Outputfunction.isapointwhenthedimensionoftheinputis$N=1$(aswesawine.g.,Example2ofthepreviousSection),alinewhen$N=2$(aswesawine.g,Example3ofthepreviousSection),andismoregenerallyforarbitray$N$ahyperplanedefinedintheinputspaceofadataset.g\left(b,\boldsymbol{\omega}\right)=\frac{1}{P}\sum_{p=1}^P\text{log}\left(1+e^{-y_p\left(b+\mathbf{x}_p^T\boldsymbol{\omega}^{\,}_{\,}\right)}\right)+\lambda\,\left\Vert\boldsymbol{\omega}\right\Vert_2^2w_1\\ThisiswhythecostiscalledSoftmax,sinceitderivesfromthegeneralsoftmaxapproximationtothemaxfunction.Inparticular-aswewillseehere-theperceptronprovidesasimplegeometriccontextforintroducingtheimportantconceptofregularization(anideawewillseeariseinvariousformsthroughouttheremainderofthetext).Thismeansthat-accordingtoequation(4)-thatforeachofour$P$pointswehavethat,$$g_p\left(\mathbf{w}\right)=\text{soft}\left(0,-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{\,}\right)=\text{log}\left(e^{0}+e^{-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{\,}}\right)=\text{log}\left(1+e^{-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{\,}}\right)Westartwithbasicsofmachinelearninganddiscussseveralmachinelearningalgorithmsandtheirimplementationaspartofthiscourse./ProcSet[/PDF/Text]44.5b,θ,representstheoffset,andhasthesamefunctionasinthesimpleperceptron-likenetworks./Resources10RTheperceptronisanalgorithmusedforclassifiers,especiallyArtificialNeuralNetworks(ANN)classifiers.Thedefaultcoloringschemeweusehere-matchingtheschemeusedinthepreviousSection-istocolorpointswithlabely_p=-1blueandy_p=+1red.Somecommonusecasesincludepredictingcustomerdefault(YesorNo),predictingcustomerchurn(customerwillleaveorstay),diseasefound(positiveornegative).Thepiecewiselinearperceptronproblemappearsasanevolutionofthepurelylinearperceptronoptimizationproblemthathasbeenrecentlyinvestigatedin[1].\mbox{soft}\left(s_{0},s_{1}\right)\approx\mbox{max}\left(s_{0},s_{1}\right).Noticethen,asdepictedvisuallyinthefigureabove,thatapropersetofweights\mathbf{w}definealineardecisionboundarythatseparatesatwo-classdatasetaswellaspossiblewithasmanymembersofoneclassaspossiblelyingaboveit,andlikewiseasmanymembersaspossibleoftheotherclasslyingbelowit.120obj<<Remember,asdetailedabove,wecanscaleanylineardecisionboundarybyanon-zeroscalarCanditstilldefinesthesamehyperplane.Becauseofthisthevalueof\lambdaistypicallychosentobesmall(andpositive)inpractice,althoughsomefine-tuningcanbeuseful.#fairness.Wecanseeherebythetrajectoryofthesteps,whicharetravelinglinearlytowardsthemininumoutat\begin{bmatrix}-\infty\\\infty\end{bmatrix},thatthelocationofthelineardecisionboundary(hereapoint)isnotchangingafterthefirststeportwo.Costfunctionofaneuralnetworkisageneralizationofthecostfunctionofthelogisticregression.Gradientdescentisanoptimizationalgorithmusedtofindthevaluesofparameters(coefficients)ofafunction(f)thatminimizesacostfunction(cost).4.Inlayman’sterms,aperceptronisatypeoflinearclassifier.Anotherlimitationarisesfromthefactthatthealgorithmcanonlyhandlelinearcombinationsoffixedbasisfunction....perceptron./Contents100RBothapproachesaregenerallyreferredtointhejargonofmachinelearningasregularizationstrategies.Intherightpanelbelowweshowthecontourplotoftheregularizedcostfunction,andwecanseeitsglobalminimumnolongerliesatinfinity.\underset{b,\,\boldsymbol{\omega}_{\,}}{\,\,\,\,\,\mbox{minimize}\,\,\,}&\,\,\,\,\frac{1}{P}\sum_{p=1}^P\text{log}\left(1+e^{-y_p\left(b+\mathbf{x}_p^T\boldsymbol{\omega}^{\,}_{\,}\right)}\right)\\$$,Nowifwetakethedifferencebetweenourdecisionboundaryanditstranslationevaluatedat$\mathbf{x}_p^{\prime}$and$\mathbf{x}_p$respectively,wehavesimplifying,Wecanalwayscomputetheerror-alsocalledthesigneddistance-ofapoint\mathbf{x}_ptoalineardecisionboundaryintermsofthenormalvector\boldsymbol{\omega}.Thecostfunctionis,sothederivativewillbe./Length207Multiplyingthecostfunctionbyascalardoesnotaffectthelocationofitsminimum,sowecangetawaywiththis.Oftendenedbythefreeparametersinalearningmodelwithaxedstructure(e.g.,aPerceptron){Selectionofacostfunction{Learningruletondthebestmodelintheclassoflearningmodels.TheL2-RegularizedcostfunctionoflogisticregressionfromthepostRegularizedLogisticRegressionisgivenby,Where$${\lambda\over2m}\sum_{j=1}^n\theta_j^2$$istheregularizationtermLaterI’llshowthatthisisgradientdescentonacostfunction,butfirstlet’sseeanapplicationofbackprop./Filter/FlateDecodeg\left(\mathbf{w}\right)=\sum_{p=1}^Pg_p\left(\mathbf{w}\right)=\underset{p=1}{\overset{P}{\sum}}\text{log}\left(1+e^{-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{\,}}\right)...butthecostfunctioncan’tbenegative,sowe’lldefineourcostfunctionsasfollows,If,-Y(X.W)>0,Ifthecostfunctionisconvex,thenalocallyoptimalpointisgloballyoptimal(providedtheoptimizationisoveraconvexset,whichitisinourcase)Optimizationcontinued.Tobegintoseewhythisnotationisusefulfirstnotehow-geometricallyspeaking-thefeature-touchingweights\boldsymbol{\omega}definethenormalvectorofthelineardecisionboundary.Themoregeneralcasefollowssimilarlyaswell.\begin{aligned}Perceptronisafunctionthatmapsitsinput“x,”whichismultipliedwiththelearnedweightcoefficient;anoutputvalue”f(x)”isgenerated.Asystem(eitherhardwareorsoftware)thattakesinoneormoreinputvalues,runsafunctionontheweightedsumof…>>endobj/Type/Pageg_p\left(\mathbf{w}\right)=\text{max}\left(0,\,-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{\,}\right)=04.Insteadoflearningthisdecisionboundaryasaresultofanonlinearregression,theperceptronderivationdescribedinthisSectionaimsatdeterminingthisideallinearydecisionboundarydirectly.orequivalentlyas\mbox{max}\left(s_{0},\,s_{1}\right)=\mbox{log}\left(e^{s_{0}}\right)+\mbox{log}\left(e^{s_{1}-s_{0}}\right).\left(\mathbf{x}_p^{\prime}-\mathbf{x}_p\right)^T\boldsymbol{\omega}=\left\Vert\mathbf{x}_p^{\prime}-\mathbf{x}_p\right\Vert_2\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2=d\,\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2,Againwecandosospecificallybecausewechosethelabelvalues$y_p\in\{-1,+1\}$.Forbinaryclassificationproblemseachoutputunitimplementsathresholdfunctionas:.\frac{b+\overset{\,}{\mathbf{x}}_{\,}^T\boldsymbol{\omega}}{\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2}=\frac{b}{\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2}+\overset{\,}{\mathbf{x}}_{\,}^T\frac{\boldsymbol{\omega}}{\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2}=0Ineachoftheepochs,thecost…w_2\\Section1.6generalizesthediscussionbyintroducingtheperceptroncostfunction,wedonotchangethenatureofourdecisionboundaryandnowourfeature-touchingweightshaveunitlengthas$\left\Vert\frac{\boldsymbol{\omega}}{\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2}\right\Vert_2=1$.Within5stepswehavereachedapointprovidingaverygoodfittothedata(hereweplotthe$\text{tanh}\left(\cdot\right)$fitusingthelogisticregressoionperspectiveontheSoftmaxcost),andonethatisalreadyquitelargeinmagnitude(ascanbeseenintherightpanelbelow).ThetransferfunctionofthehiddenunitsinMLFnetworksisalwaysasigmoidorrelatedfunction.Alternatively,youcouldthinkofthisasfoldingthe2intothelearningrate./ProcSet[/PDF/Text]Simplyput:ifalinearactivationfunctionisused,thederivativeofthecostfunctionisaconstantwithrespectto(w.r.t)input,sothevalueofinput(toneurons)doesnotaffecttheupdatingofweights.Itmakessensetoleavethe1/mterm,though,becausewewantthesamelearningrate(alpha)to…AlsolearnhowtoimplementAdalineruleinANNandtheprocessofminimizingcostfunctionsusingGradientDescentrule.\vdots\\%PDF-1.5/Contents30RAppliedMachineLearning-BeginnertoProfessionalcoursebyAnalyticsVidhyaaimstoprovideyouwitheverythingyouneedtoknowtobecomeamachinelearningexpert.\mathring{\mathbf{x}}^{T}\mathbf{w}^{\,}=0,/Type/Paged=\frac{\beta}{\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2}=\frac{b+\overset{\,}{\mathbf{x}}_{p}^T\boldsymbol{\omega}}{\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2}.activationfunction.becauseclearlyadecisionboundarythatperfectlyseparatestwoclassesofdatacanbefeature-weightnormalizedtopreventitsweightsfromgrowingtoolarge(anddivergingtooinfinity).pointisclassifiedincorrectly.130obj<<����f^ImXE�*�.SincetheReLUcostvalueisalreadyzero,itslowestvalue,thismeansthatwewouldhaltourlocaloptimizationimmediately./Length313Theabovementionedformulagivestheoverallcostfunction,andtheresidualorlossofeachhiddenlayernodeisthemostcriticaltoconstructadeeplearningmodelbasedonstackedsparsecoding./Font<</F2240R/F41110R/F2750R/F66150R/F3160R>>aclassificationalgorithmthatmakesitspredictionsbasedonalinearpredictorfunctioncombiningasetofweightswiththefeaturevector.ItnotonlyprohibitstheuseofNewton'smethodbutforcesustobeverycarefulabouthowwechooseoursteplengthparameter$\alpha$withgradientdescentaswell(asdetailedintheexampleabove)./ProcSet[/PDF/Text]>>Imaginefurtherthatweareextremelyluckyandourinitialization$\mathbf{w}^0$producesalineardecisionboundary$\mathring{\mathbf{x}}_{\,}^T\mathbf{w}^{0}=0$withperfectsepearation.Tothisend,theresidualsofthehiddenlayeraredescribedindetailbelow,andthecorrespondingrelationshipis…Wekeepsteppingthroughweightspace…CEcostfunction,softmaxoutputs,sigmoidhiddenactivationsIneachcase,applicationofthegradientdescentlearningalgorithm(bycomputingthe...Wecanthenusethechainrulesforderivatives,asfortheSingleLayerPerceptron,toBecausethesepoint-wisecostsarenonnegativeandequalzerowhenourweightsaretunedcorrectly,wecantaketheiraverageovertheentiredatasettoformapropercostfunctionas,Indeedifwemultiplyourinitialization$\mathbf{w}^0$byanyconstant$C>1$wecandecreasethevalueofanynegativeexponentialinvolvingoneofourdatapointssince$e^{-C}<1$andso,.\text{soft}\left(s_0,s_1,...,s_{C-1}\right)=\text{log}\left(e^{s_0}+e^{s_1}+\cdots+e^{s_{C-1}}\right),Wecandothisbydirectlycontrolingthesizeofjust$N$oftheseweights,anditisparticularlyconvenienttodosousingthefinal$N$featuretouchingweights$w_1,\,w_2,\,...,w_N$becausethesedefinethenormalvectortothelineardecisionboundary$\mathring{\mathbf{x}}_{\,}^T\mathbf{w}^{\,}=0$.APerceptronisanalgorithmusedforsupervisedlearningofbinaryclassifiers.Obviouslythisimplementsasimplefunctionfrommulti-dimensionalrealinputtobinaryoutput.CostFunctionofNeuralNetworks.Combiningtheactivationfunctionandcostfunctionwithasimplelinearperceptrongivesusamorecomplexderivativebecauseofallthenesting.PyCaret’sClassificationModuleisasupervisedmachinelearningmodulewhichisusedforclassifyingelementsintogroups.ThislikewisedecreasestheSoftmaxcostaswellwiththeminimumachievedonlyas$C\longrightarrow\infty$.Becausewecanalwaysfliptheorientationofanidealhyperplanebymultiplyingitby$-1$(orlikewisebecausewecanalwaysswapourtwolabelvalues)wecansaymorespecificallythatwhentheweightsofahyperplanearetunedproperlymembersoftheclass$y_p=+1$lie(mostly)'above'it,whilemembersofthe$y_p=-1$classlie(mostly)'below'it.Sincethequantity$-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{0}<0$itsnegativeexponentialislargerthanzeroi.e.,$e^{-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{0}}>0$,whichmeansthatthesoftmaxpoint-wisecostisalsononnegative$g_p\left(\mathbf{w}^0\right)=\text{log}\left(1+e^{-\overset{\,}{y}_{p}\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{0}}\right)>0$andhencetootheSoftmaxcostisnonnegativeaswell,$$Otherwise,thewholenetworkwouldcollapsetolineartransformationitselfthusfailingtoserveitspurpose.NotehoweverthatregardlessofthescalarC>1valueinvolvedthedecisionboundarydefinedbytheinitialweights\mathring{\mathbf{x}}_{\,}^T\mathbf{w}^{0}=0doesnotchangelocation,sincewestillhavethatC\,\mathring{\mathbf{x}}_{\,}^T\mathbf{w}^{0}=0(indeedthisistrueforanynon-zeroscalarC).https://sebastianraschka.com/Articles/2015_singlelayer_neurons.htmlAswehaveseenwithlogisticregressionwetreatclassificationasaparticularformofnonlinearregression(employing-withthechoiceoflabelvaluesy_p\in\left\{-1,+1\right\}-atanhnonlinearity).Theformerstrategyisstraightfoward,requiringslightadjustmentstothewaywehavetypicallyemployedlocaloptimization,butthelatterapproachrequiressomefurtherexplanationwhichwenowprovide.Notethattheperceptroncostalwayshasatrivialsolutionat\mathbf{w}=\mathbf{0},sinceindeedg\left(\mathbf{0}\right)=0,thusonemayneedtotakecareinpracticetoavoidfindingit(orapointtooclosetoit)accidentally.Synonymforloss.ThissectionprovidesabriefintroductiontothePerceptronalgorithmandtheSonardatasettowhichwewilllaterapplyit.Practicallyspeakingtheirdifferenceslieinhowwell-foraparticulardataset-onecanoptimizeeitherone,alongwith(whatisveryoftenslight)differencesinthequalityofeachcostfunction'slearneddecisionboundary.$$.endobjThispreventsthedivergenceoftheirmagnitudesinceiftheirsizedoesstarttogrowweourentirecostfunction'suffers'becauseofit,andbecomeslarge.Insimpleterms,anidentityfunctionreturnsthesamevalueastheinput.\mathring{\mathbf{x}}_{p}^T\mathbf{w}^{\,}<0&\,\,\,\,\text{if}\,\,\,y_{p}=-1.\text{soft}\left(s_0,s_1,...,s_{C-1}\right)\approx\text{max}\left(s_0,s_1,...,s_{C-1}\right)Incidentally,it'sworthnotingthatconventionsvaryaboutscalingofthecostfunctionandofmini-batchupdatestotheweightsandbiases.Thisprovidesuswithindividualnotationforthebiasandfeature-touchingweightsas,$$2.Tocomputeourdesirederrorwewanttocomputethesigneddistancebetween\mathbf{x}_panditsverticalprojection,i.e.,thelengthofthevector\mathbf{x}_p^{\prime}-\mathbf{x}_ptimesthesignof\betawhichhereis+1sinceweassumethepointliesabovethedecisionboundaryhence\beta>0,i.e.,d=\left\Vert\mathbf{x}_p^{\prime}-\mathbf{x}_p\right\Vert_2\text{sign}\left(\beta\right)=\left\Vert\mathbf{x}_p^{\prime}-\mathbf{x}_p\right\Vert_2.Howeveramorepopularapproachinthemachinelearningcommunityisto'relax'thisconstrinaedformulationandinsteadsolvethehighlyrelatedunconstrainedproblem.Whenminimizedappropriatelythiscostfunctioncanbeusedtorecovertheidealweightssatisfyingequations(3)-(5)asoftenaspossible.Thisresemblesprogress,butit'snotthesolution.The‘HowtoTrainanArtificialNeuralNetworkTutorial’focusesonhowanANNistrainedusingPerceptronLearningRule.stream/Length436"+/r��6rY��o�|���z����96���6'��K��q����~��Sl��3Z���yk�}ۋ�P�+_�7�λ��P}��rZG�G~+�C-=��`�%+R�,�ح�Q~g�}5h�݃O��5��Fұ��i���j��i3Oโ�=��i#���FA�������f��f1���$$,Sincebothformulaeareequalto$\left(\mathbf{x}_p^{\prime}-\mathbf{x}_p\right)^T\boldsymbol{\omega}$wecansetthemequaltoeachother,whichgives,.Gradientdescentisbestusedwhentheparameterscannotbecalculatedanalytically(e.g.Aperceptronconsistsofoneormoreinputs,aprocessor,andasingleoutput.\text{(bias):}\,\,b=w_0\,\,\,\,\,\,\,\,\text{(feature-touchingweights):}\,\,\,\,\,\,\boldsymbol{\omega}=>>endobjSuchaneuralnetworkiscalledaperceptron./MediaBox[00841.89595.276]Themultilayerperceptronisauniversalfunctionapproximator,asprovenbytheuniversalapproximationtheorem.Howeverwestilllearnaperfectdecisionboundaryasillustratedintheleftpanelbyatightlyfitting$\text{tanh}\left(\cdot\right)$function.Aswesawinourdiscussionoflogisticregression,inthesimplestinstanceourtwoclassesofdataarelargelyseparatedbyalineardecisionboundarywitheachclass(largely)lyingoneitherside..Anotherapproachistocontrolthemagnitudeoftheweightsduringtheoptimizationprocedureitself./Filter/FlateDecode/Font<</F2240R/F2750R/F3160R>>ParametersX{array-like,sparsematrix},shape(n_samples,n_features)Subsetofthetrainingdata.endstreamInotherwords,afterthefirstfewstepsweeachsubsequentstepissimplymultiplyingitspredecessorbyascalarvalue$C>1$.20obj<<•PerceptronAlgorithmSimplelearningalgorithmforsupervisedclassificationanalyzedviageometricmarginsinthe50’s[Rosenblatt’57].AscanbeseeninFig.WritteninthisComingbackAdaline,thiscostfunctionisJJisdefinedastheSumofsquarederrors(SSE)betweenthecalculatedoutcomebytheactivationfunctionandthetrueclasslabelNote:Heretheoutcomeisarealvalue(outputbytheactivationfunction),not{1,…2.Wemarkthispoint-to-decision-boundarydistanceonpointsinthefigurebelow,heretheinputdimension$N=3$andthedecisionboundaryisatruehyperplane.Theparameter$\lambda$isusedtobalancehowstronglywepressureonetermortheother.Liketheirbiologicalcounterpart,ANN’sarebuiltuponsimplesignalprocessingelementsthatareconnectedtogetherintoalargemesh.whichistheSoftmaxcostwesawpreviouslyderivedfromthelogisticregressionperspectiveontwo-classclassificationinthepreviousSection.10obj<<NowthatwehavesolvingODEsasjustalayer,wecanadditanywhere.Notethatweneednotworrydividingbyzeroheresinceifthefeature-touchingweights$\boldsymbol{\omega}$wereallzero,thiswouldimplythatthebias$b=0$aswellandwehavenodecisionboundaryatall.OfcoursewhentheSoftmaxisemployedfromtheperceptronperspectivethereisnoqualitativedifferencebetweentheperceptronandlogisticregressionatall.Soif-inparticular-wemultiplyby$C=\frac{1}{\left\Vert\overset{\,}{\boldsymbol{\omega}}\right\Vert_2}$wehave,.ThisrelaxedformoftheproblemconsistsinminimizingacostfunctionnthatisalinearcombinationofouroriginalSoftmaxcostthemagnitudeofthefeatureweights,Now,Iwilltrainmymodelinsuccessiveepochs.Peoplesometimesomitthe$\frac{1}{n}$,summingoverthecostsofindividualtrainingexamplesinsteadofaveraging.Matterssuchasobjectiveconvergenceandearlystoppingshouldbehandledbytheuser.
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