this point of time on the proba-

bility to create an o�spring y with OneMax(y) �OneMax(x) + d is bounded above by

�n=e

d

� � 1=nd <

1=dd.

Consider g generations of the (1+�) EA. Let x be the

current string before the �rst generation and let x0 be

the current string after the g-th generation. Let D�;g;x

denote the advance in this g generations by means

of Hamming distance, i. e. D�;g;x := OneMax(x0) �OneMax(x). Obviously, D�;g;x depends on x and

we have Prob (D�;g;x � d) � Prob (D�;g;y � d) for

all d 2 f0; 1; : : : ; ng and all x; y 2 f0; 1gn with

OneMax(x) � OneMax(y). Since the function value

of the current string of the (1 + �) EA can never de-

crease, E (D�;g;x) � g �E(D�;1;x) holds for all �, g and

x. So, we concentrate on E (D�;1;x) now.

Obviously,D�;1;x is a random variable that depends on

� and the current string x at the beginning of the gen-

eration. However, it is clear that Prob (D�;1;x � d) <

�=dd holds for all x with OneMax(x) � n � dn=ee.From now on, we always assume that OneMax(x) is

bounded above in this way.

We are interested in E (D�;1;x). Since D�;1;x 2 f0; 1,. . . , ng, we have E (D�;1;x) =

Pn

i=1 Prob (D�;1;x � d).

For d < (3 ln�)= ln ln� we use the trivial estimation

Prob (D�;1;x � d) � 1. For d with (3 ln�)= ln ln � �d < (� ln�)= ln ln� we have

dd� eln�

e((3 ln�)=(ln ln�))�((ln ln�)�ln ln ln�))<

1

and use the estimation Prob (D�;1;x � d) � 1=�.

Finally, for d � (� ln�)= ln ln � we have

�=dd � e�=e� ln� < e�� and use the estimation

Prob (D�;1;x � d) < e�� < 1=n. These three estima-

tions yield E (D�;1;x) < 4 ln�= ln ln� for the expected

advance in one generation.

Let G denote the number of generations the (1 + �)

EA needs for optimization of OneMax. Of course,

E (G) � t � Prob (G � t) holds for all values of t. As

we argued above, with probability at least 1=2 at some

point of time we have that some x withOneMax(x) 2fn � dn=ee ; : : : ; n � dn=(2e)eg as current string x.

Thus, Prob (G � t) � Prob (D�;t;x < n=e) holds, if

x is some string with at least n=e zero bits. This

yields E (G) � (t=2) � Prob (D�;t;x < n=e) = (t=2) �(1� Prob (D�;t;x � n=e)). By Markov inequation

we have E (G) � (t=2) � (1� E(D�;t;x) =(n=e)) �(t=2) � (1� e � t � E (D�;1;x) =n). Together with

our estimation for E (D�;1;x) we have E (G) �(t=2) � (1� 4e � t � ln�=(n � ln ln�)). We set t :=

(n ln ln�)=(8e ln�) and get E (G) � n ln ln �=(32e ln�)

which implies E (T ) = (n� ln ln�= ln�) for the ex-

pected optimization time E (T ), since T = G �� holds.

It is easy to see, that this implies E (T ) = !(n logn)

for � = !((lnn)(ln lnn)= ln ln lnn) as claimed.

3.3 SUMMARY

The theorems in this section provide a clear picture

of the performance of a (1 + �) EA on OneMax. It

never outperforms a (1 + 1) EA. It's performance is

about the same if � doesn't get much bigger than

(logn)(log logn)=2 log log logn, and performance de-

grades signi�cantly after that.

EVOLUTION STRATEGIES242

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4 (1 + �) EA Performance On MoreComplex Functions

The previous section showed that for simple functions

like OneMax and LeadingOnes increasing � does

not improve E (T ) over (1 + 1) EA performance. In

this section we focus on cases where increasing � does

improve performance. Intuitively, a necessary condi-

tion for it to be useful to invest more e�ort in the

random sampling in the neighborhood of the current

population is that the �tness landscape is to some de-

gree misleading in the sense that a (1+ 1) EA is more

likely to get trapped on a local peak.

In this section we show that the performance improve-

ment due to increased values of � can be tremendous.

We illustrate this with an arti�cial function for which

we can prove that increasing � from 1 to n reduces

E (T ) from en to n2 with a probability that converges

to 1 extremely fast.

The intuition for this function is quite simple as illus-

trated in Fig. 1. We want it to consist of a narrow

path that leads to the optimum. However, while fol-

lowing that path, the algorithm is confronted multiple

branch points, each with the property that from it

there are a variety of paths leading uphill, but only

the steepest one leads to the global optimum. Hence,

as we increase � we increase the likelihood of picking

the correct branch point path.

The formal de�nition of this function is more compli-

cated than the intuition. To simplify it somewhat, we

divide the de�nition into two parts. First, we de�ne

a function f : f0; 1gn ! R that realizes the main idea

but assumes that the initial string is 0n.

De�nition 1.1. For n 2 N we de�ne k := bpnc.We use jxj = OneMax(x) and de�ne the function

f : f0; 1gn ! R for all x 2 f0; 1gn by

f(x) :=

8>>>>>>>>>>><>>>>>>>>>>>:

(2i+ 3)n+ jxjif x = y0n�ik1(i�1)k with

1 � i � k=2

and y 2 f0; 1gk

(2i+ 4)n+ jxjif x = 0n�j1j with

i := dj=ke, 1 � i � k=2,

and i � k 6= j

0 otherwise

The core function f contains one main path 0i1n�i.

There are aboutpn=2 points on this path that are of

special interest. At these points it is not only bene�cial

to add another one in the right side. The function

value is also increased by ipping any of the left most

bpnc bits. Thus, at these points there are a variety

global optimum

point from Bn

point from Ln

direction of

increasing f -values

0n

1n

mainpath

mainpath

Figure 1: Core function f .

of uphill paths to chose among. One path of the form

0i1n�i leads to the global optimum, while paths of the

form y0i0

1n�i0�bpnc with y 2 f0; 1gb

pnc all lead to

local optima. Therefore we call these special points on

the path branching points.

De�nition 1.2. For n 2 N we de�ne k := bpncand call a point x 2 f0; 1gn a branching point of

dimension n i� x = 0n�(i�1)k1(i�1)k holds for some

i 2 f1; 2; : : : ; bk=2c � 1g. Let Bn denote the set of all

branching points of dimension n.

At the branching points a (1 + �) EA may proceed

toward a local optima or the global one. It is important

to know what the probabilities are for the two di�erent

possibilities:

Lemma 1.1. Let n 2 N, k := bpnc, Bn the set of

branching points of dimension n as de�ned in De�ni-

tion 1.2, x 2 Bn, and � : N ! N be a function that has

a polynomial upper bound. Consider a (1+�) EA with

current string x optimizing the function f : f0; 1gn !R as de�ned in De�nition 1.1. Let y = y1y2 � � � yn be

the �rst point with f(y) > f(x) reached by the (1 + �)

EA. The probability that OneMax(y1y2 � � � yk) > 0

holds is 1� e��(�=pn).

Proof. We consider the (1 + �) EA on f with current

string x = x1x2 � � �xn 2 Bn. Let x0 = x01x02 � � �x0n be

one individual created by mutation of x. We consider

several events concerning x0.

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The points in Bn are ordered according to the function

value. The probability that x0 has a function value

that is equal or greater to the next branching point is

obviously 2�(pn). We neglect such an event and take

care of this in our upper and lower bounds by adding

or subtracting 2�(pn).

We denote the event that OneMax(x01 � � �x0k) > 0

holds by A. We have

Prob (A) ��k

1

�� 1n��1� 1

n

�n�1

>k

en

as a lower bound and

Prob (A)

kXi=1

�k

i

��1

n

�i �1� 1

n

�n�i!+ 2�(

pn)

<

kX

i=1

�k

in

�i!+ 2�(

pn) <

k

n� k

for an upper bound. We denote the event that

OneMax(x01 � � �x0k) = 0 holds by B and have

Prob (B) � 1

n��1� 1

n

�n�1

>1

en

as a lower bound and

Prob (B) �

kXi=1

�1

n

�i

��1� 1

n

�n�i!+2�(

pn) <

1

n

for an upper bound.

We are interested in the probability of A given

that A or B occur. Since A \ B = ;, we have

Prob (AjA [ B) = Prob (A) = (Prob (A) + Prob (B)).

Obviously, for each a; b; c 2 R+ we have that a=(a +

c) � b=(b+ c) holds i� a � b. This implies that

k=(en)

k=(en) + 1=n< Prob (AjA [ B) <

k=(n� k)

k=(n� k) + 1=(en)

follows from our upper and lower bounds on

Prob (A) and Prob (B) and we have 1 � e=pn <

Prob (AjA [ B) < 1 � 1= (epn). Let C denote the

event that we are really interested in, the event that

OneMax(y1 � � � yk) > 0 holds for the �rst point

y = y1 � � � yn with f(y) > f(x) reached by the

(1 + �). Since in each generation the � o�spring

are independently generated, we have Prob (C) =

1� (1� Prob (AjA [ B))�and can conclude that

1��

epn

��

< Prob (C) < 1��

1

epn

��

holds. Since (1 � 1=x)x < e�1 < (1 � 1=x)x�1 holds

for all x 2 N, this completes the proof.

Assume that the (1 + �) EA happens to continue in

the direction of a y0i1n�i�bpnc with y 2 f0; 1gbpnc,

y 6= 0bpnc. Then it is quite likely to reach \the last

point in this direction", i. e., y0i1n�i�bpnc with y =

1bpnc. It will turn out be convenient to de�ne a set

for those points similar to Bn.

De�nition 1.3. For n 2 N we de�ne k := bpnc and

Ln :=n1k0n�ik1(i�1)k j i 2 f1; 2; : : : ; bk=2c � 1g

o:

When considering the (1+1) EA on the core function

f , it is essential that the algorithm is started with 0n

as initial string. Since the (1+1) EA choose the initial

string uniformly at random this will not be the case

with probability 1� 2�n. Therefore, we now relax the

assumption that the initial string is 0n by extending f

to g in a way that any starting point leads to the main

path de�ned by f .

De�nition 1.4. For n 2 N we de�ne m0 := bn=2c,m00 := dn=2e, and g : f0; 1gn ! R by

g(x) :=

8><>:n�OneMax(x00) if x0 6= 0m

0 ^ x00 6= 0m00

2n�OneMax(x0) if x0 6= 0m0 ^ x00 = 0m

00

f(x00) if x0 = 0m0

for all x = x1x2 � � �xn 2 f0; 1gn with x0 :=

x1x2 � � �xm0 2 f0; 1gm0

and x00 := xm0+1xm0+2 � � �xn 2f0; 1gm00

.

We double the length of the each bitstring and de�ne

the core function f only on the right half of the strings.

The �rst half is used to lead a search algorithm towards

the beginning of the main path of f . The construction

will have the desired e�ect for all search heuristics that

are eÆcient on OneMax.

Theorem 1.7. The probability that the (1+1) EA op-

timizes the function g : f0; 1gn ! R within nO(1) steps

is bounded above by 2�(pn log n).

Proof. Our proof strategy is the following. First, we

consider the expected optimization time of the (1+1)

EA on the function g : f0; 1gn ! R under the condition

that at some point of time the current string x = x0x00,

with x0 and x00 de�ned as in De�nition 1.4, is of the

form x0 = 0m0

, x00 2 Lm00 , with m0 = jx0j, m00 = jx00j,k =

jpm00k, and i 2 f1; : : : ; bk=2cg. Then, we prove

that the probability that such a string becomes current

string at some point of time is (1).

First, assume that such a string x is current string of

the (1+1) EA. Let A be the set of all such strings. Ob-

viously, this string x is di�erent from the unique global

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optimum. Moreover, due to the de�nition of g, all

points with larger function value have Hamming dis-

tance at least k and there are less than n such points.

Thus, the probability to reach such a point in one gen-

eration is bounded above by n �(1=n)k � (1=n)pn=2�2.

The probability that such an event occurs at least once

in nk generations is bounded above by n�pn=2+k+1 =

2�(pn logn).

The initial current string x = x0x00 after random ini-

tialization is some string with x0 6= 0m0

with probabil-

ity 1�2�m0

. Then, the function value is either given by

n�OneMax(x00) or 2n�OneMax(x0). With proba-

bility 1�2�(n) the all zero string 0n is reached within

O�n2 logn

�steps given that no other string y with

g(y) > g (0n) is reached before. There are less than

2k � n such strings, thus, the probability to reach such

a string within O�n2 log n

�steps is bounded above by

O

�n2 logn � 2

k � n2n

�= 2�n+O(

pn):

At each branching point, the (1+1) EA comes to a

string x = x0x00 as new current string with some

bit set to 1 within the �rst k bits of x00 with prob-

ability at least 1 � 1=(epn). This follows from the

proof of Lemma 1.1. The probability to proceed with

such strings instead of returning to a string with k

zero at these bit positions increases. In fact, it is

easy to see that with probability 1 � O (1=pn) the

(1 + �) EA reaches a point in A before reaching some

point with k zeros at these positions. Since we have

bk=2c� 1 >pn=5 branching points, we conclude that

the probability that the (1+1) EA does not reach some

point in A is bounded above by 2�(pn logn) under the

described circumstances. Combining all estimations

completes the proof.

Theorem 1.8. For all constants " > 0, the probability

that the (1 + �) EA with � = n � �0, �0 2 N optimizes

the function g : f0; 1gn ! R within O�n2 � �0� steps is

bounded below by 1� ".

Proof. First, assume that the initial string x = x0x00

is some string with x0 6= 0m0

. Given that the all zero

string is the �rst string y = y0y00 becoming current

string with y0 = 0m0

, we can conclude from Theo-

rem 1.2 that the expected number of steps the (1+�)

EA needs to reach the all zero string is O�n2 � �0�.

Similar to the proof of Theorem 1.7 we can conclude

that the all zero string is reached within cn2 � � steps

with probability at least 1� ", where c and " are pos-

itive constants. Note, that by enlarging c the failure

probability " can be made arbitrarily small. Given

that for all following current strings x = x0x00 we have

that x00 does not contain a bit di�erent from 0 within

the �rst k bits, it follows from Theorem 1.1, that the

expected number of steps until the global optimum

is reached is bounded above by O�n2 � �0�. So, un-

der the assumption that no such string is reached, we

have similarly to the reasoning above that the global

optimum is reached within O�n2 � �0� steps with prob-

ability at least 1 � ", where " is an arbitrarily small

positive constant. We know from Lemma 1.1 that

the probability to reach some x where this assump-

tion is not met is bounded above by e�(pn) at each

branching point. Since there are less thanpn branch-

ing points, we have that with probability at least

1 � pn � e�(

pn) = 1 � e�(

pn) no such point be-

comes current point x at any branching point. Com-

bining these estimations completes the proof.

At the same time, as has been seen in many other

contexts, there is no \free lunch" here in the sense that

the impressive speedup on g achieved by increasing �

is not true for all functions with local optima. For

example, it is quite easy to modify the function of the

previous section so that a (1+1) EA is eÆcient and a

(1 + �) EA with � � n fails with high probability. To

see how, consider f and all strings x = x0x00 where x00

is a branching point. Let x000 be a string of length jx00jwith x000 2 Ljx000j. that has k bits with the value 1 at

the beginning and is equal to x00 on the rest of the bits.

The proofs of Theorem 1.7 and Theorem 1.8 rely on

the fact the the (1+1) EA reaches some string x0x000

with high probability whereas the (1+�) EA does not.

De�ning a function f 0 that has all such point x0x000 as

global optimum and is equal to f on all other points is

obviously optimized within O�n2�steps by the (1+1)

EA with probability converging to 1, whereas the (1+

�) fails to optimize f 0 within a polynomial number

of steps with probability converging to 1, given that

� = (n) holds.

5 CONCLUSIONS

We have used growth curve analysis techniques to bet-

ter understand the role of o�spring population size in

(1 + �) EAs. As we have seen, increasing � on simple

functions like OneMax and LeadingOnes does not

lead to an improvement in the expected optimization

time. However, increases in � do not signi�cantly de-

grade performance as long as � < logn where n is the

dimensionality of the search space. By contrast, for

more complex functions with local optima, increasing

� can result in improvements in performance as illus-

trated in the previous section.

EVOLUTION STRATEGIES 245

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However, a note of caution needs to be raised. The

growth curve analysis techniques used here focus on

limiting behavior as n increases and view growth

curves that di�er only by a constant as equivalent.

Hence, it may be the case that speci�c values of n and

� produce \contradictory" results to the conclusions

in the previous paragraph.

In spite of this cautionary note we believe that these

insights into the role of o�spring population size pro-

vide some guidelines for the EA practitioner. Since

practitioners apply EAs to functions about which they

do not have detailed a priori knowledge, they can

\hedge their bets" by choosing � to be of order logn.

In doing so, they don't lose too much if the function

turns out to be easy and may improve their perfor-

mance on functions with local optima. An interesting

observation here is that for most practical problems,

logn is not likely to be too far away from the value

of � = 7 recommended by the ES community but de-

rived in other contexts (Schwefel 1995, p. 148). On the

other hand, if the function appears to be quite diÆcult

in the sense that each run results in convergence to a

di�erent local optima, increasing � to a value of order

n is a plausible thing to try.

This work represents a modest step along to road to

a more general EA theory. It extends some of the

techniques and results known for the (1 + 1) EA to

(1+�) EAs. Clearly, additional steps are required, in-

cluding the extension of these results to (�+ �) EAs.

From a more practical perspective, the heuristic guide-

lines proposed here for choosing values for � need to

be experimentally evaluated and tied more closely to

properties of functions.

Acknowledgments

The �rst author was supported by a fellowship within

the post-doctoral program of the German Academic

Exchange Service (DAAD).

References

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EVOLUTION STRATEGIES246

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