desiredmimiai.net

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034fNeuralNetNotesOctober28,2010Thesenotesareasupplementtomaterialpresentedinlecture.
Ilayoutthemathematicsmoreprettilyandextendtheanalysistohandlemultiple-neuronsperlayer.
Also,Idevelopthebackpropagationrule,whichisoftenneededonquizzes.
IuseanotationthatIthinkimprovesonpreviousexplanations.
Thereasonisthatthenotationhereplainlyassociateseachinput,output,andweightwithareadilyidentifiedneuron,aleft-sideoneandaright-sideone.
Whenyouarriveattheupdateformulas,youwillhavelesstroublerelatingthevariablesintheformulastothevariablesinadiagram.
Onetheotherhand,seeingyetanothernotationmayconfuseyou,soifyoualreadyfeelcomfortablewithasetofupdateformulas,youwillnotgainbyreadingthesenotes.
ThesigmoidfunctionThesigmoidfunction,y=1/(1+ex),isusedinsteadofastepfunctioninartificialneuralnetsbecausethesigmoidiscontinuous,whereasastepfunctionisnot,andyouneedcontinuitywheneveryouwanttousegradientascent.
Also,thesigmoidfunctionhasseveraldesirablequalities.
Forexample,thesigmoidfunction'svalue,y,approaches1asxbecomeshighlypositive;0asxbecomeshighlynegative;andequals1/2whenx=0.
Betteryet,thesigmoidfunctionfeaturesaremarkablysimplederivativeoftheoutput,y,withrespecttotheinput,x:dyd1=()dxdx1+exd=(1+ex)1dx=1*(1+ex)2*ex*11ex=*1+ex1+ex11+ex1=*1+ex1+ex11+ex11+ex1+ex1+ex=y(1y)Thus,remarkably,thederivativeoftheoutputwithrespecttotheinputisexpressedasasimplefunctionoftheoutput.
TheperformancefunctionThestandardperformancefunctionforgauginghowwellaneuralnetisdoingisgivenbythefollowing:1P=(dsampleosample)222wherePistheperformancefunction,dsampleisthedesiredoutputforsomespecificsampleandosampleistheobservedoutputforthatsample.
Fromthispointforward,assumethatdandoarethedesiredandobservedoutputsforaspecificsamplesothatweneednotdragasubscriptaroundasweworkthroughthealgebra.
ThereasonforchoosingthegivenformulaforPisthattheformulahasconvenientproperties.
Theformulayieldsamaximumato=dandmonotonicallydecreasesasodeviatesfromd.
Moreover,thederivativeofPwithrespecttooissimple:dPd1=[(do)2]dodo2=2*(do)1*12=doGradientascentBackpropagationisaspecializationoftheideaofgradientascent.
YouaretryingtofindthemaximumofaperformancefunctionP,bychangingtheweightsassociatedwithneurons,soyoumoveinthedirectionofthegradientinaspacethatgivesPasafunctionoftheweights,w.
Thatis,youmoveinthedirectionofmostrapidascentifwetakeastepinthedirectionwithcomponentsgovernedbythefollowingformula,whichshowshowmuchtochangeaweight,w,intermsofapartialderivative:PΔw∝wTheactualchangeisinuencedbyarateconstant,α;accordingly,thenewweight,w,isgivenbythefollowing:w=w+α*PwGradientdescentIftheperformancefunctionwere12(dsampleosample)2insteadof12(dsampleosample)2,thenyouwouldbesearchingfortheminimumratherthanthemaximumofP,andthechangeinwwouldbesubtractedfromwinsteadofadded,sowwouldbewα*wPinsteadofw+α*wP.
Thetwosignchanges,oneintheperformancefunctionandtheotherintheupdateformulacancel,sointheend,yougetthesameresultwhetheryouusegradientascent,asIprefer,orgradientdescent.
ThesimplestneuralnetConsiderthesimplestpossibleneuralnet:oneinput,oneoutput,andtwoneurons,theleftneuronandtherightneuron.
Anetwithtwoneuronsisthesmallestthatillustrateshowthederivativescanbecomputedlayerbylayer.
3xSigmoidWlplilolxSigmoidWrprorirLeftneuronRightneuronNotethatthesubscriptsindicatelayer.
Thus,il,wl,pl,andolaretheinput,weight,product,andoutputassociatedwiththeneuronontheleftwhileir,wr,pr,andoraretheinput,weight,product,andoutputassociatedwiththeneuronontheright.
Ofcourse,ol=ir.
Supposethattheoutputoftherightneuron,or,isthevaluethatdeterminesperformanceP.
TocomputethepartialderivativeofPwithrespecttotheweightintherightneuron,wr,youneedthechainrule,whichallowsyoutocomputepartialderivativesofonevariablewithrespecttoanotherintermsofanintermediatevariable.
Inparticular,forwr,youhavethefollowing,takingortobetheintermediatevariable:PPor=*wrorwrNow,youcanrepeat,usingthechain-ruletoturnworrintooprr*wprr:PPorpr=**wrorprwrConveniently,youhaveseentwoofthederivativesalready,andthethird,wprr=(wrw*rol),iseasytocompute:P=[(dor)]*[or(1or)]*[ir]wrRepeatingtheanalysisforwlyieldsthefollowing.
Eachlineisthesameasthepreviously,exceptthatonemorepartialderivativeisexpandedusingthechainrule:P=P*orwlorwl=P*or*prorprwl=P*or*pr*olorprolwl=P*or*pr*ol*plorprolplwl=[(dor)]*[or(1or)]*[wr]*[ol(1ol)]*[il]4Thus,thederivativeconsistsofproductsoftermsthathavealreadybeencomputedandtermsinthevicinityofwl.
Thisisclearerifyouwritethetwoderivativesnexttooneanother:P=(dor)*or(1or)*irwrP=(dor)*or(1or)*wr*ol(1ol)*ilwlYoucansimplifytheequationsbydefiningδsasfollows,whereeachdeltaisassociatedwitheithertheleftorrightneuron:δr=or(1or)*(dor)δl=ol(1ol)*wr*δrThen,youcanwritethepartialderivativeswiththeδs:P=ir*δrwrP=il*δlwlIfyouaddmorelayerstothefrontofthenetwork,eachweighthasapartialderivativesthatiscomputedlikethepartialderivativeoftheweightoftheleftneuron.
Thatis,eachhasapartialderivativedeterminedbyitsinputanditsdelta,whereitsdeltainturnisdeterminedbyitsoutput,theweighttoitsright,andthedeltatoitsright.
Thus,fortheweightsinthefinallayer,youcomputethechangeasfollows,whereIusefasthesubscriptinsteadofrtoemphasizethatthecomputationisfortheneuroninthefinallayer:Δwf=α*if*δfwhereδf=of(1of)*(dof)Forallotherlayers,youcomputethechangeasfollows:Δwl=α*il*δlwhereδl=ol(1ol)*wr*δrMoreneuronsperlayersOfcourse,youreallywantbackpropagationformulasfornotonlyanynumberoflayersbutalsoforanynumberofneuronsperlayer,eachofwhichcanhavemultipleinputs,eachwithitsownweight.
Accordingly,youneedtogeneralizeinanotherdirection,allowingmultipleneuronsineachlayerandmultipleweightsattachedtoeachneuron.
Thegeneralizationisanadventureinsummations,withlotsofsubscriptstokeepstraight,butintheend,theresultmatchesintuition.
Forthefinallayer,theremaybemanyneurons,sotheformula'sneedanindex,k,indicatingwhichfinalnodeneuronisinplay.
Foranyweightcontained5inthefinal-layerneuron,fk,youcomputethechangeasfollowsfromtheinputcorrespondingtotheweightandfromtheδassociatedwiththeneuron:Δw=α*i*δfkδfk=ofk(1ofk)*(dkofk)Notethattheoutputofeachfinal-layerneuronoutputissubtractedfromtheoutputdesiredforthatneuron.
Forotherlayers,theremayalsobemanyneurons,andtheoutputofeachmayinuencealltheneuronsinthenextlayertotheright.
Thechangeinweighthastoaccountforwhathappenstoallofthoseneuronstotheright,soasummationappears,butotherwiseyoucomputethechange,asbefore,fromtheinputcorrespondingtotheweightandfromtheδassociatedwiththeneuron:Δw=α*i*δliδli=oli(1oli)*wli→rj*δrjjNotethatwli→rjistheweightthatconnectsthejthright-sideneurontotheoutputoftheithleft-sideneuron.
SummaryOnceyouunderstoodhowtoderivetheformulas,youcancombineandsimplifytheminpreparationforsolvingproblems.
Foreachweight,youcomputetheweight'schangefromtheinputcorrespondingtotheweightandfromtheδassociatedwiththeneuron.
Assumingthatδisthedeltaassociatedwiththatneuron,youhavethefollowing,wherew→rjistheweightconnectingtheoutputoftheneuronyouareworkingon,theithleft-sideneuron,tothejthright-sideneuron,andδrjistheδassociatedwiththatright-sideneuron.
δo=o(1o)*(do)forthefinallayerδli=oli(1oli)*wli→rj*δrjotherwisejThatis,youcomputedchangeinaneuron'sw,ineverylayer,bymultiplyingαtimestheneuron'sinputtimesitsδ.
Theδisdeterminedforallbutthefinallayerintermsoftheneuron'soutputandalltheweightsthatconnectthatoutputtoneuronsinthelayertotherightandtheδsassociatedwiththoseright-sideneurons.
Theδforeachneuroninthefinallayerisdeterminedonlybytheoutputofthatneuronandbythedifferencebetweenthedesiredoutputandtheactualoutputofthatneuron.
6WeightsanddeltasinlayertotherightNeuronwithweighttobeadjustedw→r1wxoixxΣ∫w→rNWeighttobeadjustedxxxΣ∫δ1xxxΣ∫δΝMITOpenCourseWarehttp://ocw.
mit.
edu6.
034ArtificialIntelligenceFall2010ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.
mit.
edu/terms.