学习人工智能的深度架构

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LearningDeepArchitecturesforAI

LearningDeepArchitecturesforAI

YoshuaBengio

Dept.IRO,Universit´edeMontr´eal

C.P.6128,Montreal,Qc

Canada

yoshua.b

engio@umontreal.ca

Boston–Delft

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FoundationsandTrendsQRinMachineLearningVolume2Issue1,2009

EditorialBoard

Editor-in-Chief:

MichaelJordan

DepartmentofElectricalEngineeringandComputerScienceDepartmentofStatistics

UniversityofCalifornia,BerkeleyBerkeley,CA94720-1776

Editors

PeterBartlett(UCBerkeley)

YoshuaBengio(Universit´edeMontr´eal)AvrimBlum(CarnegieMellonUniversity)CraigBoutilier(UniversityofToronto)StephenBoyd(StanfordUniversity)

CarlaBrodley(TuftsUniversity)InderjitDhillon(UniversityofTexasatAustin)

JeromeFriedman(StanfordUniversity)KenjiFukumizu(InstituteofStatisticalMathematics)

ZoubinGhahramani(CambridgeUniversity)

DavidHeckerman(MicrosoftResearch)TomHeskes(RadboudUniversityNijmegen)GeoffreyHinton(UniversityofToronto)AapoHyvarinen(HelsinkiInstituteforInformationTechnology)

LesliePackKaelbling(MIT)MichaelKearns(UniversityofPennsylvania)

DaphneKoller(StanfordUniversity)

JohnLafferty(CarnegieMellonUniversity)MichaelLittman(RutgersUniversity)GaborLugosi(PompeuFabraUniversity)DavidMadigan(ColumbiaUniversity)PascalMassart(Universit´edeParis-Sud)AndrewMcCallum(UniversityofMassachusettsAmherst)

MarinaMeila(UniversityofWashington)AndrewMoore(CarnegieMellonUniversity)

JohnPlatt(MicrosoftResearch)

LucdeRaedt(Albert-LudwigsUniversitaetFreiburg)

ChristianRobert(Universit´eParis-Dauphine)

SunitaSarawagi(IITBombay)

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FoundationsandTrendsQRinMachineLearning,2009,Volume2,4issues.

ISSNpaperversion1935-8237.ISSNonlineversion1935-8245.Alsoavailableasacombinedpaperandonlinesubscription.

FoundationsandTrendsQRinMachineLearning

Vol.2,No.1(2009)1–127

Qc2009Y.Bengio

DOI:10.1561/2200000006

LearningDeepArchitecturesforAI

YoshuaBengio

Dept.IRO,Universit´edeMontr´eal,C.P.6128,Montreal,Qc,H3C3J7,Canada,

yoshua.bengio@umontreal.ca

Abstract

Theoreticalresultssuggestthatinordertolearnthekindofcom-plicatedfunctionsthatcanrepresenthigh-levelabstractions(e.g.,invision,language,andotherAI-leveltasks),onemayneeddeeparchitec-tures.Deeparchitecturesarecomposedofmultiplelevelsofnon-linearoperations,suchasinneuralnetswithmanyhiddenlayersorincom-plicatedpropositionalformulaere-usingmanysub-formulae.Searchingtheparameterspaceofdeeparchitecturesisadifficulttask,butlearningalgorithmssuchasthoseforDeepBeliefNetworkshaverecentlybeenproposedtotacklethisproblemwithnotablesuccess,beatingthestate-of-the-artincertainareas.Thismonographdiscussesthemotivationsandprinciplesregardinglearningalgorithmsfordeeparchitectures,inparticularthoseexploitingasbuildingblocksunsupervisedlearningofsingle-layermodelssuchasRestrictedBoltzmannMachines,usedtoconstructdeepermodelssuchasDeepBeliefNetworks.

Contents

Introduction

1

HowdoWeTrainDeepArchitectures?

4

IntermediateRepresentations:SharingFeaturesand

AbstractionsAcrossTasks

6

DesiderataforLearningAI

9

OutlineofthePaper

10

TheoreticalAdvantagesofDeepArchitectures 13

ComputationalComplexity 16

InformalArguments 18

LocalvsNon-LocalGeneralization 21

TheLimitsofMatchingLocalTemplates 21

LearningDistributedRepresentations 27

NeuralNetworksforDeepArchitectures 31

Multi-LayerNeuralNetworks 31

TheChallengeofTrainingDeepNeuralNetworks 32

UnsupervisedLearningforDeepArchitectures 40

DeepGenerativeArchitectures 41

ConvolutionalNeuralNetworks 44

Auto-Encoders 46

ix

Energy-BasedModelsandBoltzmannMachines 49

Energy-BasedModelsandProductsofExperts 49

BoltzmannMachines 54

RestrictedBoltzmannMachines 56

ContrastiveDivergence 60

GreedyLayer-WiseTrainingofDeep

Architectures 69

Layer-WiseTrainingofDeepBeliefNetworks 69

TrainingStackedAuto-Encoders 72

Semi-SupervisedandPartiallySupervisedTraining 73

VariantsofRBMsandAuto-Encoders 75

SparseRepresentationsinAuto-Encoders

andRBMs 75

DenoisingAuto-Encoders 81

LateralConnections 83

ConditionalRBMsandTemporalRBMs 84

FactoredRBMs 86

GeneralizingRBMsandContrastiveDivergence 87

StochasticVariationalBoundsforJoint

OptimizationofDBNLayers 91

UnfoldingRBMsintoInfiniteDirected

BeliefNetworks 92

VariationalJustificationofGreedyLayer-wiseTraining 94

JointUnsupervisedTrainingofAlltheLayers 97

LookingForward 101

GlobalOptimizationStrategies 101

WhyUnsupervisedLearningisImportant 107

OpenQuestions 108

Conclusion 113

Acknowledgments 115

References

117

1

Introduction

Allowingcomputerstomodelourworldwellenoughtoexhibitwhatwecallintelligencehasbeenthefocusofmorethanhalfacenturyofresearch.Toachievethis,itisclearthatalargequantityofinforma-tionaboutourworldshouldsomehowbestored,explicitlyorimplicitly,inthecomputer.Becauseitseemsdauntingtoformalizemanuallyallthatinformationinaformthatcomputerscanusetoanswerques-tionsandgeneralizetonewcontexts,manyresearchershaveturnedtolearningalgorithmstocapturealargefractionofthatinformation.Muchprogresshasbeenmadetounderstandandimprovelearningalgorithms,butthechallengeofartificialintelligence(AI)remains.Dowehavealgorithmsthatcanunderstandscenesanddescribetheminnaturallanguage?Notreally,exceptinverylimitedsettings.Dowehavealgorithmsthatcaninferenoughsemanticconceptstobeabletointeractwithmosthumansusingtheseconcepts?No.Ifweconsiderimageunderstanding,oneofthebestspecifiedoftheAItasks,wereal-izethatwedonotyethavelearningalgorithmsthatcandiscoverthemanyvisualandsemanticconceptsthatwouldseemtobenecessarytointerpretmostimagesontheweb.ThesituationissimilarforotherAItasks.

1

2 Introduction

Fig.1.1Wewouldliketherawinputimagetobetransformedintograduallyhigherlevelsofrepresentation,representingmoreandmoreabstractfunctionsoftherawinput,e.g.,edges,localshapes,objectparts,etc.Inpractice,wedonotknowinadvancewhatthe“right”representationshouldbeforalltheselevelsofabstractions,althoughlinguisticconceptsmighthelpguessingwhatthehigherlevelsshouldimplicitlyrepresent.

ConsiderforexamplethetaskofinterpretinganinputimagesuchastheoneinFigure

1.1.

WhenhumanstrytosolveaparticularAItask(suchasmachinevisionornaturallanguageprocessing),theyoftenexploittheirintuitionabouthowtodecomposetheproblemintosub-problemsandmultiplelevelsofrepresentation,e.g.,inobjectpartsandconstellationmodels

[138,

179,

197]

wheremodelsforpartscanbere-usedindifferentobjectinstances.Forexample,thecurrentstate-of-the-artinmachinevisioninvolvesasequenceofmodulesstartingfrompixelsandendinginalinearorkernelclassifier

[134,

145],

withintermediatemodulesmixingengineeredtransformationsandlearning,

3

e.g.,firstextractinglow-levelfeaturesthatareinvarianttosmallgeo-metricvariations(suchasedgedetectorsfromGaborfilters),transform-ingthemgradually(e.g.,tomaketheminvarianttocontrastchangesandcontrastinversion,sometimesbypoolingandsub-sampling),andthendetectingthemostfrequentpatterns.Aplausibleandcommonwaytoextractusefulinformationfromanaturalimageinvolvestrans-formingtherawpixelrepresentationintograduallymoreabstractrep-resentations,e.g.,startingfromthepresenceofedges,thedetectionofmorecomplexbutlocalshapes,uptotheidentificationofabstractcat-egoriesassociatedwithsub-objectsandobjectswhicharepartsoftheimage,andputtingallthesetogethertocaptureenoughunderstandingofthescenetoanswerquestionsaboutit.

Here,weassumethatthecomputationalmachinerynecessarytoexpresscomplexbehaviors(whichonemightlabel“intelligent”)requireshighlyvaryingmathematicalfunctions,i.e.,mathematicalfunc-tionsthatarehighlynon-linearintermsofrawsensoryinputs,anddisplayaverylargenumberofvariations(upsanddowns)acrossthedomainofinterest.Weviewtherawinputtothelearningsystemasahighdimensionalentity,madeofmanyobservedvariables,whicharerelatedbyunknownintricatestatisticalrelationships.Forexample,usingknowledgeofthe3Dgeometryofsolidobjectsandlighting,wecanrelatesmallvariationsinunderlyingphysicalandgeometricfac-tors(suchasposition,orientation,lightingofanobject)withchangesinpixelintensitiesforallthepixelsinanimage.Wecallthesefactorsofvariationbecausetheyaredifferentaspectsofthedatathatcanvaryseparatelyandoftenindependently.Inthiscase,explicitknowledgeofthephysicalfactorsinvolvedallowsonetogetapictureofthemath-ematicalformofthesedependencies,andoftheshapeofthesetofimages(aspointsinahigh-dimensionalspaceofpixelintensities)asso-ciatedwiththesame3Dobject.Ifamachinecapturedthefactorsthatexplainthestatisticalvariationsinthedata,andhowtheyinteracttogeneratethekindofdataweobserve,wewouldbeabletosaythatthemachineunderstandsthoseaspectsoftheworldcoveredbythesefactorsofvariation.Unfortunately,ingeneralandformostfactorsofvariationunderlyingnaturalimages,wedonothaveananalyticalunderstand-ingofthesefactorsofvariation.Wedonothaveenoughformalized

4 Introduction

priorknowledgeabouttheworldtoexplaintheobservedvarietyofimages,evenforsuchanapparentlysimpleabstractionasMAN,illus-tratedinFigure

1.1.

Ahigh-levelabstractionsuchasMANhasthepropertythatitcorrespondstoaverylargesetofpossibleimages,whichmightbeverydifferentfromeachotherfromthepointofviewofsimpleEuclideandistanceinthespaceofpixelintensities.Thesetofimagesforwhichthatlabelcouldbeappropriateformsahighlycon-volutedregioninpixelspacethatisnotevennecessarilyaconnectedregion.TheMANcategorycanbeseenasahigh-levelabstractionwithrespecttothespaceofimages.Whatwecallabstractionherecanbeacategory(suchastheMANcategory)orafeature,afunctionofsensorydata,whichcanbediscrete(e.g.,theinputsentenceisatthepasttense)orcontinuous(e.g.,theinputvideoshowsanobjectmovingat2meter/second).Manylower-levelandintermediate-levelconcepts(whichwealsocallabstractionshere)wouldbeusefultoconstructaMAN-detector.Lowerlevelabstractionsaremoredirectlytiedtoparticularpercepts,whereashigherlevelonesarewhatwecall“moreabstract”becausetheirconnectiontoactualperceptsismoreremote,andthroughother,intermediate-levelabstractions.

Inadditiontothedifficultyofcomingupwiththeappropriateinter-mediateabstractions,thenumberofvisualandsemanticcategories(suchasMAN)thatwewouldlikean“intelligent”machinetocap-tureisratherlarge.Thefocusofdeeparchitecturelearningistoauto-maticallydiscoversuchabstractions,fromthelowestlevelfeaturestothehighestlevelconcepts.Ideally,wewouldlikelearningalgorithmsthatenablethisdiscoverywithaslittlehumaneffortaspossible,i.e.,withouthavingtomanuallydefineallnecessaryabstractionsorhav-ingtoprovideahugesetofrelevanthand-labeledexamples.Ifthesealgorithmscouldtapintothehugeresourceoftextandimagesontheweb,itwouldcertainlyhelptotransfermuchofhumanknowledgeintomachine-interpretableform.

HowdoWeTrainDeepArchitectures?

Deeplearningmethodsaimatlearningfeaturehierarchieswithfea-turesfromhigherlevelsofthehierarchyformedbythecompositionof

HowdoWeTrainDeepArchitectures? 5

lowerlevelfeatures.Automaticallylearningfeaturesatmultiplelevelsofabstractionallowasystemtolearncomplexfunctionsmappingtheinputtotheoutputdirectlyfromdata,withoutdependingcompletelyonhuman-craftedfeatures.Thisisespeciallyimportantforhigher-levelabstractions,whichhumansoftendonotknowhowtospecifyexplic-itlyintermsofrawsensoryinput.Theabilitytoautomaticallylearnpowerfulfeatureswillbecomeincreasinglyimportantastheamountofdataandrangeofapplicationstomachinelearningmethodscontinuestogrow.

Depthofarchitecturereferstothenumberoflevelsofcompositionofnon-linearoperationsinthefunctionlearned.Whereasmostcur-rentlearningalgorithmscorrespondtoshallowarchitectures(1,2or3levels),themammalbrainisorganizedinadeeparchitecture

[173]

withagiveninputperceptrepresentedatmultiplelevelsofabstrac-tion,eachlevelcorrespondingtoadifferentareaofcortex.Humansoftendescribesuchconceptsinhierarchicalways,withmultiplelevelsofabstraction.Thebrainalsoappearstoprocessinformationthroughmultiplestagesoftransformationandrepresentation.Thisispartic-ularlyclearintheprimatevisualsystem

[173],

withitssequenceofprocessingstages:detectionofedges,primitiveshapes,andmovinguptograduallymorecomplexvisualshapes.

Inspiredbythearchitecturaldepthofthebrain,neuralnetworkresearchershadwantedfordecadestotraindeepmulti-layerneuralnetworks

[19,

191],

butnosuccessfulattemptswerereportedbefore2006

1

:researchersreportedpositiveexperimentalresultswithtypicallytwoorthreelevels(i.e.,oneortwohiddenlayers),buttrainingdeepernetworksconsistentlyyieldedpoorerresults.Somethingthatcanbeconsideredabreakthroughhappenedin2006:Hintonetal.atUniver-sityofTorontointroducedDeepBeliefNetworks(DBNs)

[73],

withalearningalgorithmthatgreedilytrainsonelayeratatime,exploitinganunsupervisedlearningalgorithmforeachlayer,aRestrictedBoltz-mannMachine(RBM)

[51].

Shortlyafter,relatedalgorithmsbasedonauto-encoderswereproposed

[17,

153],

apparentlyexploitingthe

1Exceptforneuralnetworkswithaspecialstructurecalledconvolutionalnetworks,dis-cussedinSection4.5.

6 Introduction

sameprinciple:guidingthetrainingofintermediatelevelsofrepresen-tationusingunsupervisedlearning,whichcanbeperformedlocallyateachlevel.OtheralgorithmsfordeeparchitectureswereproposedmorerecentlythatexploitneitherRBMsnorauto-encodersandthatexploitthesameprinciple

[131,

202]

(seeSection4).

Since2006,deepnetworkshavebeenappliedwithsuccessnotonlyinclassificationtasks

[2,

17,

99,

111,

150,

153,

195],

butalso

inregression

[160],

dimensionalityreduction

[74,

158],

modelingtex-

tures

[141],

modelingmotion

[182,

183]

,objectsegmentation

[114],

informationretrieval

[154,

159,

190],

robotics

[60],

naturallanguage

processing

[37,

130,

202],

andcollaborativefiltering[

162].

Althoughauto-encoders,RBMsandDBNscanbetrainedwithunlabeleddata,inmanyoftheaboveapplications,theyhavebeensuccessfullyusedtoinitializedeepsupervisedfeedforwardneuralnetworksappliedtoaspecifictask.

IntermediateRepresentations:SharingFeaturesandAbstractionsAcrossTasks

Sinceadeeparchitecturecanbeseenasthecompositionofaseriesofprocessingstages,theimmediatequestionthatdeeparchitecturesraiseis:whatkindofrepresentationofthedatashouldbefoundastheoutputofeachstage(i.e.,theinputofanother)?Whatkindofinterfaceshouldtherebebetweenthesestages?Ahallmarkofrecentresearchondeeparchitecturesisthefocusontheseintermediaterepresentations:thesuccessofdeeparchitecturesbelongstotherepresentationslearnedinanunsupervisedwaybyRBMs

[73],

ordinaryauto-encoders

[17],

sparseauto-encoders

[150,

153],

ordenoisingauto-encoders

[195].

Thesealgo-rithms(describedinmoredetailinSection7.2)canbeseenaslearn-ingtotransformonerepresentation(theoutputofthepreviousstage)intoanother,ateachstepmaybedisentanglingbetterthefactorsofvariationsunderlyingthedata.AswediscussatlengthinSection4,ithasbeenobservedagainandagainthatonceagoodrepresenta-tionhasbeenfoundateachlevel,itcanbeusedtoinitializeandsuccessfullytrainadeepneuralnetworkbysupervisedgradient-basedoptimization.

SharingFeaturesandAbstractionsAcrossTasks 7

Eachlevelofabstractionfoundinthebrainconsistsofthe“activa-tion”(neuralexcitation)ofasmallsubsetofalargenumberoffeaturesthatare,ingeneral,notmutuallyexclusive.Becausethesefeaturesarenotmutuallyexclusive,theyformwhatiscalledadistributedrepresen-tation

[68,

156]:

theinformationisnotlocalizedinaparticularneuronbutdistributedacrossmany.Inadditiontobeingdistributed,itappearsthatthebrainusesarepresentationthatissparse:onlyaaround1-4%oftheneuronsareactivetogetheratagiventime

[5,

113].

Sec-tion3.2introducesthenotionofsparsedistributedrepresentationandSection7.1describesinmoredetailthemachinelearningapproaches,someinspiredbytheobservationsofthesparserepresentationsinthebrain,thathavebeenusedtobuilddeeparchitectureswithsparserep-resentations.

Whereasdensedistributedrepresentationsareoneextremeofaspectrum,andsparserepresentationsareinthemiddleofthatspec-trum,purelylocalrepresentationsaretheotherextreme.Localityofrepresentationisintimatelyconnectedwiththenotionoflocalgener-alization.Manyexistingmachinelearningmethodsarelocalininputspace:toobtainalearnedfunctionthatbehavesdifferentlyindifferentregionsofdata-space,theyrequiredifferenttunableparametersforeachoftheseregions(seemoreinSection3.1).Eventhoughstatisticaleffi-ciencyisnotnecessarilypoorwhenthenumberoftunableparametersislarge,goodgeneralizationcanbeobtainedonlywhenaddingsomeformofprior(e.g.,thatsmallervaluesoftheparametersarepreferred).Whenthatpriorisnottask-specific,itisoftenonethatforcesthesolutiontobeverysmooth,asdiscussedattheendofSection3.1.Incontrasttolearningmethodsbasedonlocalgeneralization,thetotalnumberofpatternsthatcanbedistinguishedusingadistributedrepresentationscalespossiblyexponentiallywiththedimensionoftherepresentation(i.e.,thenumberoflearnedfeatures).

Inmanymachinevisionsystems,learningalgorithmshavebeenlim-itedtospecificpartsofsuchaprocessingchain.Therestofthedesignremainslabor-intensive,whichmightlimitthescaleofsuchsystems.Ontheotherhand,ahallmarkofwhatwewouldconsiderintelligentmachinesincludesalargeenoughrepertoireofconcepts.RecognizingMANisnotenough.Weneedalgorithmsthatcantackleaverylarge

8 Introduction

setofsuchtasksandconcepts.Itseemsdauntingtomanuallydefinethatmanytasks,andlearningbecomesessentialinthiscontext.Fur-thermore,itwouldseemfoolishnottoexploittheunderlyingcommon-alitiesbetweenthesetasksandbetweentheconceptstheyrequire.Thishasbeenthefocusofresearchonmulti-tasklearning

[7,

8,

32,

88,

186].

Architectureswithmultiplelevelsnaturallyprovidesuchsharingandre-useofcomponents:thelow-levelvisualfeatures(likeedgedetec-tors)andintermediate-levelvisualfeatures(likeobjectparts)thatareusefultodetectMANarealsousefulforalargegroupofothervisualtasks.Deeplearningalgorithmsarebasedonlearningintermediaterep-resentationswhichcanbesharedacrosstasks.Hencetheycanleverageunsuperviseddataanddatafromsimilartasks

[148]

toboostperfor-manceonlargeandchallengingproblemsthatroutinelysufferfromapovertyoflabelleddata,ashasbeenshownby

[37],

beatingthestate-of-the-artinseveralnaturallanguageprocessingtasks.Asimi-larmulti-taskapproachfordeeparchitectureswasappliedinvisiontasksby

[2].

Consideramulti-tasksettinginwhichtherearedifferentoutputsfordifferenttasks,allobtainedfromasharedpoolofhigh-levelfeatures.Thefactthatmanyoftheselearnedfeaturesaresharedamongmtasksprovidessharingofstatisticalstrengthinproportiontom.Nowconsiderthattheselearnedhigh-levelfeaturescanthem-selvesberepresentedbycombininglower-levelintermediatefeaturesfromacommonpool.Againstatisticalstrengthcanbegainedinasim-ilarway,andthisstrategycanbeexploitedforeverylevelofadeeparchitecture.

Inaddition,learningaboutalargesetofinterrelatedconceptsmightprovideakeytothekindofbroadgeneralizationsthathumansappearabletodo,whichwewouldnotexpectfromseparatelytrainedobjectdetectors,withonedetectorpervisualcategory.Ifeachhigh-levelcate-goryisitselfrepresentedthroughaparticulardistributedconfigurationofabstractfeaturesfromacommonpool,generalizationtounseencate-goriescouldfollownaturallyfromnewconfigurationsofthesefeatures.Eventhoughonlysomeconfigurationsofthesefeatureswouldpresentinthetrainingexamples,iftheyrepresentdifferentaspectsofthedata,newexamplescouldmeaningfullyberepresentedbynewconfigurationsofthesefeatures.

DesiderataforLearningAI 9

DesiderataforLearningAI

Summarizingsomeoftheaboveissues,andtryingtoputtheminthebroaderperspectiveofAI,weputforwardanumberofrequirementswebelievetobeimportantforlearningalgorithmstoapproachAI,manyofwhichmotivatetheresearcharedescribedhere:

Abilitytolearncomplex,highly-varyingfunctions,i.e.,withanumberofvariationsmuchgreaterthanthenumberoftrainingexamples.

Abilitytolearnwithlittlehumaninputthelow-level,

intermediate,andhigh-levelabstractionsthatwouldbeuse-fultorepresentthekindofcomplexfunctionsneededforAItasks.

Abilitytolearnfromaverylargesetofexamples:computa-

tiontimefortrainingshouldscalewellwiththenumberofexamples,i.e.,closetolinearly.

Abilitytolearnfrommostlyunlabeleddata,i.e.,toworkin

thesemi-supervisedsetting,wherenotalltheexamplescomewithcompleteandcorrectsemanticlabels.

Abilitytoexploitthesynergiespresentacrossalargenum-

beroftasks,i.e.,multi-tasklearning.ThesesynergiesexistbecausealltheAItasksprovidedifferentviewsonthesameunderlyingreality.

Strongunsupervisedlearning(i.e.,capturingmostofthesta-

tisticalstructureintheobserveddata),whichseemsessentialinthelimitofalargenumberoftasksandwhenfuturetasksarenotknownaheadoftime.

Otherelementsareequallyimportantbutarenotdirectlyconnectedtothematerialinthismonograph.Theyincludetheabilitytolearntorepresentcontextofvaryinglengthandstructure

[146],

soastoallowmachinestooperateinacontext-dependentstreamofobservationsandproduceastreamofactions,theabilitytomakedecisionswhenactionsinfluencethefutureobservationsandfuturerewards

[181]

,andtheabilitytoinfluencefutureobservationssoastocollectmorerelevantinformationabouttheworld,i.e.,aformofactivelearning

[34].

10 Introduction

OutlineofthePaper

Section2reviewstheoreticalresults(whichcanbeskippedwithouthurtingtheunderstandingoftheremainder)showingthatanarchi-tecturewithinsufficientdepthcanrequiremanymorecomputationalelements,potentiallyexponentia