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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
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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.
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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