Typical Neuron Error Back-Propagation
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1 x Mutayer Notaton y Notaton ayer of neuron abeed 1,, N neuron n ayer = vector of output from neuron n ayer nput ayer = x (the nput pattern) output ayer = y (the actua output) = weght between ayer and +1 Probem: fnd how output y vary wth weght j ( = 1,, ) 11/29/ /29/04 2 Typca Neuron Error Bac-Propagaton h e w compute E tartng wth at ayer ( = 1) and worng bac to earer ayer ( = 2,K,1) N /29/ /29/04 4 1
2 Deta Vaue Output-ayer Neuron Convenent to brea dervatve by chan rue : E = E h 1 1 h et So = E h E = 1 h 1 N h = y E t 11/29/ /29/04 6 Output-ayer Dervatve (1) Output-ayer Dervatve (2) = E h = h = d ( t ) 2 dh = 2 ( t ) ( t ) 2 ( ) d = 2 t ( ) h dh h 1 = 1 E = ( ) ( ) where = 2 t h = 1 11/29/ /29/04 8 2
3 Hdden-ayer Neuron Hdden-ayer Dervatve (1) N 1 1 h N 1 N N E Reca E = 1 h 1 = E = E +1 h +1 h h h = h +1 h h +1 m = m h h m +1 ( ) = d h = h d h ( ) = +1 h = h ( ) = h ( ) +1 11/29/ /29/04 10 Hdden-ayer Dervatve (2) h 1 = 1 E = 1 1 where = h 1 1 ( ) +1 = d = 1 j d Dervatve of Sgmod 1 Suppoe = ( h)= (ogtc gmod) 1+ exp( h) ( ) D h = D h [ 1+ exp( h) ] 1 = [ 1+ exp( h) ] 2 D h 1+ e h = ( 1+ e h ) 2 e h e h ( )= 1+ e h ( ) 2 1 e h 1+ eh = = 1+ e h h 1+ e 1+ e 1 h 1+ e h = (1 ) 11/29/ /29/
4 Summary of Bac-Propagaton Agorthm Output-ayer Computaton 1 = 1 Output ayer : = 2 ( 1 ) ( t ) E = 1 1 Hdden ayer : = 1 E = 1 1 ( ) +1 N h 1 = 2 ( 1 ) t ( ) 2 = y t 11/29/ /29/ Hdden-ayer Computaton N = 1 1 = 1 h 1 ( ) +1 11/29/ N N +1 E Tranng Procedure Batch earnng on each epoch (pa through a the tranng par), weght change for a pattern accumuated weght matrce updated at end of epoch accurate computaton of gradent Onne earnng weght are updated after bac-prop of each tranng par uuay randomze order for each epoch approxmaton of gradent Doen t mae much dfference 11/29/
5 Summaton of Error Surface E 1 E 2 E Gradent Computaton n Batch earnng E 1 E 2 E 11/29/ /29/04 18 Gradent Computaton n Onne earnng E 1 E 2 E The Goden Rue of Neura Net Neura Networ are the econd-bet way to do everythng! 11/29/ /29/
6 Compex Sytem VIII. Revew of Key Concept Many nteractng eement oca v. goba order: entropy Scae (pace, tme) Phae pace Dffcut to undertand Open ytem 11/29/ /29/04 22 Many Interactng Eement Mavey parae Dtrbuted nformaton torage & proceng Dverty avod premature convergence avod nfexbty Compementary Interacton Potve feedbac / negatve feedbac Ampfcaton / tabzaton Actvaton / nhbton Cooperaton / competton Potve / negatve correaton 11/29/ /29/
7 Emergence & Sef-Organzaton Mcrodecon ead to macrobehavor Crcuar cauaty (macro / mcro feedbac) Coevouton predator/prey, Red Queen effect gene/cuture, nche contructon, Badwn effect Pattern Formaton Exctabe meda Ampfcaton of random fuctuaton Symmetry breang Specfc dfference v. generc dentty Automatcay adaptve 11/29/ /29/04 26 Stgmergy Contnuou (uanttatve) Dcrete (uatatve) Coordnated agorthm non-confctng euentay ned Emergent Contro Stgmergy Entranment (dtrbuted ynchronzaton) Coordnated movement through attracton, repuon, oca agnment n concrete or abtract pace Cooperatve tratege nce & forgvng, but recproca evoutonary tabe trategy 11/29/ /29/
8 Attractor Cae pont attractor cycc attractor chaotc attractor Ban of attracton Imprnted pattern a attractor pattern retoraton, competon, generazaton, aocaton 11/29/04 29 ofram Cae Ca I: pont Ca II: cycc Ca III: chaotc Ca IV: compex (edge of chao) pertent tate mantenance bounded cycc actvty goba coordnaton of contro & nformaton order for free 11/29/04 30 Energy / Ftne Surface Decent on energy urface / acent on ftne urface yapunov theorem to prove aymptotc tabty / convergence Soft contrant atfacton / reaxaton Gradent (teepet) acent / decent Adaptaton & credt agnment Baed Randomne Exporaton v. expotaton Bnd varaton & eectve retenton Innovaton v. ncrementa mprovement Peudo-temperature Dffuon Mxed tratege 11/29/ /29/
9 Natura Computaton Toerance to noe, error, faut, damage Generaty of repone Fexbe repone to novety Adaptabty Rea-tme repone Optmaty econdary Student Coure Evauaton! (Do t onne) 11/29/ /29/
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