A Cost-benefit-based Adaptation Scheme for
Multimeme Algorithms
Wilfried Jakob
Forschungszentrum Karlsruhe GmbH
Institute for Applied Computer Science
P.O. Box 3640, 76021 Karlsruhe, Germany
[email protected]
Abstract. Memetic Algorithms are the most frequently used hybrid of Evolutionary Algorithms (EA) for real-world applications. This paper will deal with
one of the most important obstacles to their wide usage: compared to pure EA,
the number of strategy parameters which have to be adjusted properly is increased. A cost-benefit-based adaptation scheme suited for every EA will be introduced, which leaves only one strategy parameter to the user, the population
size. Furthermore, it will be shown that the range of feasible sizes can be reduced drastically.
1
Motivation
Almost all practical applications of Evolutionary Algorithms use some sort of hybridisation with other algorithms like heuristics or local searchers, frequently in the
form of a Memetic Algorithm (MA)1. MAs integrate local search in the offspring production part of an EA and, thus, introduce additional strategy parameters controlling
the frequency and intensity of the local search among others [2, 3]. The benefit of this
approach is a speed-up of the resulting hybrid usually in the magnitude of factors. The
draw back is the greater amount of strategy parameters which have to be adjusted
properly [2, 3, 4]. The necessary tuning of strategy parameters is one of the most important obstacles to the broad application of EAs to real-world problems. This can be
summarised by the following statement: despite the wide scope of application of Evolutionary Algorithms, they are not widely applied. To overcome this situation, a reduction of the MA strategy parameters to be adjusted manually is urgently required.
2
Introduction
To enhance the applicability of MAs, they should either adopt their strategy parameters or the parameters should be fixed to the greatest extent possible. Another point is
the usage of application-independent local searchers (LS), as they maintain the gen1
See [1] and reports about real-world applications at PPSN, ICGA, or GECCO conference series.
R.Wyrzykowski et al. (Eds.): Conf. Proc. of Parallel Processing and Applied Mathematics (PPAM 2007),
LNCS 4967, Springer-Verlag, Berlin, pp.509-519, 2008
© Springer-Verlag, Berlin, Heidelberg, 2008
eral usability of the MA. In this paper a cost-benefit-based adaptation scheme suited
for every EA shall be introduced. It will be applied to an MA for parameter optimisation, which uses two application-independent local searchers to maintain generality.
The idea of cost-benefit-based adaptation was first published in 2004 [4, 5] and in a
more elaborated form in 2006 [6]. In this paper an improved version of the adaptation
scheme shall be presented together with a more detailed analysis of the experimental
results as it was possible in [6]. To achieve this, the section about related work will be
summarised here and the interested reader is referred to the detailed discussion in [6].
Other researchers dealt with dynamic adaptation which uses some sort of coevolution to adjust the strategy parameters [7, 8] or they worked in other application
fields, especially combinatorial optimisation [8, 9]. As it is not known a priori which
local searcher or meme suits best or is feasible at all, some researches also tackled the
problem of adaptive meme construction [7] or the usage of several memes [7, 8, 10].
In [6] it is shown in detail how the work presented here fits into the gap previous
work leaves and why co-evolution is not considered the method of first choice.
Section 3 will introduce the cost-benefit-based adaptation scheme and its extension
with respect to the one used in [6]. Section 4 will contain a short introduction of the
basic algorithms used for the experiments. In Sect. 5 the test cases will be highlighted
briefly, the old and new strategy parameters compared, and the experimental results
discussed in detail. From this, a common parameterisation will be derived. The paper
shall conclude with a summary and an outlook.
3
Concept of Cost-benefit-based Adaptation
The basic idea is to use the costs measured in evaluations caused by and the benefit
measured in fitness gain obtained from an LS run to control the selection of an LS out
of a given set and the intensity of their search as well as the frequency of their usage.
Suited LS must therefore have an external controllable termination parameter like an
iteration limit or, even better, a convergence-based termination threshold.
Firstly, the adaptive mechanism is described for the parameter adjustment. For the
fitness gain a relative measure is used, because a certain amount of fitness improvement is much easier to achieve in the beginning of a search than in the end. The relative fitness gain rfg is based on a normalised fitness function in the range of 0 and
fmax, which turns every task into a maximisation problem. rfg is the ratio between the
achieved fitness improvement (fLS - fevo) and the maximum possible one (fmax - fevo), as
shown in (1), where fLS is the fitness obtained by the LS and fevo the fitness of the offspring as produced by the evolution.
rfg =
f LS − f evo
f max − f evo
(1)
∑ rfg
∑ eval
Pn ,L 1 ,i
Pn ,L 1 ,i
rfg
rfg
: ∑evalPn ,L 2 , j : ∑ Pn ,L 3 ,k
∑
Pn ,L 2 , j
∑ eval
(2)
Pn ,L 3 ,k
For each parameter Pn a set of levels is defined, each of which has a probability p
and a value v containing for each level an appropriate value of that particular parameter. Three of these levels are always active, i.e. have a probability p greater than zero.
For each active level the required evaluations eval and the obtained rfg are calculated
per LS usage and summed
up. The probabilities of the
levels are adjusted, if either
each level was used at minimum usageL,min times or they
all have been used usageL,max
in total since the last adjustment. The new relation
among the active levels L1,
L2, and L3 is calculated as
shown in (2). The sums are
reset to zero after the adjustment, such that the adaptation
Fig. 1. The three phases of a level movement. Active levis faster. If the probability of
els are marked by a grey background. p denotes the probthe lowest or highest active
ability and v the parameter value of a level
level exceeds a threshold
value of 0.5, the next lower
or higher level, respectively, is added. The level at the opposite end is dropped and its
likeliness is added to its neighbour. The new level is given a probability of 20% of the
sum of the probabilities of the other two levels. This causes a move of three consecutive levels along the scale of possible ones according to their performance determined
by the achieved fitness gain and the required evaluations. To ensure mobility in both
directions, none of the three active levels may have a probability below 0.1. An example of an upward movement of the three active levels is shown in Fig. 1. It is done,
because p of level 5 has become too large (first row). Level 3 is deactivated (p=0) and
level 4 inherits its likeliness totalling now p=0.4 (second row). Finally, the new level
6 receives 20% of the probabilities of the two others as shown in the third row.
The same procedure can be used to adjust the probabilities of the involved LSs.
Again, the required evaluations eval as well as rfg are calculated and summed up for
each LS usage. The new relation of the LS probabilities is computed in the same way
as for the active levels, if either each LS was used at minimum usageLS,min times or
there have been matingsmax matings in total since the last adjustment. If the probability
of one LS drops below Pmin for three consecutive alterations, it is ignored from then
on. To avoid premature deactivation, the probability is set to Pmin for the first time it is
lower than Pmin. For the experiments Pmin was set to 0.1. This simple LS adaptation
scheme was used for the investigations reported in [6]. As erroneous deactivations of
an LS were observed, the adaptation speed had to be reduced by comparably high
values for the re-adjustment thresholds. As a consequence, the extended LS adaptation procedure uses the old distribution by summing up one third of the old probabilities and two thirds of the newly calculated ones, thus resulting in a new likelihood of
each LS.
For EAs that create more than one offspring per mating, such as the one used here,
a choice must be made between locally optimising the best (called best-improvement)
or a fraction of up to all of these offspring (called all-improvement). This is controlled
adaptively in the following way: the best offspring always undergoes LS improvement and for its siblings the chance of being treated by the LS is adaptively adjusted
as described before, with the following peculiarities. After having processed all se-
lected offspring, fLS is estimated as the fitness of the best locally improved child and
fevo as that of the best offspring from pure evolution.
4
Basic Algorithms
As EA, GLEAM (General Learning Evolutionary Algorithm and Method) [11] is
used. It is an EA of its own, combining aspects from Evolution Strategy and Genetic
Algorithms with the concept of abstract data types for the easy formulation of genes
and chromosomes. A detailed and updated description can be found in [12]. GLEAM
uses ranking-based selection, elitist offspring acceptance, and a structured population
based on a neighbourhood model [13] that causes an adaptive balance between exploration and exploitation and avoids premature convergence. This is achieved by maintaining niches within the population for a longer period of time and thus, sustains diversity. Hence, GLEAM can be regarded a more powerful EA compared to simple
ones, which makes it harder to reach an improvement by adding local search. On the
other hand, if an improvement can be achieved by adding and applying memes adaptively, then at least the same advantage, if not better, can be expected by using a simpler EA.
As local searchers, two well-known procedures from the sixties, the Rosenbrock
and the Complex algorithms, are used, since they are known as powerful local search
procedures. As they are derivative-free and able to handle restrictions, they maintain
general usability. The implementation is based on Schwefel [14], who gives a detailed
description of both algorithms together with experimental results. Hereinafter, the algorithms will be abbreviated by R and C.
GLEAM and the two LS form HyGLEAM (Hybrid General-purpose Evolutionary
Algorithm and Method). Apart from the basic procedures, HyGLEAM contains a
simple MA (SMA), consisting of one meme (local searcher) and an adaptive Multimeme Algorithm (AMMA) using both LS. Earlier research showed that Lamarckian
evolution, where the chromosomes are updated according to the LS improvement,
performs better than without updates [3, 4, 5]. The danger of premature convergence,
which was observed by other researchers, is avoided by the neighbourhood model
used. Hence, Lamarckian evolution was used for the experiments with the AMMA.
5
Experimental Results
5.1
Test Cases
Five test functions taken from the GENEsYs collection [15] and two real-world problems [16, 17] were used, see Table 1. Due to the lack of space, they shall be described
very briefly only, and the interested reader is referred to the given literature. Rotated
versions of Shekel’s Foxholes and the Rastrigin function were employed in order to
make them harder, see [4, 5]. The scheduling task is solved largely by assigning start
times to production batches so that the combinatorial aspect is limited to solving conflicts arising from assignments of the same time.
Table 1. Important properties of the test cases used. fi are the function numbers of [15]
Para- Modality Implicit
Range
meter
Restrict.
Schwefel’s Sphere [15, f1]
30 real unimodal
no
[-5∗106, 5∗106]
Shekel’s Foxholes [15, f5]
2 real multimodal
no
[-500, 500]
Gen. Rastrigin f. [15, f7]
5 real multimodal
no
[-5.12, 5.12]
Fletcher & Powell f. [15, f16] 5 real multimodal
no
[-3.14, 3.14]
Fractal f. [15, f13]
20 real multimodal
no
[-5, 5]
Design optimisation [16]
3 real multimodal
no
Scheduling+resource opt.[17] 87 int. multimodal yes
Test Case
5.2
Target
Value
0.01
0.998004
0.0001
0.00001
-0.05
Strategy Parameters
The adaptive Multimeme Algorithm (AMMA) controls meme selection and five strategy parameters which affect the intensity of local search (thR, limitR, and limitC) and
the frequency of meme application (all-imprR and all-imprC) as shown in Table 2. thR
is a convergence-based termination threshold value of the Rosenbrock procedure,
while limitR and limitC simply limit the LS iterations. The two all-impr parameters
control the probabilities of applying the corresponding meme to the siblings of the
best offspring in case of adaptive all-improvement, see also Sect. 3.
Table 2. Adaptively controlled strategy parameters of the Multimeme Algorithm (AMMA)
Strategy Parameter
thR
limitR, limitC
all-imprR, all-imprC
Values Used for the Experiments
10-1, 10-2, 10-3, 10-4, 10-5, 10-6, 10-7, 10-8, 10-9
100, 200, 350, 500, 750, 1000, 1250, 1500, 1750, 2000
0, 0.2, 0.4, 0.6, 0.8, 1.0
The usage and matings parameters control the adaptation speed as shown in Table
3. The experiments reported in [6] motivated separate adaptation for different fitness
ranges. The analysis of these experiments showed that the amount of these ranges is
of minor influence. Consequently, the effects of unseparated and separate adaptation
using three fitness ranges (0-40%, 40-70%, and 70-100% of fmax), called separated
and common adaptation, are compared in the experiments reported here. Together
with the already introduced strategy parameters, this results in four new strategy parameters as shown
in Table 4, and a
crucial question for Table 3. Strategy parameters settings for the adaptation speed
the experiments is,
Adaptation
Meme Selection
Parameter Adaptation
whether they can be
Speed
usageLS, min matingsmax usage L,min usageL,max
set to common valfast
3
15
3
12
ues without any
medium
5
20
4
15
relevant loss of perslow
8
30
7
25
formance.
Table 4. Strategy parameters of the SMA and the AMMA. The new ones are in italic.
Strategy Parameter
population size µ
Lamarckian or Baldwinian evolution
best- or static all-improvement
best- or adaptive all-improvement
LS selection
LS iteration limit
Rosenbrock termination threshold thR
probability of the adaptive all-improvement
adaptation speed
simple or extended LS adaptation
separate or common adaptation
5.3
Relevance and Treatment
SMA
AMMA
manual
manual
manual
Lamarckian evolution
manual
manual
manual
adaptive
5000
adaptive
manual
adaptive
adaptive
manual
manual
manual
Experimental Results
An algorithm together with a setting of its strategy parameters is called a job and the
comparisons are based on the average number of evaluations from one hundred runs
per job. Where necessary, t-test (95% confidence) is used to distinguish significant
differences from stochastic ones. Only jobs are taken into account, the runs of which
are all successful, i.e. reach the target values of Table 1 or a given solution quality in
case of the real-world problems.
Table 5 compares the results for the basic algorithms2 and Table 6 shows the results of the two SMAs. An important result is that the wide range of best population
sizes ranging from 20 to 11,200 is narrowed down by the best SMAs to 5 to 70. Further results are that the choice of the best SMA as well as good values for thR are application-dependent and that static all-improvement is better in two cases. Moreover,
all-improvement is crucial to the success of the Rastrigin function. In all cases an improvement can be achieved in the magnitude of factors. In this respect the sphere
function and the design optimisation are exceptional as they are solved during the first
generation by the SMA-R or SMA-C, respectively. As most real-world applications
are of multimodal nature, the efforts are geared to this case and the sphere function is
chosen for checking the obtained results with a challenging unimodal test function.
Thus, these results are not bad. But for the design optimisation, this outcome means
that this task is of limited value for evaluating the adaptation scheme.
Next, the best jobs of adaptive adaptation are compared to those obtained from the
SMAs. What can be expected from adaptation: on the one hand, improvement because of better tuned strategy parameters and on the other hand, impairment, because
2
The results presented here differ from those reported in [3-6] in two cases. To better show the
effects of adaptation, the sphere function is parameterised now in such a way that GLEAM
can just solve it and the Rosenbrock procedure misses the 100% success rate with common
values for thR. Secondly, a different GLEAM job is used for Shekel’s foxholes as a reference,
because it has a much better confidence interval and is nearly as good as the old job.
Table 5. Best jobs of the basic algorithms together with the population size µ and thR. Abbreviations: CI: confidence interval for 95% confidence, Succ.: success rate
Test Case
µ
Sphere
120
Fletcher
600
Scheduling 1,800
Foxholes
300
Rastrigin
11,200
Fractal
20
Design
210
GLEAM
Evaluations
37,964,753
483,566
5,376,334
108,435
3,518,702
195,129
5,773
CI
974,707
191,915
330,108
8,056
333,311
20,491
610
Rosenbrock Proc.
thR Succ. Eval.
10-8 100 4.706
10-8
22 5,000
0.6
0 3,091
-4
10
5
133
0
0
10-6
15
891
Complex-Alg.
Succ. Eval.
0 5,000
26
658
0
473
0
95
0 5,000
0
781
12
102
Table 6. Best jobs of both SMAs. Confidence intervals are given for jobs better than GLEAM
only. Abbreviations: b/a: best- or static all-improvement, L: rate of Lamarckian evolution
Test Case
Sphere
Fletcher
Scheduling
Foxholes
Rastrigin
Fractal
Design
µ
20
10
5
30
70
5
10
Rosenbrock-MA (SMA-R)
Complex-MA (SMA-C)
thR b/a L
Eval.
CI
Eval.
CI
µ b/a L
10-6 b 100
5,453
395
10-4 b 100 13,535 1,586
5 b 100
4,684 812
0.6 b 100 69,448 10,343
10-2 a 100 10,710 1,751 20 b
0
8,831 2,073
10-2 a 100 315,715 46,365 150 a 100 3,882,513
10-2 b 100 30,626 3,353 10 b 100 1,065,986
10-4 b
5
4,222
724
5 b 100
1,041 141
adaptation means learning and learning causes some additional expense. And, in fact,
both effects can be observed, as is shown in Fig. 2. The Rosenbrock procedure turns
out to be the absolute dominating meme in the end phase of the runs for all test cases
with two exceptions, the Foxhole function (0.78%) and the design optimisation
(0.49%). The improved test cases all use a thR value of 0.01, while the worsened ones
require lower values, as shown in Table 6. As the adaptation starts with values of 0.1
to 0.001, a longer phase of adaptation is required to decrease thR and to increase
limitR, which is required to benefit from lower thR thresholds. This explanation of the
observed impairments is checked for the most drastic case, the sphere function, by
starting with medium levels for both parameters. As a result, the impairment factor
can be reduced to 3.3.
The effect of the extended LS adaptation as described in section 3 is also shown in
Fig. 2. Unfortunately, the only two cases with significant differences show a different
behaviour. In all other cases, a minimal, but insignificant improvement is observed in
each case. A more detailed analysis of the results as it can be presented here shows
that the danger of erroneous deactivation of a local searcher is considerably lowered,
when the extended LS adaptation is used. Thus, the adaptation speed can be increased, resulting in more adaptations and a faster and better parameter adaptation. To
interpret Fig. 2 correctly, it must be kept in mind that the performance of the SMAs is
the result of a time-consuming and cumbersome tuning by hand.
Up to now the
comparisons have
been based on alland best-improvement,
although
tasks as multimodal
as the Rastrigin
function
require
all-improvement.
Hence, a generally
applicable AMMA
must use it. Fig. 3
shows the differences of best-,
static-, and adapFig. 2. Comparison of the improvement or impairment factors based tive
all-improveon the average efforts between two variants of the AMMA and the
ment. The latter
best SMA jobs. For a better presentation impairments are shown as
outperforms
the
negative factors
two others with two
exceptions:
Fletcher’s function, where the differences are not significant, and the unimodal Sphere
function, where all-improvement is not suited at all. Fig. 3 shows that adaptive allimprovement even is superior to best-improvement is in three cases, videlicet
Shekel’s Foxholes, fractal, and in particular the Rastrigin function. But on the other
hand, it can also be oversized like in case of Fletcher’s function or the scheduling
task. Thus, the usage of adaptive all-improvement is not only motivated by its need
for the Rastrigin function.
Apart from the
performance
aspects, the stability
or robustness of the
results must be
considered
also.
Hereinafter, jobs,
where only one or
a few population
sizes yield successful runs, will be
omitted as stability-lacking jobs. A
comparison of the
remaining strategy
parameters reveals
Fig. 3. Comparison of best- and static all-improvement (SMA) and
the following facts.
adaptive all-improvement (AMMA). For a better overview, the results
Using common adof the fractal and the Sphere function are scaled by 0.1, those of the
aptation (cf. sect.
Rastrigin function and the scheduling task by 0.01. Furthermore, one
5.2), there is at the
bar of the Rastrigin function is not displayed completely
Fig. 4. Comparison of the best SMA and AMMA jobs. Empty fields
indicate an insufficient success rate (below 100%), while flat fields
denote the fulfilling of the optimisation task, but with greater effort
than the basic EA. The given improvements are the ratio between the
means of the evaluations of the basic EA and the compared algorithms.
Table 7: Results for best and recommended AMMA. Abbr.: Table 5
Test Case
µ
Sphere
5
Fletcher
10
Scheduling 20
Foxholes
30
Rastrigin
120
Fractal
10
Design
5
6
Best AMMA
Recommended AMMA
Eval.
CI
Eval.
CI
µ
54,739 3,688 30 194,653 17,094
11,451 1,644
5
12,599 2,486
251,917 22,453 20 306,843 51,595
5,467 1,086 50
5,673
916
257,911 31,818 70 300,551 73,152
19,904 2,801
5
23,075 2,562
1,383
210
5
1,581
384
minimum one test
case with jobs
lacking
stability
for all adaptation
speeds.
Using separated
adaptation according to three fitness
ranges (cf. section
5.2) instead and
extended LS adaptation (cf. section
3), best results are
reached with fast
adaptation at a
good stability. This
is together with
adaptive
all-improvement the recommended common parameterisation.
Leaving the exceptional cases of
the Sphere function and the design
task aside, 88.3%
of the performance
of the best handtuned AMMA can
be reached on an
average and 75.7%
when considering
all test cases. Fig.
4 and Table 7 summarise and compare these results.
Conclusions and Outlook
A common cost-benefit-based adaptation procedure for Memetic Algorithms was introduced, which controls both meme selection and the balance between global and local search. The latter is achieved by adapting the intensity of local search and the frequency of their usage. A common good parameterisation was given for the new strat-
egy parameters of the adaptation scheme leaving only one parameter to be adjusted
manually: the population size µ. In the experiments the useful range of µ could be
narrowed down from ranges between 20 and 11,200 to ranges between 5 and 70. A
value of 20 can be recommended to start with, provided that the task is assumed to be
not very complex with not too many suboptima.
Further investigations will take more test cases into account and aim at the extension of the approach to combinatorial tasks, namely, job scheduling and resource allocation in the context of grid computing.
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