Emphasizing Extinction in Evolutionary Programming
Garrison W. Greenwood
Dept. of Computer Science
Western Michigan University
Kalamazoo, MI 49008
[email protected]
Gary B. Fogel
Natural Selection, Inc.
3333 N. Torrey Pines Ct., Suite 200
La Jolla, CA 92037
[email protected]
AbstractEvolutionary programming typically uses tournament
selection to choose parents for reproduction. Tournaments naturally emphasize survival. However, a natural
opposite of survival is extinction, and a study of the fossil record shows extinction plays a key role in the evolutionary process. This paper presents a new evolutionary
algorithm that emphasizes extinction to conduct search
operations over a tness landscape.
1 Introduction
It is now well known that evolutionary programming
(EP) can be an eective search technique. Although
there are several versions of EP, they are all biologically
inspired and generally follow the format depicted in Figure 1.
initialize
population
evaluate
fitness
select
survivors
randomly vary
individuals
Figure 1: The canonical EP. The \select survivors" block
includes the selection of parents and the generation of
ospring.
This paper focuses on methods used for the selection
of survivors as parents for the following generation. The
most common form of selection used in EP is tournament selection, in which the tness of each individual
in the population is compared against the tness of a
randomly chosen set of other individuals in the current
population. A \win" is recorded for an individual each
time an individual's tness exceeds that of another in
the tournament set. All individuals are then ranked according to their number of wins. Those individuals with
Manuel Ciobanu
Dept. of Computer Science
Western Michigan University
Kalamazoo, MI 49008
[email protected]
the highest scores are selected as parents for the next
generation and the remainder are discarded [1].
There are several alternative views to the selection
process in general, and tournament selection in particular. Suppose we assume each individual in the population represents a distinct species. The selection process
chooses some individuals in the current population to
be parents while others will be discarded. Under the
species abstraction, we can say that selected individuals
represent species that survive, while discarded individuals represent species that become extinct. It has been
shown that extinction plays a signicant role with respect to evolution in the biological world. It is therefore
natural to ask if extinction can be exploited to increase
the eciency of EP. Here we review the role of extinction
in the fossil record and simulate this process in the context of EP. Our preliminary results suggest that extinction can enhance the performance of EP on a multimodal
function.
2 Mass Extinction and the Fossil Record
The pace of change of species in ecological systems is
closely correlated to the temporal stability of the environments in which they inhabit. During Earth's long history, environmental stresses have been prolonged and/or
severe enough to invoke widespread ecological instability.
A direct result of this stress is the simultaneous extinction of many species that were not able to adapt in ways
suitable to avoid their demise. These events as recorded
in the fossil record are known as mass extinctions [2].
Extinction has played a major role in shaping the history of life on Earth. When examining extinction rates
for a large portion of the fossil record, Raup and Sepkoski initially noted two modes of extinction: a lowlevel, continuous background extinction or a sporadic
mass extinction rate [3]. Although mass extinctions account for only a small percentage of all extinctions, they
are singularly important because they remove \resilent"
incumbent groups from niches, creating \windows of opportunity" for other species to ourish and establish an
ecological niche [4].
Five mass extinctions have been identied: terminal
Ordovician (440 Myr ago)1; late Devonian (365 Myr
1
Myr denotes one million years.
ago); late Permian (250 Myr ago); terminal Triassic
(215 Myr ago); and terminal Cretaceous (65 Myr
ago). The severity of these extinction events can be difcult to fully comprehend. For instance, it has been
estimated that 96% of all marine species on Earth went
extinct by the end of the Permian period [5], representing
52% of all marine vertebrate and invertebrate families
[3]. Several causes for these events have been proposed
including global cooling, changes in sea level, and the
impact of a meteor or comet. Some have suggested that
there is a periodicity for mass extinctions with a mean
period of 26 Myr [6, 7], although not all researchers agree
with this notion (see [8], pg 404-405).
Patterns and processes of species extinction and survival appear to be dierent between background and
mass extinction events. For instance, Jablonski [9] noted
that during background extinction intervals, Cretaceous
gastropods and bivalves with planktotrophic (planktonfeeding) larvae had longer species durations than those
with nonplanktotrophic larvae. In contrast, larval type
was not a factor during the Cretaceous mass extinction
event. It is clear that not all species were aected equally
across all extinction events. (This fact is a central principle behind our incorporation of extinction into an evolutionary algorithm.)
Wolfe and Upchurch [10] noted that one of the major
factors in the transition from predominantly evergreen
to deciduous biotas around the world was correlated to
the cooling during the Cretaceous mass extinction event.
This event led to much higher rates of extinction for
evergreens relative to deciduous plants. As Newton and
Laporte [2] noted,
One answer to the question, `How did temperate deciduous forests arise?' is, `Through
enhanced survival in the Cretaceous extinction.' Mass extinctions may have been responsible for many such cases of evolutionary
replacement in the fossil record.
Using the stratigraphical ranges of 17,621 marine genera over the last 600 Myr, Raup noted that the average
species duration was 4 Myr. Using this same data, Raup
[11] developed a \kill curve", shown in Figure 2, that
described the average spacing between extinction events
of varying intensity. For example, an extinction event
capable of killing 30% of marine species occurs every 10
Myr; an extinction event capable of killing 65% of marine
species occurs every 100 Myr. Although many assumptions were used in the development of this curve, it appears that extinctions may be continuously distributed
in a logarithmic fashion rather than separated into two
discrete categories.
80 species kill (%)
70
60
50
40
30
20
10
0
104
105
106
107
108
109
mean waiting time in years
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ......... ......... ........
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Figure 2: A kill curve showing the average spacing
between marine extinction events of specied intensity
(adopted from [11]).
3 Adding Extinction into Evolutionary Algorithms
Our extinction-based evolutionary algorithm started as
a standard evolutionary programming algorithm: reproduction performed only via mutation, a xed strategy
parameter, and tournament selection to chose survivors.
It became necessary to alter this process when modeling
extinction dynamics. For instance, not all members of
the population necessarily reproduce each generation; a
tournament does not always determine survival; and coevolution aects only a small subset of the population.
We consider an ecosystem at time t that contains a
species set (t) = fs1 ; s2 ; : : : ; sk g, each of which is characterized by a vector ~x 2 <n . These numbers represent phenotypical traits which change whenever species
evolve. All species are subjected to an external environmental stress (t), which may be severe enough to cause
some species to become extinct if their tness level is too
low. The tness of each si is measured, in part, by an objective function f (~x); minimum values of this function increase the likelihood of survival. But tness is a relative
measure, which means f (~x) alone does not completely
dene tness unless the mapping function considers the
inuence of other species in the same environment. We
t (f t ) deused the following mapping scheme. Let fmin
max
note the minimum (maximum) objective function value
of any species si 2 (t). Then the tness of si at time t
is given by
t
(1)
(s ; t) = + (1 , ) f (~xi ) , fmax
i
t , fmax
t
fmin
where 2 [0; 1]. Such a mapping forces f (~x) 7! [; 1].
(The purpose of the lower bound will be discussed
shortly.) Other types of mappings could be used; the
above method was chosen because of its simplicity.
The steps in our extinction-based evolutionary algorithm are shown below:
1. Let t = 0 and randomly initialize population (0).
Compute (si ; t) 8 i.
2. Increment t and let (t) = (t , 1).
3. Randomly choose a stress (t) 2 [0; 0:96]. Make
all si 2 (t)j(si ; t) < (t) go extinct (i.e., discard
them). Let m equal the number of species that
went extinct.
4. If (m > 0) then
fMutate each species to produce multiple new
species. Compute (; t) for all new species. Conduct a tournament among the new species and deterministically choose the m best as replacements
for those species that went extinct.g
else
f Let j = argmin i ((si ; t)). Mutate sj . Randomly
choose 5 more individuals from the current population and mutate them also. Compute (; t) for
all mutated species. g
5. If computation time not exhausted, go to step 2.
Otherwise, exit.
During each generation t, three steps take place. A
random stress (t) is applied against all species, and all
species si with tness (si ; t) < (t) are discarded to
simulate extinction. At this point, one generation of a
standard EP is conducted to nd replacement species.
By restricting the replacements to be selected only from
the new species, we model speciation. However, it is
entirely possible that all of the species may have suciently high tness so that none go extinct. In this case
the poorest t si |i.e., the species with the lowest objective function value|is mutated, which simulates a lowlevel, background evolution in (t). Whenever a species
evolves to some new state, it also aects the state and
tness other surrounding species. We simulate this coevolution by also mutating ve other randomly chosen
species.
The stress value (t) is chosen from a uniform distribution over the interval [0,0.96]. All species see the exact
same stress level. The upper limit of 0.96 sets the maximum percent of species that can go extinct in any single
generation; it is close to extinction levels seen during the
Permian mass extinction [5].
There is a rationale for choosing that species with the
lowest tness as the rst to evolve. Roberts and Newman
[12] attached a \barrier" to all species that must be overcome in order to evolve to a state of higher tness. With
respect to an evolutionary algorithm, this barrier is a region of relatively low tness and shows the mutation size
necessary to improve phenotypical behavior. Extremely
low-t individuals intuitively have very small barriers|
even a small mutation could lead to a marked improved
phenotype. Roberts and Newman chose the species with
the lowest tness to evolve rst in their model; this is
the same approach we have adopted.
There is no absolute scale of tness that can be consistently applied over all environments; tnesses can therefore lie between zero and one without any loss in generality. Our algorithm maps all tness values to the subinterval [,1], where is a user selectable parameter. The
lower bound > 0 serves a useful purpose. Suppose
= 0 so tness values can reside anywhere on the unit
interval. Furthermore, assume a very simplistic mapping
function is used|simple normalization of all tness values to the maximum tness in the current population.
If the evolutionary algorithm is working properly, the
individuals should eventually evolve to a high state of
tness, which means all tness values 1. This makes
it very unlikely that an external stress can force species
to become extinct|a behavior that directly contradicts
evidence found in the fossil record. The parameter creates a lower threshold for extinction that prevents mass
extinctions from occurring too frequently, while at the
same time not dispensing with them entirely.
Reproduction is done only via mutation:
x0i = xi + N(0,) ; xi 2 ~x
where N(0,) is a normally distributed random variable
with zero mean and standard deviation .
3
f
2
1
0.5
0.
1
0
0
1
0.5
x1
-0.5
0
x2
-0.5
-11
-1
Figure 3: The tness landscape dened by Eq. (2). The
global minimum is at (0,0).
We used our extinction-based evolutionary algorithm
to search for the global minimum of the multimodal objective function
f (x1 ; x2 ) = x21 + 2x22 , 0:3 cos(3x1 )
,0:4 cos(4x2 ) + 0:7
(2)
with x1 ; x2 2 [,1; 1]. This function is shown in Figure
3. The algorithm was run for 10,000 generations
p with
N = 100 species and = 0:8. We chose = 0:02 and
no self-adaptation was used.
The performance of our algorithm is compared
against a standard EP that was run with the same population size; the same initial population; a tournament
size of 20; and an identical method of reproduction.
Our algorithm executes 10,000 generations about three
times faster than a standard EP executes 5,000 generations. Nevertheless, our extinction-based evolutionary
algorithm consistently outperformed the standard EP in
searches conducted over the landscape depicted in Figure
3. The results are given in Table 1.
minimum f (~x)
mean
std. dev.
EA (w/extinction) standard EP
9.14010,7
6.70010,5
,
6
8.29410
4.49810,4
,
6
7.95310
3.54610,4
Table 1: Performance comparison between the evolutionary algorithm with extinction and a standard EP. Both
algorithms used the same initial population to conduct
searches in the landscape shown in Figure 3. The globally minimum value is f (0; 0) = 0. Table entries are
based on data recorded from 20 independent runs.
4 Discussion
The material from the two previous sections raises some
important philosophical issues regarding tournament selection. In order to compute the number of wins for a
target individual, a small tournament set is randomly
chosen from the current population. Fitness values are
then compared in a head-to-head manner. This presumes it is both acceptable and appropriate to compare
the tness values of distinct individuals. Tournament
selection makes sense in light of a simulation at the individual level, but dierent approaches are required when
modeling evolution at the species level. More specically,
it may not make sense to conduct head-to-head tness
comparisons when individuals in a population are interpreted to be distinct species. Indeed, inter-species tness
comparison are valid only in special circumstances.
For clarity, let represent the set of all species that
co-exist in a given environment. The phenotype of each
si 2 ultimately reects the ability of that species to
survive in that environment|it is a measure of its tness. But, comparing the phenotypes of two distinct
species may not only be dicult, it may be meaningless.
For example, in the Caprivi Game Park in northeastern Namibia, one can nd lions, antelopes, and hippos.
These species all co-exist but do not necessarily have
any direct relationship between each other. Of course
lions and antelope will since they have a predator-prey
relationship. But there is no obvious|let alone direct|
relationship between antelope and hippos nor between
lions and hippos. Does it therefore make any sense to
compare their phenotypes (equivalently, tnesses)? Yet,
that is exactly what happens during conventional tournament selection.
Our extinction-based evolutionary algorithm attempts to evolve a population to higher tness values
by emulating dynamics derived from the fossil record|
extinction must be present and tournaments are conducted among species. Although the exact cause of biotic extinction remains debatable, a number of reasons
have been proposed including excessive inbreeding [13]
or environmental stresses such as climatic change, volcanic activity, or disease [14]. Our algorithm assumes
environmental stress is culpable. Replacement species
are created by mutating existing species|emulating
speciation|and a tournament among those new candidates determines the new species that survives. This
follows Gause's Principle of Competitive Exclusion [15]
where new species must compete for the same ecological
niche. Note that this tournament does include existing
species in the tournament set even though replacements
can only come from speciation.
Extinctions (%)
100
80
60
40
20
0
7000 7200 7400 7600
Generations
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8000
Figure 4: Extinctions per generation over a portion of
the evolutionary process.
Over 25 years ago, Eldredge and Gould [16] sparked
a controversy that continues even today. They viewed
\phyletic gradualism" as a awed concept that should be
replaced by punctuated equilibrium. This concept says
that evolutionary activity is characterized by periods of
stasis, with little speciation, interspersed with episodes
of very intense speciation. Figure 4 shows that our algorithm has periods of stasis interspersed with burst of
extinctions. However, this behavior is not exactly punctuated equilibrium|we immediately replace any extinct
species whereas punctuated equilibrium does not mandate that a xed number of species be constantly maintained.
5 Conclusions & Future Work
Evolutionary change cannot be completely explained by
natural selection alone [17]. At the species level, mass
extinctions wipe out entire species for reasons unrelated
to inter-species struggles for survival. Nevertheless, this
does create windows of opportunity for other species to
ourish. Our algorithm follows a similar vein to conduct
searches: episodic mass extinctions prevent stagnation,
and rapid speciation maintains a constant number of solutions.
Our results indicate that extinction can enhance the
performance of an evolutionary search. Granted, only
one multimodal landscape was explored, but the results are certainly encouraging particularly in light of
the low execution time with respect to a standard EP.
The improved execution time is not unexpected. In
the extinction-based algorithm a tournament is not conducted each generation|only background evolution is
simulated. In fact, a tournament is conducted only after
the onset of a mass extinction. On the other hand, the
standard EP conducts a tournament every generation.
Our algorithm can easily accommodate searches over
dynamic landscapes. This will be the subject of future
studies. We have identied some other areas that can
deserve further investigation. Specically,
1. Modeling evolution
It is important to note that our algorithm, in its
present form, is not intended to serve as a model
for biotic macro-evolution. Nevertheless, it does
incorporate a number of intriguing features that
suggest it could be easily modied to perform that
function. A rst step would be to monitor the
branching and termination of species. Ideally, we
should not maintain a constant population size|
speciation should be allowed to proceed in a natural way with, perhaps, a maximum number of
species that can be supported within the environment.
Normally a species can expect to undergo a number of stresses during its lifetime and all species
co-existing in that same environment will be subjected to the same stress|but not all species are
aected to the same degree by that stress. Newman [18] put it this way:
: : : we have assumed that every stress on
the system aects every species. This
is clearly not realistic. Some stresses
will for example be localized in space,
or will not reach under the sea, or will
only reach under the sea, and so for.
The current version applies the stress (t) equally
to all species. We could specify|or even evolve|
species relationships that permits a stress factor to
aect distinct species in dierent ways. One obvious way is to let stress be analogous to a tness
penalty and let culling be done by natural selection.
2. Self-adaptive extinction
The strategy parameter sets a lower bound on
the percent of a species that can go extinct. As
! 0, a large number of mass extinctions become possible|our algorithm then approximates
a (; ) evolution strategy. Conversely, as ! 1,
virtually no extinctions occur and our algorithm
degenerates into a random walk with replacement.
It would be interesting to investigate the performance when is allowed to evolve.
3. Self-adaptive mutation
The current version of our algorithm used a xed
standard deviation () for the normal distribution.
Self-adaptation should improve the performance,
but no single method is universally applicable [19].
4. Other mutation distributions
A number of recent studies indicate that alternative probability distributions for mutation can
improve the performance over some landscapes
[20, 21]. It would be a trivial matter to incorporate
Cauchy mutations into our algorithm.
Acknowledgement
The authors would like to thank L.J. Fogel and D.B.
Fogel for their valuable comments that helped to improve
this paper.
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