Table Of ContentZhi-Hua Zhou · Yang Yu · Chao Qian
Evolutionary
Learning
Advances
in Theories
and Algorithms
Evolutionary Learning: Advances in Theories
and Algorithms
Zhi-Hua Zhou Yang Yu
(cid:129) (cid:129)
Chao Qian
Evolutionary Learning:
Advances in Theories
and Algorithms
123
Zhi-Hua Zhou Yang Yu
NanjingUniversity NanjingUniversity
Nanjing, Jiangsu,China Nanjing, Jiangsu,China
ChaoQian
NanjingUniversity
Nanjing, Jiangsu,China
ISBN978-981-13-5955-2 ISBN978-981-13-5956-9 (eBook)
https://doi.org/10.1007/978-981-13-5956-9
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Preface
Aroundtheyearof2001,thefirstauthorofthismonograph,Zhou,devel-
opedwithhiscollaboratorsapowerfulselectiveensembleapproach.This
approachisabletoproducesmall-sizedensembleswithgeneralizationper-
formanceevenstrongerthanthatoffull-sizedonesbyexploitinggenetical-
gorithm,apopularlyusedevolutionaryalgorithm(EA).Zhourealizedthat
EAs are powerful optimization techniques that can be well useful in vari-
ousmachinelearningtasks.Atthattime,however,EAswerealmostpurely
heuristic,notfavoredbythemachinelearningcommunitywithstrongthe-
oreticalflavor.ImpressedbythesuccessesofEAsinapplications,Zhoube-
lievedtheremustbesometheoreticalexplanationsbehindtheirmysteries
and decided to start this track of research. In 2004, the second author of
thismonograph,Yu,finishedhisbachelorthesisonselectiveensembleun-
derthesupervisionofZhou.YuthenjoinedZhouasaPhDstudenttaking
theoreticalaspectsofEAsashisthesistopic,andobtainedPhDdegreein
2011.In2009,Zhouacceptedthethirdauthorofthismonograph,Qian,as
hisPhDstudentwiththeoriesandalgorithmsofevolutionarylearningfor
histhesistopic,andQianobtainedhisPhDdegreein2015.Overall,thema-
jority of contents in this monograph are research results achieved by the
authors during the past two decades.
Thismonographiscomposedoffourparts.PartIbrieflyintroducesevo-
lutionarylearningandsomepreliminaries.Aimingatanalyzingtherunning
timecomplexityandtheapproximationability,thetwoessentialtheoretical
aspectsofEAs,PartIIpresentstwogeneralapproachesforderivingrunning
timeboundsandageneralframeworkforcharacterizingtheapproximation
performance.Theseresultsserveasgeneraltoolsforattainingmanytheo-
reticalresultsreportedintheremainderofthemonograph.PartIIIpresents
aseriesoftheoreticalstudies,mainlyabouttheinfluenceofmajorfactors,
such as recombination, solution representation, inaccurate fitness evalu-
ation,population,etc.,ontheperformanceofEAs.InPartIV,theauthors
comebacktoselectiveensemble,theiroriginalmotivatingtaskforstudy-
ingEAs,andpresentalgorithmsbehavingatleastaswellasstate-of-the-art
VI Preface
selectiveensemblealgorithms.Theauthorsthenstudysubsetselection,a
moregeneralproblemwidelyoccurringinvariousmachinelearningtasks,
andpresentaseriesofevolutionarylearningalgorithmswithboundedap-
proximationguarantees.
TheauthorshopethatthegeneraltheoreticaltoolspresentedinPartII
can be found useful for interested readers to explore theoretical founda-
tions of evolutionary learning, the theoretical results presented in Part III
canhelpreaderstogetmoreunderstandingaboutbehaviorsofevolution-
arylearningprocessesandoffersomeinsightforalgorithmdesign,andthe
algorithmspresentedinPartIVcanbefoundusefulinreal-worldapplica-
tionsofmachinelearning.
The authors want to thank their families, friends and collaborators.
GratitudegoestoCelineChangandAlfredHofmannwhoencouragedauthors
topublishthismonographwithSpringer.
Theauthors
February2019
Notations
R realnumber
N integer
(·)+ positive,(·)canbeRorN
(·)0+ non-negative,(·)canbeRorN
x variable
x vector
(·,·,...,·) rowvector
0,1 all-0sandall-1svectors
|·| ,|·| numberof0sand1sofavector
0 1
{0,1}n Booleanvectorspace
X matrix
(·)T transposeofavector/matrix
X set
{·,·,··· ,·}setbyenumeration
[n] set{1,2,...,n}
|·| cardinalityofaset
2X powersetofX,whichconsistsofallsubsetsofX
X\Y complementofY inX,whichconsistsofelements
inX butnotinY
VIII Notations
x stateofaMarkovchain
X statespaceofaMarkovchain
P(·),P(·|·) probabilityandconditionalprobability
π probabilitydistribution
f function
E·∼π[f(·)],E·∼π[f(·)|·]expectation and conditional expectation of f(·)
underdistributionπ,simplifiedasE[f(·)],E[f(·)|·]
whenthemeaningisclear
I(·) indicator function which takes 1 if · is true, and 0
otherwise
(cid:2)·(cid:3),(cid:4)·(cid:5) floor and ceiling functions which take the great-
est/leastintegerless/greaterthanorequaltoareal
number
OPT optimalfunctionvalue
(cid:2)
H n-thharmonicnumber,i.e., n (1/i)
n i=1
∀ forall
∃,(cid:2) thereexists,doesnotexist
Contents
PartI INTRODUCTION
1 Introduction ............................................... 3
1.1 MachineLearning....................................... 3
1.2 EvolutionaryLearning ................................... 4
1.3 Multi-objectiveOptimization............................. 6
1.4 OrganizationoftheBook................................. 9
2 Preliminaries............................................... 11
2.1 EvolutionaryAlgorithms ................................. 11
2.2 Pseudo-BooleanFunctions............................... 15
2.3 RunningTimeComplexity ............................... 17
2.4 MarkovChainModeling ................................. 19
2.5 AnalysisTools........................................... 21
PartII ANALYSISMETHODOLOGY
3 RunningTimeAnalysis:Convergence-basedAnalysis........... 29
3.1 Convergence-basedAnalysis:Framework.................. 30
3.2 Convergence-basedAnalysis:ApplicationIllustration....... 34
3.3 Summary............................................... 39
4 RunningTimeAnalysis:SwitchAnalysis....................... 41
4.1 SwitchAnalysis:Framework .............................. 41
4.2 SwitchAnalysis:ApplicationIllustration ................... 46
4.3 Summary............................................... 50
5 RunningTimeAnalysis:ComparisonandUnification .......... 51
5.1 AnalysisApproaches:Formalization....................... 51
5.2 SwitchAnalysisvs.FitnessLevel .......................... 53
X Contents
5.3 SwitchAnalysisvs.DriftAnalysis.......................... 57
5.4 SwitchAnalysisvs.Convergence-basedAnalysis............ 61
5.5 AnalysisApproaches:Unification ......................... 65
5.6 Summary............................................... 67
6 ApproximationAnalysis:SEIP................................ 69
6.1 SEIP:Framework ........................................ 70
6.2 SEIP:ApplicationIllustration ............................. 77
6.3 Summary............................................... 80
PartIII THEORETICALPERSPECTIVES
7 BoundaryProblemsofEAs................................... 83
7.1 BoundaryProblemIdentification ......................... 84
7.2 CaseStudy.............................................. 87
7.3 Summary............................................... 92
8 Recombination ............................................. 93
8.1 RecombinationandMutation ............................ 95
8.2 RecombinationEnabledMOEAs.......................... 97
8.3 CaseStudy.............................................. 100
8.4 EmpiricalVerification.................................... 106
8.5 Summary............................................... 108
9 Representation .............................................109
9.1 GeneticProgrammingRepresentation..................... 111
9.2 CaseStudy:MaximumMatchings......................... 113
9.3 CaseStudy:MinimumSpanningTrees..................... 119
9.4 EmpiricalVerification.................................... 125
9.5 Summary............................................... 128
10 InaccurateFitnessEvaluation ................................129
10.1 NoisyOptimization...................................... 130
10.2 InfluenceofNoisyFitness................................ 132
10.3 DenoisebyThresholdSelection........................... 136
10.4 DenoisebySampling .................................... 141
10.5 EmpiricalVerification.................................... 148
10.6 Summary............................................... 152
11 Population .................................................155
11.1 InfluenceofPopulation .................................. 156
11.2 RobustnessofPopulationagainstNoise ................... 160
11.3 Summary............................................... 172
