PublishedasaconferencepaperatICLR2020 DEEP BATCH ACTIVE LEARNING BY DIVERSE, UNCERTAIN GRADIENT LOWER BOUNDS JordanT.Ash ChichengZhang AkshayKrishnamurthy PrincetonUniversity UniversityofArizona MicrosoftResearchNYC JohnLangford AlekhAgarwal MicrosoftResearchNYC MicrosoftResearchRedmond ABSTRACT Wedesignanewalgorithmforbatchactivelearningwithdeepneuralnetworkmodels. Our algorithm,BatchActivelearningbyDiverseGradientEmbeddings(BADGE),samples groupsofpointsthataredisparateandhighmagnitudewhenrepresentedinahallucinated gradientspace,astrategydesignedtoincorporatebothpredictiveuncertaintyandsample diversity into every selected batch. Crucially, BADGE trades off between uncertainty anddiversitywithoutrequiringanyhand-tunedhyperparameters. Whileotherapproaches sometimes succeed for particular batch sizes or architectures, BADGE consistently performsaswellorbetter,makingitausefuloptionforrealworldactivelearningproblems. 1 INTRODUCTION Inrecentyears,deepneuralnetworkshaveproducedstate-of-the-artresultsonavarietyofimportantsuper- visedlearningtasks. However,manyofthesesuccesseshavebeenlimitedtodomainswherelargeamountsof labeleddataareavailable. Apromisingapproachforminimizinglabelingeffortisactivelearning,alearning protocolwherelabelscanberequestedbythealgorithminasequential,feedback-drivenfashion. Active learningalgorithmsaimtoidentifyandlabelonlymaximally-informativesamples,sothatahigh-performing classifiercanbetrainedwithminimallabelingeffort. Assuch,arobustactivelearningalgorithmfordeep neuralnetworksmayconsiderablyexpandthedomainsinwhichthesemodelsareapplicable. Howshouldwedesignapractical,general-purpose,label-efficientactivelearningalgorithmfordeepneural networks? Theoryforactivelearningsuggestsaversion-space-basedapproach(Cohnetal.,1994;Balcan etal.,2006),whichexplicitlyorimplicitlymaintainsasetofplausiblemodels,andqueriesexamplesforwhich thesemodelsmakedifferentpredictions. Butwhenusinghighlyexpressivemodelslikeneuralnetworks, thesealgorithmsdegeneratetoqueryingeveryexample. Further,thecomputationaloverheadoftrainingdeep neuralnetworksprecludesapproachesthatupdatethemodeltobestfitdataaftereachlabelquery,asisoften done(exactlyorapproximately)forlinearmethods(Beygelzimeretal.,2010;Cesa-Bianchietal.,2009). Unfortunately,thetheoryprovideslittleguidanceforthesemodels. Oneoptionistousethenetwork’suncertaintytoinformaquerystrategy,forexamplebylabelingsamples forwhichthemodelisleastconfident. Inabatchsetting,however,thiscreatesapathologicalscenariowhere datainthebatcharenearlyidentical,aclearinefficiency. Remedyingthisissue,wecouldselectsamples tomaximizebatchdiversity,butthismightchoosepointsthatprovidelittlenewinformationtothemodel. Forthesereasons,methodsthatexploitjustuncertaintyordiversitydonotconsistentlyworkwellacross modelarchitectures, batchsizes, ordatasets. Analgorithm thatperformswellwhen usingaResNet, for 1 0202 beF 42 ]GL.sc[ 2v17630.6091:viXra PublishedasaconferencepaperatICLR2020 example,mightperformpoorlywhenusingamultilayerperceptron. Adiversity-basedapproachmightwork wellwhenthebatchsizeisverylarge,butpoorlywhenthebatchsizeissmall. Further,whatevenconstitutes a“large”or“small”batchsizeislargelyafunctionofthestatisticalpropertiesofthedatainquestion. These weaknessesposeamajorproblemforreal,practicalbatchactivelearningsituations,wheredataareunfamiliar andpotentiallyunstructured. Thereisnowaytoknowwhichactivelearningalgorithmisbesttouse. Moreover,inarealactivelearningscenario,everychangeofhyperparameterstypicallycausesthealgorithm tolabelexamplesnotchosenunderotherhyperparameters,provokingsubstantiallabelinginefficiency. That is,hyperparametersweepsinactivelearningcanbelabelexpensive. Asaresult,activelearningalgorithms needto“justwork”,givenfixedhyperparameters,toagreaterextentthanistypicalforsupervisedlearning. Based on these observations, we design an approach which creates diverse batches of examples about which the current model is uncertain. We measure uncertainty as the gradient magnitude with respect to parameters in the final (output) layer, which is computed using the most likely label according to the model. To capture diversity, we collect a batch of examples where these gradients span a diverse set of directions. Morespecifically,webuildupthebatchofquerypointsbasedonthesehallucinatedgradients usingthek-MEANS++initialization(ArthurandVassilvitskii,2007),whichsimultaneouslycapturesboththe magnitudeofacandidategradientanditsdistancefrompreviouslyincludedpointsinthebatch. Wenamethe resultingapproachBatchActivelearningbyDiverseGradientEmbeddings(BADGE). WeshowthatBADGEisrobusttoarchitecturechoice,batchsize,anddataset,generallyperformingaswell asorbetterthanthebestbaselineacrossourexperiments,whichvaryalloftheaforementionedenvironmental conditions. Webeginbyintroducingournotationandsetting, followedbyadescriptionofthe BADGE algorithminSection3andexperimentsinSection4. WedeferourdiscussionofrelatedworktoSection5. 2 NOTATION AND SETTING Define [K] := {1,2,...,K}. Denote by X the instance space and by Y the label space. In this work we consider multiclass classification, so Y = [K]. Denote by D the distribution from which examples are drawn, by D the unlabeled data distribution, and by D the conditional distribution over labels X Y|X givenexamples. Weconsiderthepool-basedactivelearningsetup,wherethelearnerreceivesanunlabeled dataset U sampled according to D and can request labels sampled according to D for any x ∈ X Y|X U. We use E to denote expectation under the data distribution D. Given a classifier h : X → Y, D which maps examples to labels, and a labeled example (x,y), we denote the 0/1 error of h on (x,y) as (cid:96) (h(x),y) = I(h(x) (cid:54)= y). Theperformanceofaclassifierhismeasuredbyitsexpected0/1error,i.e. 01 E [(cid:96) (h(x),y)]=Pr (h(x)(cid:54)=y). Thegoalofpool-basedactivelearningistofindaclassifierwith D 01 (x,y)∼D asmallexpected0/1errorusingasfewlabelqueriesaspossible. GivenasetS oflabeledexamples(x,y), whereeachx∈S ispickedfromU,followedbyalabelquery,weuseE asthesampleaveragesoverS. S Inthispaper,weconsiderclassifiershparameterizedbyunderlyingneuralnetworksf offixedarchitecture, withtheweightsinthenetworkdenotedbyθ. Weabbreviatetheclassifierwithparametersθash sincethe θ architecturesarefixedinanygivencontext,andourclassifierstaketheformh (x)=argmax f(x;θ) , θ y∈[K] y wheref(x;θ)∈RK isaprobabilityvectorofscoresassignedtocandidatelabels,giventheexamplexand parametersθ. Weoptimizetheparametersbyminimizingthecross-entropylossE [(cid:96) (f(x;θ),y)]overthe S CE labeledexamples,where(cid:96) CE (p,y)= (cid:80)K i=1 I(y =i)ln1/pi =ln1/py . 2 PublishedasaconferencepaperatICLR2020 Algorithm1BADGE: BatchActivelearningbyDiverseGradientEmbeddings Require: Neuralnetworkf(x;θ),unlabeledpoolofexamplesU,initialnumberofexamplesM,numberof iterationsT,numberofexamplesinabatchB. 1: LabeleddatasetS ←M examplesdrawnuniformlyatrandomfromU togetherwithqueriedlabels. 2: Trainaninitialmodelθ 1 onS byminimizingE S [(cid:96) CE (f(x;θ),y)]. 3: fort=1,2,...,T:do 4: ForallexamplesxinU \S: 1. Computeitshypotheticallabelyˆ(x)=h (x). θt 2. Computegradientembeddingg = ∂ (cid:96) (f(x;θ),yˆ(x))| ,whereθ referstoparam- x ∂θout CE θ=θt out etersofthefinal(output)layer. 5: ComputeS t ,arandomsubsetofU\S,usingthek-MEANS++seedingalgorithmon{g x :x∈U \S} andqueryfortheirlabels. 6: S ←S∪S t . 7: Trainamodelθ t+1 onS byminimizingE S [(cid:96) CE (f(x;θ),y)]. 8: endfor 9: return Finalmodelθ T+1 . 3 ALGORITHM BADGE,describedinAlgorithm1,startsbydrawinganinitialsetofM examplesuniformlyatrandomfrom U andaskingfortheirlabels. Itthenproceedsiteratively,performingtwomaincomputationsateachstept: a gradientembeddingcomputationandasamplingcomputation. Specifically,ateachstept,foreveryxinthe poolU,wecomputethelabelyˆ(x)preferredbythecurrentmodel,andthegradientg ofthelosson(x,yˆ(x)) x with respect to the parameters of the last layer of the network. Given these gradient embedding vectors {g x :x∈U},BADGEselectsasetofpointsbysamplingviathek-MEANS++initializationscheme(Arthur andVassilvitskii,2007). Thealgorithmqueriesthelabelsoftheseexamples,retrainsthemodel,andrepeats. Wenowdescribethemaincomputations—theembeddingandsamplingsteps—inmoredetail. Thegradientembedding. Sincedeepneuralnetworksareoptimizedusinggradient-basedmethods,we captureuncertaintyaboutanexamplethroughthelensofgradients. Inparticular,weconsiderthemodel uncertainaboutanexampleifknowingthelabelinducesalargegradientofthelosswithrespecttothemodel parametersandhencealargeupdatetothemodel. Adifficultywiththisreasoningisthatweneedtoknowthe labeltocomputethegradient. Asaproxy,wecomputethegradientasifthemodel’scurrentpredictiononthe exampleisthetruelabel.WeshowinProposition1that,assumingacommonstructuresatisfiedbymostnatural neuralnetworks,thegradientnormwithrespecttothelastlayerusingthislabelprovidesalowerboundonthe gradientnorminducedbyanyotherlabel. Inaddition,underthatassumption,thelengthofthishypothetical gradientvectorcapturestheuncertaintyofthemodelontheexample: ifthemodelishighlycertainaboutthe example’slabel,thentheexample’sgradientembeddingwillhaveasmallnorm,andviceversaforsamples wherethemodelisuncertain(seeexamplebelow). Thus,thegradientembeddingconveysinformationboth aboutthemodel’suncertaintyandpotentialupdatedirectionuponreceivingalabelatanexample. Thesamplingstep. Wewantthenewly-acquiredlabeledsamplestoinducelargeanddiversechangesto themodel. Tothisend,wewanttheselectionproceduretofavorbothsamplemagnitudeandbatchdiversity. Specifically,wewanttoavoidthepathologyof,forexample,selectingabatchofksimilarsampleswhere evenjustasinglelabelcouldalleviateouruncertaintyonallremaining(k−1)samples. Anaturalwayofmakingthisselectionwithoutintroducingadditionalhyperparametersistosamplefrom ak-DeterminantalPointProcess(k-DPP; (KuleszaandTaskar,2011)). Thatis,toselectabatchofkpoints 3 PublishedasaconferencepaperatICLR2020 | | | | k-DPP | | k-means++ | | | | --- | --- | --- | ----- | --- | --------- | --- | --- | OpenML#6, MLP, Batch size: 100 SVHN, ResNet, Batch size: 1000 ×104 SVHN, ResNet, Batch size: 1000 | | | | 0.90 | | | 7 | | | ---- | --- | --- | ---- | --- | --- | --- | --- | | 0.90 | | | 0.80 | | | | | 6 | 0.80 | | | 0.70 | | | | | | -------- | --- | --- | -------- | --- | --- | ------ | --- | | ycaruccA | | | ycaruccA | | | 5 | | | | | | 0.60 | | | emiT 4 | | | 0.70 | | | 0.50 | | | | | 3 | 0.60 | | | 0.40 | | | | | | ---- | --- | --- | ---- | --- | --- | --- | --- | | | | | 0.30 | | | 2 | | | 0.50 | | | | | | 1 | | 0.20 0 0.40 200040006000800010000120001400016000 10000 20000 30000 40000 10000 20000 30000 40000 | | #Labels queried | | | #Labels queried | | #Labels queried | | | --- | --------------- | --- | --- | --------------- | --- | --------------- | --- | Figure1: Leftandcenter: Learningcurvesfork-MEANS++andk-DPPsamplingwithgradientembeddings fordifferentscenarios. Theperformanceofthetwosamplingapproachesnearlyperfectlyoverlaps. Right: Aruntimecomparison(seconds)correspondingtothemiddlescenario. Eachlineistheaverageoverfive | independentexperiments. | | Standarderrorsareshownbyshadedregions. | | | | | | | ----------------------- | --- | -------------------------------------- | --- | --- | --- | --- | --- | withprobabilityproportionaltothedeterminantoftheirGrammatrix. Recently,Derezin´skiandWarmuth (2018)showedthatinexperimentaldesignforleastsquarelinearregressionsettings,learningfromsamples drawnfromak-DPPcanhavemuchsmallermeansquarepredictionerrorthanlearningfromiidsamples. Inthisprocess,whenthebatchsizeisverylow,theselectionwillnaturallyfavorpointswithalargelength, whichcorrespondstouncertaintyinourspace. Whenthebatchsizeislarge,thesamplerfocusesmoreon diversitybecauselinearindependence,whichismoredifficulttoachieveforlargek,isrequiredtomake theGramdeterminantnon-zero. Unfortunately,samplingfromak-DPPisnottrivial. Manysamplingalgorithms(Kang,2013;Anarietal., 2016)relyonMCMC, wheremixingtimeposesasignificantcomputationalhurdle. Thestate-of-the-art algorithmofDerezin´ski(2018)hasahigh-orderpolynomialrunningtimeinthebatchsizeandtheembedding dimension. To overcome this computational hurdle, we suggest instead sampling using the k-MEANS++ seeding algorithm (Arthur and Vassilvitskii, 2007), originally made to produce a good initialization for k-meansclustering. k-MEANS++seedingselectscentroidsbyiterativelysamplingpointsinproportionto theirsquareddistancesfromthenearestcentroidthathasalreadybeenchosen,which,likeak-DPP,tends toselectadiversebatchofhigh-magnitudesamples. Forcompleteness,wegiveaformaldescriptionofthe k-MEANS++seedingalgorithminAppendixA. Example: multiclassclassificationwithsoftmaxactivations. Consideraneuralnetworkf wherethelast nonlinearityisasoftmax,i.e. σ(z) =ezi/(cid:80)K ezj. Specifically,f isparametrizedbyθ =(W,V),where i j=1 θ = W = (W ,...,W )(cid:62) ∈ RK×d aretheweightsofthelastlayer,andV consistsofweightsofall | out | 1 | K | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | previouslayers. Thismeansthatf(x;θ) = σ(W ·z(x;V)), wherez isthenonlinearfunctionthatmaps aninputxtotheoutputofthenetwork’spenultimatelayer. Letusfixanunlabeledsamplexanddefine | p i =f(x;θ) | i . Withthisnotation,wehave | | | | | | | | ----------- | --------------------------- | --- | --- | --- | --- | --- | --- | | | | | |  |  | | | K (cid:88) | | | (cid:96) (f(x;θ),y)=ln | | eWj·z(x;V) | −W | ·z(x;V). | | | --- | --- | ----------------------- | --- | ---------- | --- | -------- | --- | | | | CE | | | y | | | j=1 Definegy = ∂ (cid:96) (f(x;θ),y)foralabelyandg =gyˆasthegradientembeddinginouralgorithm,where | x | CE | | | x x | | | | | --- | --- | --- | --- | --- | --- | --- | --- | ∂W yˆ=argmax i∈[K] p i . Thenthei-thblockofg x (i.e. thegradientscorrespondingtolabeli)is ∂ | | | (g ) = | (cid:96) (f(x;θ),yˆ)=(p | | −I(yˆ=i))z(x;V). | | (1) | | --- | --- | ------ | ----------------------- | --- | ---------------- | --- | --- | | | | x i | CE | | i | | | | | | | ∂W i | | | | | Basedonthisexpression,wecanmakethefollowingobservations: 4 PublishedasaconferencepaperatICLR2020 1. Eachblockofg isascalingofz(x;V),whichistheoutputofthepenultimatelayerofthenetwork. x In this respect, g x captures x’s representation information similar to that of Sener and Savarese (2018). 2. Proposition1belowshowsthatthenormofg x isalowerboundonthenormofthelossgradient induced by the example with true label y with respect to the weights in the last layer, that is (cid:107)g (cid:107)≤(cid:107)gy(cid:107). Thissuggeststhatthenormofg conservativelyestimatestheexample’sinfluenceon | | x | x | | | x | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | thecurrentmodel. 3. Ifthecurrentmodelθishighlyconfidentaboutx,i.e. vectorpisskewedtowardsastandardbasis vector e , then yˆ = j, and vector (p −I(yˆ = i))K has a small length. Therefore, g has a | | j | | | i | i=1 | | | x | | --- | --- | --- | --- | --- | --- | --- | --- | --- | smalllengthaswell. Suchhigh-confidenceexamplestendtohavegradientembeddingsofsmall magnitude,whichareunlikelytoberepeatedlyselectedbyk-MEANS++atiterationt. | Proposition1. | Forally | ∈{1,...,K},letgy | | = ∂ (cid:96) | (f(x;θ),y). | Then | | | | ------------- | ------- | ---------------- | --- | ------------ | ----------- | ---- | --- | --- | | | | | | x ∂W CE | | | | | K | | | | | (cid:16)(cid:88) | (cid:17) | | | | | --- | --- | --- | --------------------- | ---------------- | -------- | -------------------------- | --- | --- | | | | | (cid:107)gy(cid:107)2 | = p2+1−2p | | (cid:107)z(x;V)(cid:107)2. | | | | | | | x | i | y | | | | i=1 | Consequently,yˆ=argmin | | | (cid:107)gy(cid:107). | | | | | | | ---------------------- | --- | ----- | --------------------- | --- | --- | --- | --- | --- | | | | y∈[K] | x | | | | | | Proof. ObservethatbyEquation(1), | | | K | | | | K | | | | --------------------------------------------- | --------------------- | --------------- | ------ | ----------------------------- | ---------------- | ------- | -------------------------- | --- | | | | (cid:88)(cid:0) | | (cid:1)2 | (cid:16)(cid:88) | | (cid:17) | | | | (cid:107)gy(cid:107)2 | = | p −I(y | =i) (cid:107)z(x;V)(cid:107)2 | = | p2+1−2p | (cid:107)z(x;V)(cid:107)2. | | | | x | | i | | | i | y | | | | | i=1 | | | i=1 | | | | | Thesecondclaimfollowsfromthefactthatyˆ=argmax | | | | | | p . | | | y∈[K] y Thissimplesamplertendstoproducediversebatchessimilartoak-DPP.AsshowninFigure1,switching betweenthetwosamplersdoesnotaffecttheactivelearner’sstatisticalperformancebutgreatlyimproves itscomputationalperformance. AppendixGcomparesruntimeandtestaccuracyforbothk-MEANS++and k-DPPbasedsamplingbasedonthegradientembeddingsoftheunlabeledexamples. Figure2illustratesthebatchdiversityandaveragegradientmagnitudeperselectedbatchforavarietyof sampling strategies. As expected, both k-DPPs and k-MEANS++ tend to select samples that are diverse (as measured by the magnitude of their Gram determinant) and high magnitude. Other samplers, such asfurthest-firsttraversalfork-Centerclustering(FF-k-CENTER),donotseemtohavethisproperty. The FF-k-CENTER algorithm is the sampling choice of the CORESET approach to active learning, which we describeintheproceedingsection(SenerandSavarese,2018). AppendixFdiscussesdiversitywithrespect touncertainty-basedapproaches. AppendixBprovidesfurtherjustificationforwhy BADGE yieldsbetterupdatesthanvanillauncertainty | samplinginthespecialcaseofbinarylogisticregression(K | | | | | =2andz(x;V)=x). | | | | | ---------------------------------------------------- | --- | --- | --- | --- | --------------- | --- | --- | --- | 4 EXPERIMENTS WeevaluatetheperformanceofBADGEagainstseveralalgorithmsfromtheliterature. Inourexperiments, weseektoanswerthefollowingquestion:Howrobustarethelearningalgorithmstochoicesofneuralnetwork architecture,batchsize,anddataset? To ensure a comprehensive comparison among all algorithms, we evaluate them in a batch-mode active learningsetupwithM =100beingthenumberofinitialrandomlabeledexamplesandbatchsizeBvarying from {100,1000,10000}. The following is a list of the baseline algorithms evaluated; the first performs representative sampling, the next three are uncertainty based, the fifth is a hybrid of representative and uncertainty-basedapproaches,andthelastistraditionalsupervisedlearning. 5 PublishedasaconferencepaperatICLR2020 | | | k-DPP k-means++ | Rand | FF k-center | | | --- | --- | --------------- | ---- | ----------- | --- | OpenML #6, MLP, Batch size: 100 SVHN, ResNet, Batch size: 1000 OpenML #6, MLP, Batch size: 100 | hctab fo tnanimreted goL | | hctab fo tnanimreted goL 0 | | hctab ni mron 35 | | | ------------------------ | --- | -------------------------- | --- | ----------------- | --- | 500 | 250 | | 5000 | | 30 | | | --- | --- | ----- | --- | --- | --- | | 0 | | 10000 | | 25 | | 20 | 250 | | 15000 | | | | | --- | --- | ----- | --- | --- | --- | | 500 | | 20000 | | 15 | | 2 | 750 | | 25000 | | egarevA 10 | | | ---- | --- | ----- | --- | ----------- | --- | | 1000 | | 30000 | | 5 | | 0 | 1250 | | 35000 | | | | | ---- | --- | ----- | --- | --- | --- | 0 2000 4000 6000 8000100001200014000 0 10000 20000 30000 40000 50000 0 2000 4000 6000 8000 100001200014000 | | #Labels queried | | #Labels queried | | #Labels queried | | --- | --------------- | --- | --------------- | --- | --------------- | Figure2: Acomparisonofbatchselectionalgorithmsusingourgradientembedding. Leftandcenter: Plots showingthelogdeterminantoftheGrammatrixoftheselectedbatchofgradientembeddingsaslearning progresses. Right: The average embedding magnitude (a measurement of predictive uncertainty) in the selectedbatch. TheFF-k-CENTERsamplerfindspointsthatarenotasdiverseorhigh-magnitudeasother samplers. Notice also that k-MEANS++ tends to actually select samples that are both more diverse and higher-magnitudethanak-DPP,apotentialpathologyofthek-DPP’sdegreeofstochastisity. Standarderrors areshownbyshadedregions. 1. CORESET: Adiversity-basedapproachusingcoresetselection. Theembeddingofeachexample iscomputedbythenetwork’spenultimatelayerandthesamplesateachroundareselectedusing agreedyfurthest-firsttraversalconditionedonalllabeledexamples(SenerandSavarese,2018). 2. CONF (Confidence Sampling): An uncertainty-based active learning algorithm that selects B exampleswithsmallestpredictedclassprobability,maxK f(x;θ) (e.g.WangandShang,2014). i=1 i 3. MARG(MarginSampling):Anuncertainty-basedactivelearningalgorithmthatselectsthebottomB examplessortedaccordingtotheexample’smulticlassmargin,definedasf(x;θ) −f(x;θ) ,where yˆ y(cid:48) yˆandy(cid:48)aretheindicesofthelargestandsecondlargestentriesoff(x;θ)(RothandSmall,2006). 4. ENTROPY: An uncertainty-based active learning algorithm that selects the top B examples according to the entropy of the example’s predictive class probability distribution, defined as (cid:80)K | H((f(x;θ) | )K ),whereH(p)= | | p ln1/pi (WangandShang,2014). | | | | --------- | --------------- | --- | ----------------------------- | --- | --- | | | y y=1 | | i=1 i | | | 5. ALBL(ActiveLearningbyLearning): Abandit-stylemeta-activelearningalgorithmthatselects betweenCORESETandCONFateveryround(HsuandLin,2015). 6. RAND: Thenaivebaselineofrandomlyselectingkexamplestoqueryateachround. We consider three neural network architectures: a two-layer Perceptron with ReLU activations (MLP), an 18-layer convolutional ResNet (He et al., 2016), and an 11-layer VGG network (Simonyan and Zisserman, 2014). We evaluate our algorithms using three image datasets, SVHN (Netzer et al., 2011), 1, CIFAR10 (Krizhevsky, 2009) and MNIST (LeCun et al., 1998) and four non-image datasets from the OpenMLrepository(#6,#155,#156,and#184). 2 Westudyeachsituationwith7activelearningalgorithms, includingBADGE,makingfor231totalexperiments. Fortheimagedatasets,theembeddingdimensionalityintheMLPis256. FortheOpenMLdatasets,the embeddingdimensionalityoftheMLPis1024,asmorecapacityhelpsthemodelfittrainingdata. Wefit 1BecauseMNISTisadatasetthatisextremelyeasytoclassify,weonlyuseMLPs,ratherthanconvolutionalnetworks, tobetterstudythedifferencesbetweenactivelearningalgorithms. 2TheOpenMLdatasetsarefromopenml.organdareselectedontwocriteria: first, theyhaveatleast10000 samples;second,neuralnetworkshaveasignificantlysmallertesterrorratewhencomparedtolinearmodels. 6 PublishedasaconferencepaperatICLR2020 ALBL Conf Coreset BADGE Entropy Marg Rand 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 5000 10000 15000 20000 25000 30000 35000 #Labels queried ycaruccA SVHN, ResNet, Batch size: 100 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 5000 10000 15000 20000 25000 30000 35000 #Labels queried ycaruccA SVHN, ResNet, Batch size: 100 0.90 0.85 0.80 0.75 0.70 0.65 500 1000 1500 2000 2500 3000 3500 4000 #Labels queried (a) ycaruccA OpenML#156, MLP, Batch size: 1000 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 500010000150002000025000300003500040000 #Labels queried (b) ycaruccA CIFAR10, VGG, Batch size: 10000 (c) Figure3: Activelearningtestaccuracyversusthenumberoftotallabeledsamplesforarangeofconditions. Standarderrorsareshownbyshadedregions. modelsusingcross-entropylossandtheAdamvariantofSGDuntiltrainingaccuracyexceeds99%. We usealearningrateof0.001forimagedataandof0.0001fornon-imagedata. Weavoidwarmstartingand retrainmodelsfromscratcheverytimenewsamplesarequeried(AshandAdams,2019). Allexperimentsare repeatedfivetimes. Nolearningrateschedulesordataaugmentationareused. Baselinesuseimplementations fromthelibactlibrary(Yangetal.,2017). AllmodelsaretrainedinPyTorch(Paszkeetal.,2017). Learningcurves. Hereweshowexamplesoflearningcurvesthathighlightsomeofthephenomenawe observerelatedtothefragilityofactivelearningalgorithmswithrespecttobatchsize,architecture,anddataset. Often,weseethatinearlyroundsoftraining,itisbettertododiversitysampling,andlaterintraining,itis bettertodouncertaintysampling. ThiskindofeventisdemonstratedinFigure3a,whichshowsCORESET outperformingconfidence-basedmethodsatfirst,butthendoingworsethanthesemethodslateron. Overall(33) Inthisfigure,BADGEperformsaswellasdiversity BADGE ALBL Coreset Conf Marg Entropy Rand samplingwhenthatstrategydoesbest,andaswell as uncertainty sampling once those methods start BADGE 0.0 9.18 10.97 12.56 3.88 13.16 10.04 12 outpacing CORESET. This suggests that BADGE isagoodchoiceregardlessoflabelingbudget. ALBL 0.34 0.0 5.18 3.15 0.31 6.81 4.95 10 Separately,wenoticethatdiversitysamplingonly Coreset 1.65 2.02 0.0 6.55 3.08 8.13 6.56 seems to work well when either the model has good architectural priors (inductive biases) built 8 in, or when the data are easy to learn. Otherwise, Conf 0.54 2.78 7.61 0.0 0.33 5.96 6.08 penultimatelayerrepresentationsarenotmeaning- 6 ful, and diverse sampling can be deleterious. For Marg 0.96 7.14 10.87 9.34 0.0 11.33 9.34 this reason, CORESET often performs worse than randomonsufficientlycomplexdatawhennotusing Entropy 0.31 1.79 6.05 1.25 0.35 0.0 5.12 4 a convolutional network (Figure 3b). That is, the diversityinducedbyunconditionalrandomsampling Rand 0.84 5.63 7.61 8.67 2.98 10.65 0.0 2 canoftenyieldabatchthatbetterrepresentsthedata. Even when batch size is large and the model has 0.66 4.08 6.9 5.93 1.56 8.01 6.01 helpfulinductivebiases,theuncertaintyinformation 0 in BADGE can give it an advantage over pure Figure4: Apairwisepenaltymatrixoverallexperiments. diversity approaches (Figure 3c). Comprehensive ElementP correspondsroughlytothenumberoftimes i,j plots of this kind, spanning architecture, dataset, algorithmioutperformsalgorithmj.Column-wiseaverages andbatchsizeareinAppendixC. atthebottomshowoverallperformance(lowerisbetter). 7 PublishedasaconferencepaperatICLR2020 Pairwisecomparisons. Wenextshowacomprehensivepairwisecomparisonofalgorithmsoveralldatasets (D),batchsizes(B),modelarchitectures(A),andlabelbudgets(L). Fromthelearningcurves,itcanbeob- servedthatwhenlabelbudgetsarelargeenough,allalgorithmseventuallyreachsimilarperformance,making thecomparisonbetweenthemuninterestinginthelargesamplelimit. Forthisreason,foreachcombinationof (D,B,A),weselectasetoflabelingbudgetsLwherelearningisstillprogressing.Weexperimentedwiththree differentbatchsizesandelevendataset-architecturepairs,makingthetotalnumberof(D,B,A)combinations 3×11=33. Specifically,wecomputen 0 ,thesmallestnumberoflabelswhereRAND’saccuracyreaches | | | | | | | | | (cid:8) +2m−1B | | | (cid:9) | | --- | --- | --- | --- | --- | --- | --- | --- | -------------- | --- | --- | ------- | 99% of its final accuracy, and choose label budget L from M :m∈[(cid:98)log((n −M)/B)(cid:99)] . 0 The calculation of scores in the penalty matrix P follows the following protocol: For each (D,B,A,L) combination and each pair of algorithms (i,j), we have 5 test errors (one for each repeated run), | (cid:8) e1 | ,...,e5 | (cid:9) | (cid:8) e1 ,...,e5 | (cid:9) | | | | | | √ | | | ---------- | ------- | ------- | ------------------ | ------- | --- | --- | --- | --- | --- | --- | --- | and respectively. We compute the t-score as t = 5µˆ/σˆ, where | i | | i | j | j | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | Overall (cid:118) | | | 5 | | (cid:117) | 5 | | | 1.0 | | | | | --- | --------- | -------- | --- | ------------------- | ------ | ------ | --- | -------------------- | --- | --- | --- | | | 1(cid:88) | | | (cid:117) 1(cid:88) | | | | ycneuqerf evitalumuC | | | | | | µˆ= | (el−el), | | σˆ = (cid:116) | (el−el | −µˆ)2. | | | | | | | | 5 | i | j | 4 | i | j | | 0.8 | | | | | | l=1 | | | | l=1 | | | | | | | 0.6 | We | use the | two-sided | t-test | to compare | | pairs of | algo- | | | | | | ------- | --------- | --------- | --------- | ---------- | --------- | -------- | ----- | --- | --- | --- | --- | | rithms: | algorithm | | i is said | to beat | algorithm | j in | this | 0.4 | | | | | setting | if | t > 2.776 | (the | critical | point of | p-value | be- | | | | | 0.2 | ing | 0.05), | and similarly | algorithm | | j beats | algorithm | i | | | | | | ----------- | ------ | --------------------------------- | --------- | --- | -------------- | --------- | --- | ------- | ---------------- | ----------- | ------- | | ift<−2.776. | | Foreach(D,B,A)combination,suppose | | | | | | 0.0 | | | | | | | | | | | | | 0.3 0.4 | 0.5 0.6 0.7 | 0.8 0.9 1.0 | 1.1 1.2 | | therearen | | differentvaluesofL. | | | Then,foreachL, | | | | | | | | | | D,B,A | | | | | | | Normalized error | | | ifalgorithmibeatsalgorithmj,weaccumulateapenalty of 1/n to P ; otherwise, if algorithm j beats al- BADGE Coreset Marg Rand | | D,B,A | i,j | | | | | | | | | | | ---------------------------------- | ----- | --- | --- | --- | --- | --------- | ----- | ---- | ---- | ------- | --- | | | | | | | | | | ALBL | Conf | Entropy | | | gorithmi,weaccumulateapenaltyof1/n | | | | | | D,B,A toP | j,i . | | | | | Thechoiceofthepenaltyvalue1/n D,B,A istoensurethat Figure5: Thecumulativedistributionfunctionof every(D,B,A)combinationisassignedequalinfluence normalizederrorsforallacquisitionfunctions. | intheaggregatedmatrix. | | | Therefore,thelargestentryofP | | | | | | | | | | ---------------------- | --- | --- | ---------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | isatmost33,thetotalnumberof(D,B,A)combinations. Intuitively,eachrowiindicatesthenumberofsettingsinwhichalgorithmibeatsotheralgorithmsandeach columnj indicatesthenumberofsettingsinwhichalgorithmj isbeatenbyanotheralgorithm. ThepenaltymatrixinFigure4summarizesallexperiments,showingthatBADGEgenerallyoutperforms baselines. MatricesgroupedbybatchsizeandarchitectureinAppendixDshowasimilartrend. Cumulativedistributionfunctionsofnormalizederrors. Foreach(D,B,A,L)combination,wecom- putetheaverageerrorforeachalgorithmiase¯ = 1(cid:80)5 el. Toensurethattheerrorsofthesealgorithms | | | | | | | i | 5 l=1 | i | | | | | --- | --- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- | are on the same scale in all settings, we compute the normalized error of every algorithm i, defined as ne i =e¯ i /e¯ r ,whereristheindexoftheRANDalgorithm. Bydefinition,thenormalizederrorsoftheRAND algorithmareidentically1inallsettings.Likewithpenaltymatrices,foreach(D,B,A)combination,weonly | | | | | | (cid:8) | | | | (cid:9) | | | | --- | --- | --- | --- | --- | ------- | --- | --- | --- | ------- | --- | --- | considerasubsetofLvaluesfromtheset M +2m−1B :m∈[(cid:98)log((n −M)/B)(cid:99)] . Weassignaweight 0 proportionalto1/n D,B,A toeach(D,B,A,L)combination,wheretherearen D,B,A differentLvaluesfor thiscombinationof(D,B,A). Wethenplotthecumulativedistributionfunctions(CDFs)ofthenormalized errorsofallalgorithms: foravalueofx,theyvalueisthetotalweightofsettingswherethealgorithmhas normalizederroratmostx;ingeneral,analgorithmthathasahigherCDFvaluehasbetterperformance. WeplotthegeneratedCDFsinFigures5,22and23. WecanseefromFigure5thatBADGEhasthebest overallperformance. Inaddition,fromFigures22and23inAppendixE,wecanconcludethatwhenbatch sizeissmall(100or1000)orwhenanMLPisused,bothBADGEandMARGperformbest. However,inthe regimewhenthebatchsizeislarge(10000), MARG’sperformancedegrades,whileBADGE,ALBLand CORESETarethebestperformingapproaches. 8 PublishedasaconferencepaperatICLR2020 5 RELATED WORK Activelearningisabeenwell-studiedproblem(Settles,2010;Dasgupta,2011;Hanneke,2014). Thereare twomajorstrategiesforactivelearning—representativesamplinganduncertaintysampling. Representative sampling algorithms select batches of unlabeled examples that are representative of the unlabeledsettoaskforlabels. Itisbasedontheintuitionthatthesetsofrepresentativeexampleschosen, once labeled, can act as a surrogate for the full dataset. Consequently, performing loss minimization on thesurrogatesufficestoensurealowerrorwithrespecttothefulldataset. Inthecontextofdeeplearning, SenerandSavarese(2018);GeifmanandEl-Yaniv(2017)selectrepresentativeexamplesbasedoncore-set construction,afundamentalproblemincomputationalgeometry. Inspiredbygenerativeadversariallearning, Gissin and Shalev-Shwartz (2019) select samples that are maximally indistinguishable from the pool of unlabeledexamples. On the other hand, uncertainty sampling is based on a different principle—to select new samples that maximallyreducetheuncertaintythealgorithmhasonthetargetclassifier. Inthecontextoflinearclassifi- cation,TongandKoller(2001);SchohnandCohn(2000);Turetal.(2005)proposeuncertaintysampling methodsthatqueryexamplesthatlieclosesttothecurrentdecisionboundary. Someuncertaintysampling approacheshavetheoreticalguaranteesonstatisticalconsistency(Hanneke,2014;Balcanetal.,2006). Such methodshavealsobeenrecentlygeneralizedtodeeplearning. Forinstance, Galetal.(2017)useDropout asanapproximationoftheposteriorofthemodelparameters,anddevelopinformation-baseduncertainty reductioncriteria;inspiredbyrecentadvancesonadversarialexamplesgeneration, DucoffeandPrecioso (2018) use the distance between an example and one of its adversarial examples as an approximation of itsdistancetothecurrentdecisionboundary,andusesitasthecriterionoflabelqueries. Anensembleof classifierscouldalsobeusedtoeffectivelyestimateuncertainty(Beluchetal.,2018). Thereareseveralexistingapproachesthatsupportahybridofrepresentativesamplinganduncertaintysam- pling. Forexample, Barametal.(2004);HsuandLin(2015)presentmeta-activelearningalgorithmsthat cancombinetheadvantagesofdifferentactivelearningalgorithms. Inspiredbyexpectedlossminimization, Huangetal.(2010)developlabelquerycriteriathatbalancesbetweentherepresentativenessandinforma- tivenessofexamples. AnothermethodforthisisActiveLearningbyLearning(HsuandLin,2015),which canselectwhethertoexerciseadiversitybasedalgorithmoranuncertaintybasedalgorithmateachroundof trainingasasequentialdecisionprocess. Thereisalsoalargebodyofliteratureonbatchmodeactivelearning,wherethelearnerisaskedtoselecta batchofsampleswithineachround(GuoandSchuurmans,2008;WangandYe,2015;ChenandKrause; Weietal.,2015;Kirschetal.,2019). Intheseworks,batchselectionisoftenformulatedasanoptimization problemwithobjectivesbasedon(upperboundsof)averagelog-likelihood,averagesquaredloss,etc. Adifferentquerycriterionbasedonexpectedgradientlength(EGL)hasbeenproposedintheaswell(Settles etal.,2008). Inrecentwork,Huangetal.(2016)showthattheEGLcriterionisrelatedtotheT-optimality criterioninexperimentaldesign.TheyfurtherdemonstratethatthesamplesselectedbyEGLareverydifferent from those by entropy-based uncertainty criterion. Zhang et al. (2017a) use the EGL criterion in active sentenceanddocumentclassificationwithCNNs. TheseapproachesdiffermostsubstantiallyfromBADGE inthattheydonottakeintoaccountthediversityoftheexamplesqueriedwithineachbatch. Thereisawidearrayoftheoreticalarticlesthatfocusontherelatedproblemofadaptivesubsamplingfor fully-labeleddatasetsinregressionsettings(Hanetal.,2016;Wangetal.,2018;TingandBrochu,2018). Empiricalstudiesofbatchstochasticgradientdescentalsoemployadaptivesamplingto“emphasize”hardor representativeexamples(Zhangetal.,2017b;Changetal.,2017). Theseworksaimatreducingcomputation costsorfindingabetterlocaloptimalsolution,asopposedtoreducinglabelcosts. Nevertheless,ourworkis inspiredbytheirsamplingcriteria,whichalsoemphasizesamplesthatinducelargeupdatestothemodel. 9 PublishedasaconferencepaperatICLR2020 As mentioned earlier, our sampling criterion has resemblance to sampling from k-determinantal point processes (Kulesza and Taskar, 2011). Note that in multiclass classification settings, our gradient-based embeddingofanexamplecanbeviewedastheouterproductoftheoriginalembeddinginthepenultimate layerandaprobabilityscorevectorthatencodestheuncertaintyinformationonthisexample(seeSection3). In this view, the penultimate layer embedding characterizes the diversity of each example, whereas the probabilityscorevectorcharacterizesthequalityofeachexample. Thek-DPPisalsoanaturalprobabilistic toolforsamplingthattradesoffbetweenqualityanddiversity(SeeKuleszaetal.,2012,Section3.1). We remarkthatconcurrenttoourwork, Bıyıketal.(2019)developsk-DPPbasedactivelearningalgorithms basedonthisprinciplebyexplicitlydesigningdiversityanduncertaintymeasures. 6 DISCUSSION WehaveestablishedthatBADGEisempiricallyaneffectivedeepactivelearningalgorithmacrossdifferent architecturesandbatchsizes,performingsimilartoorbetterthanotheractivelearningalgorithms. Afunda- mentalremainingquestionis: "Why?"Whiledeeplearningisnotoriouslydifficulttoanalyzetheoretically, thereareseveralintuitivelyappealingpropertiesofBADGE: 1. Thedefinitionofuncertainty(alowerboundonthegradientmagnitudeofthelastlayer)guarantees someupdateofparameters. 2. It optimizes for diversity as well as uncertainty, eliminating a failure mode of choosing many identicaluncertainexamplesinabatch,anddoessowithoutrequiringanyhyperparameters. 3. Therandomizationassociatedwiththek-MEANS++ initializationsamplerimpliesthat, evenfor adversariallyconstructeddatasets,iteventuallyconvergestoagoodsolution. Thecombinationofthesepropertiesappearstogeneratetherobustnessthatweobserveempirically. REFERENCES DavidCohn,LesAtlas,andRichardLadner.Improvinggeneralizationwithactivelearning.Machinelearning, 1994. Maria-FlorinaBalcan,AlinaBeygelzimer,andJohnLangford. Agnosticactivelearning. InInternational ConferenceonMachineLearning,2006. Alina Beygelzimer, Daniel J Hsu, John Langford, and Tong Zhang. 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Cheng Zhang, Hedvig Kjellstrom, and Stephan Mandt. Determinantal point processes for mini-batch diversification. UncertaintyinArtificialIntelligence,2017b. 12 PublishedasaconferencepaperatICLR2020 Haw-ShiuanChang,ErikLearned-Miller,andAndrewMcCallum. Activebias: Trainingmoreaccurateneural networksbyemphasizinghighvariancesamples. InNeuralInformationProcessingSystems,2017. AlexKulesza,BenTaskar,etal. Determinantalpointprocessesformachinelearning. FoundationsandTrends inMachineLearning,2012. ErdemBıyık,KennethWang,NimaAnari,andDorsaSadigh. Batchactivelearningusingdeterminantalpoint processes. arXivpreprint,2019. StephenMussmannandPercySLiang. Uncertaintysamplingispreconditionedstochasticgradientdescent onzero-oneloss. InNeuralInformationProcessingSystems,2018. A THE k-MEANS++ SEEDING ALGORITHM Herewebrieflyreviewthek-MEANS++ seedingalgorithmby(ArthurandVassilvitskii,2007). Itsbasic idea is to perform sequential sampling of k centers, where each new center is sampled from the ground set with probability proportional to the squared distance to its nearest center. It is shown in (Arthur and Vassilvitskii, 2007) that the set of centers returned is guaranteed to approximate the k-means objective functioninexpectation,thusensuringdiversity. Algorithm2Thek-MEANS++seedingalgorithm(ArthurandVassilvitskii,2007) Require: GroundsetG⊂Rd,targetsizek. Ensure: CentersetC ofsizek. C ←{c },wherec issampleduniformlyatrandomfromG. 1 1 1 fort=2,...,k:do DefineD (x):=min (cid:107)x−c(cid:107) . t c∈Ct−1 2 c ←SamplexfromGwithprobability Dt(x)2 . t (cid:80) x∈G Dt(x)2 C ←C ∪{c }. t t−1 t endfor return C . k B BADGE FOR BINARY LOGISTIC REGRESSION We consider instantiating BADGE for binary logistic regression, where Y = {−1,+1}. Given a linear classifierw,wedefinethepredictiveprobabilityofwonxasp (y|x,θ)=σ(yw·x),whereσ(z)= 1 w 1+e−z isthesigmoidfunciton. Recallthatyˆ=yˆ(x)isthehallucinatedlabel: (cid:26) +1, p (+1|x,θ)>1/2, yˆ(x)= w −1, p (+1|x,θ)≤1/2. w Thebinarylogisticlossofclassifierwonexample(x,y)isdefinedas: (cid:96)(w,(x,y))=ln(1+exp(−yw·x)). Now,givenmodelwandexamplex,wedefinegˆ = ∂ (cid:96)(w,(x,yˆ))=(1−p (yˆ|x,θ))·(−yˆ·x)astheloss x ∂w w gradientinducedbytheexamplewithhallucinatedlabel,andg˜ = ∂ (cid:96)(w,(x,y))=(1−p (y|x,θ))·(−y·x) x ∂w w asthelossgradientinducedbytheexamplewithtruelabel. 13 PublishedasaconferencepaperatICLR2020 SVHN, ResNet, Batch size: 100 0.90 OpenML#6, MLP, Batch size: 100 OpenML#6, MLP, Batch size: 1000 OpenML#6, MLP, Batch size: 10000 | 0.80 | | | 0.90 | | | 0.90 | | | | ---- | --- | --- | ---- | --- | --- | ---- | --- | --- | 0.70 0.90 ycaruccA | ycaruccA 0.60 0.80 | | | ycaruccA 0.80 | | | ycaruccA 0.80 | | | | ------------------ | --- | --- | ------------- | --- | --- | ------------- | --- | --- | | 0.50 0.70 | | | 0.70 | | | 0.70 | | | 0.40 | 0.60 | | | 0.60 | | | 0.60 | | | | ---- | --- | --- | ---- | --- | --- | ---- | --- | --- | 0.30 | 0.50 | | | | | | 0.50 | | | | --------- | ---------------- | ----------------- | ---------- | --- | --- | ---- | --- | --- | | 0.20 | | | 0.50 | | | | | | | 0.10 0.40 | 5000 10000 15000 | 20000 25000 30000 | 35000 0.40 | | | 0.40 | | | 200040006000800010000120001400016000 2000 4000 6000 800010000120001400016000 2000 4000 6000 8000 10000 | | #Labels queried #Labels queried | | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | ---- | ------- | --- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | | BADGE | Entropy | Marg | Rand | Figure6: FulllearningcurvesforOpenML#6withMLP. SVHN, ResNet, Batch size: 100 0.90 OpenML#155, MLP, Batch size: 100 OpenML#155, MLP, Batch size: 1000 OpenML#155, MLP, Batch size: 10000 | 1.00 | | | 1.00 | | | 1.00 | | | | ---- | --- | --- | ---- | --- | --- | ---- | --- | --- | 0.80 | 0.95 | | | 0.95 | | | 0.95 | | | | ---- | --- | --- | ---- | --- | --- | ---- | --- | --- | 0.70 | ycaruccA ycaruccA 0.90 | | | ycaruccA 0.90 | | | ycaruccA 0.90 | | | | ---------------------- | --- | --- | ------------- | --- | --- | ------------- | --- | --- | | 0.60 0.85 | | | 0.85 | | | 0.85 | | | 0.50 | 0.80 | | | 0.80 | | | 0.80 | | | | --------- | --- | --- | ---- | --- | --- | ---- | --- | --- | | 0.40 0.75 | | | 0.75 | | | 0.75 | | | | 0.30 0.70 | | | 0.70 | | | 0.70 | | | | 0.20 | | | 0.65 | | | | | | | 0.65 | | | | | | 0.65 | | | | 0.10 0.60 | | | 0.60 | | | 0.60 | | | 5000 10000 10000 20000 15000 20000 30000 25000 40000 30000 50000 35000 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 | | #Labels queried #Labels queried | | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | ---- | ------- | --- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | | BADGE | Entropy | Marg | Rand | Figure7: FulllearningcurvesforOpenML#155withMLP. Suppose that BADGE only selects examples from region S w = {x:w·x=0}, then as p w (+1|x,θ) = 1,wehavethatforallxinS p (−1|x,θ)= ,gˆ =s ·g forsomes ∈{±1}. Thisimpliesthat,sampling | w | 2 | | | w x | x x | x | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | fromaDPPinducedbygˆ ’sisequivalenttosamplingfromaDPPinducedbyg ’s. ItisnotedinMussmann | | | x | | | | | x | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | andLiang(2018)thatuncertaintysampling(i.e. samplingfromD )implicitlyperformspreconditioned |Sw stochasticgradientdescentontheexpected0-1loss. Inaddition,ithasbeenshownthatDPPsamplingover gradientsmayreducethevarianceofthemini-batchstochasticgradientupdates(Zhangetal.,2017b);this suggeststhatBADGE,whenrestricteditssamplingoverlow-marginregions(S w ),improvesoveruncertainty samplingbycollectingexamplesthattogetherinducelower-varianceupdatesonthegradientdirectionof expected0-1loss. | C ALL | LEARNING | CURVES | | | | | | | | ----- | -------- | ------ | --- | --- | --- | --- | --- | --- | Weplotalllearningcurves(testaccuracyasafunctionofthenumberoflabeledexamplequeried)inFigures6 to12. Inaddition, wezoomintoregionsofthelearningcurvesthatdiscriminatestheperformanceofall algorithmsinFigures13to19. D PAIRWISE | | COMPARISONS | | OF | ALGORITHMS | | | | | | --- | ----------- | --- | --- | ---------- | --- | --- | --- | --- | InadditiontoFigure4inthemaintext,wealsoprovidepenaltymatrices(Figures20and21),wherethe results are aggregated by conditioning on a fixed batch size (100, 1000 and 10000) or on a fixed neural networkmodel(MLP,ResNetandVGG).Foreachpenaltymatrix,theparenthesizednumberinitstitleisthe 14 PublishedasaconferencepaperatICLR2020 SVHN, ResNet, Batch size: 100 0.90 OpenML#156, MLP, Batch size: 100 OpenML#156, MLP, Batch size: 1000 OpenML#156, MLP, Batch size: 10000 | 0.80 0.90 | | | 0.90 | | 0.90 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | 0.70 | ycaruccA 0.85 | | | 0.85 | | 0.85 | | | | ------------- | --- | --- | -------- | --- | -------- | --- | --- | | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | 0.80 | | | 0.80 | | | | | | 0.50 | | | | | 0.80 | | | 0.40 | 0.75 | | | 0.75 | | 0.75 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.30 | 0.70 | | | 0.70 | | | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | | 0.20 | | | | | 0.70 | | | | 0.65 | | | 0.65 | | | | | 0.10 5000 10000 10000 20000 15000 20000 30000 25000 40000 30000 50000 35000 10000 20000 30000 40000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | ---- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | Figure8: FulllearningcurvesforOpenML#156withMLP. SVHN, ResNet, Batch size: 100 0.90 OpenML#184, MLP, Batch size: 100 OpenML#184, MLP, Batch size: 1000 OpenML#184, MLP, Batch size: 10000 | 0.80 0.80 | | | 0.80 | | 0.80 | | | | ------------------ | --- | --- | -------- | --- | -------- | --- | --- | | ycaruccA 0.70 0.70 | | | 0.70 | | 0.70 | | | | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | 0.60 | | | 0.60 | | 0.60 | | | 0.50 | 0.50 | | | 0.50 | | 0.50 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.40 | 0.30 0.40 | | | 0.40 | | 0.40 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.20 0.30 | | | 0.30 | | 0.30 | | | 0.10 0.20 5000 5000 10000 10000 15000 20000 15000 25000 20000 30000 35000 2500500075001000012500150001750020000 2500 5000 75001000012500150001750020000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | ---- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | Figure9: FulllearningcurvesforOpenML#184withMLP. 15 PublishedasaconferencepaperatICLR2020 SVHN, ResNet, Batch size: 100 SVHN, MLP, Batch size: 100 SVHN, MLP, Batch size: 1000 SVHN, MLP, Batch size: 10000 0.90 | 0.80 0.80 | | | 0.80 | | | 0.80 | | | | ------------------ | --- | --- | ------------- | --- | --- | -------- | --- | --- | | 0.70 0.70 | | | 0.70 | | | 0.70 | | | | ycaruccA | | | | | | ycaruccA | | | | ycaruccA 0.60 0.60 | | | ycaruccA 0.60 | | | 0.60 | | | | 0.50 0.50 | | | 0.50 | | | 0.50 | | | 0.40 | 0.40 | | | 0.40 | | | 0.40 | | | | --------- | --- | --- | ---- | --- | --- | ---- | --- | --- | | 0.30 | | | | | | 0.30 | | | | 0.30 | | | 0.30 | | | | | | | 0.20 | | | | | | 0.20 | | | | 0.10 0.20 | | | 0.20 | | | | | | 5000 10000 15000 20000 25000 30000 35000 10000 20000 30000 40000 50000 | | 5000 10000 | 15000 20000 25000 | 30000 10000 | 20000 30000 | 40000 | 50000 | | | | --- | ------------------------------- | ----------------- | ----------- | --------------- | ----- | ----- | --------------- | --- | | | #Labels queried #Labels queried | | | #Labels queried | | | #Labels queried | | SVHN, ResNet, Batch size: 100 0.90 SVHN, ResNet, Batch size: 100 SVHN, ResNet, Batch size: 1000 SVHN, ResNet, Batch size: 10000 | 0.90 | | | 0.90 | | | 0.90 | | | | ---- | --- | --- | ---- | --- | --- | ---- | --- | --- | 0.80 | 0.70 0.80 | | | 0.80 | | | 0.80 | | | | ------------- | --- | --- | -------- | --- | --- | -------- | --- | --- | | ycaruccA 0.70 | | | 0.70 | | | 0.70 | | | | ycaruccA 0.60 | | | ycaruccA | | | ycaruccA | | | | 0.60 | | | 0.60 | | | 0.60 | | | 0.50 | 0.50 | | | 0.50 | | | 0.50 | | | | --------- | --- | --- | ---- | --- | --- | ---- | --- | --- | | 0.40 0.40 | | | 0.40 | | | 0.40 | | | 0.30 | 0.30 | | | 0.30 | | | 0.30 | | | | ---- | --- | --- | ---- | --- | --- | ---- | --- | --- | 0.20 0.20 | | | | 0.20 | | | 0.20 | | | | --------- | --- | --- | ---- | --- | --- | ---- | --- | --- | | 0.10 0.10 | | | 0.10 | | | 0.10 | | | 5000 10000 10000 20000 15000 20000 30000 25000 40000 30000 50000 35000 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 | | #Labels queried #Labels queried | | | #Labels queried | | | #Labels queried | | | --- | ------------------------------- | --- | --- | --------------- | --- | --- | --------------- | --- | SVHN, ResNet, Batch size: 100 0.90 SVHN, VGG, Batch size: 100 SVHN, VGG, Batch size: 1000 SVHN, VGG, Batch size: 10000 | 0.80 0.90 | | | 0.90 | | | 0.90 | | | | ------------------ | --- | --- | ------------- | --- | --- | ------------- | --- | --- | | ycaruccA 0.70 0.80 | | | 0.80 | | | 0.80 | | | | ycaruccA 0.60 0.70 | | | ycaruccA 0.70 | | | ycaruccA 0.70 | | | | 0.50 0.60 | | | 0.60 | | | 0.60 | | | | 0.50 | | | | | | 0.50 | | | | 0.40 | | | 0.50 | | | | | | | 0.40 | | | 0.40 | | | 0.40 | | | 0.30 | 0.20 0.30 | | | 0.30 | | | 0.30 | | | | --------- | --- | --- | ---- | --- | --- | ---- | --- | --- | | 0.20 | | | | | | 0.20 | | | | 0.10 | | | 0.20 | | | | | | 5000 5000 10000 10000 15000 15000 20000 20000 25000 25000 30000 35000 10000 20000 30000 40000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | | #Labels queried | | | --- | ------------------------------- | ---------------------------------------------- | ------- | --------------- | --- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | | Entropy | Marg | Rand | | | Figure10: | FulllearningcurvesforSVHNwithMLP,ResNetandVGG. | | | | | | | SVHN, ResNet, Batch size: 100 0.90 MNIST, MLP, Batch size: 100 MNIST, MLP, Batch size: 1000 MNIST, MLP, Batch size: 10000 0.80 | 0.95 | | | 0.95 | | | 0.95 | | | | ---- | --- | --- | ---- | --- | --- | ---- | --- | --- | ycaruccA 0.70 | ycaruccA 0.60 0.90 | | | ycaruccA 0.90 | | | ycaruccA 0.90 | | | | ------------------ | --- | --- | ------------- | --- | --- | ------------- | --- | --- | 0.85 | 0.50 0.85 | | | | | | 0.85 | | | | --------- | --- | --- | --- | --- | --- | ---- | --- | --- | 0.80 | 0.40 0.80 | | | | | | 0.80 | | | | --------- | --- | --- | --- | --- | --- | ---- | --- | --- | 0.75 | 0.30 0.75 | | | | | | 0.75 | | | | --------- | --- | --- | ---- | --- | --- | ---- | --- | --- | | 0.20 | | | 0.70 | | | | | | | 0.70 | | | | | | 0.70 | | | | 0.10 | | | 0.65 | | | | | | 5000 10000 10000 20000 15000 20000 30000 25000 40000 30000 50000 35000 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 | | #Labels queried #Labels queried | | | #Labels queried | | | #Labels queried | | | --- | ------------------------------- | --------- | ---------------------------------- | --------------- | --- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | | Entropy | Marg | Rand | | | | Figure11: | FulllearningcurvesforMNISTwithMLP. | | | | | | 16 PublishedasaconferencepaperatICLR2020 SVHN, ResNet, Batch size: 100 CIFAR10, MLP, Batch size: 100 CIFAR10, MLP, Batch size: 1000 CIFAR10, MLP, Batch size: 10000 0.90 | 0.80 0.50 | | | 0.50 | | 0.50 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | 0.70 | ycaruccA 0.45 | | | 0.45 | | 0.45 | | | | ------------- | --- | --- | -------- | --- | -------- | --- | --- | | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | 0.40 | | | 0.40 | | 0.40 | | | 0.50 | 0.40 0.35 | | | 0.35 | | 0.35 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.30 0.30 | | | 0.30 | | 0.30 | | | | 0.20 0.25 | | | 0.25 | | 0.25 | | | 0.10 | | 5000 10000 15000 | 20000 25000 30000 | 35000 | | | | | | --- | ---------------- | ----------------- | ----- | --- | --- | --- | --- | 10000 20000 30000 40000 50000 10000 20000 30000 40000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | --- | --- | --------------- | --- | --------------- | --- | SVHN, ResNet, Batch size: 100 0.90 CIFAR10, ResNet, Batch size: 100 CIFAR10, ResNet, Batch size: 1000 CIFAR10, ResNet, Batch size: 10000 | | | | 0.60 | | 0.60 | | | | --- | --- | --- | ---- | --- | ---- | --- | --- | 0.80 | 0.70 0.50 | | | 0.50 | | 0.50 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | ycaruccA | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | ------------- | --- | --- | -------- | --- | -------- | --- | --- | | 0.40 | | | 0.40 | | 0.40 | | | 0.50 | 0.30 | | | 0.30 | | 0.30 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.40 0.30 | 0.20 | | | 0.20 | | 0.20 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.20 | 0.10 | | | 0.10 | | 0.10 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.10 5000 10000 10000 20000 15000 20000 30000 25000 40000 30000 35000 10000 20000 30000 40000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | --- | --- | --------------- | --- | --------------- | --- | SVHN, ResNet, Batch size: 100 0.90 CIFAR10, VGG, Batch size: 100 CIFAR10, VGG, Batch size: 1000 CIFAR10, VGG, Batch size: 10000 | 0.80 0.80 | | | 0.80 | | 0.80 | | | | ------------------ | --- | --- | ------------- | --- | -------- | --- | --- | | ycaruccA 0.70 0.70 | | | 0.70 | | 0.70 | | | | ycaruccA 0.60 0.60 | | | ycaruccA 0.60 | | ycaruccA | | | 0.60 | 0.50 0.50 | | | 0.50 | | 0.50 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.40 0.40 | | | 0.40 | | 0.40 | | | | 0.30 | | | 0.30 | | 0.30 | | | 0.30 | 0.20 0.20 | | | 0.20 | | 0.20 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.10 0.10 | | | 0.10 | | 0.10 | | | 5000 5000 10000 10000 15000 15000 20000 20000 25000 25000 30000 30000 35000 10000 20000 30000 40000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | ------------------------------------------------- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | | | Figure12: | FulllearningcurvesforCIFAR10withMLP,ResNetandVGG. | | | | | | SVHN, ResNet, Batch size: 100 0.90 OpenML#6, MLP, Batch size: 100 OpenML#6, MLP, Batch size: 1000 OpenML#6, MLP, Batch size: 10000 0.80 | 0.90 | | | 0.90 | | 0.90 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | ycaruccA 0.70 | ycaruccA 0.60 0.80 | | | ycaruccA 0.80 | | ycaruccA 0.80 | | | | ------------------ | --- | --- | ------------- | --- | ------------- | --- | --- | | 0.50 0.70 | | | 0.70 | | 0.70 | | | 0.40 | 0.60 | | | 0.60 | | 0.60 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.30 | 0.20 0.50 | | | 0.50 | | 0.50 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.10 0.40 | | | 0.40 | | 0.40 | | | 5000 2000 10000 4000 15000 6000 20000 8000 25000 10000 30000 35000 2000 4000 6000 8000 10000 12000 2000 4000 6000 8000 10000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | --------- | ------------------------------------------ | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | | | | Figure13: | Zoomed-inlearningcurvesforOpenML#6withMLP. | | | | | 17 PublishedasaconferencepaperatICLR2020 SVHN, ResNet, Batch size: 100 OpenML#155, MLP, Batch size: 100 OpenML#155, MLP, Batch size: 1000 OpenML#155, MLP, Batch size: 10000 | 0.90 | | | | | 1.00 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.80 1.00 | | | 1.00 | | | | | | 0.95 | | | 0.95 | | 0.95 | | | 0.70 | ycaruccA 0.90 | | | 0.90 | | 0.90 | | | | ------------- | --- | --- | ------------- | --- | ------------- | --- | --- | | ycaruccA 0.60 | | | ycaruccA 0.85 | | ycaruccA 0.85 | | | 0.50 0.85 | 0.80 | | | 0.80 | | 0.80 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.40 | | | 0.75 | | | | | | 0.75 | | | | | 0.75 | | | | 0.30 | | | 0.70 | | 0.70 | | | | 0.70 | | | 0.65 | | | | | | 0.20 | | | | | 0.65 | | | | 0.10 0.65 | | | 0.60 | | 0.60 | | | 1000 5000 2000 10000 3000 15000 4000 20000 5000 25000 6000 30000 7000 35000 100020003000400050006000700080009000 10000 20000 30000 40000 50000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | -------------------------------------------- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | | | Figure14: | Zoomed-inlearningcurvesforOpenML#155withMLP. | | | | | | SVHN, ResNet, Batch size: 100 0.90 OpenML#156, MLP, Batch size: 100 OpenML#156, MLP, Batch size: 1000 OpenML#156, MLP, Batch size: 10000 | 0.80 0.90 | | | 0.90 | | 0.90 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | 0.70 | ycaruccA 0.85 | | | 0.85 | | 0.85 | | | | ------------- | --- | --- | -------- | --- | -------- | --- | --- | | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | 0.80 | | | 0.80 | | | | | | 0.50 | | | | | 0.80 | | | | 0.40 0.75 | | | 0.75 | | 0.75 | | | 0.30 | 0.70 | | | 0.70 | | | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | | 0.20 | | | | | 0.70 | | | | 0.65 | | | 0.65 | | | | | 0.10 500 5000 10000 1000 15000 1500 20000 2000 25000 2500 30000 3000 35000 500 1000 1500 2000 2500 3000 3500 4000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | -------------------------------------------- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | | | Figure15: | Zoomed-inlearningcurvesforOpenML#156withMLP. | | | | | | SVHN, ResNet, Batch size: 100 OpenML#184, MLP, Batch size: 100 OpenML#184, MLP, Batch size: 1000 OpenML#184, MLP, Batch size: 10000 0.90 | 0.80 0.80 | | | 0.80 | | 0.80 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.70 0.70 | | | 0.70 | | 0.70 | | | ycaruccA | ycaruccA 0.60 0.60 | | | ycaruccA | | ycaruccA 0.60 | | | | ------------------ | --- | --- | -------- | --- | ------------- | --- | --- | | 0.50 | | | 0.60 | | | | | | 0.50 | | | | | 0.50 | | | | 0.40 | | | 0.50 | | | | | 0.40 | 0.30 | | | 0.40 | | 0.40 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.30 | 0.20 | | | 0.30 | | 0.30 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.10 0.20 5000 5000 10000 10000 15000 20000 15000 25000 20000 30000 35000 5000 10000 15000 20000 2500 5000 75001000012500150001750020000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | -------------------------------------------- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | | | Figure16: | Zoomed-inlearningcurvesforOpenML#184withMLP. | | | | | | 18 PublishedasaconferencepaperatICLR2020 SVHN, ResNet, Batch size: 100 SVHN, MLP, Batch size: 100 SVHN, MLP, Batch size: 1000 SVHN, MLP, Batch size: 10000 0.90 | 0.80 0.80 | | | 0.80 | | 0.80 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.70 0.70 | | | 0.70 | | 0.70 | | | ycaruccA | ycaruccA 0.60 0.60 | | | ycaruccA 0.60 | | ycaruccA 0.60 | | | | ------------------ | --- | --- | ------------- | --- | ------------- | --- | --- | | 0.50 0.50 | | | 0.50 | | 0.50 | | | 0.40 | 0.40 | | | 0.40 | | 0.40 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.30 | 0.30 | | | 0.30 | | 0.30 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.20 | 0.10 0.20 | | | 0.20 | | 0.20 | | | | --------- | ---------------- | ----------------- | ----- | --- | ---- | --- | --- | | | 5000 10000 15000 | 20000 25000 30000 | 35000 | | | | | 2500 5000 75001000012500150001750020000 5000 10000 15000 20000 25000 10000 20000 30000 40000 50000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | --- | --- | --------------- | --- | --------------- | --- | SVHN, ResNet, Batch size: 100 0.90 SVHN, ResNet, Batch size: 100 SVHN, ResNet, Batch size: 1000 SVHN, ResNet, Batch size: 10000 | 0.90 | | | 0.90 | | 0.90 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.80 | 0.70 0.80 | | | 0.80 | | 0.80 | | | | ------------- | --- | --- | -------- | --- | -------- | --- | --- | | ycaruccA 0.70 | | | 0.70 | | 0.70 | | | | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | 0.60 | | | 0.60 | | 0.60 | | | 0.50 | 0.50 | | | 0.50 | | 0.50 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.40 0.40 | | | 0.40 | | 0.40 | | | 0.30 | 0.30 | | | 0.30 | | 0.30 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.20 0.20 | | | | 0.20 | | 0.20 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.10 0.10 | | | 0.10 | | 0.10 | | | 5000 5000 10000 10000 15000 15000 20000 20000 25000 25000 30000 30000 35000 35000 500010000150002000025000300003500040000 10000 20000 30000 40000 50000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | --- | --- | --------------- | --- | --------------- | --- | SVHN, ResNet, Batch size: 100 0.90 SVHN, VGG, Batch size: 100 SVHN, VGG, Batch size: 1000 SVHN, VGG, Batch size: 10000 | 0.80 0.90 | | | 0.90 | | 0.90 | | | | ------------------ | --- | --- | ------------- | --- | ------------- | --- | --- | | ycaruccA 0.70 0.80 | | | 0.80 | | 0.80 | | | | ycaruccA 0.60 0.70 | | | ycaruccA 0.70 | | ycaruccA 0.70 | | | | 0.50 0.60 | | | 0.60 | | 0.60 | | | | 0.50 | | | | | 0.50 | | | | 0.40 | | | 0.50 | | | | | | 0.40 | | | 0.40 | | 0.40 | | | 0.30 | 0.20 0.30 | | | 0.30 | | 0.30 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.20 | | | | | 0.20 | | | | 0.10 | | | 0.20 | | | | | 5000 5000 10000 10000 15000 15000 20000 20000 25000 25000 30000 30000 35000 5000 10000 15000 20000 25000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | --------------------------------------------------- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | | | Figure17: | Zoomed-inlearningcurvesforSVHNwithMLP,ResNetandVGG. | | | | | | SVHN, ResNet, Batch size: 100 0.90 MNIST, MLP, Batch size: 100 MNIST, MLP, Batch size: 1000 MNIST, MLP, Batch size: 10000 0.80 | 0.95 | | | 0.95 | | 0.95 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | ycaruccA 0.70 | ycaruccA 0.60 0.90 | | | ycaruccA 0.90 | | ycaruccA 0.90 | | | | ------------------ | --- | --- | ------------- | --- | ------------- | --- | --- | 0.85 | 0.50 0.85 | | | | | 0.85 | | | | --------- | --- | --- | --- | --- | ---- | --- | --- | 0.80 | 0.40 0.80 | | | | | 0.80 | | | | --------- | --- | --- | --- | --- | ---- | --- | --- | 0.75 | 0.30 0.75 | | | | | 0.75 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.20 | | | 0.70 | | | | | | 0.70 | | | | | 0.70 | | | | 0.10 | | | 0.65 | | | | | 250050007500100001250015000175002000022500 5000 10000 15000 20000 25000 30000 35000 2500 5000 75001000012500150001750020000 10000 20000 30000 40000 50000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | ---- | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | Figure18: Zoomed-inlearningcurvesforMNISTwithMLP. 19 PublishedasaconferencepaperatICLR2020 SVHN, ResNet, Batch size: 100 CIFAR10, MLP, Batch size: 100 CIFAR10, MLP, Batch size: 1000 CIFAR10, MLP, Batch size: 10000 0.90 | 0.80 0.50 | | | 0.50 | | 0.50 | | | | ------------- | --- | --- | -------- | --- | -------- | --- | --- | | 0.70 | | | 0.45 | | | | | | ycaruccA 0.45 | | | | | 0.45 | | | | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | 0.40 | | | 0.40 | | 0.40 | | | 0.50 | 0.40 0.35 | | | 0.35 | | 0.35 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.30 0.30 | | | 0.30 | | 0.30 | | | | 0.20 0.25 | | | 0.25 | | 0.25 | | | 0.10 | | 5000 10000 15000 | 20000 25000 30000 | 35000 | | | | | | --- | ---------------- | ----------------- | ----- | --- | --- | --- | --- | 10000 #Labels queried 20000 30000 40000 5000 100001500020000250003000035000 500010000150002000025000300003500040000 | | #Labels queried | | | #Labels queried | | #Labels queried | | | --- | --------------- | --- | --- | --------------- | --- | --------------- | --- | SVHN, ResNet, Batch size: 100 0.90 CIFAR10, ResNet, Batch size: 100 CIFAR10, ResNet, Batch size: 1000 CIFAR10, ResNet, Batch size: 10000 | 0.55 | | | 0.60 | | 0.60 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.80 | 0.70 0.50 | | | 0.50 | | 0.50 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | ycaruccA 0.45 | ycaruccA 0.60 | | | ycaruccA | | ycaruccA | | | | ------------- | --- | --- | -------- | --- | -------- | --- | --- | | 0.40 | | | 0.40 | | 0.40 | | | 0.50 | 0.35 | | | 0.30 | | 0.30 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.40 0.30 0.30 | 0.25 | | | 0.20 | | 0.20 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.20 | 0.20 | | | 0.10 | | 0.10 | | | | ---- | --- | --- | ---- | --- | ---- | --- | --- | 0.10 5000 10000 10000 20000 15000 20000 30000 25000 40000 30000 35000 10000 20000 30000 40000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | --- | --- | --------------- | --- | --------------- | --- | SVHN, ResNet, Batch size: 100 0.90 CIFAR10, VGG, Batch size: 100 CIFAR10, VGG, Batch size: 1000 CIFAR10, VGG, Batch size: 10000 | 0.80 0.80 | | | 0.80 | | 0.80 | | | | ------------------ | --- | --- | ------------- | --- | -------- | --- | --- | | ycaruccA 0.70 0.70 | | | 0.70 | | 0.70 | | | | ycaruccA 0.60 0.60 | | | ycaruccA 0.60 | | ycaruccA | | | 0.60 | 0.50 0.50 | | | 0.50 | | 0.50 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.40 0.40 | | | 0.40 | | 0.40 | | | | 0.30 | | | 0.30 | | 0.30 | | | 0.30 | 0.20 0.20 | | | 0.20 | | 0.20 | | | | --------- | --- | --- | ---- | --- | ---- | --- | --- | | 0.10 0.10 | | | 0.10 | | 0.10 | | | 5000 10000 10000 20000 15000 20000 30000 25000 40000 30000 35000 500010000150002000025000300003500040000 500010000150002000025000300003500040000 | | #Labels queried #Labels queried | | | #Labels queried | | #Labels queried | | | --- | ------------------------------- | ------------------------------------------------------ | ------- | --------------- | ------- | --------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | | | Figure19: | Zoomed-inlearningcurvesforCIFAR10withMLP,ResNetandVGG. | | | | | | 20 PublishedasaconferencepaperatICLR2020 Batch size: 100(11) Batch size: 1000(11) Batch size: 10000(11) BADGEALBLCoresetConf MargEntropyRand BADGEALBLCoresetConf MargEntropyRand BADGEALBLCoresetConf MargEntropyRand SVHN, ResNet, Batch size: 100 BADGE 0.0 4.44 5.35 5.36 0.76 5.81 4.09 BADGE 0.0 3.58 3.96 4.37 1.96 4.85 4.37 BADGE 0.0 1.17 1.67 2.83 1.17 2.5 1.58 2.5 | 0.90 | | 5 | | | | | | ---- | --- | --- | --- | --- | --- | --- | ALBL 0.2 0.0 2.71 1.8 0.31 3.19 1.49 ALBL 0.14 0.0 2.14 1.02 0.0 2.12 2.2 4 ALBL 0.0 0.0 0.33 0.33 0.0 1.5 1.25 0.80 2.0 Coreset 0.2 0.34 0.0 2.71 0.87 3.28 1.76 4 Coreset 0.62 1.19 0.0 2.0 1.04 3.02 2.8 Coreset 0.83 0.5 0.0 1.83 1.17 1.83 2.0 | 0.70 | | | | 3 | | | | ---- | --- | --- | --- | --- | --- | --- | ycaruccA 0.21 0.67 3.02 0.0 0.0 2.08 2.18 0.0 1.12 3.26 0.0 0.0 2.71 2.65 0.33 1.0 1.33 0.0 0.33 1.17 1.25 | 0.60 Conf | | Conf | | Conf | | 1.5 | | --------- | --- | ---- | --- | ---- | --- | --- | 3 Marg 0.46 3.9 4.77 4.82 0.0 5.25 3.77 Marg 0.17 2.24 4.1 3.51 0.0 4.58 3.99 Marg 0.33 1.0 2.0 1.0 0.0 1.5 1.58 | 0.50 | | | | 2 | | | | ---- | --- | --- | --- | --- | --- | --- | | | | 2 | | | | 1.0 | 0.40 Entropy 0.31 0.59 2.6 0.33 0.1 0.0 1.58 Entropy 0.0 0.62 2.45 0.33 0.0 0.0 1.95 Entropy 0.0 0.58 1.0 0.58 0.25 0.0 1.58 1 0.30 Rand 0.2 2.26 3.53 3.45 0.3 4.53 0.0 1 Rand 0.14 2.2 2.5 3.05 1.84 3.95 0.0 Rand 0.5 1.17 1.58 2.17 0.83 2.17 0.0 0.5 0.20 0.23 1.74 3.14 2.64 0.33 3.45 2.12 0.15 1.56 2.63 2.04 0.69 3.03 2.57 0.29 0.77 1.13 1.25 0.54 1.52 1.32 | | | 0 | | 0 | | 0.0 | | --- | --- | --- | --- | --- | --- | --- | 0.10 | 5000 | 10000 15000 | 20000 25000 30000 | 35000 | | | | | ---- | ----------- | ----------------- | ----- | --- | --- | --- | #Labels queried Figure20: Pairwisepenaltymatricesofthealgorithms,groupedbydifferentbatchsizes. Theparenthesized numberinthetitleisthetotalnumberof(D,B,A)combinationsaggregated,whichisalsoanupperbound onallitsentries. Element(i,j)correspondsroughlytothenumberoftimesalgorithmibeatsalgorithmj. Column-wiseaveragesatthebottomshowaggregateperformance(lowerisbetter). Fromlefttoright: batch size=100,1000,10000. | | MLP(21) | | ResNet(6) | | VGG(6) | | | --- | ------- | --- | --------- | --- | ------ | --- | BADGEALBLCoresetConf MargEntropyRand BADGEALBLCoresetConf MargEntropyRand BADGEALBLCoresetConf MargEntropyRand SVHN, ResNet, Batch size: 100 BADGE 0.0 8.31 10.67 11.0 2.77 10.94 8.39 BADGE 0.0 0.29 0.3 1.03 0.82 1.3 0.64 BADGE 0.0 0.59 0.0 0.53 0.29 0.92 1.02 1.6 | | | 10 | | 1.6 | | | | --- | --- | --- | --- | --- | --- | --- | 0.90 ALBL 0.0 0.0 5.18 2.33 0.11 5.2 3.26 ALBL 0.34 0.0 0.0 0.49 0.1 0.88 0.39 1.4 ALBL 0.0 0.0 0.0 0.33 0.1 0.73 1.3 1.4 0.80 | | | 8 | | | | 1.2 | | --- | --- | --- | --- | --- | --- | --- | Coreset 1.17 1.64 0.0 5.32 2.4 5.9 4.02 Coreset 0.2 0.0 0.0 1.03 0.53 1.28 0.84 1.2 Coreset 0.29 0.39 0.0 0.2 0.14 0.96 1.69 0.70 | ycaruccA | | | | | | 1.0 | | -------- | --- | --- | --- | --- | --- | --- | 0.60 Conf 0.21 2.31 7.12 0.0 0.0 5.96 4.77 6 Conf 0.0 0.14 0.34 0.0 0.0 0.0 0.64 1.0 Conf 0.33 0.33 0.14 0.0 0.33 0.0 0.68 | | | | | 0.8 | | 0.8 | | --- | --- | --- | --- | --- | --- | --- | 0.50 Marg 0.62 7.0 10.2 8.65 0.0 10.34 7.45 Marg 0.0 0.0 0.34 0.49 0.0 0.59 0.64 Marg 0.33 0.14 0.33 0.2 0.0 0.4 1.25 | | | 4 | | | | 0.6 | | --- | --- | --- | --- | --- | --- | --- | Entropy 0.11 1.1 5.81 0.67 0.0 0.0 3.66 Entropy 0.1 0.58 0.24 0.58 0.35 0.0 0.83 0.6 Entropy 0.1 0.1 0.0 0.0 0.0 0.0 0.63 0.40 | | | | | 0.4 | | 0.4 | | --- | --- | --- | --- | --- | --- | --- | 0.30 Rand 0.5 4.8 6.82 6.44 1.49 7.92 0.0 2 Rand 0.1 0.29 0.59 1.4 1.2 1.75 0.0 Rand 0.24 0.54 0.2 0.82 0.29 0.98 0.0 | | | | | 0.2 | | 0.2 | | --- | --- | --- | --- | --- | --- | --- | 0.20 0.37 3.59 6.54 4.92 0.97 6.61 4.51 0.11 0.19 0.26 0.72 0.43 0.83 0.57 0.19 0.3 0.1 0.3 0.16 0.57 0.94 | | | 0 | | 0.0 | | 0.0 | | --- | --- | --- | --- | --- | --- | --- | 0.10 | 5000 | 10000 15000 | 20000 25000 30000 | 35000 | | | | | ---- | ----------- | ----------------- | ----- | --- | --- | --- | #Labels queried Figure21: Pairwisepenaltymatricesofthealgorithms,groupedbydifferentneuralnetworkmodels. The parenthesizednumberinthetitleisthetotalnumberof(D,B,A)combinationsaggregated,whichisalsoan upperboundonallitsentries. Element(i,j)correspondsroughlytothenumberoftimesalgorithmibeats algorithmj. Column-wiseaveragesatthebottomshowaggregateperformance(lowerisbetter). Fromleftto right: MLP,ResNetandVGG. totalnumberof(D,B,A)combinationsaggregated;asdiscussedinSection4,thisisalsoanupperboundon allitsentries. Itcanbeseenthatuncertainty-basedmethods(e.g. MARG)performwellonlyinsmallbatch sizeregimes(100)orwhenusingMLPmodels; representativesamplingbasedmethods(e.g. CORESET) onlyperformwellinlargebatchsizeregimes(10000)orwhenusingResNetorVGGmodels. Incontrast, BADGE’sperformanceiscompetitiveacrossallbatchsizesandneuralnetworkmodels. 21 PublishedasaconferencepaperatICLR2020 | | Batch size: 100 | | | Batch size: 1000 | | Batch size: 10000 | | | --- | --------------- | --- | --- | ---------------- | --- | ----------------- | --- | MLP | ycneuqerf evitalumuC 1.0 1.0 | | | ycneuqerf evitalumuC 1.0 | | ycneuqerf evitalumuC 1.0 | | | | ---------------------------- | --- | --- | ------------------------ | --- | ------------------------ | --- | --- | ycneuqerf evitalumuC | 0.8 | | | 0.8 | | 0.8 | | | | --- | --- | --- | --- | --- | --- | --- | --- | 0.8 | 0.6 0.6 | | | 0.6 | | 0.6 | | | | ------- | --- | --- | --- | --- | --- | --- | --- | | 0.4 | | | 0.4 | | | | | | 0.4 | | | | | 0.4 | | | | 0.2 0.2 | | | 0.2 | | 0.2 | | | | 0.0 0.0 | | | 0.0 | | 0.0 | | | 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1.0 1.0 1.1 1.1 1.2 1.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 | | Normalized error Normalized error | | | Normalized error | | Normalized error | | | --- | --------------------------------- | ---- | ------- | ---------------- | ------- | ---------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | Figure22: CDFsofnormalizederrorsofthealgorithms,groupbydifferentbatchsizes. HigherCDFindicates | betterperformance. | | Fromlefttoright: | batchsize=100,1000,10000. | | | | | | ------------------------ | --- | ---------------- | ------------------------- | ------ | -------------------- | --- | --- | | | | MLP | | ResNet | | VGG | | | 1.0 | | MLP | 1.0 | | 1.0 | | | | ycneuqerf evitalumuC 1.0 | | | ycneuqerf evitalumuC | | ycneuqerf evitalumuC | | | ycneuqerf evitalumuC | 0.8 | | | 0.8 | | 0.8 | | | | --- | --- | --- | --- | --- | --- | --- | --- | 0.8 | 0.6 0.6 | | | 0.6 | | 0.6 | | | | ------- | --- | --- | --- | --- | --- | --- | --- | | 0.4 | | | 0.4 | | 0.4 | | | 0.4 | 0.2 0.2 | | | 0.2 | | 0.2 | | | | ------- | --- | --- | --- | --- | --- | --- | --- | 0.0 | 0.0 | | | 0.0 | | 0.0 | | | | --- | --- | --- | --- | --- | --- | --- | --- | 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1.0 1.0 1.1 1.1 1.2 1.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 | | Normalized error Normalized error | | | Normalized error | | Normalized error | | | --- | --------------------------------- | ---- | ------- | ---------------- | ------- | ---------------- | ---- | | | ALBL | Conf | Coreset | BADGE | Entropy | Marg | Rand | Figure23: CDFsofnormalizederrorsofthealgorithms,groupbydifferentneuralnetworkmodels. Higher | CDFindicatesbetterperformance. | | | Fromlefttoright: | MLP,ResNetandVGG. | | | | | ------------------------------ | --- | --- | ---------------- | ----------------- | --- | --- | --- | E CDFS | | OF | NORMALIZED | ERRORS | OF DIFFERENT | ALGORITHMS | | | | --- | --- | ---------- | ------ | ------------ | ---------- | --- | --- | InadditiontoFigure5thataggregatesoverallsettings, weshowheretheCDFsofnormalizederrorsby conditioningonfixedbatchsizes(100,1000and10000)inFigure22,andshowtheCDFsofnormalized errorsbyconditioningonfixedneuralnetworkmodels(MLP,ResNetandVGG)inFigure23. | F BATCH | UNCERTAINTY | | AND DIVERSITY | | | | | | ------- | ----------- | --- | ------------- | --- | --- | --- | --- | Figure24givesacomparisonofsamplingmethodswithgradientembeddingintwosettings(OpenML#6, MLP,batchsize100andSVHN,ResNet,batchsize1000),intermsofuncertaintyanddiversityofexamples selectedwithinbatches. Thesetwopropertiesaremeasuredbyaverage(cid:96) 2 normanddeterminantoftheGram matrixofgradientembedding,respectively. Itcanbeseenthat,k-MEANS++(BADGE)inducesgoodbatch diversityinbothsettings. CONFgenerallyselectsexampleswithhighuncertainty,butinsomeiterationsof OpenML#6,thebatchdiversityisrelativelylow,asevidencedbythecorrespondinglogGramdeterminant being−∞. TheseareasareindicatedbygapsinthelearningcurveforCONF. Situationswherethereare 22 PublishedasaconferencepaperatICLR2020 OpenML #6, MLP, Batch size: 100 SVHN, ResNet, Batch size: 1000 | hctab fo tnanimreted goL 500 | | | | hctab fo tnanimreted goL 0 | | | ---------------------------- | --- | --- | --- | -------------------------- | --- | | 250 | | | | 5000 | | | 0 | | | | 10000 | | | 250 | | | | 15000 | | | 500 | | | | 20000 | | | 750 | | | | 25000 | | 1000 30000 1250 35000 0 2000 4000 6000 8000 100001200014000 0 10000 20000 30000 40000 50000 | | #Labels queried | | | | #Labels queried | | --- | --------------- | --- | --- | --- | --------------- | | | (a) | | | | (b) | OpenML #6, MLP, Batch size: 100 SVHN, ResNet, Batch size: 1000 | hctab ni mron 50 | | | | hctab ni mron | | | ----------------- | --- | --- | --- | -------------- | --- | 10 40 8 30 6 | 2 20 | | | | 2 | | | ---- | --- | --- | --- | --- | --- | 4 | egarevA | | | | egarevA | | | -------- | --- | --- | --- | -------- | --- | | 10 | | | | 2 | | | 0 | | | | 0 | | 0 2000 4000 6000 8000 100001200014000 0 10000 20000 30000 40000 50000 | | #Labels queried | | | | #Labels queried | | --- | --------------- | --------- | ---- | ----------- | --------------- | | | (c) | | | | (d) | | | k-DPP | k-means++ | Rand | FF k-center | Conf | Figure 24: A comparison of batch selection algorithms in gradient space. Plots a and b show the log determinantsoftheGrammatricesofgradientembeddingswithinbatchesaslearningprogresses. Plotsc anddshowtheaverageembeddingmagnitude(ameasurementofpredictiveuncertainty)intheselected batch. Thek-centerssamplerfindspointsthatarenotasdiverseorhigh-magnitudeasothersamplers. Notice alsothatk-MEANS++tendstoactuallyselectsamplesthatarebothmorediverseandhigher-magnitudethan ak-DPP,apotentialpathologyofthe k-DPP’sdegreeofstochastisity. Amongallalgorithms, CONF has thelargestaveragenormofgradientembeddingswithinabatch;however,inOpenML#6,andthefirstfew interationsofSVHN,somebatcheshavealogGramdeterminantof−∞(shownasgapsinthecurve),which showsthatCONFsometimesselectsbatchesthatareinferiorindiversity. manygapsintheCONFplotseemtocorrespondtosituationsinwhichCONFperformspoorlyintermsof accuracy(seeFigure13forthecorrespondinglearningcurve). Bothk-DPPandFF-k-CENTER(analgorithm thatapproximatelyminimizesk-centerobjective)selectbatchesthathavelowerdiversitythank-MEANS++ (BADGE). | G COMPARISON | OF | k-MEANS++ | AND k-DPP | IN BATCH | SELECTION | | ------------ | --- | --------- | --------- | -------- | --------- | InFigures25to31,wegiverunningtimeandtestaccuracycomparisonsbetweenk-MEANS++andk-DPP forselectingexamplesbasedongradientembeddinginbatchmodeactivelearning. Weimplementthek-DPP samplingusingtheMCMCalgorithmfrom(Kang,2013),whichhasatimecomplexityofO(τ ·(k2+kd)) 23 PublishedasaconferencepaperatICLR2020 OpenML#6, MLP, Batch size: 100 ×104OpenML#6, MLP, Batch size: 100 OpenML#6, MLP, Batch size: 1000 ×105OpenML#6, MLP, Batch size: 1000 | | 2.5 | | 2.0 | | ---- | --- | ---- | --- | | 0.90 | 2.0 | 0.90 | | 1.5 | ycaruccA 0.80 | emiT 1.5 | ycaruccA 0.80 | emiT | | ------------- | -------- | ------------- | ---- | | 0.70 | | 0.70 | 1.0 | 1.0 | 0.60 | | 0.60 | | | ---- | --- | ---- | --- | | 0.50 | 0.5 | 0.50 | 0.5 | | 0.40 | 0.0 | 0.40 | 0.0 | 200040006000800010000120001400016000 #Labels queried 200040006000800010000120001400016000 #Labels queried 200040006000800010000120001400016000 #Labels queried 200040006000800010000120001400016000 #Labels queried k-DPP k-means++ Figure25: LearningcurvesandrunningtimesforOpenML#6withMLP. OpenML#155, MLP, Batch size: 100 ×104OpenML#155, MLP, Batch size: 100 OpenML#155, MLP, Batch size: 1000 ×10O5penML#155, MLP, Batch size: 1000 | 1.00 | 6 | 1.00 | | | -------- | ---- | -------- | ---- | | 0.95 | 5 | 0.95 | 2.0 | | 0.90 | | 0.90 | | | ycaruccA | 4 | ycaruccA | 1.5 | | 0.85 | emiT | 0.85 | emiT | | 0.80 | 3 | 0.80 | 1.0 | | 0.75 | 2 | 0.75 | | | 0.70 | | 0.70 | 0.5 | | 0.65 | 1 | 0.65 | | | | 0 | 0.60 | 0.0 | 0.60 10000 20000 30000 40000 10000 20000 30000 40000 10000 20000 30000 40000 10000 20000 30000 40000 #Labels queried #Labels queried #Labels queried #Labels queried k-DPP k-means++ Figure26: LearningcurvesandrunningtimesforOpenML#155withMLP. andspacecomplexityofO(k2 +kd),whereτ isthenumberofsamplingsteps. Wesetτ as(cid:98)5klnk(cid:99)in ourexperiment. Thecomparisonsforbatchsize10000arenotshownhereastheimplementationofk-DPP samplingrunsoutofmemory. Itcanbeseenfromthefiguresthat,althoughk-DPPandk-MEANS++arebasedondifferentsamplingcriteria, theclassificationaccuraciesoftheirinducedactivelearningalgorithmaresimilar. Inaddition,whenlarge batchsizesarerequired(e.g. k =1000),therunningtimesofk-DPPsamplingaregenerallymuchhigher thanthoseofk-MEANS++. OpenML#156, MLP, Batch size: 100 ×105OpenML#156, MLP, Batch size: 100 OpenML#156, MLP, Batch size: 1000 ×10O5penML#156, MLP, Batch size: 1000 6 | 0.90 | | 0.90 | 2.0 | | ---- | --- | ---- | --- | 5 | ycaruccA 0.85 | 4 | ycaruccA 0.85 | 1.5 | | ------------- | ---- | ------------- | ---- | | 0.80 | emiT | 0.80 | emiT | | | 3 | | 1.0 | | 0.75 | | 0.75 | | | | 2 | | 0.5 | | 0.70 | 1 | 0.70 | | 0.65 | 0.65 | 0 | | 0.0 | | ---- | --- | --- | --- | 10000 #Labels queried 20000 30000 40000 10000 #Labels queried 20000 30000 40000 10000 #Labels queried 20000 30000 40000 10000 #Labels queried 20000 30000 40000 k-DPP k-means++ Figure27: LearningcurvesandrunningtimesforOpenML#156withMLP. 24 PublishedasaconferencepaperatICLR2020 OpenML#184, MLP, Batch size: 100 ×106OpenML#184, MLP, Batch size: 100 OpenML#184, MLP, Batch size: 1000 ×10O4penML#184, MLP, Batch size: 1000 | | 1.2 | 0.80 | 8 | | -------- | -------- | -------- | ---- | | 0.80 | 1.0 | | | | 0.70 | | 0.70 | 6 | | ycaruccA | 0.8 | ycaruccA | | | 0.60 | emiT 0.6 | 0.60 | emiT | | 0.50 | | 0.50 | 4 | 0.4 | 0.40 | 0.2 | 0.40 | 2 | | ---- | --- | ---- | --- | | 0.30 | | 0.30 | | | | 0.0 | | 0 | 2000 4000 6000 8000 100001200014000 2000 4000 6000 8000 100001200014000 2000 4000 6000 8000100001200014000 2000 4000 6000 8000100001200014000 #Labels queried #Labels queried #Labels queried #Labels queried k-DPP k-means++ Figure28: LearningcurvesandrunningtimesforOpenML#184withMLP. | | ×105 | | ×105 | | --- | ---- | --- | ---- | SVHN, MLP, Batch size: 100 SVHN, MLP, Batch size: 100 SVHN, MLP, Batch size: 1000 SVHN, MLP, Batch size: 1000 | 0.80 | | 0.80 | 1.2 | | ------------- | -------- | ------------- | -------- | | 0.70 | 3.0 | | | | | 2.5 | 0.70 | 1.0 | | ycaruccA 0.60 | emiT 2.0 | ycaruccA 0.60 | emiT 0.8 | | 0.50 | | 0.50 | 0.6 | 1.5 | 0.40 | 1.0 | 0.40 | 0.4 | | ---- | --- | ---- | --- | | 0.30 | 0.5 | 0.30 | 0.2 | | 0.20 | 0.0 | 0.20 | 0.0 | 5000 10000 15000 20000 25000 5000 10000 15000 20000 25000 5000 10000 15000 20000 25000 30000 5000 10000 15000 20000 25000 30000 #Labels queried #Labels queried #Labels queried #Labels queried SVHN, ResNet, Batch size: 100 ×105 SVHN, ResNet, Batch size: 100 SVHN, ResNet, Batch size: 1000 ×104SVHN, ResNet, Batch size: 1000 | 0.90 | 2.00 | 0.90 | 7 | | ------------- | --------- | ------------- | ------ | | 0.80 | 1.75 | 0.80 | | | 0.70 | 1.50 | 0.70 | 6 | | ycaruccA 0.60 | emiT 1.25 | ycaruccA 0.60 | emiT 5 | | | 1.00 | | 4 | | 0.50 | 0.75 | 0.50 | 3 | | 0.40 | | 0.40 | 2 | | 0.30 | 0.50 | 0.30 | | | 0.20 | 0.25 | 0.20 | 1 | | 0.10 | 0.00 | | 0 | 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 10000 20000 30000 40000 10000 20000 30000 40000 #Labels queried #Labels queried #Labels queried #Labels queried k-DPP k-means++ Figure29: LearningcurvesandrunningtimesforSVHNwithMLPandResNet. MNIST, MLP, Batch size: 100 ×104 MNIST, MLP, Batch size: 100 MNIST, MLP, Batch size: 1000 ×104 MNIST, MLP, Batch size: 1000 | | 8 | 0.95 | 8 | | --- | --- | ---- | --- | 0.95 | ycaruccA 0.90 | 6 | ycaruccA 0.90 | 6 | | ------------- | ---- | ------------- | ---- | | | emiT | | emiT | | 0.85 | 4 | 0.85 | 4 | | 0.80 | | 0.80 | | | 0.75 | 2 | 0.75 | 2 | 0.70 | 0.70 | 0 | | 0 | | ---- | --- | --- | --- | 10000 #Labels queried 20000 30000 40000 50000 10000 #Labels queried 20000 30000 40000 50000 10000 #Labels queried 20000 30000 40000 10000 #Labels queried 20000 30000 40000 k-DPP k-means++ Figure30: LearningcurvesandrunningtimesforMNISTwithMLP. 25 PublishedasaconferencepaperatICLR2020 | | ×105 | | ×104CIFAR10, MLP, Batch size: 1000 | | ----------------------------- | ----------------------------- | ------------------------------ | ---------------------------------- | | CIFAR10, MLP, Batch size: 100 | CIFAR10, MLP, Batch size: 100 | CIFAR10, MLP, Batch size: 1000 | | | 0.50 | 1.50 | 0.50 | 8 | | 0.45 | 1.25 | 0.45 | 6 | | ycaruccA 0.40 | emiT 1.00 | ycaruccA 0.40 | emiT | 4 | 0.35 | 0.75 | 0.35 | | | ---- | ---- | ---- | --- | 0.50 | 0.30 | 0.25 | 0.30 | 2 | | ---- | ---- | ---- | --- | | 0.25 | | 0.25 | | | | 0.00 | | 0 | 10000 20000 30000 40000 10000 20000 30000 40000 10000 20000 30000 40000 10000 20000 30000 40000 #Labels queried #Labels queried #Labels queried #Labels queried CIFAR10, ResNet, Batch size: 100 ×105CIFAR10, ResNet, Batch size: 100 0.60 CIFAR10, ResNet, Batch size: 1000 ×104CIFAR10, ResNet, Batch size: 1000 | | 2.5 | | 8 | | ------------- | ---- | ------------- | ---- | | 0.50 | | 0.50 | | | | 2.0 | | 6 | | ycaruccA 0.40 | emiT | ycaruccA 0.40 | emiT | 1.5 | 0.30 | 1.0 | 0.30 | 4 | | ---- | --- | ---- | --- | | 0.20 | 0.5 | 0.20 | 2 | | 0.10 | 0.0 | 0.10 | 0 | 10000 20000 30000 40000 10000 20000 30000 40000 10000 20000 30000 40000 10000 20000 30000 40000 #Labels queried #Labels queried #Labels queried #Labels queried k-DPP k-means++ Figure31: LearningcurvesandrunningtimesforCIFAR10withMLPandResNet. 26