ArchitectureandEquilibra结构和平衡刘瑞华罗雪梅导师曾

ArchitectureandEquilibra

结构和平衡

刘瑞华罗雪梅

导师：曾平Chapter6

2004.11.101Chapter6ArchitectureandEquilibriaPerface

lyaoynovstabletheorem2004.11.102Chapter6ArchitectureandEquilibria

6.1NeutralNetworkAsStochasticGradientsystemClassifyNeutralnetworkmodelBytheirsynapticconnectiontopolgiesandbyhowlearningmodifiestheirconnectiontopologies

synapticconnectiontopolgieshowlearningmodifiestheirconnectiontopologies2004.11.103Chapter6ArchitectureandEquilibria

6.1NeutralNetworkAsStochasticGradientsystem2004.11.104Chapter6ArchitectureandEquilibria

6.1NeutralNetworkAsStochasticGradientsystemThreestochasticgradientsystemsrepresentthethreemaincategories:1)Feedforwardsupervisedneuralnetworkstrainedwiththebackpropagation(BP)algorithm.2)Feedforwardunsupervisedcompetitivelearningoradaptivevectorquantization(AVQ)networks.3)Feedbackunsupervisedrandomadaptivebidirectionalassociativememory(RABAM)networks.2004.11.105Chapter6ArchitectureandEquilibria

6.2GlobalEquilibra:convergenceandstabilityNeuralnetwork:synapses,neuronsthreedynamicalsystems:synapsesdynamicalsystems

neuonsdynamicalsystemsjointsynapses-neuronsdynamicalsystemsHistorically,Neuralengineersstudythefirstorsecondneuralnetwork.Theyusuallystudylearningin

feedforwardneuralnetworksandneuralstabilityinnonadaptivefeedbackneuralnetworks.RABAMandARTnetworkdependonjointequilibrationofthesynapticandneuronaldynamicalsystems.2004.11.106Chapter6ArchitectureandEquilibria

6.2GlobalEquilibra:convergenceandstabilityEquilibriumissteadystate.Convergenceissynapticequilibrium.Stabilityisneuronalequilibrium.Moregenerallyneuralsignalsreachsteadystateeventhoughtheactivationsstillchange.WedenotesteadystateintheneuronalfieldNeuronfluctuatefasterthansynapsesfluctuate.Stability-Convergencedilemma:Thesynapsedslowlyencodetheseneuralpatternsbeinglearned;butwhenthesynapsedchange,thistendstoundothestableneuronalpatterns.2004.11.107Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithmsWeshallprovethat:CompetitveAVQsynapticvectorconvergetopattern-classcentroid.TheyvibrateaboutthecentroidinaBrowmianmotionCompetitvelearningadpatively

qunatizestheinputpatternspace

charcaterizesthecontinuousdistributionsofpattern.2004.11.108Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithmsTheRandomIndicatorfunction

Supervisedlearningalgorithmsdependexplicitlyontheindicatorfunctions.Unsupervisedlearningalgorthmsdon’trequirethispattern-classinformation.Centriod

ComptetiveAVQStochasticDifferentialEquations2004.11.109Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithmsTheStochasticunsupervisedcompetitivelearninglaw:WewanttoshowthatatequilibriumWeassumeTheequilibriumandconvergencedependonapproximation(6-11),so6-10reduces:2004.11.1010Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithmsCompetitiveAVQAlgorithms1.Initializesynapticvectors:2.Forrandomsample,findthecloset(“winning”)synapticvector3.UpdatethewiningsynapticvectorsbytheUCL,SCL,orDCLlearningalgorithm.2004.11.1011Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithmsUnsupervisedCompetitiveLearning(UCL)definesaslowlydeceasingsequenceoflearningcoefficientSupervisedCompetitiveLearning(SCL)2004.11.1012Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithmsDifferentialCompetitiveLearning(DCL)denotesthetimechangeofthejthneuron’scompetitivesignal.Inpracticeweonlyusethesignof(6-20)StochasticEquilibriumandConvergenceCompetitivesynapticvectorcovergetodecsion-classcentrols.Maycovergetolocallymaxima.2004.11.1013Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithmsAVQcentroidtheorem:ifacompetitiveAVQsystemconverges,itconvergetothecentroidofthesampleddecisionclass.Proof.SupposethejthneuroninFywinstheactitvecompetition.SupposethejthsynapticvectorcodesfordecisionclassSupposethesynapticvectorhasreachedequilibrium2004.11.1014Chapter6ArchitectureandEquilibria

6.3Synapticconvergencetocentroids:AVQAlgorithms2004.11.1015Chapter6ArchitectureandEquilibria

6.4AVQConvergenceTheoremAVQConvergenceTheorem:Stochasticcompetitivelearningsystemsareasymptoticallystable,andsynapticvectorsconvergetocentroids.Competitivesynapticvectorsconvergeexponentiallyquiklytopattern-classcentroids.Proof.ConsidertherandomquadraticformLThepatternvectorsxdonotchangeintime.2004.11.1016Chapter6ArchitectureandEquilibria

6.4AVQConvergenceTheoremTheaverageE[L]asLyapunovfunctionforthesochastic

competiticedynamicalsystem.Assume:Noiseprocessiszero-meanandindependenceofthenoiseprocesswith“signal”process2004.11.1017Chapter6ArchitectureandEquilibria

6.4AVQConvergenceTheoremSo,onaveragebythelearninglaw6-12,Ifanysynapticvectormovealongitstrajetory.So,thecompetitiveAVQsystemisasymtotically

stabel,andingereralconvergesexponentiallyquicklytoalocallyequilibrium.Suppose

TheneverysynapticvectorhasReachedequilibriumandisconstant.2004.11.1018Chapter6ArchitectureandEquilibria

6.4AVQConvergenceTheoremSincep(x)isanonnegativeweigthfunction.Theweightedintegralofthelearningdifferencemustequalzero:Soequilibriumsynapticvectorequalcentroids.Q.E.D2004.11.1019Chapter6ArchitectureandEquilibria

6.5GlobalstabilityoffeedbackneuralnetworksGlobalstabilityisjointlyneuronal-synapticssteadystate.Globalstabilitytheoremsarepowerfulbutlimited.Theirpower:theirdimensionindependencenonlineargeneralitytheirexponentiallyfastconvergencetofixedpoints.Theirlimitation:donottelluswheretheequilibriaoccurinthestatespace.2004.11.1020Chapter6ArchitectureandEquilibra

6.5GlobalstabilityoffeedbackneuralnetworksStability-ConvergenceDilemmaStability-ConvergenceDilemmaarisefromtheasymmetryinneounalandsynapticfluctuationrates.Neuronschangefasterthansynapseschange.Neuronsfluctuateatthemillisecondlevel.Synapsesfluctuateatthesecondorevenminutelevel.Thefast-changingneuronsmustbalancetheslow-changingsynapses.2004.11.1021Chapter6ArchitectureandEquilibria

6.5GlobalstabilityoffeedbackneuralnetworksStability-ConvergenceDilemma1.Asymmetry:NeuronsinandfluctuatefasterthanthesynapsesinM.2.stability:(patternformation).3.Learning:4.Undoing:theABAMtheoremoffersageneralsolutiontostability-convergencedilemma.2004.11.1022Chapter6ArchitectureandEquilibria

6.6TheABAMTheoremTheABAMTheorem(Adaptive

bidirectionalassociativememory)TheHebbianABAMandcompetitiveABAMmodelsaregloballystabel.HebbianABAMmodel:CompetitiveABAMmodel,replacing6-35with6-362004.11.1023Chapter6ArchitectureandEquilibria

6.6TheABAMTheoremIfthepositivityassumptionsThen,themodelsareasymptoticallystable,andthesquaredactivationandsynapticvelocitiesdecreaseexponentiallyquicklytotheirequilibriumvalues:Proof.

theproofusestheboundedlyapunovfunction

L2004.11.1024Chapter6ArchitectureandEquilibria

6.6TheABAMTheoremMakethedifferenceto6-37:2004.11.1025Chapter6ArchitectureandEquilibria

6.6TheABAMTheoremToproveglobalstabilityforthecompetitvelearninglaw6-36WeprovethestrongerasymptoticstableoftheABAMmodelswiththepositivityassumptions.2004.11.1026Chapter6ArchitectureandEquilibria

6.6TheABAMTheoremAlongtrajectoriesforanynonzerochangeinanyneuronalactivationoranysynapse.Trajectoriesendinequilibriumpoints.Indeed6-43implies:Thesquaredvelocitiesdeceaseexponentiallyquicklybecauseofthestrictnegativityof(6-43)and,toruleoutpathologies.Q.E.D2004.11.1027Chapter6ArchitectureandEquilibria

6.7structuralstabilityofunsuppervisedlearningandRABAMIsunsupervisedlearningstructuralstability?StructuralstabilityisinsensivitytosmallperturbationsStructuralstabilityignoresmanysmallperturbations.Suchperturbationspreservequalitativeproperties.Basinsofattractionsmaintaintheirbasicshape.2004.11.1028Chapter6ArchitectureandEquilibria

6.7StructuralstabilityofunsuppervisedlearningandRABAMRandomAdaptiveBidirectionalAssociativeMemoriesRABAMBrowiandiffusionsperturbRABAMmodel.Thedifferentialequationsin6-33through6-35nowbecomestochasticdifferentialequations,withrandomprocessesassolutions.ThediffusionsignalhebbianlawRABAMmodel:2004.11.1029Chapter6ArchitectureandEquilibria

6.7StructuralstabilityofunsuppervisedlearningandRABAMWiththestochasticcompetitiveslaw:2004.11.1030Chapter6ArchitectureandEquilibria

6.7Structuralstabilityofunsuppervise