基于元學(xué)習(xí)和對稱性的數(shù)據(jù)高效深度學(xué)習(xí)探索 Towards data-efficient deep learning with meta-learning and symmetries

TowardsData-EfficientDeepLearningwithMeta-LearningandSymmetries

JinXu

BalliolCollege

UniversityofOxford

AthesissubmittedforthedegreeofDoctorofPhilosophyinStatistics

Trinity2023

2

Acknowledgements

Firstandforemost,Iwanttoexpressmydeepgratitudetomysupervisors,Prof.Yee

WhyeTehandDr.TomRainforth.Theirunwaveringsupport,carefulguidance,andconstantinspirationhavebeeninvaluablethroughoutmyPhDjourney.Ithasbeenaprivilegetobementoredbythem,whoIregardasresearchrolemodels.Theirdepthandbreadthofknowledgehavebeenbothhumblingandenlightening.SpecialacknowledgementgoestoYeeWhye,whohasalwaysbeenconsiderateandreadytohelpintoughtimes.MyheartfeltthanksgotoTomforhisguidanceduringthechallengingtimesbroughtonbythepandemic.

IwouldliketoextendmygratitudetoallmycollaboratorsHyunjikKim,Jean-FrancoisTon,AdamKosiorek,EmilienDupont,andKasparM?rtens.TheirexpertiseandfeedbackhavebeencrucialinimprovingmyworkandIlearnagreatdealfromthem.AbigthankyoutoProf.RyanAdamsfromPrincetonUniversityandtomyinternshiphosts,JamesHensmanandMaxCrociatMicrosoftResearch.TheirmentorshipoutsideofmyPhDlifehasbeenanindispensablepartofmyresearchexperience.

Moreover,Ifeelextremelyfortunatetobesurroundedbyamazingandcaringfriendswhosenamesarenotpossibletoenumeratehere.AmongthemareEmilienDupont,Jean-FrancoisTon,CharlineLeLan,BobbyHe,SheheryarZaidi,QinyiZhang,GuneetDhillon,AndrewCampbell,ChrisWilliams,CarloAlfano,FaaizTaufiq,AnnaMenacherandothersfromourlovelyoffice1.17,HanwenXing,YanzhaoYang,NingMiao,ChaoZhang,Yutonglu,YixuanHe,XiLin,YuanZhou,FanWu,BohaoYaofromthedepartmentofstatistics,DunhongJin,SihanZhou,SijiaYao,HuiningYang,KevinWang,NataliaHong,HangYuan,KangningZhang,ChengyangWangandmanyothersfromotherdepartmentsatOxford,DenizOktay,SulinLiu,JennyZhanandothersfromPrincetonUniversity,internshippeersatMicrosoftResearchincludingAlexanderMeulemans,SalehAshkboosfromETH.

Aspecialthankstoalluniversityanddepartmentstaff,especiallyChrisCullenforhiskindandpatientsupportduringdifficulttimes,andtoJoannaStoneham,Stuart

3

McRobert,andotherswhoensuredasmoothPhDexperience.

Finally,aboveall,mydeepestthanksgotoYifanYuforherloveandcompanionship.SheimmenselyenrichedmytimeinOxford,bringingcolourandjoytomylife.Additionally,IameternallygratefultomyparentsChengxiangXuandFengChenforgivingmethefreedomtopursuemypassionsandfortheirunquestioningsupportthroughoutthisjourney.

4

Abstract

Recentadvancesindeeplearninghavebeensignificantlypropelledbytheincreasingavailabilityofdataandcomputationalresources.Whiletheabundanceofdataenablesmodelstoperformwellincertaindomains,therearereal-worldapplications,suchasinthemedicalfield,wherethedataisscarceordifficulttocollect.Furthermore,therearealsoscenarioswherethelargedatasetisbetterviewedaslotsofrelatedsmalldatasets,andthedatabecomesinsufficientforthetaskassociatedwithoneofthesmalldatasets.Itisalsonoteworthythathumanintelligenceoftenrequiresonlyahandfulofexamplestoperformwellonnewtasks,emphasizingtheimportanceofdesigningdata-efficientAIsystems.Thisthesisdelvesintotwostrategiestoaddressthischallenge:meta-learningandsymmetries.Meta-learningapproachesthedata-richenvironmentasacollectionofmanysmall,individualdatasets.Eachofthesesmalldatasetsrepresentsadistincttask,yetthereisunderlyingsharedknowledgebetweenthem.Harnessingthissharedknowledgeallowsforthedesignoflearningalgorithmsthatcanefficientlyaddressnewtaskswithinsimilardomains.Incomparison,symmetryisaformofdirectpriorknowledge.Byensuringthatmodels’predictionsremainconsistentdespiteanytransformationtotheirinputs,thesemodelsenjoybettersampleefficiencyandgeneralization.

Inthesubsequentchapters,wepresentnoveltechniquesandmodelswhichallaimatimprovingthedataefficiencyofdeeplearningsystems.Firstly,wedemonstratethesuccessofencoder-decoderstylemeta-learningmethodsbasedonConditionalNeuralProcesses(cnps).Secondly,weintroduceanewclassofexpressivemeta-learnedstochasticprocessmodelswhichareconstructedbystackingsequencesofneuralparameterisedMarkovtransitionoperatorsinfunctionspace.Finally,weproposegroupequivariantsubsampling/upsamplinglayerswhichtacklesthelossofequivarianceinconventionalsubsampling/upsamplinglayers.Theselayerscanbeusedtoconstructend-to-endequivariantmodelswithimproveddata-efficiency.

i

Contents

1Introduction

1

1.1Motivation

1

1.2Thesisoutline

3

1.3Papers

4

2Background

6

2.1Meta-learning

6

2.1.1Conventionalsupervisedlearningandmeta-learning

6

2.1.2Differentviewsofmeta-learning

8

2.1.3Commonapproachestometa-learning

10

2.2Neuralprocesses

11

2.2.1Stochasticprocesses

12

2.2.2Neuralprocessesasstochasticprocesses

12

2.2.3Neuralprocesstrainingobjectives

13

2.2.4Ameta-learningperspective

14

2.3Symmetriesindeeplearning

15

2.3.1Group,cosetandquotientspace

15

2.3.2Grouphomomorphism,groupactionsandgroupequivariance

.16

2.3.3Homogeneousspacesandliftingfeaturemaps

16

2.3.4FeaturemapsinG-CNNs

17

2.3.5Groupequivariantneuralnetworks

18

3MetaFun:Meta-LearningwithIterativeFunctionalUpdates

20

3.1Introduction

20

3.2MetaFun

22

3.2.1Learningfunctionaltaskrepresentation

23

3.2.2MetaFunforregressionandclassification

26

3.3Relatedwork

27

ii

3.4Experiments

31

3.4.11-Dfunctionregression

31

3.4.2Classification:miniImageNetandtieredImageNet

33

3.4.3Ablationstudy

36

3.5Conclusionsandfuturework

37

3.6Supplementarymaterials

38

3.6.1Functionalgradientdescent

38

ReproducingkernelHilbertspace

38

Functionalgradients

39

Functionalgradientdescent

40

3.6.2Experimentaldetails

40

4DeepStochasticProcessesviaFunctionalMarkovTransitionOpera-

tors

44

4.1Introduction

44

4.2Background

46

4.3Markovneuralprocesses

47

4.3.1AmoregeneralformofNeuralProcessdensityfunctions

47

4.3.2Markovchainsinfunctionspace

48

4.3.3Parameterisation,inferenceandtraining

49

4.4Relatedwork

52

4.5Experiments

54

4.5.11Dfunctionregression

54

4.5.2Contextualbandits

55

4.5.3Geologicalinference

56

4.6Discussion

58

4.7Supplementarymaterials

59

4.7.1Proofs

59

60

4.7.2Implementationdetails

63

4.7.3Data

63

Modelarchitecturesandhyperparameters

65

Computationalcostsandresources

66

4.7.4Broaderimpacts

67

iii

5GroupEquivariantSubsampling

68

5.1Introduction

68

5.2Equivariantsubsamplingandupsampling

70

5.2.1TranslationequivariantsubsamplingforCNNs

70

5.2.2Groupequivariantsubsamplingandupsampling

72

5.2.3ConstructingΦ

75

5.3Application:Groupequivariantautoencoders

75

5.4Relatedwork

77

5.5Experiments

79

5.5.1Basicproperties:Equivariance,disentanglementandout-of-

distributiongeneralization

80

5.5.2Singleobject

81

5.5.3Multipleobjects

82

5.6Conclusions,limitationsandfuturework

83

5.7Supplementarymaterials

84

5.7.1Equivariantsubsamplingandupsampling

84

ConstructingΦ

84

Multiplesubsamplinglayers

85

5.7.2Groupequivariantautoencoders

87

5.7.3Proofs

88

5.7.4Implementationdetails

93

Data

93

Modelarchitectures

94

Hyperparameters

95

Computationalresources

95

6ConclusionsandFutureOutlook

96

Bibliography

99

1

Chapter1

Introduction

1.1Motivation

Recentbreakthroughsindeeplearningcanbelargelyattributedtothevastamountofdataavailableandtheadvancementofcomputationalresources[

Dengetal.,

2009,

Rainaetal.,

2009,

Silveretal.,

2016,

Jumperetal.,

2021,

Brownetal.,

2020a]

.Whiletrainingonlargedatasetsenablesdeeplearningmodelstoexcelincertaintasks,manyreal-worldapplicationsonlyprovidelimiteddataforaspecifictask.Forinstance,inmedicalfields,obtainingdata,especiallyforrarediseases,ischallengingandoftenexpensive.Indrugdevelopmentorrecommendationsystems,therewillalwaysbeinsufficientdatafornewdrugs/users,eventhoughabundantdataexistsforotherdrugsorusers.Therefore,toapplydeeplearningtothesefields,itisvitaltodevelopsystemsthataredata-efficient.Moreover,foradvancedAIsystems,data-efficiencycanbeacrucialingredient:Firstly,AIsystemsshouldbeabletogeneralizebeyondspecificdatadistributionswithoutrelyingondata;forinstance,animagerecognitionsystemshouldrecognizeobjectsregardlessoftheirpositionororientation.Secondly,humanintelligencecanoftensolvenewtaskswithjustafewexamples.Thus,forAItoemulatehuman-likeintelligence,itshouldalsohavesuchcapability.

FromaBayesianperspective,learninginvolvesupdatingourbeliefsaboutamodel(representedbyθ)giventhedata,i.e.p(θ|Ddata).Foramodeltolearnefficientlyfromasmallamountofdata,it’simportanttostartwithagoodinitialguessor"prior"p(θ).Inthispaper,welookattwodirectionstoobtainsuchpriorfordata-efficientlearning:Thefirstismeta-learning,whichlearnstheprior(orthesharedknowledge)from

2

similartasks.Itcanbeunderstoodas"learningtolearnmoreefficiently".Thesecondissymmetriesindeeplearning,whichservesasaknownpriorforcertainproblems.Symmetry,afundamentalconceptinphysics,representsaformofpriorknowledgethatisubiquitouslyobservedthroughoutourphysicalworld.

Meta-learningtacklesaspecificscenarioinwhichthevastpoolofdatacanbeviewedasmanysmalldatasets,eachrepresentingadistincttask.Yet,thesetaskscontainunderlyingsharedknowledgethatcanbeharnessedtoaddressnewtaskswithinthesamecategory.Thisscenarioisprevalentinmanyapplications.Take,forinstance,anonlineretailcompanywithdatafromcustomersworldwide.Thedataassociatedwitheachuseristypicallysparse.Inthiscontext,predictingbehavioursforeachuserconstitutesanindividualtask,butpatternsamongdifferentusersoftenexhibitsimilarities.Meta-learningalgorithmsaredesignedtohandlesuchcircumstances.Thegoalofmeta-learningistolearndata-efficientlearningalgorithmsthatcanlaterbeappliedtoaparticulartask.Thetrainingdataformeta-learningcomprisesnumerousrelatedtasks,eachwithalimitedsetofdatapoints.Afterthemeta-learningphase,thelearnedlearningalgorithmscansolveanewtaskinadata-efficientmanner.Incontrast,theaimofconventionalsupervisedlearningisjusttolearnapredictivemodel.

Meta-learningproblemscanbetackledfromvariousperspectives,andtheseap-proachescanbeunderstoodthroughdifferentviewpointssuchasoptimization-basedap-proaches[

RaviandLarochelle,

2016,

Finnetal.,

2017a

],metric-basedapproaches[

Koch,

2015

,

Vinyalsetal.,

2016,

Sungetal.,

2018,

Snelletal.,

2017],andmodel-based

approaches[

Santoroetal.,

2016,

Mishraetal.,

2018,

Garneloetal.,

2018a

],amongothers.Notethattheseviewsarenotexclusive.Forexample,methodssuchasprototypicalNetworks[

Snelletal.,

2017

],MAML[

Finnetal.,

2017a

],ML-PIP[

Gordon

etal.

,

2018

]etc.canbereformulatedunderamodel-basedframeworkthatusesanencoder-decodersetup.Inthissetup,theencoderproducesataskrepresentationusingtrainingdata,andthedecoderthenmakespredictionsbasedonthetaskrep-resentation.Theseapproachestransformthemeta-learningchallengetoresemblearegularlearningprobleminvolvingsequences,anditisalsomorecomputationallyefficientifnogradientcomputationisinvolvedinboththeencoderandthedecoderlikecnp-typemodels[

Garneloetal.,

2018a]

.OurstudyinChapter

3

explicitlyadoptsthisencoder-decoderframeworkformeta-learning.Byusingafunctionaltaskrepresentation,anditerativelyupdatingtherepresentationdirectlyinfunctionspace,

3

wedemonstratethatencoder-decoderapproacheswithoutgradientinformationcanalsobecompetitivewithotherapproaches,whichhasnotbeenshownbefore.

Furthermore,becausetrainingdataforeachtaskinmeta-learningisoftenlimited,uncertaintyestimationbecomescrucial.StochasticProcesses(sps)(e.g.GaussianProcesses(gps))canbeusedtomakepredictionswithuncertaintyestimation.Thus,learningtheseprocessescanbeseenasawaytoapproachmeta-learningwithuncer-taintyinmind.InChapter

4

,weproposeanewframeworktoconstructexpressiveneuralparameterisedspsbyparameterisingMarkovtransitionsinfunctionspace.

Unlikemeta-learningabove,whichdiscoverssharedknowledgefromrelatedtasks,symmetryservesasadirectformofpriororinductivebias,integratedintodeeplearningmodelswithouttheneedforpre-training.Symmetriesrefertotransformationsthatmaintaincertainpropertiesofanobjectofinterestunchanged.Theseincludetransformationssuchasimagetranslation,rotation,orpermutationofsetelements.Byincorporatingthesesymmetriesintodeeplearningmodels,ensuringthattheoutputsremainconsistent(thesameorundergothecorrespondingtransformation)despiteinputtransformations,themodelinherentlygeneralizestotransformedinputs.Consequently,deeplearningmodelsequippedwiththesesymmetriesnotonlybecomemoredata-efficientbutalsogeneralizebetter.AsimpleexampleofthisisConvolutinalNeuralNetworks(cnns),whichareinvarianttoinputtranslationsforclassificationtasks,andperformsignificantlybettercomparedtoplainfeed-forwardnetworks.Earlierresearchhasintroducedmanymethodstobuildconvolutional[

Cohenand

Welling,

2016,

2017,

Cohenetal.,

2019]andattentionblocks[Hutchinsonetal.,

2021,

Fuchsetal.,

2020

]thatareequivariantw.r.t.tovarioussymmetries.However,thepoolinglayersorsubsampling/upsamplinglayerscommonlyusedinvariousdeeplearningarchitecturesbreakthesesymmetries[

Zhang,

2019]

.InChapter

5,wepresent

groupequivariantsubsampling/upsamplinglayersthathaveexactequivariance.

1.2Thesisoutline

InChapter

2

,weprovideashortintroductiontometa-learning,neuralprocessesandsymmetriesindeeplearning,tosetthestageforlaterchapters.

InChapter

3

,weintroduceaniterativefunctionalencoder-decodermethodforsu-pervisedmeta-learning,whichisbasedonNeuralProcesses(nps)[

Garneloetal.,

4

2018a

,b]

.Onstandardfew-shotclassificationbenchmarkslikeminiImageNetandtieredImageNet,itisdemonstratedthatmeta-learningmethodsbasedontheneuralprocessfamilycanbecompetitiveorevenoutperformgradient-basedmethodssuchasMAML[

Finnetal.,

2017a

]andLEO[

Rusuetal.,

2019]

.

InChapter

4

,weintroduceMarkovNeuralProcesses(MNPs),anewclassofStochasticProcesses(SPs)whichareconstructedbystackingsequencesofneuralparameterisedMarkovtransitionoperatorsinfunctionspace.Therefore,theproposediterativeconstructionaddssubstantialflexibilityandexpressivitytotheoriginalframeworkofNeuralProcesses(NPs)withoutcompromisingconsistencyoraddingrestrictions.OurexperimentsdemonstrateclearadvantagesofMNPsoverbaselinemodelsonavarietyoftasks.It’snoteworthythatspmodelscanbeviewedthroughameta-learninglens.Sotheproposedmethodcanalsobeseenasameta-learningapproachwithprincipleduncertaintyestimation.

Chapter

5

,wefirstintroducetranslationequivariantsubsampling/upsamplinglayersthatcanbeusedtoconstructexacttranslationequivariantCNNs.Wethengeneralisetheselayersbeyondtranslationstogeneralgroups,thusproposinggroupequivariantsubsampling/upsampling.Weusetheselayerstoconstructgroupequivariantautoen-coders(GAEs)thatallowustolearnlow-dimensionalequivariantrepresentations.Weempiricallyverifyonimagesthattherepresentationsareindeedequivarianttoinputtranslationsandrotations,andthusgeneralisewelltounseenpositionsandorienta-tions.WefurtheruseGAEsinmodelsthatlearnobject-centricrepresentationsonmulti-objectdatasets,andshowimproveddataefficiencyanddecompositioncomparedtonon-equivariantbaselines.

InChapter

6

,wesummarizeourfindingsandexplorepotentialavenuesforfutureresearchtofurtheradvancethefield.

1.3Papers

Thisisanintegratedthesisandincludesthefollowingpublishedpapers:Chapter3contains:

Xu,J.,Ton,J.F.,Kim,H.,Kosiorek,A.,&Teh,Y.W.Metafun:Meta-

5

learningwithiterativefunctionalupdates.InternationalConferenceon

MachineLearning(ICML),2020[

Xuetal.,

2020]

Chapter4contains:

Xu,J.,Kim,H.,Rainforth,T.,&Teh,Y.(2021).Groupequivariantsub-sampling.AdvancesinNeuralInformationProcessingSystems(NeurIPS),2021[

Xuetal.,

2021]

Chapter5contains

Xu,J.,Dupont,E.,M?rtens,K.,Rainforth,T.,&Teh,Y.W.(2023).DeepStochasticProcessesviaFunctionalMarkovTransitionOperators.AdvancesinNeuralInformationProcessingSystems(NeurIPS),2023[

Xu

etal.

,

2023]

6

Chapter2

Background

2.1Meta-learning

2.1.1Conventionalsupervisedlearningandmeta-learning

Inconventionalsupervisedlearning,theobjectiveistolearnafunctionfthatmapsaninputfeaturevectorx∈Xtoanoutputlabely∈Y.Learningisbasedonexampleinput-outputpairsinatrainingsetDtrain={(xi,yi.Commontypesofsupervisedlearningtasksincluderegressionwhereoutputlabelsarereal-valued,andclassificationwheretheoutputlabelsrepresentdifferentclasses.Thefunctionf,oftenreferredto

asthepredictivemodel,isamemberofahypothesisclass,H:={f|f(x;?),?∈Rdφ}.

Foreachtask,thereisariskfunction?(y,f(x))whichmeasurespredictionerror.Asanexample,inthecontextofaregressiontask,?oftentakestheformofasquarederror,?(y,f(x))=(y?f(x))2.Thetrainingprocessofthemodelftranslatestosolvinganoptimizationproblemdefinedasfollows:

ItiscalledempiricalriskminimizationbecausethisobjectiveisanestimationofthepopulationriskE(xi,yi)~p(x,y)[?(yi,f(xi))]basedontheempiricaldistributionoftrainingdata.

7

Aftertraining,themodelshouldgeneralizeeffectivelywhenpresentedwithatestset,denotedasDtest={(xi,yim+1.Themodel’sperformancecanbeassessedusing

thetestrisk(f;Dtest)whichservesasanestimateoftheoverallpopulationrisk

usingunseendata.

Figure2.1:Dataforameta-classificationproblem.Boththemeta-trainingandmeta-testsetsconsistoftasks(redrectangles)andarepresumedtocomefromthesametaskdistributionp(T).Eachofthesetasksencompassesitsowntask-specifictrainingandtestsets,whicharecommonlyreferredtoasthecontext(yellowlabels)andthetarget(greylabels)respectively.

Inpractice,itiscommontohavescenarioswherelotsofsupervisedlearningtasksarerelatedtoeachother,yetthenumberofdatapointsforeachindividualtaskislimited.Meta-learningemergesasanewlearningparadigmtoaddresssuchchallenges.

Specifically,wehaveameta-trainingsetdefinedasMtrain={(Dt(a)in,Dt(s)t,?(j)

andameta-testsetgivenbyMtest={(Dt(a)in,Dt(s)t,?(j)M+1.Eachelementinthese

meta-datasetsisatupleconsistingofatrainingset(calledthecontext),atestset(calledthetarget)andariskfunction(typicallythesamewithinameta-dataset).This3-tuplecharacterizesataskTj(seeFigure

2.1

illustration).Insupervisedlearning,weusetrainingdatatotrainapredictivemodel,hopingitcangeneralizeacrosstheentiredatadistribution.Inmeta-learning,theassumptionisthatthereisacommontaskdistribution,denotedasp(T),fromwhichboththemeta-trainingsetandthemeta-testsetaredrawn.Meta-learningalgorithmsaimtousemeta-trainingdatatodiscoverlearningalgorithmsthatcangeneralizeacrosstheentiretaskdistribution.

Morespecifically,alearningalgorithmforasupervisedlearningtasktakesinatraining

8

setDtrain,ariskfunction?andoutputsapredictivemodel,writtenas:

=ΦA(chǔ)LGO(Dtrain,?).(2.2)

Since?isusuallyfixed,wewillomitthedependencyonitinsubsequentdiscussions.Foraparticulartask,thelearningalgorithmΦA(chǔ)LGOcanbeevaluatedbythetestriskofthelearnedpredictivemodel,denotedas:

(;Dtest).(2.3)

Meta-learningfindsalearningalgorithmbasedontasksfromthemeta-trainingsetMtrain,sothatthislearningalgorithmcanbemoreefficientlyappliedtonewtasks,andgeneralizesacrossthetaskdistributionp(T).Themeta-learningalgorithmcanberepresentedas:

ΦA(chǔ)LGO=MetaAlgo(Mtrain).(2.4)

Toevaluatethemeta-learningalgorithm,wecancompute:

Whileitresemblesthetestlossinsupervisedlearning,theaggregatedtestriskforataskreplacesthetraditionalriskfunctionforadatapoint.

Itisworthnotingthatwhilewefocusonsupervisedlearningtaskshere,meta-learningcanbeextendedtounsupervisedlearning[

EdwardsandStorkey,

2016,

Reedetal.,

2018

,

Hsuetal.,

2018]orreinforcementlearning[

Wangetal.,

2016,

Finnetal.,

2017a

,b]

.

2.1.2Differentviewsofmeta-learning

Bi-leveloptimizationviewLetusassumeboththepredictivemodelfandthelearningalgorithmΦA(chǔ)LGOcanbeparameterised,andtheparametersaredenotedas?andθaccordingly.Thatistosay,thelearningalgorithmcanbewrittenas:

?=ΦA(chǔ)LGO(Dtrain;θ).(2.6)

9

Meta-learningcanbeformulatedasthefollowingbi-leveloptimizationproblem:

wheretask-specificparameter?jdependsonθthroughtheinner-loopoptimization:

?j(θ)=ΦA(chǔ)LGO(Dt(a)in;θ)(2.8)

Manymeta-learningalgorithmsaredevelopedbasedonthisbi-leveloptimizationview,suchas

Finnetal.

[2017a],

Nicholetal.

[2018],

RaviandLarochelle

[2016]

.

HierarchicalmodelviewFromaprobabilisticperspective,thegenerativeprocessforeachtaskTjcanbeexpressedas:

θ～p(θ),?j～p(?j|θ),yi(j)～p(yi(j)|xi(j)?j,θ)(2.9)

BoththetrainingsetDt(a)inandthetestsetDt(s)tfollowthesamedistribution(as

illustratedinFigure

2.2

).Thiscanbeseenasaprobabilistichierarchicalmodelwhereθindicatesthehigh-levelglobalparametersforalltasksand?jdenotesthelow-levellocalparametersforeachtask.Inthiscontext,meta-learningisaboutinferringθfromlotsoftasksinthemeta-trainingset,thatisp(θ|Mtrain).Learning,ontheother

hand,infers?jgiventhetrainingsetDt(a)infortaskTj,thatisp(?j|θ,Dt(a)in).

(j)i

j=1,...

Figure2.2:Meta-learningashierarchicalmodels(AremakeofFigure1in

Gordon

etal.

[2018])

.Task-specificparameter?jdependsontheglobalparameterθ.Datapointsinboththecontextandthetargethavethesamegenerativeprocess,whichdependonbothθand?j.

Notethatp(?j|θ)canbeseenasapriorfortaskTjconditionedonθ.Therefore,meta-learningcanbeseenaslearninganempiricalpriorfromthemeta-trainingset.

Finnetal.

[2018],

Requeimaetal.

[2019]adoptsthisview

.

10

Model-basedviewAlearningalgorithmf=ΦA(chǔ)LGO(Dtrain)canbeseenasafunctionthattakesintheentiretrainingsetandoutputsapredictivemodel.ThemodelisthenusedtomakepredictionsontestdatainDtest.Thelearningandpredictionprocessescanthusbeconceptualizedassequence-to-sequencemappings.Forthesakeofbrevity,let’suseaconcisenotationfordatasequences,suchasx1:n={x1,x2,...,xn}.ForaspecifictaskTj,makingpredictionsfortestsetdatapointsbasedonthosefromthetrainingsetcanbedescribedasthefollowinginferencetask

p(ym+1:n|xm+1:n,x1:m,y1:m).(2.10)

Fromthisperspective,meta-learningisaboutcreatingthisconditionalmodel.Meta-learningonlydiffersfromconventionalsupervisedlearninginthatboththeinp