Original paper
Soft Computing 6 (2002) 449 ± 455 Ó Springer-Verlag 2002 DOI 10.1007/s00500-001-0160-8
Mixed size tournament selection R. Huber, T. Schell
Abstract In genetic algorithms, tournament schemes are often applied as selection operators. The advantage is simplicity and ef®ciency. On the other hand, major de®ciencies related to tournament selection are the coarse scaling of the selection pressure and the poor sampling accuracy. We introduce a new variant of tournament selection which provides an adjustable probability distribution, a ®ne-tuning facility for the selection pressure and an improved sampling accuracy at the cost of a minimal increase of the complexity and with almost no loss of ef®ciency. Keywords Genetic algorithms, Selection methods, Sampling
1 Introduction Tournament selection (TS) is a popular selection scheme within genetic algorithms. Mainly, this is because of its ef®ciency and its simple implementation. But, one of the major de®ciencies of TS is the coarse scaling of the selection pressure, i.e. increasing the tournament size causes a severe increase of the selection pressure and vice versa. In the past, new tournament schemes have been introduced to provide a ®ner scaling facility for the selection pressure, these methods are called probabilistic tournament selection (pTS). pTS enables a ®ne scaling of the selection pressure but at the expense of increased complexity and reduced ef®ciency. Furthermore, the sampling accuracy of pTS is ®xed to the accuracy of roulette wheel sampling which is somehow the lower bound of sampling
accuracy. We will assess the sampling accuracy by the chi-square goodness-of-®t test. In this work, we introduce a new variant of TS with a facility to ®ne-tune the selection pressure. Our TS-scheme provides a ¯exible probability distribution which can be used to adapt the probability distribution of our scheme to any other scheme. With a speci®c implementation, we obtain a sampling accuracy which is clearly better than the accuracy of roulette wheel sampling. And, our scheme is almost as ef®cient as the original TS. In a generational genetic algorithm, this is achieved by introducing tournaments of varying size during one selection step. We will call this type of TS mixed size tournament selection (msTS). The article is organized as follows: In Sect. 2, we will review various variants of pTS. In Sect. 3, we will introduce msTS in detail. The adaptability of msTS will be demonstrated by simulating two ranking schemes with msTS and the appropriate parameterization of msTS will be given. Finally, some practical observations concerning sampling accuracy will be presented.
2 Equivalences of tournament selection to other selection methods In recent work, authors have identi®ed equivalences among selection schemes. At ®rst, equivalences of selection schemes with speci®c parameter settings have been noted. For example, Goldberg et al. [4] demonstrated that binary TS with replacement is equivalent to linear ranking selection (LRk) with g 1=N. The parameter g determines the slope of the probability distribution of LRk [2]. BaÈck presented in his book [1] the equivalence of TS with t > 2 where t is the number of contestants and exponential ranking selection (ERk) with speci®c R. Huber parameterization. Advanced Computer Vision Julstrom [5] has shown the equivalence of binary pTS Austrian Research Centers GmbH-ARC and LRk over the full range 0 g 1. Wohllebengasse 6, A-1040 Vienna, Austria e-mail:
[email protected] Recently, Julstrom et al. [6] have introduced weighted k-tournament selection (wTS) which is a speci®c type of T. Schell (&) pTS. wTS enables a simulation of ERk for an almost Department of Scienti®c Computing, University of Salzburg, arbitrary setting of c. Hellbrunnerstraûe 34, A-5020 Salzburg, Austria In the following sections, we will revisit pTS and wTS. e-mail:
[email protected] We will study their distribution of selection probabilities Thomas Schell was supported by the FWF-Project P13732-MAT. and their sampling accuracy. The authors would like to thank Gunther Leobacher and Friedrich Pillichshammer, both with the University of Linz, Austria, for their mathematical advice concerning the optimization problem. The authors are grateful for the statistical contribution provided by Stefan Wegenkittl of Proceryon Biosciences, Salzburg, Austria.
2.1 Probabilistic tournament selection One popular notion to overcome the in¯exibility of TS concerning selection pressure was the introduction of
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probabilistic tournaments. After the tournament, the winner and the losers are assigned probabilities such that each participant has a chance to survive the tournament even if it is not the best individual. The winner of the tournament is determined by sampling the probabilities of the participants. The balancing of the probabilities between winner and losers provides a ®ne-tuning facility for the selection pressure. Besides, modi®cations of the distribution of the selection probabilities are enabled. wTS is based on the idea of probabilistic tournaments and was introduced by Julstrom [6]. In contrast to pTS, the selection probabilities among the winner and the losers of a tournament are exponentially normalized. wTS is a hybrid selection method which consists of t-TS and ERk. The t-TS chooses t candidates form the population. From the t candidates one individual is selected with a ``mini''-ERk for the mating pool. The ¯exibility of ``mini''-ERk provides the improved adaptability of wTS. The two-step process of selecting individuals for a tournament and performing a ``mini''-ERk among these individuals results in increased complexity compared to standard TS. We noticed some problems with wTS as presented in [6]. Pn The basic requirement for a probability distribution i1 pi 1 is violated. Speci®cally, the summation of all pi results in a value less than 1. Besides, the selection probabilities pi contain indeterminate expressions like 00 for i 1; n. In the parameter tables for wTS, the absolute error is given. We found that for low selection probabilities the low absolute error is contrasted by a high relative error ranging from 30 to 50%. The repercussions of the latter deviations of wTS from the simulated distribution of selection probabilities need to be clari®ed. In the next section, we introduce a new variant of TS which provides a ¯exible distribution of selection probabilities and which is almost as ef®cient and simple as TS.
3 Motivation for a new tournament selection variant The major shortcomings of TS are the rather coarse scaling of the selection pressure and the in¯exible distribution of selection probabilities. In the past several approaches have been presented to provide an improved scalability of the selection pressure in TS. For the range of low selection pressure the introduction of probabilistic binary tournaments adds scalability to the tournament scheme but at the expense of low sampling accuracy, i.e. probabilistic binary TS operates at the sampling accuracy of roulette wheel sampling. The aforementioned wTS provides a ®ne adjustment of the selection pressure and a ¯exible distribution of selection probabilities but at the expense of low sampling accuracy and high computational complexity. Speci®cally, during a tournament an additional ``mini''-ERk has to be performed. In Sect. 3.1 we introduce msTS which can be implemented ef®ciently and provides scalable selection pressure over the entire range from low to high pressures. In Sect. 3.2, the accurate parameterization of a msTS to ®t the selection probabilities of a given selection method is de®ned as an optimization problem. In Sect. 3.3 we derive an analytical result to ®t msTS to LRk and the search heuristic for msTS parameters to ®t ERk will be presented. The
parameter tables of msTS to ®t various instances of LRk and ERk with the according absolute error are displayed in Sect. 3.4. In Sect. 3.5, we compare the sampling accuracy of wTS, msTSA and msTSB .
3.1 Mixed size tournament selection The most basic tournament schemes are unary TS and binary TS. Obviously, unary TS features uniformly distributed selection probabilities. In contrast, binary TS provides a linear distribution of selection probabilities. The step from 1-TS to 2-TS causes a drastic increase of the selection pressure. To obtain a smooth increase of selection pressure from 1-TS to 2-TS one can mix 1- and 2-tournaments such that the portion of 1-tournaments is decreased while at the same time the portion of 2-tournaments is increased. The number of tournaments is equivalent to the single size TS but during the selection phase tournaments of varying size do occur. By admitting tournament sizes t 1 any distribution of selection probabilities can be approximated by msTS at a high level of accuracy. To demonstrate the adaptability of our approach, we will simulate LRk and ERk with appropriately adjusted msTS. Moreover, compared to the sampling accuracy of roulette wheel sampling the accuracy of msTS can be improved by applying type B tournaments. (TS type B will be described in the next section.) 3.2 Definition of the optimization problem To ®t msTS to an arbitrary selection scheme we need at ®rst the distribution of the selection probabilities of the scheme in consideration. We denote the selection probability of an individual i using a selection method X by pXi . The distribution of the selection probabilities of the entire population is denoted by the vector pX pX1 pXN . In our work, we deal with two different variants of TS. Speci®cally, we denote TS with replacement by TSA . While TSB refers to TS with partial replacement which was introduced in [8]. For TSB , the population is copied t times and each copy is randomly permutated. The copies are concatenated to a tournament-list of size t N. Subsequently, for each tournament, t elements are successively removed from the tournament-list and the winner will be inserted into the mating pool. Each individual will participate in exactly t tournaments, therefore it is said that the individuals are partially replaced. The selection probabilities pi of the tournament schemes A and B are de®ned by i 1 it
i 1t A B t 1 pi;t and pi;t N Nt t The de®nition above includes the population size N 2 N, an index to the sorted individuals i (1 i N) in increasing order of ®tness and the tournament size t 1. The vector pYt pY1;t pYN;t consists of the selection probabilities of the entire population. The superscript Y 2 fA; Bg is used to denote either type A or type B tournaments and t indicates the size of the tournament. We approximate an arbitrary selection scheme X with a weighted sum of TS of varying size t. The parameters of
msTS which need to be identi®ed are the maximal tournament size n and the weights at of the involved tournament schemes.
f
a1 ; . . . ; an : pX
n X t1
at pYt ;
nN
1
We need to minimize the function f
a1 ; . . . ; an and we aim for minimal differences of the selection probabilities for each single individual subject to the following constraints:
at 0; t 2 f1; . . . ; ng n X at 1 t1
at N 2 N0 The ®rst constraint determines the impact of a single tournament scheme. The second constraint guarantees the same number of tournament winners as in the single size case and the third condition speci®es that the number of tournaments of size t has to be integer-valued.
3.3 Parameter adjustment of msTS In the following we will consider how one may achieve an identical msTS distribution of selection probabilities for an arbitrarily parameterized ranking scheme, i.e. LRk or ERk. Firstly, we present an analytical result how to adjust the msTS-parameters such that equivalence of msTS and LRk is obtained. Secondly, regarding ERk and msTS equivalence, we describe an algorithm for the adjustment of the msTS-parameters. 3.3.1 Fitting msTS to LRk Firstly, we consider the simpler case of how msTS can be adapted to LRk. The basic idea is: LRk can be simulated by msTS using tournaments of size 1 and 2. In fact, 1-tournaments provide a constant function and 2-tournaments provide a linear function with variable slope provided the weighting factor is accordingly modi®ed. In our derivation, we use the notation of Blickle et al. [3] even though LRk was introduced by Baker [2]. The population is sorted in increasing order and numbered from 1 which corresponds to the least ®t individual to N which corresponds to the most ®t individual. For LRk we use the selection probabilities pLi which are de®ned by 1 N i g
g g pLi N N 1 The parameters of LRk are constrained by 0 g 1 and g 2 g . At ®rst, we use as components of msTS TSA . The selection probabilities have been derived by BaÈck [1]. The ordering of the individuals is consistent with the de®nitions presented above. Now, we derive the weights a1 , and a2 in dependence of g , and g .
i 1 i2
i 12 a2 N2 N By comparing the endpoints of the lines (probability distribution) of LRk and msTS we obtain the following expressions to determine the weights of msTS in accordance with LRk. A pmsTS a1 i
a1 g a2
i
g g 2
N 1
g g 2
1 N1
In a next step, we can easily exchange the components of msTS. Instead of TSA , we apply TSB . B pmsTS a1 i
1 2
i 1 a2 N N
N 1
With the same technique presented above, we obtain the expressions to determine the weights in accordance with LRk.
a1 g g
g 2 A detailed derivation of the weights (a1 , a2 ) can be found in the Appendix of this paper. a2
3.3.2 Fitting msTS to ERk The selection probability of the i-th ranked individual using ERk is given by [7] c 1 N i c ; i 2 f1; . . . ; Ng : pEi N 1 c The parameter c is in the range 0; 1 and is typically very close to 1. Determination of the weight parameters at involves solving the minimization problem already de®ned in Sect. 3.2. The function to be minimized is observed to be a convex function. We introduce the following convex combination
" #
n X
E A A A f
k1 ; . . . ; kn p p1 kt
pt p1 ;
t1 1 Pn where n N; kt 0 and t1 kt 1. Obviously, the definition of the minimization problem for TSB is analogous. The at -parameters are Pnderived from the kt -parameters simply by a1 1 t2 kt and at kt ; t 2. A gradient descent method is applied to the latter function. Although, convexity ensures theoretically the ®nding of the global optimum, in practice numerical problems occurred close to the global optimum, where the gradient gets very small and eventually changes its sign; hence, the global optimum was not always found. Therefore, we applied exhaustive searches for identifying weight parameters.
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452
3.4 Estimated msTS parameters We present results of msTS simulating LRk and ERk. Table 1 shows the approximation of LRk with a ®xed number of individuals N 100. The parameter g is varied from 1.0 (uniform selection) to 2.0 (linear ranking). The maximum absolute difference D between LRk and msTS increases with growing g , which can be seen from the table. The weighting parameters a1 and a2 for msTS have been derived analytically dependent on g , g , and N. The derivations are presented in the Appendix. Basically, an absolute difference D is introduced when rounding the real-valued products a1 N and a2 N to integers. Therefore, the difference D decreases in general with growing population size N. We also note, that for the limiting case N ! 1 the weighting parameters for msTSA become a1 g and a2
g g =2, i.e. equal to those for msTSB . Table 2 shows results for ®tting msTS to ERk. The number of individuals is ®xed to N 100 and the parameter c for ERk is varied over a typical range from 0:95
to 0:99. The number n corresponding to the maximum tournament size is calculated using [6]
N
1 c n 1 cN
The mixture of tournaments is indicated in column at N, e.g. ERk with c 0:98 is approximated by msTSA through a mixture of 31 times 1-tournaments (uniform selection), 34 times 2-tournaments, 7 times 3-tournaments and 28 times 4-tournaments. Figure 1a shows the selection probabilities for ranked individuals and 1- to 4-tournaments with replacement. In Fig. 1b the resulting mixture selection probability is plotted. ERk is very well approximated. The region of the largest absolute difference D 0:00008 is shown in the zoom window (the dashed line corresponds to ERk and the solid line is msTSA ). The results of ERk approximations with population sizes N larger than 100 are presented in Table 3. In this case the parameter c was ®xed to 0.99. It is observed that D is in the order of magnitude of the measured D for N 100. For ERk, the weighting parameters at for speci®c setTable 1. Weighting parameters a1 ; a2 and maximum differences tings of c and N are found by solving the optimization D between LRk and msTS (with n 2), N 100 and varying g problem introduced in Sect. 3.3.1. using either msTSA (with replacement) or msTSB (with partial replacement) g
+
msTS
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
A
msTS
B
a1
a2
D
a1
a2
D
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.00 0.00003 0.00006 0.00009 0.00012 0.00015 0.00016 0.00017 0.00018 0.00019 0.0003
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.00 0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 0.00014 0.00016 0.00018 0.0002
3.5 Sampling accuracy As an example, we illustrate the sampling accuracy of wTS, msTS with type A and B tournaments simulating ERk (c 0:975). For the analysis of the sampling accuracy, we apply the statistical methods presented in [8]. Basically, selection in
Table 2. Weighting parameters at N and maximum differences D between ERk and msTS with N 100, varying c and n using either msTSA (with replacement) or msTSB (with partial replacement) c
n
at N
3 4 4 4 5 6
58 31 23 20 11 5
26 34 29 12 15 14
16 7 0 1 0 3
28 48 67 1 0
3 4 4 4 5 6
59 36 24 21 13 3
25 0 29 3 7 19
16 64 0 21 7 3
0 47 55 11 0
D
A
msTS 0.99 0.98 0.975 0.97 0.96 0.95
73 0
78
0.000069 0.00008 0.00021 0.000469 0.000467 0.00049
74
0.000074 0.00048 0.000187 0.000564 0.000585 0.00042
B
msTS 0.99 0.98 0.975 0.97 0.96 0.95
62 1
Fig. 1. Selection probabilities: a TSA (with replacement) and varying t, b msTSA with a1 0:31; a2 0:34; a3 0:07; a4 0:28 (solid line) approximating ERk with c 0:98 (dashed line). The zoom window shows the region of maximum D 0:00008, which occurs for rank index i 100
a GA is a two-step procedure. In the ®rst step, P the selection probabilities p
p1 ; . . . ; pN , pi 2 0; 1, Ni1 pi 1 are calculated for all individuals i, 1 i N, of the current population on the basis of their ®tness. The distribution p of the selection probabilities gives the expected number for the offspring rate ei , ei N pi , of every individual in the next generation. In the second step, N individuals are drawn from the current population by a sampling algorithm. This process is controlled by the selection probabilities in such a way that the number of offspring of individual i, oi , has expectation Eoi ei . On the average, each individual is thus represented proportional to its assigned selection probability PN in the mating pool for the next generation. Clearly, i1 oi N. All characteristics of a sampling algorithm are contained in the vector
o1 ; . . . ; oN . The chi-square goodnessof-®t statistic provides a measure which is well calibrated for the two extreme sampling methods, stochastic universal sampling (SUS, very high sampling accuracy) and roulette wheel sampling (RWS, low sampling accuracy). We group the individuals into some d 10 mutually distinct classes I1 ; . . . ; Id , Ij f1; . . . ; Ng, P [dj1 Ij f1; . . . ; Ng, and let Ej i2Ij ei and P Oj i2Ij oi denote the overall expectation and the overall observation in each class. Our measure for the sampling accuracy is the chi-square statistic
v :
d X
Ej
Oj 2 Ej
:
which give a good approximation of the accuracy and spread in p terms of the expectation Ev and standard deviation Vv. Finally, we calculate the empirical distribution function (EDF)
1 EDF
t : #f1 l s : vl tg; t 2 0; 1 ; s which can be compared to the theoretical chi-square distribution v2d 1 of RWS by means of a two-sided Kolmogorov±Smirnov test. When simulating ERk (c 0:975) with wTS, we obtain the according parameters k 3, w 0:13 for wTS from Table 2 in [6]. The value of the two-sided Kolmogorov± Smirnov (KS) test statistics in Table 4 indicates that wTS exceeds the limit of 1:63 and hence we reject the hypothesis that the intrinsic sampling of wTS and RWS are equivalent at the 0:01 level of signi®cance. The high mean value e^ compared to the mean value e 9:0 of the v2d 1 distribution indicates a sampling accuracy below that of RWS. In Fig. 2a, we plot the empirical distribution function EDF
t of wTS (thick line). The thin line shows the v2d 1 distribution and the dotted lines represent the 99% con®dence band which has been derived from the Kolmogorov±Smirnov distribution under the hypothesis of RWS. To illustrate high sampling accuracy, we consider SUS. SUS' EDF resembles almost the EDFp of perfect sampling 0 t 2
1; 0 EDFp
t 1 t 2 0; 1 :
Con®rming the results of Table 4, the EDF of wTS is below 2 For our empirical study, we generate s 256 independent the vd 1 distribution indicating lower sampling accuracy than RWS. At t 13, the EDF of wTS is outside of the 99% replications of the vector
o1 ; . . . ; oN for each of the con®dence band. If the EDF is not completely inside the inspected selection methods. From the latter sequence, con®dence band, the statistical evidence against the we calculate the sample mean hypothesis of RWS is equal to or greater than 99%. s In Fig. 2b we present the EDF of msTS with type A 1X vl ; e^ tournaments. The EDF for msTSA perfectly lies within the s l1 99% con®dence band which indicates a sampling accuracy equivalent to RWS. The values reported in Table 4 con®rm and sample standard deviation s the latter observations. s 1 X The EDF of msTSB presented in Fig. 2c proves the su2 ^
vl e^ ; r perior sampling accuracy of type B tournaments compared s 1 l1 to RWS which is indicated by an EDF far above the EDF of RWS. The EDF of msTSB is approaching the EDF of SUS with its high sampling accuracy. The higher sampling Table 3. Weighting parameters at N and maximum differences D between ERk and msTS with c = 0.99 and varying N and n using accuracy of msTSB is also con®rmed by the low e^, r ^ values either msTSA (with replacement) or msTSB (with partial displayed in Table 4. Nevertheless, some of msTSB 's replacement) sampling accuracy compared to a pure (single size) TSB was lost. This loss of sampling accuracy is due to the N n at N D modi®cations which were required to adapt type B msTSA tournaments to msTS. j1
200 300 400 500
msTSB 200 300 400 500
5 6 7 7 5 6 7 7
62 47 31 16 63 50 32 12
63 63 36 11 62 81 24 39
46 99 145 189
7 22 23 19 32 56 20 1
44 12 19 5 132 4 183 0 27 128 69 1
49 8 6
28 71 0
92 257
63 251
0.00005 0.000052 0.0000097 0.000019 0.000012 0.000047 0.000015 0.00002
Table 4. Sample mean and standard deviations for wTS, msTSA, and msTSB for c = 0.975
^e ^ r KS
wTS
msTSA
msTSB
10.172 5.042 1.681
8.804 4.109 0.982
5.373 2.404 6.818
453
tournaments (with replacement). (note for LRk: 0 g 1 and g 2 g )
N 1X g N i1
a1
g
N X i
N g N
i 1
N 2 X
i 1 i a2 N i1
i1
i 12 N2
considering only one term of the summation we obtain:
N i 2i 1 a1 a2 N 1 N we consider the speci®c case i : 1: 1 g a1 a2 N and the case i : N: 1 g a1 a2 2 N
454
g
g
g
from the equations above we yield the formulas to determine the weights (a1 , a2 ) from the LRk-parameters (g , g ):
g g 2
N 1
a1 g a2
g g 2
1 N1
If we use type B tournaments (with partial replacement) instead of type A tournaments the derivation is analogous and even simpler.
N 1X g Fig. 2. Empirical distribution functions for a wTS, b msTSA , and N B i1
c msTS simulating ERk (c = 0.975)
4 Conclusions The popularity of TS in evolutionary computing is due to its ef®ciency and simplicity. In this paper we have revisited approaches to overcome the major de®ciencies of TS which are speci®cally the coarse scaling of the selection pressure and the in¯exible distribution of the selection probabilities. We have introduced a new variant of TS i.e. msTS which provides a ¯exible probability distribution and a ®netuning facility for the selection pressure. Besides, we have demonstrated the adaptability of msTS by simulating LRk and ERk. With type B tournaments as basic components of msTS we were able to obtain improved sampling accuracy compared to roulette wheel sampling. As a result, we would like to note that the aforementioned improvements were achieved at a minimal increase of complexity while preserving the ef®ciency of the original TS.
a1
g
N X 1 i1
N
a2
N g N
i 1
N X 2
i 1 N
N 1 i1
now we consider only terms of the above equation with identical index i:
N i 2
i a1 a2 N 1 N for case i : N we obtain: g
g
g
1 1
g a1 2a2 and for case i : 1
g a1 as before, the weights (a1 , a2 ) of msTS are determined by the LRk-parameters (g , g ).
References
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