赢得netflix推荐系统大奖的算法

VI.EXTENSIONSTORESTRICTEDBOLTZMANN

MACHINES

A.RBMswithconditionalvisibleunits

WeextendedtheRestrictedBoltzmannMachines(RBM)modelsuggestedby[12].Theoriginalworkshowedasigni -cantperformanceboostbymakingthehiddenunitsconditionalonwhichmoviesthecurrentuserhasrated.Wehavefoundthatasimilarlysigni cantperformanceboostisachievedbyconditioningthevisibleunits.

Intuitivelyspeaking,eachoftheRBMvisibleunitscorre-spondstoaspeci cmovie.Thustheirbiasesrepresentmovie-biases.However,weknowthatotherkindsofbiasesaremoresigni ly,user-bias,single-dayuserbias,andfrequency-basedmoviebias.Therefore,weaddthosebiasestothevisibleunitsthroughconditionalconnections,whichdependonthecurrentlyshownuseranddate.

Letusborrowtheoriginalnotation[12],whichusesaconditionalmultinomialdistributionformodelingeachcolumnoftheobservedvisiblebinaryratingmatrixV:

p(vkexp(bki+∑Fj=1hjWiki

=1|h)=

j)∑l=1exp(bl

i+∑j=1hjWilj)

(19)

Weextendthisconditionaldistributionasfollows.Letthecurrentinstancerefertouseruanddatet.Weaddauser/datebiasterm:p(vkexp(bkut+bku+bki+∑Fj=1hjWiki

=1|h,u,t)=

j)∑l=1exp(blut+blu+bli+∑j=1hjWilj)

(20)

wherebkuisauser-speci cparameter,andbkruleis

utisauser×date-speci cvariable.Thelearning bku=ε1( vki data vki T), bkut=ε2( vki data vki T).

Wehavefounditconvenientheretodeviatefromthemini-batchlearningschemesuggestedin[12],andtolearnbkandbkweupdateeachu

immediatelyutinafullyonlinemanner.Thatis,afterobservingacorrespondingtraininginstance.Theusedlearningratesare:ε1=0.0025andε2=0.008.Noticethatunlessotherwisestated,weusethesameweightdecaysuggestedintheoriginalpaper,whichis0.001.

Consideringthesigni canteffectoffrequencies,wecanfurtherconditiononthemhere.Letthecurrentinstancerefertouseruanddatetwithassociatedfrequencyf.Theresultingconditionaldistributionisasfollows:

p(vkexp(bkif+bkut+bku+bki+∑Fj=1hjWikj)

i

=1|h,u,t,f)=

∑l=1exp(bif+but+blu+bi+∑j=1hjWij)

(21)

wherebkrulewillifisamovie×frequency-speci cvariable.Itslearning

be

bkif=ε3( vki data vk

i T).

whereε3=0.0002.Onlinelearningisusedaswell.

Whenusingfrequencieswealsoemployedthefollowingmodi cationtothevisible-hiddenweights,whichwasbroughttoourattentionbyourcolleaguesatthePragmaticTheory

7

team.InsteadofusingtheoriginalweightsWikj,wewilluse

frequency-dependentweightsWikjf,whicharefactoredas

Wikjf=Wikj·(1+Cfj).

WeuseonlinelearningforthenewparametersCfj,withlearningrateof1e-5.

Asitturnedout,thisextensionoftheweightsbarelyimprovesperformancewhenfrequency-biasesarealreadypresent,whilebeingsomewhatonerousintermsofrunningtime.Thus,weareunlikelytorecommendit.Still,itispartofourfrequency-awareRBMimplementation.B.RBMswithday-speci chiddenunits

2Motivated

bytheday-speci cuserfactor(14),wealso

triedtocreateday-speci cRBMhiddenunits.OntopoftheFhiddenunits,wealsoaddGday-speci cunits.Forauserthatratedonrdifferentdays,wecreaterparallelcopiesoftheGday-speci cunits.Allthoseparallelcopiessharethesamehidden-visibleweights,hiddenbiases,andconditionalconnections.Also,eachparallelcopyisconnectedonlytothevisibleunitscorrespondingtotheratingsgiveninitsrespectiveday.

Toputthisformally,foraday-speci chiddenunitindexedbyj,withacorrespondingratingdatet,weusetheindicatorvectorrt∈{0,1}ntodenotewhichmoviesthecurrentuserratedondatet.Then,theBernoullidistributionformodelinghiddenuserfeaturesbecomesn

n

p(hj=1|V,rt)=σ(bj+∑

riviWik

j+

Dij).

(22)

i=1k∑

5

tk=1

i∑rit=1

Inourimplementationweusedthismodeltogetherwith

thefrequency-biasedRBM.Allparametersassociatedwiththeday-speci cunitswerelearnedinmini-batches,astheirnonday-speci ccounterparts,butwithalearningrateof0.005andaweightdecayof0.01.Resultswerenotencouraging,andfurtherre nementisstillneeded.Stillasinglevariantofthisschemecontributestotheblend.C.What’sintheblend?

Firstanoteonover-training.OurparametersettingmadetheRBMtypicallyconvergeatlowestQuizRMSEwith60–90iterations.However,fortheoverallblenditwasbene cialtocontinueover ttingthetrainingset,andlettheRBMrunformanyadditionaliterations,aswillbeseeninthefollowing.WeincludeintheblendfourvariantsoftheRBMmodelfollowing(20):

1)F=200,#iterations=52,RMSE=0.89512)F=400,#iterations=62,RMSE=0.89423)F=400,#iterations=82,RMSE=0.89444)F=400,#iterations=100,RMSE=0.8952

TherearealsotwovariantsoftheRBMwithfrequencies(21):

1)F=200,#iterations=90,RMSE=0.89282)F=200,#iterations=140,RMSE=0.8949

2An

ideadevelopedtogetherwithMartinPiotte