Outline (1/2) 2.1 Formulation of the Learning Problem. Outline (2/2) Problem Statement, Classical Approaches, and Adaptive Learning

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1 Outle /. Mathematcal ormulato o rectve lear roblem Classcato reresso est estmato vector quatzato Problem Statemet Classcal Aroaches a Aatve Lear. Classcal statstcal aroaches to estmato rom ata Parametrc moel ML ERM: Ver r assumto about ukow eeec screac Noarametrc moel: Lear wth lare samles asmtotc case Lear From Data Chater Outle /.3 Aatve Methos A ror assumto Choos a moel o otmal comlet or te ata How to cororate a ror assumto to lear? How to measure moel comlet? How to a otmal balace betwee the ata a a ror kowlee Lear. Formulato o the Lear Problem Process o estmat a ukow eeec or structure o a sstem us a lmte umber o observatos Three comoets ➀ Geerator: raom ut vector ➁ Sstem: returs a outut or a ve ut vector ➂ Lear mache: estmates a ukow ma o the sstem rom the observe samles

2 . Problem Formulato. Problem Formulato Geerator Prouces raom vectors R raw eeetl rom a e robablt est whch s ukow observatoal A moeler has ha o cotrol over whch ut values were sule to the sstem Sstem. Problem Formulato Prouces a outut value or ever ut vector accor to the e cotoal est whch s also ukow. Cha uobserve uts o the outut o sstem s characterze as raom a reresete as a rob. strbuto Lear Mache : etermstc : reresso ormulato Imlemet a set o uctos Ω where Ω s a set o abstract arameters use ol to e the set o uctos Eamle. Parametrc Reresso Fe Deree Polomal The set o uctos s sece as a olomal o e eree The tra ata have a sle rector varable w M w Ω { w [w...w M- ] } M : e

3 Eamle. Eamle.3 Semarametrc Reresso Polomal o Arbtrar Deree Remove the restrcto o e olomal eree Deree becomes aother arameter m w m m w Ω { w m [w...w m- ] } m : arbtrar Noarametrc Reresso Kerel Smooth Kerel avera [... ] w α w K K α α where s umber o samles K α s a kerel ucto wth bawth α Eamle.3: Noarametrc Reresso Proertes R ➀ K takes o ts mamum value whe ➁ K ecreases wth - ➂ K s a ucto o varables Raall smmetrc: - to oe varable Gaussa K α ' ' e α Ω takes vector orm [ α w...w ] : the umber o samles. Problem Formulato Choce o Aromat Fuctos Relects a ror kowlee about the sstem Nee to cororate a ror kowlee to the lear metho wth a alrea ve set o aromat uctos Lear arameters vs. olear arameters Troometrc easo lear arameterzato v m m w m m j v j s j w cos j w Multlaer etwork olear arameterzato w V w m m w j v j vj j j

4 .. Role o the Lear Mache Selects a ucto that best aromates the sstem s resose Jot robablt est ucto Tra Data... Loss ucto: measures the qualt o aromato Loss: screac betwee the outut rouce b the sstem a the lear mache or a ve ot Rsk uctoal The eecte value o the loss Lear L R L Is the rocess o estmat the ucto w whch mmzes the rsk uctoal over the set o uctos suorte b the lear mache us tra ata s ukow. Estmates w* the otmal soluto obtae wth te ata set us some lear roceure. Iuctve rcle How shoul a lea mache use tra ata? A eeral rescrto or obta a estmate w* o the true eeec the class o aromat uctos rom the avalable te tra ata. Tell us what to o wth the ata. Lea metho Seces how to obta a estmate. A costructve mlemetato o a uctve rcle or select a estmate w* rom a artcular set o uctos w. For a ve uctve rcle there are ma lear methos correso to a eret set o uctos o a lear mache... Commo Lear Tasks Classcato Outut o lear mache ee ol take o two values. Loss ucto: classcato error w L w Rsk uctoal: Quates the robablt o msclasscato R L w Lear: To the cator ucto w mmz the rob. o msclasscato.

5 Reresso Process o estmat a real-value ucto base o a te set o os samles. R L R [ ] E R [ ] E E R Nose has zero mea : ose varace Sce We have Sce the rst termsose varace oes ot ee o w mmz the seco termucto aromato error s equvalet to mmz Thus obta smallest recto rsk s equvalet to the most accurate estmato o the ukow ucto b a lear mache E R R Dest Estmato Outut o the sstem s ot use. The outut o the lear mache reresets est.e. becomes a set o estes Loss ucto Rsk uctoal Mmz us the tra ata leas to the est estmato. R L l l Ω R...

6 Cluster a Vector Quatzato Otmal artto o the ukow strbuto -sace to a resece umber o reos. Future samles ca be aromate b a sle otcluster ceter or local rotote. The set o vector-value uctos Ω are vector quatzers Loss ucto: Rsk uctoal: c L w w w R w square error storto Vector quatzer mmz R otmall quatzes uture ata eerate rom eerate rom Dmesoalt reucto low-mesoal mas o a hhmesoal strbuto. Summar Loss ucto Classcato w L w Reresso L Dest Estmato L l Cluster a Vector Quatzato L Rsk Fuctoal Classcato Reresso Dest Estmato R L w R R l Cluster a Vector Quatzato R

7 ..3 Scoe o the Lear Problem Formulato Iorme art o lear sstems Selecto o ut/outut varables Data eco/reresetato Icororat a ror kowlee to the sstem The cocetual rae o the ormal lear moel a the role o the huma artcat ur a ormal stae Do relmar work Dee the sstem eleate ts scoe b select ut varable Choose outut base o useuless a easblt..3 Scoe o the Lear Problem Formulato Have cotrol over the eerator or saml rate or strbuto Select the most sutable set o uctos Tcal assumtos mose o the eerator strbuto Prouce eeetl raw samles rom a e robablt strbuto Eceto: tme seres recto roblemsamles eerate b a amcal sstem Ucha eerator strbuto Eceto: actve lear. Classcal Aroaches Two arts o the lear roblemclasscal ➀ Seccato: eterme the arametrc orm o the ukow uerl strbuto ➁ Estmato: eterme arameters that characterze the sece strbutos Two uctve rcles commo classcal lear Emrcal rsk mmzato ERM Mamum lkelhoo ML: a secc orm o ERM.. Dest Estmato The classcal aroach restrcts the class o est uctos suorte b the lear mache to a arametrc set. w w Ω s a set o estes where w s a M-mesoal vector Ω s R M M s e Lkelhoo ucto Gve a set o... tra ata X [... ] the robablt o X as a ucto o w s P X w w

8 Mamum lkelhoo uctve rcle We shoul choose the arameters w whch mamze the lkelhoo ucto correso to choos a w*.e. moel w* whch s most lkel to eerate X to make roblem more tractable lo lkelhoo ucto s use ML rsk uctoal R ML w l w to be mmze Emrcal rsk mmzato uctve rcle emrcall estmates the rsk uctoal us tra ata Emrcal rsk: averae rsk or the tra ata mmze b choos the arorate arameters Eecte rskest estmato R w L w Ths eectato s estmate b tak a averae o the rsk over the tra ata R em w L w The otmum arameter values w* are ou b mmz the emrcal rsk wth resect to w ERM s more eeral tha ML sce t oes ot sec the artcular orm o the loss ucto. ERM s equvalet to ML or est estmato the loss ucto s Lw -l w Eamle.4 Estmat the Parameters o the Normal Dstrbuto Us Fte Data samles o eote b... were eerate accor to the ormal strbuto mea µ a varace are the two ukow arameters µ e π Lo lkelhoo ucto or ths roblem s P X µ lπ l µ

9 Ths ca be mamze b tak artal ervatves lea to the estmates ˆµ ˆ µ Eamle.5 Mture o Normals Vak 995 The estmato or a more comlcate est Let samles o eote b... be eerate accor to the strbuto Lo lkelhoo ucto e e π µ π P e e l π µ π µ X We ca show that or certa values o a there oes ot est a lobal mamumcat that ML als to rove a ete soluto Seccall s set to the value o a tra ata ot the there s o value o that ves a lobal mamum. Evaluate the lkelhoo or the choce Coser a lower bou b assum that some o the terms take o ther mmum values µ µ µ e l P π π µ X e e l π µ π > P e l l π π µ X > P l l l π µ X The lower bou o the lkelhoo cotues to crease or ecreas whch meas that a lobal mamum oes ot est.

10 .. Classcato Dscrmat Aalss Classcal Classcato Cotoal estes or each class a are estmate b classcal arametrc est estmato a the ML uctve rcle ML estmates arametrc α * β * Pror Probabltes The robablt o occurrece o each class P P Posteror Probabltes The robablt o observato belo to each class Costruct a scrmat rule that escrbes how a observato shoul be classe so as to mmze the rob. o error Choose outut class hav mamum osteror rob. Baes rule s use to calculate the osteror rob. or each class * α P P * β P P Deomator : ormalz costat o ee to comute sce the ecso rule s a comarso o relatve matues o osteror robabltes α* P β* P Oce osteror robabltes are eterme the scrmat ucto s use to class : α* P > β* P otherwse * * P I l β l α l > P where I s the cator ucto that takes ts arumet s true a otherwse Cotoal class estes Determ the arameters α * a β * us ML or ERM Al the ERM rectl rst to estmate the est Use them to ormulate the ecso rule Ths ers rom al the ERM rcle rectl to mmze the emrcal rsk R em w I w B estmat R or classcato us averae o the rsk over the ata.

11 ..3 Reresso Classcal ormulato o the reresso roblem Seek to estmate a vector o arameters o a ukow ucto w b mak measuremets o the ucto wth error at a ot w where error s eeet o a s strbute accor to a kow est. O ata Z {...} lkelhoo s P Z w l w Lkelhoo or Gaussa error wth zero mea a e varace P Z w Mamz the lkelhoo s equvalet to mmz uctoal R em whch s the rsk uctoal obtae b ERM or the square rom ucto. Square loss ucto Arorate ol or aussa ose. w l π w w Ote use ractcal alcato where the ose s ot aussa...4 Stochastc Aromato Stochastc Aromato Robbs a Moroe 95 Parameters a aromat ucto are estmate sequetall. For each vual ata samle resete a ew arameter estmate s rouce. As samles becomes lare rsk coveres to mmum. Geeral eecte rsk uctoal R L z z z stochastc aromato roceure k k γ ra L z k k where z...z s sequece o ata samles Geeral coto lm γ k Ital motvato k k k Geerate arameter estmates a real-tme asho as ata are collecte Beets Lare amout ata ee ot be store at oe tme. Aat to slowl cha ata-eerat sstem. γ k k k γ k <

12 Reccl each ccle eoch Store batch o ata s resete sequetall. Prouces a asmtotcall lare tra sequece. Comutatoall less comlcate tha batch oe. Whe to sto the uat rocess Motor the raet or each resete samles Check threshol Obe ERM ot use lear s halte earl beore small raets are see..5 Solv Problems wth Fte Data Do ot attemt to solve a sece roblem b rectl solv a harer eeral roblem as a termeate ste More eeral lear roblem larer samles requre Secc task shoul be solve rectl Estmate eatures o the est that are crtcal or solv our artcular roblem. Eamle.6 Soluto to ot eouh ata To mose artcal costrat Lear ecso rule Dscrmat Aalss Bul a two-class classer rom ata ata are eerate accor to the multvarate ormal robablt strbuto N µ a N µ estmate µ us ML base o the tra ata µ otmal ecso rule T T I µ µ µ µ c et Σ P c l l et Σ P > T T T P I µ µ Σ µ Σ µ µ Σ µ l > P estmate two meas a ol oe covarace matr The smler lear ecso rule ote erorms better tha the quaratc ecso rule eve whe Demostrato ata samles er class accor to Class est s ausa meas a covarace matrces are ukow Quaratc ecso rule vs. lear ecso rule Lear ecso rule oes ot match uerl class strbuto Frst-orer moel roves the lowest classcato error F..

13 ..6 Noarametrc Methos Develomet o oarametrc metho Attemt to eal wth the ma shortcom o classcal techques Sec the arametrc orm o ukow strbutos a eeeces Requre ew assumtos but lare umber o samles Noarametrc Dest Estmato Hstoram Dve samle sace to bs o costat wth Determe umber o samles all to each b F..3

14 Drawback : result est s scotuous Sl wow kerel ucto Results a smooth estmate Geeral rcle or oarametrc est estmator. Solv the teral equato e the est u u F where F s cumulatve strbuto ucto: c Sce the c s ukow t s aromate b the emrcal c covere to true c as samles tes to t F I Caot be solve b strahtorwar sce the emrcal c has scotutes. A aroach to the cotuous soluto to the est Relace the Drac ucto wth a cotuous ucto aromate the est as a sum o kerel uctos at each ata ot Kα K α ' where s a kerel ucto ee 9 Drawbacks oor scal roertes or hh-mesoal ata a volume eclos eouh ata ot s ot local a more raus o volume ca be a scat racto o total rae o ata Asmtotc assumtos classcal oarametrc methos ot ese or small umber o samles oor results ractcal stuatos wth lmte ata.3 Aatve Lear: Cocets a Iuctve Prcles Fleble or aatve lear methos Fleblt A metho s caablt to estmate arbtrar eeeces rom te ata Prevous roblem Parametrc methos Imose ver stret assumto Are lkel to al the true arametrc orm o a eeec s ot kow Classcal oarametrc methos Fal or hh-mesoal roblems wth te samles

15 Uversal aromato roert Use leble ver we class o aromat uctos Aromate a cotuous ucto wth a resece accurac Fteess o avalable ata Set o uctos ees to be costrae to rouce a uque soluto Iuctve rcle Prove a ramework or select a uque soluto rom a we class o uctos us te ata.3. Phlosoh Major Cocets a Issues Two stes rectve lear F..4 ➀ Lea ukow eeec rom samles Iucto ➁ Us eeec estmate to rect outut or uture ut values Deucto Trasuctve aroach Estmate oututs o ukow ucto or several ots o terest rectl rom the tra ata Local Estmato A secal case o trasucto The recto s mae at a sle ot. Lea to local rsk ormulato Derece betwee trasucto a local estmato Assume that the trasucto reers to rectos at two or more ut values smultaeousl. Prectve lear Frst ste: challe oe Its soluto requres a ror kowlee ato to ata Seco ste: sml calculat the value o ucto All lear methos use a ror kowlee the orm o the class o aromat uctos o a lear mache Parametrc Metho Use a ver restrcte set o aromat uctos o resece arametrc orm

16 Aatve Methos Atoal a ror kowlee s eee or mos atoal costrats ealt o a otetal o a ucto to be a soluto to the lear roblem Two tes o a ror kowlee ➀ Choos a we leble set o aromat uctos o a lear mache ➁ Imos atoal costrats o the uctos wth ths set Seco te to a ror kowlee Requre ollows to orm a uque eeralzato moel rom te ata ➀ A set o aromat uctos. ➁ A ror kowlee or costrats ➂ A uctve rcle or comb a ror kowlee wth ata ➃ A lear metho o uctve rcle or a ve class o aromat uctos Iuctve Prcle vs. Lear Methos For a ve uctve rcle there ma be ma lear methos correso to eret classes o aromat uctos a/or eret otmzato techques.3. A Pror Kowlee a Moel Comlet No more ths shoul be resume to est tha are absolutel ecessar Occam s razor rcle attrbute to W. Occam c Geeral bele For aatve lear methos wth te samles the best recto erormace s rove b a moel o otmum comlet Moel Selecto Seek smler moels over comle oes. Otmze the trae-o betwee moel comlet a the accurac o moel s escrto t to the tra ata. Moel Comlet Usuall cotrolle b a ror kowlee A Pror Kowlee Caot assume a moel o e comlet. Shoul ot be automatcall use or rectve lear wth te samles eve the true arametrc orm o a moel s kow a ror.

17 Eamle.7 Frst-orer olomal vs. seco-orer olomal Frst-orer olomal roves the lowest rsk F..5 Parametrc Estmato or Fte Data Parametrc reresso roblem ata ots are eerate accor to where the ose s aussa wth zero mea a varace.5 quatt has uorm strbuto [ ]. Assume Kow: a olomal o seco orer has eerate the ata Ukow: the coecets o the olomal Remark: For te ata t s ot the valt o the assumtos but the comlet o the moel that etermes recto accurac. Two Coclusos ➀ A otmal trae-o betwee the moel comlet a avalable te ata are mortat eve whe the arametrc orm o the moel s kow. 5 tra ata -> seco-orer olomal 5 samles -> a mea estmate zero-orer olomal ➁ A ror kowlee ca be useul or lear rectve moels ol t cotrols elctl or mlctl the moel comlet. Two ssues How to ee a measure the moel comlet? How to rove oo arameterzato or a aml o aromat uctos o a lear mache?.3.3 Iuctve Prcles Classcal moel Moel s ve rst a arameters are estmate rom ata us emrcal rsk mmzato uctve rcle. Aatve moel Uerl moel s ot kow a t s estmate us a lare umber o caate moels. Ma ssue: Choos the caate moel o the rht comlet to escrbe the tra ata as state b Occam s razor.

18 Iuctve Prcles Prove eret quattatve ormulato o Occam s rcle Der terms o Reresetato eco o a ror kowlee Alcablt whe the true moel oes ot belo to the set o aromat uctos Mechasm or comb a ror kowlee wth tra ata Avalablt o costructve roceures or a ve rcle Fve uctve rcles revew Pealzatoreularzato uctve rcle Earl sto rules Structural rsk mmzatosrm Baesa erece Mmum Descrto LethMDL Pealzato Reularzato Iuctve Prcle Assumes a Fleble class o aromat uctos w w Ω where Ω s a set o abstract arameters. Pealzato reularzato term s use to restrct solutos R e R λφ[ ] em R em : usual emrcal rsk Φ[ ] : ealtoeatve eeet o ata λ> : reularzato arameter A ror kowlee s clue ealt orm. Reularzato arameter λ Cotrols the streth o ror kowlee Ver lare λ -> result o mmz R em oes ot ee o ata Small λ -> al moel oes ot ee o ealt uctoal Otmal value o λ s chose us resaml methos Otmal moel estmate s ou as a result o a trae-o betwee tt the ata a a ror kowlee.e. a ealt term.

19 Earl Sto Rules Sto Rules A heurstc uctve rcle ote use the alcatos o eural etworks. Avo overtt wth overarameterze moels. Ca be terrete as a mlct orm o ealzato A ealt s ee o a ath correso to the successve moel estmates obtae ur raet-escet tra. Solutos are ealze accor to the umber o raet escet stes take alo the ath.e. the stace rom the start ot. Dcult to cotrol a terret ealzato va earl sto rule Frema 994 Structural Rsk Mmzato SRM Rsk Mmzato Aromat uctos are orere accor to ther comlet orm a este structure S S S Eamle: Class o olomal aromat uctos The elemets o a structure are olomals o a ve eree. Polomals o eree m are a subset o olomals o eree m. Goal o lear Choose a otmal elemet o a structure.e. olomal eree a estmate ts coecets rom a ve tra set.... Baesa Ierece Comlet Number o ree arameters lear uctos VC-meso olear uctos Otmal Choce o Moel Comlet Proves the mmum o the eecte rsk. Statstcal lear theor roves aaltc uer-bou estmates or eecte rsk. These estmates are use or moel selecto choos a otmal elemet o a structure uer the SRM uctve rcle. Baesa Ierece Uses atoal a ror ormato about aromat uctos. Obta a uque rectve moel rom te ata. Pror Probablt Dstrbuto Probablt o a ucto be the true ucto Relects subjectve eree o bele Proves a eectve wa o eco ror kowlee. Base o the classcal Baes ormula or uat ror robablt us the evece rove b the ata: P[ata moel] P[moel] P [moel ata] P[ata]

20 Where P[moel]: Pror robablt beore the ata are observe P[ata]: The robablt o observ tra ata P[moelata]: A osteror robablt o a moel ve the ata P[atamoel]: The robablt that the ata are eerate b a moel also kow as the lkelhoo Dest estmato w w Ω s a set o estes w: m-mesoal vector o ree arameters m s e w belos to ths class X [... ] Probablt o see X as a ucto o w A ror est ucto P [moel] w Baes ormula P X w w P w X P X Pror strbuto s take rather broal relect eeral ucertat about correct arameter values Posteror strbuto coverte rom ror robablt ater hav observe the ata more arrow be cosstet wth the observe ata P[ata moel] P W w w Two Baesa aroaches or obta a estmate o ukow... True Baesa aroach Averae over all ossble moels Θ X w w X w where wx s ve b the Baes ormula Maralzato : volves terat out reuat varables Fal moel s a wehte sum o all ossble rectve moels wth wehts ve b the evece that each moel s correct Multmesoal terato resets a challe roblem Staar umercal terato s mossble Whe aussa assumtos o ot hol varous orms o raom saml such as Mote Carlo methos have bee roose MAP aroach Choose a estmate w* mamz osteror robablt wx Mamum a osteror robablt MAP estmate Equvalet to the ealzato ormulato as show below Reresso ormulato Tra ata eerate accor to w

21 Z [X ] where X [... ] a [... ] w s a ror est PZ w w w Z PZ Z w P X w R ma w l w w N R l ma w l w w l w l w w Equvalet to ealzato ormulato wth reularzato arameter relect the kowlee o ose varace Estmat os varace Estmat reularzato arameter Choce o a ealt term <-> ror ormato strbuto Choce o the reularzato arameter <-> kowlee estmate o the amout o ose Choos the value o reularzato arameterf a oo ror Talor rors to the ata Te II mamum lkelhoo techque Cotracts the oral oto o ata-eeet ror kowlee Comarso o Baesa a Pealzato Methos Baes: To ecoe a ror kowlee about multle eeral user-ee characterstcs o the taret ucto. Pealzato: To erorm comlet cotrol b eco a ror kowlee about ucto smoothess terms o a ealt uctoal. Baesa moel selecto: Pealze more comle moels choos the moel Not uaratee the best eerato erormace Pealzato a SRM: elct mmzato: elct mmzato o the recto rsk Baesa Moel Comarso Two moels: MDLs wth a eret o. o he uts. M w a M w Problem: choose the best moel to escrbe a ve ata set Z Estmate relatve lausbltes o two moels us Baes actor P M P M Evece o moel M: PZ M Z PZ M P M Z PZ M P M PZw M w PZ w M w M w

22 Mmum Descrto Leth MDL MDL rcle Base o the ormato-theoretc aalss o the raomess cocet Rears moels as coes as ecos o tra ata Ma ea A ata set ca be aroratel ecoe a ts coe leth reresets a heret roert o the ata whch s rectl relate to the eeralzato caablt o the moel.e. coe Characterstcs o raomess o a ata set Be the shortest bar coe escrb the ata Relate to the leth o the bar coe MDL uctve rcle Tra ata set:... { } s -mesoal eature vector Problem Gve a ata object X... s a bar str... raom? Ecoe the outut str b a ossbl shorter coe Moel: coe leth Lmoel Error term: coe leth Lata moel Total leth: l l Lmoel Lata moel Coecet o comresso: Kmoel Small coecet -> str s ot raom Summarz Proertes o Iuctve Prcle T X Lmo el lo m e loc loe loloe Lata moel R T < K T l lη

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