Efficient Online Relative Comparison Kernel Learning

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by  Eric Heim

Efficient Online Relative Comparison Kernel Learning
EricHeim∗ MatthewBerger† LeeM.Seversky† MilosHauskrecht∗
Abstract In learning a kernel from human supervision, it is
Learning a kernel matrix from relative comparison human importanttoobtainfeedbackwhichisintuitivefortheuser
feedbackisanimportantproblemwithapplicationsincollab- to provide and informative for a learning algorithm to use.
5 orativefiltering,objectretrieval,andsearch. Forlearninga Forinstance,naiveformsofsupervisionsuchasnumerical
1 kerneloveralargenumberofobjects,existingmethodsface judgmentsbetweenpairsofobjectshavebeenshowntobe
0 significantscalabilityissuesinhibitingtheapplicationofthese very noisy [24]. A relative comparison, the response to a
2
methodstosettingswhereakernelislearnedinanonlineand queryoftheform“IsobjectAmoresimilartoobjectBorC?”,
n timelyfashion. Inthispaperweproposeanovelframework iswellknownasanintuitivemechanismforsolicitinghuman
a
J calledEfficientonlineRelativecomparisonKernelLEarning feedback and an effective way of learning similarity [11].
2 (ERKLE),forefficientlylearningthesimilarityofalargeset Recentworksaddressingfine-grainedcategorization[27]and
1 ofobjectsinanonlinemanner. Welearnakernelfromrela- perceptualvisualizationdesign[6]haveshownthepracticality
tivecomparisonsviastochasticgradientdescent,onequery andbenefitoflearningkernelsfromrelativecomparisons.

] response at a time, by taking advantage of the sparse and ManyRCKLmethods[1,26]learnakernelbysolving
G
low-rankpropertiesofthegradienttoefficientlyrestrictthe asemidefiniteprogram(SDP)inbatch, whereallobtained
L kernel to lie in the space of positive semidefinite matrices. relativecomparisonsarerequiredtolearnthekernel. How-
s. In addition, we derive a passive-aggressive online update ever,inmanypracticalapplications,abatchapproachisnot
c forminimallysatisfyingnewrelativecomparisonsastonot appropriateduetotheonlineanddynamicnatureoftheappli-
[
disrupttheinfluenceofpreviouslyobtainedcomparisons. Ex- cation. Forexampleincrowdsourcing,itisoftenofinterest

2 perimentally, we demonstrate a considerable improvement tominimizethenumberofdispatchedtasksandthusthecost
v inspeedwhileobtainingimprovedorcomparableaccuracy ofthecrowdbyleveragingactivelearningtechniques[25,9]
2
4 comparedtocurrentmethodsintheonlinelearningsetting. toadaptivelyselectthemostinformativerelativecomparison
query. Thesuccessofthesetechniquesdependsonmaintain-
2
1 Keywords inganuptodatemodelsoastoensurethemostinformative
0 queryisselected,aswellasanefficientlearningmethodto
OnlineLearning,KernelLearning,RelativeComparisons.
. quicklyupdatethemodelsothatnocrowdparticipantisidle.
1
0 Likewise,recommendationsystemsforonlinemarketplaces
1 Introduction
5 obtaincontinuousfeedbackintheformofclick-throughdata
1 Learningasimilaritymodeloverasetofobjectsfromhuman viauserinteraction. Inorderforthelearnedkerneltobeupto
v: feedbackisimportanttomanyapplicationsincollaborative dateandreflectthelatestuserfeedback,thelearningmethod
i filtering,documentandmultimediaretrieval,andvisualiza- mustbeabletoquicklyincorporatefeedbackasitisreceived.
X
tion.Ithasbeenshownthatbyincorporatinghumanfeedback, Thesescenariosmotivatetheneedforanefficientandon-
ar theoverallperformanceofsuchapplicationscanbegreatly linemethodforlearningfromlarge-scalerelativecomparison
improved [12, 10, 13, 15, 30]. In this work we focus on data. Batchmethodspoorlyscaleforlargeobjectcollections
learning a similarity model from human feedback through primarilybecausetheymustensuretheirsolutionsarePSD.
relativecomparisons. Morespecifically,wefocusontherela- Withoutanypriorassumptionsonthedatathisoperationis
tivecomparisonkernellearning(RCKL)problem,inwhich ofO(n3)timecomplexityfornobjects,whichforlargenis
thegoalistolearnapositivesemidefinite(PSD)kernelma- prohibitivelyslowfortheaforementionedapplications.
trix from relative comparisons given by humans. Kernels ThisworkintroducesanovelonlineRCKLframework
areusedformodelingobjectrelationshipsinmanylearning calledEfficientonlineRelativecomparisonKernelLEarning
techniques[23],andhenceareapplicabletomanymethods (ERKLE) that sequentially updates a kernel one query
thatutilizekernelsfortheseapplications. response at a time in O(n2) complexity. ERKLE employs
stochasticgradientdescent[3]forRCKL,takingadvantage
∗UniversityofPittsburgh,Department.ofComputerScience ofthesparseandlow-rankstructureoftheRCKLgradient
{eric,milos}@cs.pitt.edu
over a single comparison to devise fast updates that only
†AirForceResearchLaboratory,InformationDirectorate
requirefindingthesmallesteigenvectorandeigenvalueofa
{matthew.berger.1,lee.seversky}@us.af.mil
suitablematrix. Weshowthatthegradientstructure,which shows how to devise efficient update schemes for solving
enables such an efficient update, generalizes several well- SDPswhenthegradientoftheobjectivefunctionislow-rank.
knownconvexRCKLmethods[1,26]. Thestructureofthe Webuilduponandimprovetheefficiencyofthiswork, by
gradient also reveals a simple way to bound the smallest takingadvantageofthesparseandlow-rankstructureofthe
eigenvalue after each gradient step, which often allows gradientcommoninconvexRCKLformulations.
updatestobeperformedinconstanttime. Motivatedbywork Our passive-aggressive step size procedure is similar
in online learning [5], we also derive a passive-aggressive to that which is introduced in [5] for other online learning
versionofERKLEtoensurelearnedkernelsmodelthemost problems. In their work, the authors create a passive-
recentlyobtainedrelativecomparisonswithoutover-fitting. aggressiveonlineupdateruleforclassicSVMformulations
Insummary,ourmaincontributionsare: used in problems such as binary/multi-class classification
and regression. In deriving such an update for different
1. AnonlineRCKLframeworkforlarge-scalesimilarity RCKLlossfunctions,werelatehowdifferentmethodscan
learningthatgeneralizesmanycurrentRCKLmethods. beutilizedunderacommonpassive-aggressiveframework.
2. An efficient kernel update method with O(n2) time Toourknowledge,suchanupdateforRCKLproblemsand
complexitythatexploitstheuniquestructureofRCKL theassociatedanalysisofRCKLmethodshasnotbeendone.
stochasticgradientswhenstochasticgradientstepsmay
3 Preliminaries
resultinanon-PSDmatrix.
3. Apassive-aggressiveupdateprocedureforonlinerela- Inthissection,weformallydefineRCKLandprovideabrief
tivecomparisonkernellearning overview of RCKL methods. Let Sn be the set of n×n
+
4. AnexperimentalevaluationthatshowsERKLEhasboth PSDmatrices,andMabbetheentryatrowacolumnbofa
improvedperformanceandfasterruntimescomparedto matrixM. ThegoalofRCKListolearnaPSDkernelmatrix
batchRCKLmethods. K∈Sn overnobjects,givenasetT oftriplets:
+
(3.1) T ={(a,b,c)|aismoresimilartobthanc}
2 RelatedWork
Theproblemoflearningakernelmatrix,drivenbyrelative suchthatsquareddistanceconstraintsaresatisfied:
comparisonfeedback,hasbeenthefocusofmuchrecentwork.
Mostrecenttechniquesprimarilydifferbythechoiceofloss ∀ : d2 (a,b)<d2 (a,c)
(a,b,c)∈T K K
(3.2)
function. Forinstance,GeneralizedNon-metricMultidimen- where d2 (a,b)=Kaa+Kbb−2Kab.
sionalScaling[1]employshingeloss,CrowdKernelLearn- K
ing [25] uses a scale-invariant loss, and Stochastic Triplet WesayakernelKsatisfiesatriplett = (a ,b ,c ) ∈ T if
i i i i
Embedding[26]usesalogisticlossfunction. theconstraintin(3.2)correspondingtot issatisfied.
i
TheaforementionedRCKLmethodscanbeviewedas In this work, we consider triplets that are answers to
solving a kernelized special case of the classic non-metric relative comparison queries posed to one or more people.
multidimensionalscalingproblem[14],wherethegoalisto We define a query q to have three components, a “head”
find an embedding of objects in Rd such that they satisfy object h to be compared with two objects o1 and o2. A
givenEuclideandistanceconstraints. Incontrasttomanyof queryq =(cid:0)h,(cid:8)o1,o2(cid:9)(cid:1)canbeansweredbyeitherthetriplet
thekernel-learningformulations,theiranalogousembedding- (cid:0)h,o1,o2(cid:1)or(cid:0)h,o2,o1(cid:1),indicatingthathismoresimilarto
learningcounterpartsarenon-convexoptimizationproblems, o1thano2orhismoresimilartoo2thano1,respectively. It
whichonlyguaranteeconvergencetoalocalminimum. Inthe isdesirabletolearnakernelthatnotonlysatisfiesobserved
typicalnon-convexbatchsetting,multiplesolutionsarefound triplets,butalsothatgeneralizestounseentriplets,leadingto
with different initializations and the best is chosen among alearnedkernelthatmodelsamorecompletenotionofthe
them. This strategy is poorly suited for the online setting desiredhumansimilarityspace.
where triplets are being observed sequentially, and which
solutionisbestmaychangeasfeedbackisreceived. 3.1 RCKL Formulation Many RCKL methods can be
In this work we consider the online RCKL problem, generalizedbythefollowingSDP:
where one is sequentially acquiring relative comparisons
among a large collection of objects. Stochastic gradient min L(K,T)+τTrace(K)
(3.3) K
descent techniques [21] are a popular class of methods for
s.t. K(cid:23)0.
onlinelearningofhigh-dimensionaldataforaverygeneral
class of functions, where recent techniques [29, 22] have Theobjectivefunctioniscomposedoftwoterms. Thefirst
demonstratedcompetitiveperformancewithbatchtechniques. term is a function L measuring how much loss K incurs
Inparticular,recentmethods[8,16]havedevelopedefficient for not satisfying triplets in T. The second term is a trace
methodstosolveSDPsinanonlinefashion. Theworkof[4] regularizationonKweightedbyahyperparameterτ. Trace
regularizationisusedasaconvexapproximationofthenon- stochasticgradientdescenttechniques[3]. Asshownin(3.4),
convexrankfunction. Highervaluesofτ enforcethat(3.3) thelossfunctionLnaturallydecomposesintothesumover
produceslower-complexitysimilaritymodels. Finally,Kis losses l defined on individual observations (triplets in our
constrainedtobePSD. case). Fromthisdecomposition,ERKLEfirstperformsthe
The loss function in the objective can be decomposed followingstochasticgradientstep:
intothesumoflossesoverindividualtriplets:
(4.9) K(cid:48) ←K −δ ∇l(K ,t ),
j j−1 j j−1 j
(cid:88)
(3.4) L(K,T)= l(K,t).
where triplets t ,...,t have been observed, K is the
1 j−1 j−1
t∈T
onlinesolutionafterobservingthej−1triplet,
Existing RCKL methods differ in the choice of the loss PerformingastochasticoptimizationgivesERKLEan
functionl.TheStochasticTripletEmbedding(STE)approach advantage over current RCKL methods that perform batch
of [26] defines l(K,t) = −logpK as the loss function, optimizations. Batch methods attempt to minimize a loss
t
wherepKistheprobabilitythatatripletissatisfied: function over a training set. This is known to minimize
t
empiricalriskwithrespecttotheparticulartrainingsamples,
exp(−d2 (a,b)) whichisusedasanestimateofexpectedriskovertheground
(3.5) pK = K .
t=(a,b,c) exp(−d2 (a,b))+exp(−d2 (a,c)) truthdistributionoverallsamples. Obtainingtripletsinan
K K
online fashion from a source can be viewed as sampling
GeneralizedNonmetricMultidimensionalScaling(GNMDS)
triplets from a ground truth distribution at random. As
[1]usesahingeloss,wherel(K,t=(a,b,c))isdefinedas:
such,takingstochasticstepsoversamplesdirectlyminimizes
expected risk with respect to the ground truth distribution
(3.6) max(0,d2 (a,b)−d2 (a,c)+1).
K K of triplets, not empirical risk with respect to the training
instances. Thepracticalimpactofthischaracteristicisthat
Foreitherlossfunctionl,(3.3)isaconvexoptimization
stochasticmethodstendtogeneralizebettertounobserved
problem and the globally optimal solution is found by
samples. For more discussion on this characteristic of
performingprojectedgradientdescent,whichconsistsoftwo
stochasticmethodssee[3].
updatesteps. Thefirststepisasimpledescendingstepalong
Notethatouronlineformulationdoesnotincludetrace
thegradientoftheobjective:
regularization. Although this may impact our method
(3.7) K(cid:48) =K −δ (∇L(K ,T)+τI), in generalizing to unseen triplets, our online formulation
i i−1 i i−1
achievesgoodgeneralizationthroughcarefullyconstructed,
whereidenotesthecurrentiteration,δi isthelearningrate. data-dependentstepsizesδj,asdetailedinSection4.3.
Thesecondstepprojectstheresultofthefirstgradientstep
ontothePSDcone: 4.2 Efficient Projection In order to retain positive semi-
definiteness, after taking a stochastic gradient step the
(3.8) Ki =ΠS+(K(cid:48)i). resulting matrix K(cid:48)j must be projected onto the PSD cone.
FollowingtheprocedureofΠ isprohibitivelyexpensive
Thesestepsareiterateduntilconvergence. S+
forouronlinesetting. Instead, forRCKLmethodswecan
take advantage of the sparse and low-rank nature of the
4 Efficient Online Relative Comparison Kernel
gradienttodeviseanefficientprojectionscheme. Tothisend,
Learning(ERKLE)
we introduce a canonical gradient matrix G over a triplet
ThemaincomputationalbottleneckoftraditionalRCKLmeth- t=(a,b,c)),wheretheentriesaredefinedas:
odsistheprojectionontothePSDcone,Π .Thisprojection
S+ 
iois.fec.KoKm(cid:48)i mi=o=VnlVΛyVf[ΛouT]n+adVnbdty,swfietrhtsietnrgteaka[·li]ln+gneitsghaedteievfiiegneeendigdeeenncvotramyl-upweosissitetoioa0ns, (4.10) Gij =−212 iiifffiii===aab,,,jjj===bbcoorrii==bc,,jj ==aa
[tfΛhoeriist]ht+reuc=pturomrejeaocxft(ioK0n,(cid:48)i.Λ,iiStis)i.nfucAlelbetshieginsetniodsefacanonmyOpp(onrsi3oit)riooknpnieosrwanlteeiocdengs,esatohrnye −01 oifthie=rwci,sje.=c
projectionsteprendersbatchmethodscomputationallypro-
Nowconsiderthefollowingchoiceforthestochasticstep:
hibitiveforlearningthesimilarityofalargenumberofobjects
inanonlinemanner. (4.11) ∇l(K,t)=f(K,t)G,
4.1 Stochastic Gradient Step To create an efficient and wheref isareal-valuedfunctiondefinedbelow. With(4.11)
online framework for RCKL – ERKLE – we leverage asthegradientin(4.9),K isupdatedbyincreasingentries
j−1
correspondingtothesimilaritybetweenobjectsaandband Algorithm1EfficientPSDProjection
decreasing the similarity between a and c by a factor of 1: procedureΠ1(K)
+
f(Kj−1,tj). 2: Findλ↓andv↓fromK
Thefunctionf canbedefinedsuchthatwerecoverthe 3: ifλ↓ <0then
gradientsoflfordifferentconvexRCKLformulations. The 4: returnK−λ↓v↓v↓T
stochasticgradientforSTEcanbeobtainedbydefiningf as: 5: else
6: returnK
(4.12) f(K,t)=1−pK
t 7: endif
8: endprocedure
Similarlybydefiningf tobe:
(cid:26) 1 ifd2 (a,b)+1<d2 (a,c)
(4.13) f(K,t)= K K λˆ0. Initially,λˆ0 ← λ0. Itisupdatedeachiterationwithit’s
0 otherwise j 0 0
lowerboundλˆ0 ← λˆ0 −3γ . Ifλˆ0 < 0, thenAlg. 1is
j j−1 j j
thestochasticgradientforGNMDSisobtained. Note,that used to project onto the PSD cone and λˆ0 ← max(0,λ ).
this not only generalizes these two methods for use in our j ↓
Otherwise, no projection is performed. In the case where
onlineframeworkbutalsosuggestsasimplewaytocreate λ0 >>−3γ ,thissimplelower-boundingprocedurecansave
new online RCKL methods by designing a function f that 0 j
manyeigenvalue/eigenvectorcomputationsuntilaprojection
weighsthecontributionofindividualtriplets.
maybenecessary.
Decomposingtheonlineupdatesinsuchawayreveals
akeyinsightintohowtoperformefficientprojectionsonto
4.3 Passive-Aggressive Updates A key difference be-
thePSDconeafterthestochasticstep. Algorithm1outlines
tween the batch and stochastic RCKL updates is the mag-
the procedure for efficient projection in ERKLE. Here, λ
↓ nitude of the updates. For both methods the magnitude of
andv arethesmallesteigenvalueandeigenvectorofmatrix
↓ theupdateswithrespecttoasingletriplettisafunctionofa
K,respectively. ThisprocedurehasatimecomplexityO(n2)
learningrateandhowwelltheprevioussolutionsatisfiest. In
due to finding λ and v . To show that Algorithm 1 does
↓ ↓ theprevioussectionwedenotedthemagnitudeofanERKLE
indeedperformaprojectionontothePSDcone,weprovethe
updateasγ . Inthebatchsetting,thesamelearningrateδ
j i
followingtheorem:
isusedforalltripletsinagivenstep. Incontrast,stochastic
methodstypicallyusedifferentlearningratesδ fordifferent
THEOREM4.1. Algorithm1resultsinaPSDmatrixKj that j
isclosesttoK(cid:48) intermsofFrobeniusdistance. tripletstj,whichcanresultinfasterconvergencerates. To
j
takeadvantageoffasterconvergence,thelearningratesmust
Proof. Let K ∈ Sn (i.e. identity). We use this as our satisfy certain conditions. Early work [3] on the topic of
0 +
base case and show inductively that after each iteration of learningratessuggestthatδj shouldsatisfytwoconstraints:
the main loop, Kj remains PSD. Let γj = δjf(Kj−1,tj) (cid:80)∞j=1δj2 < ∞and(cid:80)∞j=1δj = ∞. Forexampleδj = 1/j
be the magnitude of an update. By (4.11), the update in satisfiestheseconstraints. Laterwork[18]suggestsamore
√
Equation (4.9) can be written as Kj−1 − γjG. The only aggressivesettingofδj =1/ j.
nonzero eigenvalues of −γ G are λ = 3γ and λ = However, in the online setting there is no reason to
j 1 j 2
−3γ . It follows from Weyl’s inequality that the matrix believethatatripletshouldhavelessinfluenceonthekernel
j
K(cid:48) = K −γ G has at most one negative eigenvalue. thanthoseobtainedbeforeit. Ontheotherhand,wedonot
j j−1 j
IfK(cid:48) hasnonegativeeigenvalues,thenitisPSD(line6of wishtoover-fittothemostrecentlyobtainedtriplets. Itisthis
j
Algorithm(1)). IfK(cid:48) hasonenegativeeigenvalue,line4of observationthatmotivatesPassive-Aggressive(PA)Online
j
Algorithm1resultsinaPSDmatrixK thatisclosesttoK(cid:48) Learning[5]. IntheRCKLsetting,thegeneralideaisthatif
j j
intermsofFrobeniusdistancebyCase2ofTheorem4in[4]. theprevioussolutionKj−1satisfiesanewlyobtainedtriplet
t =(a,b,c)byamarginof1,thendonotupdatethekernel
j
TheimportantimplicationofThm. 4.1isthatERKLE (passive). Otherwise,updatethekernelsothatthekernelis
can incorporate a triplet into a kernel in O(n2) time by changedtheminimalamount,butt issatisfiedbyamarginof
j
performingtheefficientprojectionoutlinedinAlgorithm1. 1(aggressive). Afortunatesideeffectofchoosingminimally
Furthermore,ifastepissufficientlysmall,thennoprojection sizedupdatesisthatupdatesarelesslikelytoresultinnon-
is needed at all. Let λ0j be the smallest eigenvalue of Kj. PSDmatricesthanlargersteps,thuspotentiallyreducingthe
ByWeyl’sinequality,ifλ0−3γ ≥ 0,thenalleigenvalues numberofprojectionsontothePSDconeviaourconservative
j j
ofK(cid:48) aregreaterthanorequalto0. Thiscanbeusedto eigenvalueestimate(Section4.2).
j+1
skiptheprojectionstepwhentheupdateisknowntoresult Toderiveapassive-aggressiveupdateforERKLE,we
in a PSD matrix. In our algorithm, we lower bound the wish to learn a magnitude of a stochastic step γ =
j
smallesteigenvaluebymaintainingaconservativeestimate δ f(K ,t ) with passive-aggressive properties. f as de-
j j−1 j
finedbyGNMDSin(4.13)isinherentlypassive,butifK updates using STE and GNMDS are very similar. Setting
j−1
doesnotsatisfythemarginconstraint,ittakesastepindepen- P = e in(4.20)recoverstheGNMDSpassive-aggressive
1+e
dentofhowclosetheprevioussolutionistosatisfyingt . As step in (4.18), and changing the margin in (4.18) recovers
j
such,wewishtofindaδ thattakesanaggressivestep. We differentsettingsofP.
j
dothisbysolvingthefollowingoptimizationproblem: Notethatusing(4.18)asastepsizeresultsinaK(cid:48) with
j
theintendedpassive-aggressiveproperty,notnecessarilythe
(4.14) mδjin δj2 kernelKj aftertheprojection. Wechoosetofindapassive-
s.t. d2 (a,b)+1≤d2 (a,c),δ ≥0 aggressivestepsizeinsteadofafullupdateforcomputational
K(cid:48) K(cid:48) j
j j efficiency Finding a true passive-aggressive step size with
respecttoK wouldrequireiterativelyprojectingontothe
By(4.11)and(4.13),thefirstconstraintcanberewrittenas: j
PSDcone,whichiscomputationallyprohibitiveintheonline
(4.15) d2 (a,b)−d2 (a,c)−10δ +1≤0 setting. In practice, d2 is a good approximation to d2 ,
Kj−1 Kj−1 j K(cid:48)j Kj
as their difference is dependent on the magnitude of the
Withtheassumptionthatthetripletisnotsatisfiedbyamargin (potentially) negative eigenvalue of K(cid:48), which tends to be
j
of one in K , no update is required; otherwise, only a
j−1 quitesmall.
positive value of δ can satisfy (4.15), making the positive
j Evenforapropersettingofδ ,ithasbeenshownthat
j
constraintonδ redundant. Also,thesmallestδ thatsatisfies
j j stochastic methods perform best when multiple rounds of
(4.15)istheonethatmakesthelefthandsideexactlyzero.As
updatesorpassesareperformedontheobservedsamples[2,
aresult,theinequalityconstraintcanbehandledasequality.
20,28]. Forourproblemsetting,thisindicatesthatERKLE
TofindtheoptimumwefirstwritetheLagrangianL(δ ,α):
j maybenefitfromrevisitingtripletsthatwerepreviouslyused
(cid:16) (cid:17) to update the kernel. In our experiments we perform a
(4.16) δj2+α d2Kj−1(a,b)−d2Kj−1(a,c)−10δj +1 simplemulti-passschemewhereforeachnewtriplet,ERKLE
not only steps over the most recently obtained triplet, but
Taking the partial derivative of (4.16) with respect to δj, also a number of randomly sampled triplets from the set
setting it to 0, and solving for δj results in δj = 5α. of previously obtained triplets. We denote the number of
Substitutingthisbackinto(4.16)makestheLagrangian: “passes”ERKLEperformseachtimeanewtripletisobserved
(cid:16) (cid:17) asβ. Algorithm2inSec. Bdescribesthisprocessinmore
(4.17) −25α2+α d2Kj−1(a,b)−d2Kj−1(a,c)+1 detail. Thissimpleapproachissufficientformaintaininghigh
accuracywhilestillensuringcomputationalefficiencyforthe
Taking the partial derivative of (4.17) with respect to α, onlinesetting.
settingitto0,solvingforαandthensubstitutingthisback
intoδj = 5αresultsintheminimumstepsizethatsatisfies 5 Experiments
themarginconstraint:
Inthissection,weevaluateERKLEbycomparingittobatch
d2 (a,b)−d2 (a,c)+1 RCKLmethods. Batchmethodsarenottrulyapplicableto
(4.18) δ = Kj−1 Kj−1 theonlinelearningsetting,butcanbeappliedinwhatisoften
j 10
called “mini-batches”. In the mini-batch learning setting,
Asimilarpassive-aggressiveupdatecanbederivedusing every time a new batch of m triplets are received, batch
theprobabilityofatripletbeingsatisfiedinSTE.Consider RCKLisrunonallobtainedtripletssofar. Thus,wecompare
thefollowingoptimization: ERKLEtorunningtheirbatchcounterpartsinmini-batches.
Weevaluateeachmethodonfourdifferentdatasets,each
min δ2 with its own challenges. First, we start with a small-scale
j
(4.19) δj K(cid:48) syntheticexperimenttoevaluatehowthemethodsperform
s.t. p j ≥P,δ ≥0
tj j in an idealized setting. Second, a large-scale synthetic
experimentisruntoshowhowERKLEandbatchcomparein
In (4.19) the minimal step size is chosen such that the
termsofpracticalruntime. Third,adatasetoftripletsover
probabilitythatatripletissatisfiedaftertheupdateisgreater
popular music artists is used to evaluate how the methods
than or equal to a given probability P ∈ (0.5,1). Using
performinareal-worldsettingwithmoderatetripletnoise.
(4.19),wederivethefollowingstepsize:
Finally,ERKLEandbatchRCKLareevaluatedonadataset
d2 (a,b)−d2 (a,c)+κ oftripletsoversceneimages,whichconsistofasmallnumber
(4.20) δ = Kj−1 Kj−1 oftriplets,thusfocusingontheperformanceofthesemethods
j 10
withverylittlefeedback.
whereκ=log(P)−log(1−P).Thefullderivationisgiven Fortheseexperiments,wewishtoseehowthelearned
in Sec. A. Both derivations reveal that passive-aggressive kernelsgeneralizetoheldouttripletsastripletsareobtained.
0.5 0.5 0.32
STE−ERKLE (δ = 1/j) STE−ERKLE (β = 1, δ = 1) ERKLE−STE (δ = 1)
GNMDS−ERKLE (δ = 1/j) STE−ERKLE (β = 10, δ = 1) 0.3 ERKLE−PA
0.45 SGTNEM−DESR−KELREK (Lδ E= (1δ/ j=^ (11//j2^)()1/2)) 0.45 PPAA−−EERRKKLLEE ((ββ == 11)0) 0.28 SGTNEM−DBSa−tcBhatch
malized test error0.03.54 SPTAE−E−ERRKKLELE (δ = 1) malized test error0.003..534 SGCTKNELM−−DBBSaa−ttccBhhatch malized test error000...222246
Nor 0.3 Nor Nor 0.2
0.25
0.18
0.25 0.2
0.16
0.20 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1 2 3 4 5 6 7 8 9 10
Number of observed triplets Number of observed triplets Number of effective passes on observed triplets
(a)ComparisonofERKLElearningratesδ (b)TestErrorvs.#ofobservedtriplets (c)TestErrorvs.#ofeffectivepasses
Figure1: Resultsfromexperimentsonthesmallsyntheticdataset(10trials)
Thisisimportantinreal-worldapplicationswherethegoal betweenpoints,weansweredallpossiblerelativecomparison
is to accurately model all the relationships among objects, querieswhichresultedin485100triplets. 10000tripletswere
notjusttheobservedones. Becauseofthis,oneofourmain usedasthetrainsetandtherestwereusedasthetestset.
evaluationmetricsisnormalizedtesterror,whichisdefined Discussion: Figure1ashowstheeffectthatthelearning
as the total number of unsatisfied test triplets by a learned rateparameterδ hasontheperformanceofERKLEasmore
j
kerneldividedbythetotalnumberoftesttriplets. triplets are observed in an online fashion. For a setting
Unlessotherwisenoted,theexperimentswererunwith of 1/j, the learning rate decays too rapidly to improve
the following specifications. Each method started with an performancesignificantlyafterj =3000. Thelearningrate
√
initial kernel set to identity in order to give no method an 1/ j performs better, but still levels off, quicker than the
advantage(allmethodsinitiallysatisfynotriplets). Allbatch finaltwomethods. Thelasttwomethodshavelearningrates
methodswereterminatedafteramaximumof1000iterations thatareindependentofthenumberofobservedtriplets. STE-
orwhenthechangeinobjectivebetweeniterationswasless ERKLEwithaconstantlearningrateandPA-ERKLEtake
than 10−7. We denote the batch methods with the suffix stepssolelybasedonhowwellthecurrentsolutionsatisfies
“-Batch”(e.g. STE-Batch)andtheERKLEvariantswith“- the observed triplet, and vastly outperform the alternative
ERKLE”(e.g. STE-ERKLE).Wedenotepassive-aggressive learningratesbasedonnumberofobservations. Thisresult
ERKLEasPA-ERKLE,andusethestepsizethatsatisfiesthe indicates that reducing the influence of a triplet because it
marginby1asin(4.18). Themini-batchsizeis100,andall wasobservedlaterhasanadverseeffectontheabilityofa
methodsareevaluatedevery100observedtriplets. learnedkerneltogeneralizetounobservedtriplets.
We used the batch STE, GNMDS, and CKL (Crowd Figure1bshowstheperformanceofSTE-ERKLE(with
KernelLearning[25])MATLABimplementationsspecified δ setto1),andPA-ERKLEcomparedtothreebatchRCKL
j
by[26]inwhichtheeigMATLABfunctionisusedtofindthe methods. Theτ hyperparameterwaschosenbyselectingthe
alleigenvaluesandeigenvectorsforprojectionontothePSD bestsettingoverchoicesasevaluatedonthetestset. With
cone. ERKLEwasalsoimplementedinMATLAB,wherethe asinglepassoverthedata(β = 1), bothERKLEmethods
eigsfunctionisusedtofindasingleeigenvalue/eigenvector outperformed all batch methods slightly. With ten passes
pair with the smallest eigenvalue for the projections. The overthedata,theERKLEmethodsoutperformedthebatch
τ hyperparameter was chosen to be the best performing methodsbyalargemargin. Inaddition,thebatchmethods
settingovertenvaryingoptions. Thetimedexperimentswere leveloffmorequicklythantheERKLEmethods,indicating
[email protected] that if more triplets were obtained, the ERKLE methods
16GBofRAMandthesinglethreadoptionenabled. Each wouldfurtheroutperformeventhebatchmethods.Webelieve
experimentwasperformedwithtentrials,eachwithdifferent, thattheseresultsshowthatbyminimizingtheexpectedrisk
randomlychosentest,trainandvalidationsets. Theerrorbars directly,ERKLEisabletolearnamoregeneralkernelthan
inthegraphsrepresentthe95%confidenceinterval. batchmethodsthatminimizeempiricalrisk.
Figure1cshowstheperformanceoftwoERKLEmeth-
5.1 Small-ScaleSyntheticData Ourfirstexperimentisto odsandtwobatchRCKLmethodsasafunctionofhowmany
testeachmethodonanideal,small-scale,syntheticdataset. effective“passes”eachmethodperformedonthedata. For
Wecreatedthesyntheticdatasetbyfirstgenerating100data ERKLE,thisamountstothesettingoftheβ parameter. For
points(n=100)inR50 fromN (0,1). Usingthedistances thebatchRCKLmethods,thisisthenumberoffullgradient
7000 0.85
PA−ERKLE (β = 1) PA−ERKLE (β = 1)
GNMDS−Batch (1 step) 0.8 GNMDS−Batch (1 step)
6000 GNMDS−Batch (2 steps) GNMDS−Batch (2 steps)
0.75
5000 0.7
Run time (seconds)34000000 Normalized test error00..056.556
2000 0.5
0.45
1000
0.4
00 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.350 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of observed triplets Number of observed triplets
(a)Runtime(seconds)vs.numberofobservedtriplets (b)Testerrorvs.#ofobservedtriplets
Figure2: Resultsfromexperimentsonthelarge-scalesyntheticdataset(5trials)
stepsittakes. Eachmethodwasrunoveralltrainingtriplets havingfasterruntime.
with the step size δ validated on the test set for the batch
j
methods. Thiseffectivelymeasurestrainingcostasafunc- 5.3 MusicArtistSimilarity Forthelasttwoexperiments
tionofpassesthroughthedata,andthus,isindependentof weperformedevaluationsonreal-worlddatasets. First,we
implementation. Clearly,ifonlyfewpassesthroughthedata performedanexperimentusingrelativecomparisonsamong
canbeperformed,thenERKLEisthebetterchoice. popular music artists gathered from a web survey. The
aset400 data set [7] contains 16,385 relative comparisons
5.2 Large-ScaleSyntheticData Next,weevaluatedhow over 412 artists. We randomly chose 10000 triplets as the
PA-ERKLEcomparedtobatchGNMDSintermsofpractical trainset,1000asthevalidationsetfortheτ parameter,and
runtimeonalargescaleexperiment. Forthisexperiment,we the rest were used as the test set. The aset400 data set
generated5000datapointsinthesamemannerinwhichthe presentsachallengenotpresentinthesyntheticdata: Ithasa
small-scalesyntheticdatawasgenerated. 10000randomly moderateamountofconflictingtriplets,thusmethodsusedin
generated triplets were used as the train set and 50000 theevaluationmustdealwithnoisewithinthetriplets.
were used as the test set. The batch methods were run in Discussion: Figure 3a shows how ERKLE and batch
mini-batches of 500 triplets due to time constraints. The RCKL methods generalize to the test set. STE-ERKLE
hyperparameter τ and the step size δ were chosen as the performsconsiderablyworsethantheothermethods,most
i
settingsthatbestperformedonthetestset. Thisexperiment likelyduetothenoiseintheobservedtriplets.Theprobability
wasrunover5trials,eachwithadifferenttrainandtestset. pK usedinSTE-ERKLEdecaysrapidly. Thus,tripletsthat
t
Discussion: Figure2ashowsthecumulativeruntimeof are in agreement with previously obtained triplets do not
onepassofPA-ERKLE,and1and2stepsofbatchGNMDS. influencethelearnedkernelgreatly. However,aconflicting
Thetimesshownforthebatchmethodsareforthebestchosen triplet will make STE-ERKLE perform a relatively more
τ andnotforthetotaltimeittooktofindit. Thefigureshows drasticupdate. PA-ERKLE,however,ismuchmorerobust
thatasinglepassofPA-ERKLEisoftensignificantlyfaster tonoiseduetotheminimalstepsizetakentosatisfyatriplet.
thanasinglegradientstepofbatchGNMDS.Twostepsof Becauseofthis, PA-ERKLEperformsaswellasthebatch
GNMDS takes even longer. ERKLE can perform online methodsandoftenbetterwhenmultiplepassesaretaken.
updatesmuchfasterduetotheefficientprojectionprocedure Figure3bshowsthetrainingerrorsofeachmethod. This
aswellastheabilitytoskipcertainprojectionsbyestimating figurehighlightshowwelleachmethodfitstotheobserved
the lower bound. In this experiment, the mean number of triplets. TheSTE-ERKLEmodelsaregreatlyeffectedbythe
eigenvalue/eigenvector computations over the 5 trials was presenceofconflictsinthattheydonotlearnakernelthatfits
724.2withastandarddeviationof3.7. HencePA-ERKLE toalargenumberoftheobservedtriplets. PA-ERKLE,onthe
wasabletoskiptheprojectionsteproughly93%ofthetime. otherhand,isabletofitbettertothesetofobservedtriplets,
Figure2bdepictsthetesterrorsofeachmethod. Initially, thusresultinginbettertestaccuracy,aswell.
thebatchmethodsperformbetter,butaround2500triplets, Aspreviouslydiscussed,dissimilarfrombatchmethods
PA-ERKLEoutperformsthebatchmethods. Thisexperiment ERKLE does not use trace regularization. Experimentally,
indicates that PA-ERKLE can achieve competitive results however,weneverthelessfindthatourmethodoutperforms
withbatchmethodsinasinglepassoverthedata,andproduce batchmethodsthatusetraceregularization,ineitherproduc-
trulyonlinesolutionsinsteadofmini-batchsolutionswhile inglow-rankorhigh-rankkernels. Todemonstratethis, in
0.55 0.18 250
STE−ERKLE (δ = 1 β = 1)
0.5 SPTAE−E−ERRKKLELE (β ( δ= =1 )1 β = 5) 0.16
0.45 PSATE−E−BRaKtLcEh (β = 5) 0.14 200
GNMDS−Batch
Normalized test error0.003..534 Normalized train error000...0001.6821 Learned kernel rank110500
STE−ERKLE (δ = 1 β = 1)
0.25 0.04 SPTAE−E−ERRKKLELE (β ( δ= =1 )1 β = 5) 50
0.2 0.02 PSATE−E−BRaKtLcEh (β = 5) STE−Batch
GNMDS−Batch GNMDS−Batch
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 00 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 00 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of observed triplets Number of observed triplets Number of observed triplets
(a)Testerrorvs.numberofobservedtriplets (b)Trainerrorvs.#ofobservedtriplets (c)Kernelrankvs.#ofobservedtriplets
Figure3: Resultsfromexperimentsontheaset400dataset(10trials)
Figure3cweplottheranksofthekernelslearnedbythebatch errorcomparedtoallothermethods,indicatingthatevenin
methods. Inourexperiments,therangeofpotentialτ values such a challenging scenario, the passive-aggressive update
wassetsothatthebatchmethodsneverchoseeithertheup- schememinimallyinterfereswithpreviouslyobtainedtriplets.
perorlowerbound. Wedidthistoensurethattherangeof
regularizationoptionsweresufficientlystrictorlenient. We 6 ConclusionandFutureWork
observethatthebatchmethodsgenerallyproducelow-rank Inthiswork,wedevelopedamethodtolearnaPSDkernel
kernelsunderasmallnumberoftriplets,butasthenumber matrixfromrelativecomparisonsgiveninanonlinefashion.
of triplets are observed the rank increases. Our method is By taking advantage of the sparse and low-rank structure
abletobettergeneralizewithoutusingtraceregularization, of the online formulation, we show how to take stochastic
regardlessofthepreferredrank,duetothePAupdatesonly gradient descent updates of complexity O(n2). We show
satisfyingtripletstothenecessaryextent. howpassive-aggressiveonlinelearningbenefitsourmethod
intermsofgeneralizingtounseentriplets,andinconjunction
5.4 OutdoorSceneSimilarity Ourfinalexperimentused withthestochasticgradientstructure,enablesustoperform
tripletsover200randomlychosenimagesofscenesfromthe a small number of necessary PSD projections in practice.
Outdoor Scene Recognition (OSR) data set [19]. Relative Experimentally,weshowonsyntheticandreal-worlddatathat
comparison queries were posed to 20 people via an online ourmethodlearnskernelsthatgeneralizeaswellandoften
system. After an initial 1200 randomly chosen queries bettertoheldoutrelativecomparisonsthanbatchmethods,
were answered (every object appeared as the head of a whiledemonstratingimprovedrun-timeperformance.
triplet6times),20“rounds”of200tripletswereadaptively For future work, we wish to improve online RCKL
chosenaccordingtotheadaptiveselectioncriterionin[25], in three ways. First, will explore the use of online trace
resulting in 3600 total triplets. For each trial of this regularization. If trace regularization is naively applied to
experiment, 1000 triplets were randomly chosen as the the stochastic gradient in (4.9), the update becomes full-
test set, 1000 as the train set, and 600 were used as the rankandourefficientprojectionprocedurecannotbeused.
validation set for the τ parameter. This experiment is However,anefficientupdateschemeshouldbepossibleifthe
especially challenging for two reasons. First, this is the kernelitselfislow-rank. Wewillinvestigatenovelmethods
smallest experiment in terms of triplets, highlighting how for appropriately weighting the trace in an online manner,
the methods perform with little feedback. In addition, the so that we are consistent with the parameter-free property
adaptive selection algorithm chooses relative comparison ofPA-ERKLE.Second,PA-ERKLEperformedwellinour
queries with the highest information gain, meaning, the experiments with moderate triplet noise, however, it could
tripletsareintentionallychosentogivedisparateinformation be beneficial to explicitly handle conflicting triplets when
abouttherelationshipsamongobjects. they are observed. This can be done out of model using a
Discussion: Figure 4a depicts test errors on each denoising method [17], or in model using a threshold on
method. WeobservethatSTE-ERKLEconsistentlyoutper- the passive-aggressive learning rate. Finally, one of the
formsSTE-Batch,andinparticularSTE-ERKLEperforms mainbenefitsofhavinganonlinelearningalgorithmisthe
well under a small number of triplets relative to all other natural application of active learning methods. Prior work
methods. PA-ERKLEiscomparableoroutperformsitsbatch hasproposedanadaptiveselectionschemewhichoperatesin
counterpartinGNMDS-Batch,givenenoughtriplets(atleast mini-batches[25];however,suchaschemeistooexpensive
500). However,PA-ERKLEperformsquitewellintraining
0.6 0.1
STE−ERKLE (δ = 1 β = 1) STE−ERKLE (δ = 1 β = 1)
0.58 PA−ERKLE (β = 1) 0.09 PA−ERKLE (β = 1)
STE−Batch STE−Batch
GNMDS−Batch 0.08 GNMDS−Batch
0.56
0.07
Normalized test error00..055.245 Normalized train error000...000456
0.03
0.48
0.02
0.46 0.01
0.440 100 200 300 400 500 600 700 800 900 1000 00 100 200 300 400 500 600 700 800 900 1000
Number of observed triplets Number of observed triplets
(a)Testerrorvs.#ofobservedtriplets (b)Trainerrorvs.#observedtriplets
Figure4: ResultsfromexperimentsontheOSRdataset(10trials)
tobeappliedonline.Wewillinvestigatenoveladaptivetriplet B ERKLEwithMultiplePasses
selectionmethodswhicharebothefficientandinformative.
Algorithm2ERKLEwithMultiplePasses
A DerivationofSTEPassive-AggressiveStepSize Input: β : #oftripletssteppedover
ToderivetheSTEversionofthepassive-aggressivestepsize 1: K0 ←I
wewishtosolvethefollowingoptimization(4.19): 2: forj =1,2,...do
msδ.jitn. δpj2K(cid:48)j ≥P,δ ≥0 534::: iKKfj(cid:48)jj ←←>2KΠβ1Sjt+−h1(cid:0)eKn−(cid:48)jδ(cid:1)j∇l(Kj−1,tj)
tj j 6: fork =1,2,...,β−1do
AswiththeGNMDSderivationwiththeassumptionthatthe 7: Randomlyselectt(cid:48)from{t1,t2,...,tj}
tripletisnotsatisfiedbyaprobabilitygreaterthanorequalto 8: K(cid:48)j ←Kj −δj+k∇l(Kj−1,t(cid:48))
Pm,akoinnlgytahpeopsoitsiivteivveacluoensotrfaδijntcoannsδajtirsefdyutnhdeafintr.stIncoandsdtirtaiionnt,, 190:: endKfojr←Π1S+(cid:0)K(cid:48)j(cid:1)
thesmallestδj thatsatisfiestheremainingconstraintisthe 11: endif
onethatmakesthelefthandsideexactlyzero. Asaresult, 12: endfor
theinequalityconstraintcanbehandledasequality. Next,we
taketheLagrangian: Algorithm 1 is much like the original ERKLE algorithm.
δ2 +α(log(P)−log(1−P)) Here,afterasufficientnumberoftripletshavebeenobtained
j (cid:0) (cid:1) (inourexperiments,wechose2β),β−1tripletsareselected
+α d (a,b)−d (a,c) −10δ α
Kj−1 Kj−1 j every iteration from all previously observed triplets (for a
TakingthepartialderivativeoftheLagrangianwithrespect totalofβ updatesperiteration). Thesetripletsarestepped
to δ , setting it to 0, and solving for δ results in δ = 5α. over as done in the original ERKLE algorithm. For our
j j j
SubstitutingthisbackintotheLagrangianmakesit: randomselectionusedinourexperiments,wesimplyselected
uniformly at random with replacement from the obtained
−25α2 +α(log(P)−log(1−P))
(cid:0) (cid:1) triplets. More sophisticated random selection procedures
+α d (a,b)−d (a,c)
Kj−1 Kj−1 may be used in order ensure triplets obtained initially do
TakingthepartialderivativeoftheLagrangianwithrespect notgetselecteddrasticallymoretimesthanthoseobtained
toα,settingitto0,andsolvingforαresultsin: later. For instance, when a triplet gets chosen on line 7,
onecouldreducetheprobabilityofthattripletbeingchosen
log(P)−log(1−P)+d (a,b)−d (a,c)
α= Kj−1 Kj−1 subsequently.
50
Substitutingthisintothesolutionforδj givesus: References
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