SVMs for regression Multilayer neural networks

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1 Lecture SVMs for regresson Muter neur netors Mos Husrecht 539 Sennott Squre Support vector mchne SVM SVM mmze the mrgn round the seprtng hperpne. he decson functon s fu specfed suset of the trnng dt the support vectors.

2 he decson oundr: ˆ he decson: Support vector mchnes ˆ α SV ˆ sgn αˆ SV!!: Decson on ne requres to compute the nner product eteen the empes Smr the optmzton depends on n n J α α α α Nonner cse he ner cse requres to compute he non-ner cse cn e hnded usng set of fetures. Essent e mp nput vectors to rger feture vectors φ It s posse to use SVM formsm on feture vectors Kerne functon φ φ ' Cruc de: If e choose the erne functon se e cn compute ner seprton n the feture spce mpct such tht e eep orng n the orgn nput spce!!!! K ' φ φ '

3 Kerne functon empe Assume [ nd feture mppng tht mps the nput ] nto qudrtc feture set φ [ ] Kerne functon for the feture spce: K ' φ ' φ ' ' ' ' ' ' ' ' ' he computton of the ner seprton n the hgher dmenson spce s performed mpct n the orgn nput spce Nonner etenson Kerne trc Repce the nner product th erne A e chosen erne eds to n effcent computton

4 Ponom erne Kerne functons Lner erne K ' ' [ ] ' K ' Rd ss erne K ' ep ' Kernes Kernes defne smrt mesure : defne dstnce n eteen to oects Desgn crter: e nt ernes to e vd Stsf Mercer condton of postve semdefnteness good emod the true smrt eteen oects pproprte generze e effcent the computton of K s fese NP-hrd proems ound th grphs

5 Kernes Reserch hve proposed ernes for comprson of vret of oects: Strngs rees Grphs Coo thng: SVM gorthm cn e no pped to cssf vret of oects Regresson fnd functon tht fts the dt. A dt pont m e rong due to the nose Ide: Error from ponts hch re cose shoud count s vd nose Lne shoud e nfuenced the re dt not the nose. Support vector mchne for regresson

6 Lner mode rnng dt: n {... } R R Our go s to fnd functon f tht hs t most devton from the ctu otned trget for the trnng dt. f Lner mode Lner functon: f We nt functon tht s: ft: mens tht one sees sm dt ponts re thn ts neghorhood he proem cn e formuted s conve optmzton proem: mnmze suect to A dt ponts re ssumed to e n the neghorhood

7 f Lner mode Re dt: not dt ponts s f nto the neghorhood Ide: penze ponts tht f outsde the neghorhood f Lner mode Lner functon: Ide: penze ponts tht f outsde the neghorhood suect to mnmze C

8 -ntensve oss functon ntensve oss functon otherse for Lner mode Lgrngn tht soves the optmzton proem Optmzton C L η η η η Suect to Prm vres

9 L L C L η Optmzton C L η Dervtves th respect to prm vres C C L η η ω ω Optmzton

10 C C C C L ω η η ω η η Optmzton ] [ : suect to - C L L - Optmzton Mmze the du

11 SVM souton L We cn get: f t the optm souton the Lgrnge mutpers re non-zero on for ponts outsde the nd. Muter neur netors Or nother of modeng nonnertes for regresson nd cssfcton proems

12 Lner unts Lner regresson f d d f Logstc regresson f p g d z d f p d On-ne grdent updte: α f d he sme On-ne grdent updte: α f α f α f Lmttons of sc ner unts Lner regresson f d Logstc regresson f p g d f z p d d d d Functon ner n nputs!! Lner decson oundr!!

13 Regresson th the qudrtc mode. Lmtton: ner hper-pne on non-ner surfce cn e etter Cssfcton th the ner mode. Logstc regresson mode defnes ner decson oundr Empe: csses ue nd red ponts Decson oundr

14 Lner decson oundr ogstc regresson mode s not optm ut not tht d When ogstc regresson fs? Empe n hch the ogstc regresson mode fs

15 Lmttons of ner unts. Logstc regresson does not or for prt functons - no ner decson oundr ests Souton: mode of non-ner decson oundr f Etensons of smpe ner unts use feture ss functons to mode nonnertes Lner regresson m φ φ φ φ - n rtrr functon of Logstc regresson f g φ m d φ m m

16 f Lernng th etended ner unts Feture ss functons mode nonnertes Lner regresson m φ φ φ Logstc regresson f m g φ d φ m m Importnt propert: he sme proem s ernng of the eghts for ner unts the nput hs chnged ut the eghts re ner n the ne nput Proem: too mn eghts to ern Mut-ered neur netors An terntve to ntroduce nonnertes to regresson/cssfcton modes Ke de: Cscde sever smpe neur modes th ogstc unts. Much e neuron connectons.

17 Muter neur netor Aso ced muter perceptron MLP d Cscdes mutpe ogstc regresson unts Empe: er cssfer th non-ner decson oundres z z z p Input er Hdden er Output er Muter neur netor Modes non-nert through ogstc regresson unts Cn e pped to oth regresson nd nr cssfcton proems Input er d Hdden er z z Output er regresson f f z cssfcton f p opton

18 Muter neur netor Non-nertes re modeed usng mutpe hdden ogstc regresson unts orgnzed n ers he output er determnes hether t s regresson or nr cssfcton proem Input er Hdden ers Output er regresson f f cssfcton d opton f p Lernng th MLP Ho to ern the prmeters of the neur netor? Grdent descent gorthm Weght updtes sed on the error: J D α J D We need to compute grdents for eghts n unts Cn e computed n one crd seep through the net!!! he process s ced c-propgton

19 Bcpropgton --th eve -th eve -th eve z z z - output of the unt on eve - nput to the sgmod functon on eve z g z - eght eteen unts nd on eves - nd

20 Bcpropgton δ u u n u f K δ Updte eght usng dt pont D J α D J z δ Let hen: z z D J D J δ S.t. s computed from nd the net er δ δ δ Lst unt s the sme s for the regur ner unts: It s the sme for the cssfcton th the og-ehood mesure of ft nd ner regresson th est-squres error!!! } { > < D Lernng th MLP Grdent descent gorthm Weght updte: D J α z z D J D J δ αδ δ - -th output of the - er - dervtve computed v c-propgton α - ernng rte

21 Lernng th MLP Onne grdent descent gorthm Weght updte: α J onne D u J onne Du z J onne Du z δ αδ - -th output of the - er δ - dervtve computed v cpropgton α - ernng rte Onne grdent descent gorthm for MLP Onne-grdent-descent D numer of tertons Intze eghts for :: numer of tertons do seect dt pont D u <> from D set ernng rte α compute outputs for ech unt compute dervtves δ v cpropgton updte eghts n pre αδ end for return eghts

22 Xor Empe. ner decson oundr does not est Xor empe. Lner unt

23 Xor empe. Neur netor th hdden unts Xor empe. Neur netor th hdden unts

24 MLP n prctce Optc chrcter recognton dgts Automtc sortng of ms 5 er netor th mutpe output functons outputs 9 er Neurons Weghts nputs

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