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EVOLUTION STRATEGIES232
0.2 0.3 0.4 0.5Expansion Probability
0.5
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0.6
0.65
0.7
0.75
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EVOLUTION STRATEGIES 237
An Analysis of the Role of O�spring Population Size in EAs
Thomas Jansen
Krasnow Institute
George Mason University
Fairfax, VA 22030
Kenneth De Jong
Krasnow Institute
George Mason University
Fairfax, VA 22030
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
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
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
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).
EVOLUTION STRATEGIES 241
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
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.
EVOLUTION STRATEGIES 243
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
EVOLUTION STRATEGIES244
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
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).
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EVOLUTION STRATEGIES246
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EVOLUTION STRATEGIES254
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EVOLUTION STRATEGIES 255
EVOLUTION STRATEGIES256