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Benchmarking sep-CMA-ES on the BBOB-2009 Function Testbed
Raymond Ros
To cite this version:
Raymond Ros. Benchmarking sep-CMA-ES on the BBOB-2009 Function Testbed. GECCO, Jul 2009, Montréal, Canada. �inria-00377087�
Benchmarking sep-CMA-ES on the BBOB-2009 Function Testbed
Raymond Ros
Univ. Paris-Sud, LRI
UMR 8623 / INRIA Saclay, projet TAO F-91405 Orsay, France
raymond.ros@lri.fr
ABSTRACT
ApartlytimeandspaelinearCMA-ESisbenhmarkedon
theBBOB-2009 noiselessfuntiontestbed. Thisalgorithm
with a multistart strategy with inreasing population size
solves17funtionsoutof24in20-D.
Categories and Subject Descriptors
G.1.6 [Numerial Analysis℄: Optimization|globalopti-
mization, unonstrained optimization; F.2.1 [Analysis of
Algorithmsand Problem Complexity ℄: NumerialAl-
gorithmsandProblems
General Terms
Algorithms
Keywords
Benhmarking,Blak-box optimization,Evolutionaryom-
putation,Covarianematrixadaptation,Evolutionstrategy
1. INTRODUCTION
The sep-CMA-ES algorithm introdued in[7℄ is a vari-
antof the ovarianematrixadaptationevolution strategy
(CMA-ES)[5℄thatislinearintimeandspae.Thisproperty
ombinedwithafasterlearningratemakessep-CMA-ESap-
propriate for separable funtionand larger dimensions. A
mixedstrategyof usingsep-CMA-ESandCMA-ESis pro-
posedhereandbenhmarkedonanoiselessfuntiontestbed.
2. ALGORITHM PRESENTATION
Initsdesign, thesep-CMA-ESdiersfrom theCMA-ES
by two aspets: rst, the ovarianematrixis onstrained
to be diagonal at eahof its update, seond, the learning
rate is inreased by a fator of n+3=2
3
, where n is the di-
mension ofthe searh spae 1
. These modiations result
1
Pleasenotethatthefatorforthelearning rateissmaller
thantheonein[7℄.
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GECCO’09, July 8–12, 2009, Montréal Québec, Canada.
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inanalgorithm that tradesmodelomplexitywith atime
andspaesalingthatisbetterthantheoriginalCMA-ES.
The(=
w
;)-sep-CMA-EShasbeenshowntooutperform
(=w;)-CMA-ESonseparablefuntions.
Weproposeherewhatwouldbethebestoftwoworlds: to
usesep-CMA-ESfortherstfewiterationsandthenswith
toCMA-ES.Atthetimeoftheswith,allparametersarere-
tainedexeptforthelearningratethatisdereasedbakto
itsdefaultvalue. Thisimplies thediagonal ovarianema-
trixaquired usingsep-CMA-ESis diretlyused byCMA-
ES. Thismixedstrategy istherefore expetedto befaster
than CMA-ES onseparable funtions. Ongoingwork has
also shownthat for some test funtionsthe rstiterations
usingsep-CMA-ESwouldnotdisadvantagethelatteruseof
CMA-ESinanyway.Inotherterms,theostofinitiallyus-
ing sep-CMA-ESwouldnotindueapenaltyintheostof
solvingthefuntionwithCMA-ESafterwards. Theauthor
admitssomefuntionsouldinduesuhapenalty.
Asforthemultistartstrategy,weusetheinreasingpop-
ulation sizeIPOP-CMA-ES[1℄. Thoughthis approahhas
shown itslimits [6℄, independent restart may improve the
probability of the algorithm reahing a given target fun-
tionvalue.
3. EXPERIMENTAL PROCEDURE
TheMatlabimplementationoftheCMA-ES(version3.23beta)
isused 2
. Weusethe(=
w
;)-IPOP-CMA-ESvariantwith
aninitialdefaultpopulationsize=4+b3ln(n)inreas-
ingtwieateahrestart. Exeptthelearningrate,allother
algorithm parameters are set to their default values. The
ovarianematrixisonstrainedtobediagonalonlyforthe
rst1+100n=
p
iterationsoftherststart. Amaximum
of8independentrestartsisonduted. Restartsourafter
100+300n p
n=iterationsorifanyofthedefaultstopping
riterionis met. Theinitial stepsizehas beensetto 2and
the starting point has been hosen uniformly in [ 4;4℄
n
.
The maximum number of funtion evaluations was set to
10 4
times the dimension. No parametertuning was done,
theCrE[3℄ isomputedtozero.
4. RESULTS AND DISCUSSION
Resultsfromexperimentsaording to [3℄ onthe benh-
mark funtions given in [2, 4℄ are presented in Figures 1
and 2 andin Table 1. The algorithm solves17out of the
24funtions in20-D.Thealgorithm performswellonuni-
2
Latest version available here:http://www.lri.fr/
~hansen/maesintro.html
2 3 5 10 20 40 0
1 2 3
4 1 Sphere
+1 +0 -1 -2 -3 -5 -8
2 3 5 10 20 40
0 1 2 3 4
5 2 Ellipsoid separable
2 3 5 10 20 40
0 1 2 3 4 5 6 7
13 4
3 Rastrigin separable
2 3 5 10 20 40
0 1 2 3 4 5 6 7 2
4 Skew Rastrigin-Bueche separable
2 3 5 10 20 40
0 1 2
3 5 Linear slope
2 3 5 10 20 40
0 1 2 3 4
5 6 Attractive sector
2 3 5 10 20 40
0 1 2 3 4 5
6 7 Step-ellipsoid
2 3 5 10 20 40
0 1 2 3 4 5
6 8 Rosenbrock original
2 3 5 10 20 40
0 1 2 3 4 5
6 9 Rosenbrock rotated
2 3 5 10 20 40
0 1 2 3 4
5 10 Ellipsoid
2 3 5 10 20 40
0 1 2 3 4
5 11 Discus
2 3 5 10 20 40
0 1 2 3 4
5 12 Bent cigar
2 3 5 10 20 40
0 1 2 3 4 5
6 13 Sharp ridge 14
2 3 5 10 20 40
0 1 2 3 4 5
6 14 Sum of different powers
2 3 5 10 20 40
0 1 2 3 4 5 6 7
14 12 15 Rastrigin 1
2 3 5 10 20 40
0 1 2 3 4 5 6 7
11 7 16 Weierstrass
2 3 5 10 20 40
0 1 2 3 4 5
6 17 Schaffer F7, condition 10 14
2 3 5 10 20 40
0 1 2 3 4 5 6 7
5 18 Schaffer F7, condition 1000
2 3 5 10 20 40
0 1 2 3 4 5 6 7
14 13
2
19 Griewank-Rosenbrock F8F2
2 3 5 10 20 40
0 1 2 3 4 5 6 7
7
20 Schwefel x*sin(x)
2 3 5 10 20 40
0 1 2 3 4 5 6 7
14 13 11
3 3
21 Gallagher 101 peaks
2 3 5 10 20 40
0 1 2 3 4 5 6 7
12 8
1 22 Gallagher 21 peaks
2 3 5 10 20 40
0 1 2 3 4 5 6 7
9
23 Katsuuras
2 3 5 10 20 40
0 1 2 3 4 5 6 7 3
24 Lunacek bi-Rastrigin
+1 +0 -1 -2 -3 -5 -8
Figure1: ExpetedRunningTime(ERT,)toreahf
opt
+f andmediannumberoffuntionevaluationsof
suessfultrials (+),shown forf =10;1;10 1
;10 2
;10 3
;10 5
;10 8
(the exponentisgivenin the legendoff1
and f
24
) versus dimension in log-logpresentation. The ERT(f) equalsto #FEs (f) divided by the number
ofsuessful trials, whereatrial is suessfulif f
opt
+f wassurpassed during the trial. The #FEs (f) are
thetotalnumberoffuntionevaluationswhilefopt+f wasnotsurpassedduringthetrialfromallrespetive
trials(suessfulandunsuessful),andf
opt
denotestheoptimalfuntionvalue. Crosses()indiatethetotal
number offuntion evaluations #FEs ( 1). Numbers above ERT-symbolsindiatethe numberof suessful
trials. Annotated numbers on the ordinate are deimal logarithms. Additional grid lines show linear and
