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1961 1958 1959 1964 1957 1955 1962 1960 1965 1963 1956 1954 UnderstandingHierarchicalMethodsforDifferentiallyPrivateHistogramsWahbehQardaji,WeiningYang,NinghuiLiPurdueUniversity305N.UniversityStreet,WestLafayette,IN47907,USAfwqardaji,yang469,ninghuig@cs.purdue.eduABSTRACTInrecentyears,manyapproachestodi erentiallyprivatelypublishhistogramshavebeenproposed.Severalapproach-esrelyonconstructingtreestructuresinordertodecreasetheerrorwhenanswerlargerangequeries.Inthispaper,weexaminethefactorsa ectingtheaccuracyofhierarchicalap-proachesbystudyingthemeansquarederror(MSE)whenansweringrangequeries.Westartwithone-dimensionalhis-tograms,andanalyzehowtheMSEchangeswithdi eren-tbranchingfactors,afteremployingconstrainedinference,andwithdi erentmethodstoallocatetheprivacybudgetamonghierarchylevels.Ouranalysisandexperimentalre-sultsshowthatcombiningthechoiceofagoodbranchingfactorwithconstrainedinferenceoutperformthecurrents-tateoftheart.Finally,weextendouranalysistomulti-dimensionalhistograms.Weshowthatthebene tsfromemployinghierarchicalmethodsbeyondasingledimensionaresigni cantlydiminished,andwhenthereare3ormoredimensions,itisalmostalwaysbettertousetheFlatmethodinsteadofahierarchy.1.INTRODUCTIONAhistogramisanimportanttoolforsummarizingdata.Inrecentyears,signi cantimprovementshavebeenmadeintheproblemofpublishinghistogramswhilesatisfyingd-i erentiallyprivacy.Theutilitygoalistoensurethatrangequeriesovertheprivatehistogramcanbeansweredasaccu-ratelyaspossible.Thenaiveapproach,whichwecallthe\ratmethod,istoissueacountqueryforeachunitbininthehis-togram,andanswerthesequeriesusingtheLaplacianmech-anismintroducedbyDworketal.[8].Becauseeachsuchqueryhassensitivity1,themagnitudeoftheaddednoiseissmall.Whilethisworkswellwithhistogramsthathaveasmallnumberofunitbins,theerrorduetothismethodin-creasessubstantiallyasthenumberofthebinsincreases.Inparticular,whenansweringrangequeriesusingtheprivatehistogram,themeansquarederrorincreaseslinearlyinthesizeofthequeryrange.Permissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprotorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitationontherstpage.Tocopyotherwise,torepublish,topostonserversortoredistributetolists,requirespriorspecicpermissionand/orafee.ArticlesfromthisvolumewereinvitedtopresenttheirresultsatThe39thInternationalConferenceonVeryLargeDataBases,August26th30th2013,RivadelGarda,Trento,Italy.ProceedingsoftheVLDBEndowment,Vol.6,No.14Copyright2013VLDBEndowment21508097/13/14...$10.00.Hayetal.[14]introducedthehierarchicalmethodforop-timizingdi erentiallyprivatehistograms.Inthisapproach,inadditiontoaskingforcountsofunit-lengthintervals,onealsoasksforcountsoflargerintervals.Conceptually,onecanarrangeallqueriedintervalsintoatree,wheretheunit-lengthintervalsaretheleaves.Thebene tofthismethodisthatarangequerywhichincludesmanyunitbinscanbean-sweredusingasmallnumberofsub-intervalswhichexactlycoverthequeryrange.Thetradeo overthe\ratmethodisthatwhenqueriesareissuedatmultiplelevels,eachquerymustsatisfydi erentialprivacyforasmaller.Werefertothisastheprivacybudgetallocatedto(orconsumedby)thequery.Hayetal.[14]alsointroducednovelconstrainedinferencetechniquestoimproveoverthebasichierarchicalmethod,whichexploitstheobservationthatqueryresult-satdi erentlevelsshouldsatisfycertainconsistencyre-lationshipstoobtainimprovedestimates.Thishierarchicalmethodwithconstrainedinferencehasbeenadoptedbyoth-erresearchers[5,4].Anumberofothermethodshavealsobeendevelopedandappliedtothehistogramprogram.Forexample,Xiaoetal.[21]introducedthePrivletmethod,whichdecomposestheoriginalhistogramusingtheHaarwavelet,thenaddsnoisetothedecomposedcoecients,and nallyreconstruct-sthehistogramusingthenoisycoecients.Thebene tofthismethodisthat,whenansweringrangequeries,noiseaddedtothecoecientscanbepartiallycancelledbecauseofthenatureofHaarwaveletprocessing.Lietal.[15]in-troducedthematrixmechanism,whichtriestooptimizean-swersfora xedsetofcountingqueries.Assolvingtheoptimizationproblemiscomputationallyinfeasible,severalapproximationtechniqueshavebeendeveloped,suchastheeigen-selectalgorithm[16]andthelow-rankapproximationalgorithm[24].Weobservethathierarchicalmethodscanbeparameter-izedinseveraldi erentways,anddi erentparametervaluesmaysigni cantlya ecttheaccuracyoftheresults.Suchpa-rametersincludethebranchingfactorforthehierarchy,aswellasthedivisionofprivacybudgetamongitslevels.Fur-thermore,severalotherparametersalsoa ecttherelativestrengthofeachmethod,includingthetotalnumberofhis-togrambins,thenumberofdimensions,andthedistributionofthequeries.Tothoroughlyunderstandthestrengthsandweaknessesofthesemethods,weneedtounderstandhowtheseparametersa ecttheaccuracy.Inthispaper,wecombinecombinatorialanalysisofhowthedi erentparametersa ecttheaccuracyofeachmethod,withexperimentalevaluationthatvalidatesthetheoretical