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extinction event were removed quickly, generating new. “windows of opportunity” for other species to flourish. The temporal stability of the environment is closely.

Evolutionary Computation with Extinction: Experiments and Analysis Gary B. Fogel

Garrison W. Greenwood

Kumar Chellapilla

Natural Selection, Inc. 3333 N. Torrey Pines Ct., Suite 200 La Jolla, CA 92037 [email protected]

Portland State University Dept. of Elec. & Computer Engr. Portland, OR 97207 [email protected]

Natural Selection, Inc. 3333 N. Torrey Pines Ct., Suite 200 La Jolla, CA 92037 [email protected]

Abstract- Under a species-level abstraction of classical evolutionary programming, the standard tournament selection model is not appropriate. When viewed in this manner, it is more appropriate to consider two modes of life histories: background evolution and extinction. The utility of this approach as an optimization procedure is evaluated on a series of test functions relative to the performance of classical evolutionary programming and fast evolutionary programming. The results indicate that on some smooth, convex landscapes and over noisy, highly multimodal landscapes, extinction evolutionary programming can outperform classical and fast evolutionary programming. On other landscapes, however, extinction evolutionary programming performs considerably worse than classical and fast evolutionary programming. Potential reasons for this variability in performance are indicated.

1 Introduction The general format of evolutionary programming (EP) follows a two-step process of selection and variation in a population. Following initialization of a population, the fitness of each individual in the population is scored with respect to an arbitrary fitness function. In general, selection is applied as a tournament wherein the fitness of each individual in the population is compared against the fitness of a random set of other individuals in the same population. A “win” is recorded for an individual each time that individual’s fitness equals or exceeds that of another in the tournament set. Individuals are then ranked with respect to the number of wins and those with the highest number of wins over some threshold are selected as parents for the next generation. Parents are randomly varied to generate offspring and the fitness of each member in the population is re-evaluated. This process is repeated for a user-specified number of generations (Blickle, 1997). If one makes the interpretation that individuals in the population represent species in a community, the validity of tournament selection is suspect. This is due to the fact that in natural communities, distinct species do not necessarily interact in direct competition with one another. Yet in tournament selection, every individual in the current

population has an equal probability of tournament participation. More specifically, interspecies fitness comparisons only are valid in special circumstances. Under the assumption that “individuals” in the population are interpreted as species, a new modeling approach is required. To address this situation, Greenwood et al. (1999) proposed a new form of EP that models evolution at the species level rather than at the level of the individual. This process was demonstrated as a potential optimization procedure. In modeling species evolution, processes of extinction and background evolution were employed. In this new “extinction EP” algorithm, not all members of the population necessarily reproduced each generation; a tournament did not always determine survival; and coevolution affected only a small subset of the population. This procedure was tested successfully as a potential optimization procedure on only one test function. In this paper, the extinction EP has been modified to tie it closer to natural evolution. This improved version was tested on eight functions reported in Yao (1999). The results were compared to that given for fast EP and classical EP (Yao et al., 1999) to test robustness of this strategy. Our results suggest that with an equivalent number of function evaluations, the improved extinction EP algorithm can significantly outperform both fast EP and classical EP on some functions, whereas on others it lacks required performance. The potential reasons for this limited success are discussed in the conclusions of this article. 1.1 Mass Extinction Extinction has been shown to play a significant role in the history of life on Earth. Five mass extinctions are recorded in the fossil record (Ordovician, ~440 million years ago (Mya); late Devonian, ~365 Mya; late Permian, ~250 Mya; terminal Triassic, ~215 Mya; and terminal Cretaceous, ~65 Mya). These five mass extinctions have extinction rates that are considerably higher than “background” levels of extinction. For instance, evidence from the fossil record indicates that 96% of all marine species on Earth went extinct during the Permian extinction, representing 52% of all marine vertebrate and invertebrate families (Raup, 1986). Extinctions of this extreme order rapidly “rearranged” the

playing field, generating new niches and destroying others. Species that persisted for millions of years prior to such an extinction event were removed quickly, generating new “windows of opportunity” for other species to flourish. The temporal stability of the environment is closely correlated to the rate at which species in natural systems change over time. On occasion, environmental stress has been so severe or prolonged so as to invoke widespread ecological instability. The causal agent of the ecological instability can be considered as the instigator of widespread (roughly) simultaneous extinction of many species. In rapid succession, species that were once well suited for their particular environmental niche were only barely able to survive. Other species, unable to adapt or be particularly “pre-adapted” in some regard to the new environmental conditions, met their demise. Events of this magnitude are recorded in the fossil record as mass extinctions. It is important to note that this process of extinction differs significantly from natural selection at the level of the individual in that the otherwise fittest species may easily perish during extinction intervals. Under natural selection, organisms that are least fit are culled predominantly from the population. In evolutionary computation, extinction would serve to periodically remove the “best” organisms from the population at random intervals during an optimization procedure. Using the stratigraphical ranges for 17,621 marine genera over the past 600 Myr, Raup (1991) developed a “kill curve,” that described the average time between extinction events of varying intensity. For example, an extinction event capable of killing 30% of all marine species on Earth occurred every 10 Myr and an extinction event capable of killing 65% of all marine species occurred every 100 Myr. The kill curve suggests that the intensity of extinction events is continuously distributed in a logarithmic fashion over time than as a binary classification of “background” and “mass” extinctions. Extinctions are thought to arise as the result of either biotic interactions (food web interactions, etc.) or abiotic events (bolide impact, etc.) (Newton and Laporte, 1989). For the purposes of this investigation, we have chosen to model mass extinction as the sole result of abiotic events and not incorporate any direct ecological interactions. Such interactions will be the focus of future research.

part, by an objective function f (x) minimum values of this function increase the likelihood of survival. But fitness is a relative measure, which means f (x) alone does not completely define fitness unless the mapping function considers the influence of other species in the same environment. We used the following mapping scheme. Let fmin(t) (fmax(t)) denote the minimum (maximum) objective function value at generation t among all species si ∈ Ω(t). The fitness of si at time t is given by

Φ (si , t ) = α + (1 − α ) ⋅

where α ∈ [0,1]. Such a mapping forces f (x ) → [α,1]. (The purpose of the lower bound α will be discussed shortly.) Other types of mappings could be used; the above method was chosen because of its simplicity. The steps in our updated extinction-based evolutionary algorithm are shown below: 1.

Let t=0 and randomly initialize population Ω(0). Compute Φ(si,t) for all i.

2.

Increment t and let Ω(t) = Ω(t−1).

3.

Randomly choose a stress η(t) ∈ [0,0.96]. Make all si ∈ Ω(t)| Φ(si,t) < η(t) go extinct (i.e., discard them). Let m equal the number of species that go extinct.

4.

If (m>0) then { Mutate the top 10% of the species to produce multiple new species. Compute Φ(⋅,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. } else { Mutate 10% of the survivors and compute Φ(⋅,t) for the new species. Replace the old species with its new species if the latter has higher fitness. }

5.

If computation time not exhausted, go to step 2.

2 Methods We consider an ecosystem at time t that contains species set Ω(t)={s1, s2,..., sk}, where each si is characterized by the vector x ∈ ℜ . These numbers represent phenotypical traits that change as species evolve. All species are subjected to an external environmental stress η(t), which may be severe enough to cause some species to become extinct if their fitness level is too low. Such a stress is analogous to the rapid environmental change following a large bolide impact. The fitness of each si is measured, in

f ( x i ) − f max (t ) f min (t ) − f max (t )

n

It is important to highlight the differences between this updated algorithm and our previous implementation. The previous extinction-based EP (Greenwood et al., 1999) operated as follows. During each generation T, three steps took place. A random stress was applied to all species, and all species si with fitness Φ(si, t) < η(t) were discarded to simulate extinction. At this point, one generation of a

Table 1. Test functions used to evaluate extinction EP (adopted from Yao et al., 1999). In all cases, n=30 dimensions, S ⊆ ℜn and fmin=0.

Test Function

Function Name

S

Sphere model

[−100,100] n

Schwefel’ Problem 1.2

[−100,100] n

Schwefel’s Problem 2.21

[−30,30] n

Generalized Rosenbrock Function

[−100,100] n

Step Function

[−100,100] n

Noisy Quartic

[−1.28,1.28] n

Generalized Rastrigin’s Function

[−5.12,5.12] n

Griewank Function

[−600,600] n

f1 ( x ) = ∑i =1 xi2 n

f 2 ( x) = ∑i =1 (∑ j =1 x j ) 2 n

i

f 3 ( x) = max{ xi , 1 ≤ i ≤ n} i

[

f 4 ( x) = ∑i =1 100( xi +1 − xi2 ) 2 + ( xi − 1)2 n

]

f 5 ( x ) = ∑ i =1 (xi + 0.5)

2

n

f 6 ( x ) = ∑i =1 ixi4 + random[0,1) n

[

f 7 ( x) = ∑i =1 xi2 − 10 cos(2πxi ) + 10 n

]

x  n n 1 f 8 ( x) = x 2 − ∏i cos i  + 1 ∑ i =1 i 4000  i classical EP was conducted to generate new species. If no species went extinct, the lowest fit species was mutated. Five other randomly chosen species were also mutated to simulate species interaction and evolution. If one or more species went extinct, then the surviving species created multiple offspring and tournament selection was used to choose the replacement species. The previous algorithm deviated from natural evolution in the following ways: 1) During background evolution, the lowest fit species was mutated. The rationale for doing this was predicated on attaching a ‘‘barrier’’ to all species that must be overcome in order to evolve to a state of higher fitness. But nature has no such rule that states speciation—i.e., mutation—should be limited to the least-fit species. Furthermore, in the context of optimization, this concept inhibits the optimization process. In the new algorithm, the top 10% of the existing species are chosen at random for speciation and a tournament among all the species decides the survivors. 2) Following mass extinction in the previous algorithm, each survivor created multiple new species and a tournament chose the replacements. In the new algorithm, each survivor generates γ new species each. A tournament is conducted among all members of the population (parents + offspring) and the top m are selected from the tournament for replacement. This procedure can be thought of as simulating sympatric speciation. Each run of the extinction-based EP evolved a population of 100 species. Speciation was emulated by adding a random variable to each object parameter. All

tournaments were conducted with a tournament size equal to 10% of the survivors. In the case of mass extinctions, each survivor generated a minimum of γ=2 offspring (γ was increased automatically if an insufficient number of offspring would be produced to maintain a constant population size). The number of generations processed in each run depended on the objective function. This is explained in greater detail in the next section.

3 Results The evaluation of our updated algorithm used the same suite of functions that were used for the fast EP evaluation (Yao et al., 1999). These functions are described in Table 1. The results for fast EP and classical EP results from Yao et al. (1999) are shown for comparison to those of extinction EP in Table 2. All extinction-based EP results are taken over 25 runs. We observed that in some instances a bimodal distribution produced the best results while in others a Cauchy distribution was better (all functions with an “∗” used a Cauchy distribution). The number of generations processed depended on which objective function was being evaluated. Yao et al. (1999) identified how many generations were processed for each of the listed functions. We could not use the same criteria because the amount of processing in a generation depended on not only whether or not a mass extinction occurred, but also on the intensity of the extinction. Consequently, a run in the extinction-based EP was terminated when an equivalent number of function evaluations to Yao et al. (1999) were processed.

Table 2. Performance comparison between the fast EP, classical EP (adopted from Yao et al., 1999) and our extinction-based EP. The optimal value for all functions is fmin= 0. Function

Fast EP (mean) 5.7 x 10-4

Classic EP (mean) 2.2 x 10-4

Extinction EP (mean) 4.2 x 10-4

Extinction EP (best) 7.9 x 10-6

f2

1.6 x 10-2

5.0 x 10-2

6.7 x 103

7.9 x 102

f3

3.0 x 10-1

2.0

6.4 x 101

3.4 x 101

f4

5.1

6.2

6.7 x 102

2.5

f 5*

0.0

5.8 x 102

1.2 x 103

0.0

-2

-4

f

* 1

-3

f6

7.6 x 10

1.8 x 10

2.9 x 10

6.5 x 10-7

f 7*

4.6 x 10-2

8.9 x 101

2.2 x 101

9.9

f8

1.6 x 10-2

2.2 x 10-2

2.8 x 10-1

1.8 x 10-3

∗ denotes Cauchy distribution was used 4 Discussion Our extinction-based evolutionary algorithm attempts to evolve a population to higher fitness (lower error) values by emulating dynamics derived from the fossil record. Within such a system, 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 (Frankham, 1998) or environmental stresses such as climatic change, volcanic activity, or disease (Skelton, 1993). Our algorithm assumes environmental stress is culpable. Existing species are mutated to create replacement species (emulating speciation) and a tournament among these new candidates ultimately determines the survivors. This follows Gause’s Principle of Competitive Exclusion (Gause, 1934) 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. Our results show the extinction-based EP works quite well over smooth, convex surfaces and over noisy, highly multimodal surfaces—literally the two extremes of fitness landscape topologies. Although the extinction-based EP performed considerably worse in f2, in most cases it outperformed the CEP and in several cases outperformed FEP. Notice that in several instances (e.g., f4 and f5) the best value and mean value are orders of magnitude different. We observed that most runs produced very similar final results. However, in a small number of runs the extinction EP algorithm quickly stagnated at optima far removed from the global optima. Although this occurred infrequently, the final fitness value was large enough to skew the mean. For example, in 20 runs of f5 the final fitness value was less than 10-4. However, two runs stagnated with fitness values of over 2000.

No standard deviation or statistical significance tests were conducted for the extinction-based EP. We specifically precluded such analysis because it is meaningless whenever extinction is present. Extinction is a radical form of selection and, in our algorithm, mass extinction removes even highly fit species (recall that as many as 96% of the species may go extinct). This suggests that a large percentage of the population can be eliminated at any given time, which leads to a loss of genetic diversity. An immediate consequence is that exploration of some regions in the search space abruptly ceases. But mathematically this loss alters the distribution of species thereby biasing any statistical measurements. Statistical information could be extracted if the number of extinctions in each generation were taken from some probability distribution, but this is not the case. The random stress value η(t) helps to choose which (if any) individuals go extinct during generation t. However, the stress values themselves do not determine extinctions; η(t) only defines the threshold for extinctions and only those species whose fitness levels are less than that threshold go extinct. Since fitness levels may differ considerably in distinct runs, so will the number of extinctions. Therefore, it is expected that the statistical measures will vary significantly across an ensemble of runs. The anomaly described above for function f5 is a perfect example of this and it explains why the mean is three orders of magnitude higher than the best value. It should also be noted that the stress value η(t) is applied equally across all species and that species fitnesses are used to determine which species go extinct relative to that stress. Clearly, this is an oversimplification of real macro-evolutionary processes. For example, some stresses might be localized in geography or ecology. Within the context of an optimization procedure, however, the notion of a stress applied equally to all species appears to be successful.

The strategy parameter α sets a lower bound on the percentage of species that can go extinct. It would be interesting to observe the performance of this algorithm as an optimization procedure where α is allowed to evolve. Similarly, in the current methods, a fixed standard deviation (σ) was used for the Cauchy or bimodal distribution. σ could be made to evolve and may improve the performance (Saravanan et al., 1995).

20

18

16

14

Extinctions

12

10

8

6

4

2

0 3000

3100

3200

3300

3400

3500 3600 Generations

3700

3800

3900

4000

Figure 1. Extinctions during a portion of a run of Rastrigin’s function. Figure 1 shows the number of extinctions that occurred during a run of f7. Notice that there are periods of intense evolutionary activity interspersed with periods of stasis where only background evolution takes place. This is precisely the type of dynamics some evolutionary biologists believe takes place in nature (Eldredge and Gould, 1972; Marshall, 1998).

might have played a significant role in the biotic replacements observed in the fossil record. Pimm et al. (1995) noted that species most prone to the extinction at the present are those that are rare and local. Simulations that incorporate a notion of geographical space and absolute number might be useful in understanding these trends. Newman (2000), generated a model of extinction that assumed extinction was caused entirely by stresses on the ecosystem by external (abiotic) factors. No interaction between species was introduced in this model. Such a model runs contrary to that of Bak and Sneppen (1993) wherein it was assumed that extinction is caused entirely by coevolutionary effects arising from the interaction of species in an environment. Our model combines both of these approaches in the framework of evolutionary computation. During extinction, an external stress is applied equally to all species and those that do not have sufficient fitness to meet this stress are culled. During background evolution, species interactions (through tournament selection) help to specify newly generated species. We have incorporated extinction into the framework of evolutionary programming in an effort to enhance optimization performance. This performance was quite varied, but in some cases increased relative to that of classical and fast methods of evolutionary programming. A similar method could be used not simply as an optimization procedure but as a method of understanding observed patterns in the fossil record.

Acknowledgements The authors would like to thank David M. Raup and Wirt Atmar for their valuable discussions.

References Atmar, J. W. (1976). Speculation on the Evolution of Intelligence and its Possible Realization in Machine Form. Doctoral Dissertation, New Mexico State University.

5 Future Work We have specifically identified other areas of research deserving further investigation with respect to modeling extinction. The optimization procedure outlined in this paper in not intended to serve as a model of biotic macroevolution. However, it would be interesting to tag and monitor the duration at diversification of species over time. Atmar (1976) demonstrated a similar technique by generating phylogenies of evolving finite-state machines, arriving at results that were extremely similar to those presented by Raup (1972) for the diversity of marine invertebrates over the Phanerozoic. Similar investigations might be useful for understanding observed trends in the fossil record. A constant population size need not be maintained; rather the population could be made to vary with respect to some notion of environmental capacity. Briggs (1998) noted that ecological/community interactions

Bak, P. and Sneppen, K. (1993). “Punctuated Equilibrium and Criticality in a Simple Model of Evolution,” Physical Review Letters, 71(24):4083-4086. Blickle, T. (1997). “Tournament selection,” in Handbook of Evolutionary Computation, T. Back, D. B. Fogel, and Z. Michalewicz (eds.). Oxford Univ. Press, New York. C2.3:14. Briggs, J. C. (1998). “Biotic replacements—extinction or clade interaction?,” BioScience, 48(5):389-395. Eldredge, N. and Gould, S. J. (1972). “Punctuated Equilibria: An Alternative to Phyletic Gradualism,” Models of Paleobiology, T. Schopf (ed.), pp. 82-115.

Frankham, R. and Ralls, K. (1998). “Inbreeding leads to extinction,” Nature, 392:441-442. Gause, G. F. (1934). The Struggle for Existence. Williams and Wilkins, Baltimore. Greenwood, G. W., Fogel, G. B., and Ciobanu, M. (1999). “Emphasizing Extinction in Evolutionary Programming,” Proceedings of the 1999 Congress on Evolutionary Computation, IEEE Press, Washington, D.C., pp. 666-671.

Saravanan, N., Fogel, D. B., and Nelson, K. (1995) “A comparison of methods for self-adaptation in evolutionary algorithms.” Biosystems. 36:157-166. Skelton, P. (1993). Evolution: A Biological and Paleontological Approach. Addison-Wesley, Wokingham, England. Raup, D. (1972). “Taxonomic Diversity During the Phanerozoic,” Science, 177:1065-1071.

Marshall, C. R. (1998) “Mass Extinction Probed,” Nature, 392:17-20.

Raup, D. (1986). “Biological extinction in earth history,” Science, 231:1528-1533.

Newman, M. (2000). “Simple Models of Evolution and Extinction,” IEEE Computing in Science and Engineering. Jan./Feb.:80-86.

Raup, D. (1991). Extinction: Bad Genes or Bad Luck? W. W. Norton, New York.

Newton, C. and Laporte, L. (1989). Ancient Environments. Prentice Hall, New Jersey. Pimm, S. L., Russell, G. J., Gittleman, J. L. and Brooks, T. M. (2000). “The Future of Biodiversity,” Science, 269:347350.

Yao, X., Liu, Y. and Lin, G (1999). “Evolutionary programming made faster,” IEEE Trans. on Evol. Comput., 3(2):82-102.

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