minimumnumberq=q(k;)ofqueriesrequiredforanyalgorithmAto-testk-juntas - PDF document

minimumnumberq=q(k;)ofqueriesrequiredforanyalgorithmAto-testk-juntas?Background.TherstresultrelatedtotestingjuntaswasobtainedbyBellare,Goldreich,andSudan[2]inthecontextoftestinglongcodes.ThatresultwasgeneralizedbyParnas,Ron,andSamorodnitsky[18]toobtainanalgorithmfor-testing1-juntaswithonlyO(1=)queries.Thenextimportantstepintestingk-juntaswastakenbyFischer,Kindler,Ron,Safra,andSamorodnitsky[10],whodevelopedmultiplealgorithmsfortest-ingk-juntaswithpoly(k)=queries.Thosealgorithmswereparticularlysignif-icantforshowingexplicitlythattestingjuntascanbedonewithaquerycom-plexityindependentofthetotalnumberofvariables.Themostquery-ecientalgorithmstheypresentedrequire~O(k2)=queries1to-testk-juntas.Fischeretal.[10]alsogavetherstnon-triviallowerboundonthequerycomplexityforthetestingjuntasproblem.Theyshowedthatanynon-adaptivealgorithmfor-testingk-juntasrequiresatleast~ (p k)queriesandconjecturedthatthetruequerycomplexityfornon-adaptivealgorithmsisk2=queries.ChocklerandGutfreund[8]improvedthelowerboundfortestingjuntasbyshowingthatallalgorithms{adaptiveornon-adaptive{for-testingk-juntasrequire (k)queries.Thisresultappliesforallvaluesof1=8,butthebounditselfdoesnotincreaseasdecreases.Ourresultsandtechniques.Ourmainresultisanimprovementontheupperboundforthequerycomplexityofthejuntatestingproblem.Theorem1.1.Thepropertyofbeingak-juntacanbe-testedbyanon-adaptivealgorithmwith~O(k3=2)=queries.Thenewalgorithmpresentedinthisarticleistherstfortestingjuntaswithanumberofqueriessub-quadraticink,anddisprovesthelowerboundconjectureofFischeretal.OuralgorithmisbasedonanalgorithmofFischeretal.fortestingjuntas[10,x4.2]).TheobservationthatledtothedevelopmentofthenewalgorithmisthatthealgorithmofFischeretal.canbebrokenupintotwoseparatetests:a\blocktest"andasimple\samplingtest".Inthisarticle,wegeneralizethesamplingtest,andweestablishastructuralLemmaforfunctionsthatare-farfrombeingk-juntastoshowhowthetwotestscanbecombinedto-testk-juntasmoreeciently.Oursecondresultisanimprovedlowerboundonthenumberofqueriesrequiredfortestingjuntaswithnon-adaptivealgorithms.Thenewboundistherstlowerboundforthequerycomplexityofthejuntatestingproblemthatincorporatestheaccuracyparameter. 1Hereandintherestofthisarticle,the~O(
)notationisusedtohidepolylogfactors.(i.e.,~O�f(x)
=O�f(x)logcf(x)
and~ �f(x)
= �f(x) logcf(x)
forsomec0.) TheIndependenceTest.AfunctionfissaidtobeindependentofasetS[n]ofcoordinatesifVrf(S)=0.Thedenitionofvariationsuggestsanat-uraltestforindependence:IndependenceTest[10]:Givenafunctionf:f0;1gn!f0;1gandasetS[n],generatetwoinputsx;y2f0;1gnindependentlyanduniformlyatrandom.Iff(x)=f(xSyS),thenaccept;otherwise,reject.LetusdeneIndependenceTest(f,S,m)tobethealgorithmthatrunsminstancesoftheIndependenceTestonfandSandacceptsifandonlyifeveryinstanceoftheIndependenceTestaccepts.Bythedenitionofvariation,thisalgorithmacceptswithprobability�1�Vrf(S)
m.Inparticular,thistestalwaysacceptswhenfisindependentofthesetSofcoordinates,andrejectswithprobabilityatleast1�whenVrf(S)ln(1=)=m.3TheAlgorithmforTestingJuntasInthissection,wepresentthealgorithmfor-testingk-juntaswith~O(k3=2)=queries.Thealgorithmhastwomaincomponents:theBlockTestandtheSamplingTest.WeintroducetheBlockTestinSection3.1andtheSam-plingTestinSection3.2.Finally,inSection3.3weshowhowtocombinebothteststoobtainanalgorithmfortestingjuntas.3.1TheBlockTestThepurposeoftheBlockTestistoacceptk-juntasandrejectfunctionsthathaveatleastk+1coordinateswith\large"variation.TheBlockTestrstrandomlypartitionsthecoordinatesin[n]intossetsI1;:::;Is.ItthenappliestheIndependenceTesttoblocksofthesesetstoidentifythesetsofcoordinatesthathavelowvariation.ThetestacceptsifallbutatmostkofthesetsI1;:::;Isareidentiedashavinglowvariation.ThefullalgorithmispresentedinFig.1.TheBlockTestisbasedonFischeretal.'snon-adaptivealgorithmfortestingjuntas[10,x4.2],whichusesaverysimilartest.3AsthefollowingtwoPropositionsshow,withhighprobabilitytheBlockTestacceptsk-juntasandrejectsfunctionswithk+1coordinateswithvariationatleast.Proposition3.1(Completeness).Fix�0,andletf:f0;1gn!f0;1gbeak-junta.ThentheBlockTestacceptsfwithprobabilityatleast1�.Proof.LetIjbeasetthatcontainsonlycoordinatesiwithvariationVrf(i)=0.Inagivenround,theprobabilitythatIjisincludedinBTandnoneofthesets 3TheprincipaldierencebetweenourversionoftheBlockTestandFischeretal.'sversionofthetestisthatin[10],thesetTisgeneratingbyincludingexactlykindiceschosenatrandomfrom[s]. SamplingTest(f,k,l,,)Additionalparameters:r=d128k2ln(2=)=l2e,m=dln(2r=)=e1.Initializethesuccesscounterc 0.2.Foreachofrrounds,2.1.PickarandomsubsetT[n]byincludingeachcoordinateindependentlywithprobability1=k.2.2.IfIndependenceTest(f,T,m)accepts,setc c+1.3.Acceptfifc=r(1�1=k)k�l=16k;otherwiserejectf. Fig.2.Thealgorithmforthesamplingtest.3.2TheSamplingTestThepurposeoftheSamplingTestistoacceptk-juntasandrejectfunctionsthathavealargenumberofcoordinateswithnon-zerovariation.TheSamplingTest,asitsnameimplies,usesasamplingstrategytoesti-matethenumberofcoordinateswithnon-negligiblevariationinagivenfunctionf.ThesamplingtestgeneratesarandomsubsetT[n]ofcoordinatesineachround,andusestheIndependenceTesttodetermineiffisindependentofthecoordinatesinT.Thetestacceptswhenthefractionofroundsthatpasstheindependencetestisnotmuchsmallerthantheexpectedfractionofroundsthatpassthetestwhenfisak-junta.ThedetailsofthealgorithmarepresentedinFig.2.Proposition3.3(Completeness).Fix�0,l2[k].Letf:f0;1gn!f0;1gbeak-junta.ThentheSamplingTestacceptsfwithprobabilityatleast1�.Proof.Whenfisak-junta,theprobabilitythatthesetTinagivenroundcontainsonlycoordinatesiwithvariationVrf(i)=0isatleast(1�1=k)k.Whenthisoccurs,thesetTalsohasvariationVrf(T)=0.LettbethenumberofroundsforwhichthesetTsatisesVrf(T)=0.ByHoeding'sbound,Prt r(1�1=k)k�l 16ke�2r
(l=16k)2=2whenr128k2ln(2=)=l2.EverysetTwithvariationVrf(T)=0alwayspassestheIndependenceTest,soctandthecompletenessclaimfollows.utProposition3.4(Soundness1).Fix�0,l2[k].Letf:f0;1gn!f0;1gbeafunctionforwhichthereisasetS[n]ofsizejSj=k+lsuchthateverycoordinatei2ShasvariationVrf(i).ThentheSamplingTestrejectsfwithprobabilityatleast1�.Proof.Inagivenround,theprobabilitythattherandomsetTdoesnotcontainanyofthek+lcoordinateswithlargevariationis(1�1=k)k+l.Whenlk,(1�1=k)l1�l=2k,andwhenk2,(1�1=k)k1=4.Sotheprobabilitythat Fact3.8(Fischeretal.[10])Foranyfunctionf:f0;1gn!f0;1gandsetsofcoordinatesS;T[n],thefollowingtwopropertieshold:(i)Urf;S(T)Vrf(T),and(ii)Urf;S([n])=Vrf(S).WearenowreadytocompletetheproofofProposition3.5.Proof(ofProposition3.5).TherearetwowaysinwhichtheSamplingTestcanacceptf.Thetestmayacceptfifatleasta(1�1=k)k�1=16fractionoftherandomsetsThavevariationVrf(T).Alternatively,thetestmayalsoacceptifsomeofthesetsTwithvariationVrf(T)passtheIndepen-denceTest.Byourchoiceofmandtheunionbound,thislattereventhappenswithprobabilityatmost=2.SotheproofofProposition3.5iscompleteifwecanshowthattheprobabilityoftheformereventhappeningisalsoatmost=2.LettrepresentthenumberofroundswheretherandomsetThasvariationVrf(T).WewanttoshowthatPrt=r(1�1=k)k�1=16=2.Infact,since(1�1=k)k1=4forallk2,itsucestoshowthatPr[t=r3=16]=2.Inagivenround,theexpecteduniquevariationoftherandomsetTwithrespecttoSinfisE[Urf;S(T)]=Xi2[n]1 kUrf;S(i)=Urf;S([n]) k=Vrf(S) k 2k;wherethethirdequalityusesFact3.8(ii).ByProperty(i)oftheProposition,Urf;S(T)isthesumofnon-negativevariablesthatareboundedaboveby.SowecanapplyLemma3.6with=1=32HktoobtainPr[Urf(T)]e 2eke 32Hk�1:ByFact3.8(i)andthefactthate 2eke 32Hk�11=8forallk1,wehavethatE[t=r]=Pr[Vrf(T)]1=8:ThenalresultfollowsfromanapplicationofHoeding'sinequalityandthechoiceofr.utTheSamplingTestalgorithmmakes2mqueriestofineachround,sothetotalquerycomplexityofthealgorithmis2rm=O(k2log(k=l)=l2).3.3TheJuntaTestIntheprevioustwosubsections,wedenedtwotests:theBlockTestthatdistinguishesk-juntasfromfunctionswithk+1coordinateswithlargevariation,andtheSamplingTestthatdistinguishesk-juntasfromfunctionsthathavesomevariationdistributedoveralargenumberofcoordinates.ThefollowingstructuralLemmaonfunctionsthatare-farfrombeingk-juntasshowsthatthesetwotestsaresucientfortestingjuntas. andtheunionbound,k-juntasareacceptedbytheJuntaTestwithprobabilityatleast2=3.Letfbeanyfunctionthatis-farfrombeingak-junta.IffsatisesProp-erty(i)ofLemma3.9withparametert=64,considertheminimumintegerl02[k]forwhichthereisasetS[n]ofsizek+l0suchthateverycoordinatei2ShasvariationVrf(i) 64Hkl0.Ifl0dk1=2e,thenbyProposition3.2,theBlockTestrejectsfwithprobability1�&#x]TJ/;༔ ; .96;& T; 10;&#x.758;&#x 0 T; [0;2=3.Ifl0k1=2,thenbyPropo-sition3.4,theSamplingTestwiththeparameterlthatsatisesll02lrejectsthefunctionwithprobability1�&#x]TJ/;༔ ; .96;& T; 10;&#x.758;&#x 0 T; [0;2=3.IffsatisesProperty(ii)ofLemma3.9,byProposition3.5,thelastSam-plingTestrejectsthefunctionwithprobability1�&#x]TJ/;༔ ; .96;& T; 10;&#x.758;&#x 0 T; [0;2=3.SinceLemma3.9guaranteesthatanyfunction-farfrombeingak-juntamustsatisfyatleastoneofthetwopropertiesoftheLemma,thiscompletestheproofofsoundnessoftheJuntaTest.TocompletetheproofofTheorem1.1,itsucestoshowthattheJuntaT-estisanon-adaptivealgorithmandthatitmakesonly~O(k3=2)=queriestothefunction.Thenon-adaptivityoftheJuntaTestisapparentfromthefactthatallqueriestotheinputfunctioncomefromindependentinstancesoftheInde-pendenceTest.ThequerycomplexityoftheJuntaTestalsofollowsfromtheobservationthateachinstanceoftheBlockTestortheSamplingTestinthealgorithmrequires~O(k3=2)=queries.SincethereareatotalofO(logk)callstothosetests,thetotalquerycomplexityoftheJuntaTestisalso~O(k3=2)=.ut4TheLowerBoundInthissection,weshowthateverynon-adaptivealgorithmfor-testingk-juntasmustmakeatleastmin�~ (k=);2k=k
queriestothefunction.ToproveTheorem1.2,weintroducetwodistributions,DyesandDno,overfunctionsthatarek-juntasandfunctionsthatare-farfromk-juntaswithhighprobability,respectively.Wethenshowthatnodeterministicnon-adaptivealgo-rithmcanreliablydistinguishbetweenfunctionsdrawnfromDyesandfunctionsdrawnfromDno.Thelowerboundonallnon-adaptivealgorithmsfor-testingk-juntasthenfollowsfromanapplicationofYao'sMinimaxPrinciple[20].AcentralconceptthatweuseextensivelyintheproofofTheorem1.2isChocklerandGutfreund'sdenitionoftwins[8].Denition4.1.Twovectorsx;y2f0;1gnarecalledi-twinsiftheydierex-actlyintheithcoordinate(i.e.,ifxi6=yiandxj=yjforallj2[n]nfig).Thevectorsx;yarecalledtwinsiftheyarei-twinsforsomei2[n].WenowdenethedistributionsDyesandDno.TogenerateafunctionfromthedistributionDno,werstdeneafunctiong:f0;1gk+1!f0;1gbysettingthevalueg(x)foreachinputx2f0;1gk+1independentlyatrandom,withPr[g(x)=1]=6.Wethenextendthefunctionoverthefulldomainbydeningf(x)=g(x[k+1])foreveryx2f0;1gn.ThedistributionDyesisdenedtobe ThedistributionsHjandHj�1arenearlyidentical.Theonlydierencebe-tweenthetwodistributionsisthatjisconstrainedtotakethevaluejinHj,whileitisanindependent6-biasedrandomvariableinHj�1.SothestatisticaldistancebetweenHjandHj�1istwicetheprobabilitythatj6=jinHj�1.Thus,PyHj(y)�Hj�1(y)24(1�6)24andtheLemmafollows.utWithLemma4.3,wecannowboundthestatisticaldistancebetweentheresponsesobservedwhentheinputfunctionisdrawnfromDyesorDno.Lemma4.4.LetQbeasequenceofqqueriescontainingtpairsoftwins.LetRyesandRnobethedistributionsoftheresponsestothequeriesinQwhentheinputfunctionisdrawnfromDyesorDno,respectively.ThenXy2f0;1gqRyes(y)�Rno(y)24t k+1:Proof.SinceRyesisamixturedistributionoverR(1)yes;:::;R(k+1)yes,thenXyRyes(y)�Rno(y)=Xyk+1Xi=1R(i)yes(y) k+1�Rno(y)1 k+1k+1Xi=1XyR(i)yes(y)�Rno(y):ByLemma4.3,theaboveequationisupperboundedby1 k+1Pk+1i=124ti,wheretirepresentsthenumberofi-twinsinQ.Lemma4.4thenfollowsfromthefactthatt=Pk+1i=1ti.utThepreviousLemmaboundsthestatisticaldistancebetweentheresponsesobservedfromafunctiondrawnfromDyesorDnowhenwehaveaboundonthenumberoftwinsinthequeries.ThefollowingLemmashowsthatthenumberofpairsoftwinsinasequenceofqqueriescannotbelargerthanqlogq.Lemma4.5.Letfx1;:::;xqgf0;1gnbeasetofqdistinctqueriestoafunc-tionf:f0;1gn!f0;1g.Thenthereareatmostqlogqpairs(xi;xj)suchthatxiandxjaretwins.Proof.Anaturalcombinatorialrepresentationforaqueryx2f0;1gnisasavertexonthen-dimensionalbooleanhypercube.Inthisrepresentation,apairoftwinscorrespondstoapairofverticesconnectedbyanedgeonthehypercube.Sothenumberofpairsoftwinsinasetofqueriesisequaltothenumberofedgescontainedinthecorrespondingsubsetofverticesonthehypercube.TheLemmathenfollowsfromtheEdge-IsoperimetricInequalityofHarper[13],Bernstein[3],andHart[14](seealso[7,x16]),whichstatesthatanysubsetSofqverticesinthebooleanhypercubecontainsatmostqlogqedges.5utWecannowcombinetheaboveLemmastoproveTheorem1.2. 5TheresultofHarper,Bernstein,andHartisslightlytighter,givingaboundofPqi=1h(i),whereh(i)isthenumberofonesinthebinaryrepresentationofi. Acknowledgments.TheauthorwishestothankRyanO'Donnellformanyvaluablediscussionsandsuggestionsduringthecourseofthisresearch.TheauthoralsothanksAnupamGupta,YiWu,andtheanonymousrefereesformanyhelpfulsuggestionsonearlierdraftsofthisarticle.References1.AlpAtcandRoccoA.Servedio.Quantumalgorithmsforlearningandtestingjuntas.QuantumInformationProcessing,6(5):323{348,2007.2.MihirBellare,OdedGoldreich,andMadhuSudan.Freebits,PCPsandnon-approximability{towardstightresults.SIAMJ.Comput.,27(3):804{915,1998.3.ArthurJ.Bernstein.Maximallyconnectedarraysonthen-cube.SIAMJ.Appl.Math.,15(6):1485{1489,1967.4.AvrimBlum.Relevantexamplesandrelevantfeatures:thoughtsfromcomputa-tionallearningtheory.InAAAIFallSymposiumon`Relevance',1994.5.AvrimBlum.Learningafunctionofrrelevantvariables.InProc.16thConferenceonComputationalLearningTheory,pages731{733,2003.6.AvrimBlumandPatLangley.Selectionofrelevantfeaturesandexamplesinmachinelearning.ArticialIntelligence,97(2):245{271,1997.7.BelaBollobas.Combinatorics.Cambridge,1986.8.HanaChocklerandDanGutfreund.Alowerboundfortestingjuntas.InformationProcessingLetters,90(6):301{305,2004.9.IliasDiakonikolas,HominK.Lee,KevinMatulef,KrzysztofOnak,RonittRubin-feld,RoccoA.Servedio,andAndrewWan.Testingforconciserepresentations.InProc.48thSymposiumonFoundationsofComputerScience,pages549{558,2007.10.EldarFischer,GuyKindler,DanaRon,ShmuelSafra,andAlexSamorodnitsky.Testingjuntas.J.Comput.Syst.Sci.,68(4):753{787,2004.11.MiraGonenandDanaRon.Onthebenetsofadaptivityinpropertytestingofdensegraphs.InProc.11thWorkshopRANDOM,pages525{539,2007.12.DavidGuijarro,JunTarui,andTatsuieTsukiji.FindingrelevantvariablesinPACmodelwithmembershipqueries.InProc.10thConferenceonAlgorithmicLearningTheory,pages313{322,1999.13.LarryH.Harper.Optimalassignmentsofnumberstovertices.SIAMJ.Appl.Math.,12(1):131{135,1964.14.SergiuHart.Anoteontheedgesofthen-cube.Disc.Math.,14:157{163,1976.15.JeKahn,GilKalai,andNatiLinial.Thein uenceofvariablesonbooleanfunc-tions.InProc.29thSym.onFoundationsofComputerScience,pages68{80,1988.16.RichardJ.Lipton,EvangelosMarkakis,AranyakMehta,andNisheethK.Vishnoi.OntheFourierspectrumofsymmetricbooleanfunctionswithapplicationstolearningsymmetricjuntas.InProc.20thConferenceonComputationalComplexity,pages112{119,2005.17.ElchananMossel,RyanO'Donnell,andRoccoA.Servedio.Learningfunctionsofkrelevantvariables.J.Comput.Syst.Sci.,69(3):421{434,2004.18.MichalParnas,DanaRon,andAlexSamorodnitsky.Testingbasicbooleanformu-lae.SIAMJ.Discret.Math.,16(1):20{46,2003.19.RonittRubinfeldandMadhuSudan.Robustcharacterizationsofpolynomialswithapplicationstoprogramtesting.SIAMJ.Comput.,25(2):252{271,1996.20.AndrewC.Yao.Probabilisticcomputations:towardsauniedmeasureofcom-plexity.InProc.18thSym.onFoundationsofComput.Sci.,pages222{227,1977.