Linear regression (cont.) Linear methods for classification
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1 CS 75 Mache Lear Lecture 7 Lear reresso cot. Lear methods for classfcato Mlos Hausrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Lear Coeffcet shrae he least squares estmates ofte have lo bas but hh varace he predcto accurac ca be ofte mproved b sett some coeffcets to zero Icreases the bas reduces the varace of estmates Solutos: Subset selecto Rde reresso Prcpal compoet reresso Net: rde reresso CS 75 Mache Lear
2 Rde reresso Error fucto for the stadard least squares estmates: J.. * We see: ar m Rde reresso: J + λ Where.. d.. ad λ What does the e error fucto do? CS 75 Mache Lear Rde reresso Stadard reresso: J Rde reresso: J d.. + λ.. pealzes o-zero ehts th the cost proportoal to λ a shrae coeffcet If a put attrbute j has a small effect o mprov the error fucto t s shut do b the pealt term Icluso of a shrae pealt s ofte referred to as reularzato CS 75 Mache Lear
3 Supervsed lear ata: { d d.. d} a set of eamples d < > s put vector ad s desred output ve b a teacher Objectve: lear the mapp f : X Y s.t. f for all.. o tpes of problems: Reresso: Y s cotuous Eample: ears product orders compa stoc prce Classfcato: Y s dscrete Eample: temperature heart rate dsease oda: bar classfcato problems: CS 75 Mache Lear Bar classfcato o classes Y {} Our oal s to lear to classf correctl to tpes of eamples Class labeled as Class labeled as We ould le to lear f : X { } Zero-oe error loss fucto f Error f Error e ould le to mmze: E Error Frst step: e eed to devse a model of the fucto CS 75 Mache Lear
4 scrmat fuctos Oe coveet a to represet classfers s throuh scrmat fuctos Wors for bar ad mult-a classfcato Idea: For ever class defe a fucto mapp X R Whe the decso o put should be made choose the class th the hhest value of So hat happes th the put space? Assume a bar case. CS 75 Mache Lear scrmat fuctos CS 75 Mache Lear
5 scrmat fuctos CS 75 Mache Lear scrmat fuctos CS 75 Mache Lear
6 efe decso boudar. scrmat fuctos CS 75 Mache Lear Quadratc decso boudar 3 ecso boudar CS 75 Mache Lear
7 Lostc reresso model efes a lear decso boudar scrmat fuctos: here z / + e z f - s a lostc fucto Iput vector z f d Lostc fucto d CS 75 Mache Lear Lostc fucto fucto z z + e also referred to as a smod fucto Replaces the threshold fucto th smooth stch taes a real umber ad outputs the umber the terval [] CS 75 Mache Lear
8 Lostc reresso model scrmat fuctos: z Where z / + e - s a lostc fucto Values of dscrmat fuctos var [] Probablstc terpretato f p Iput vector d d z p CS 75 Mache Lear Lostc reresso Istead of lear the mapp to dscrete values f : X {} e lear a probablstc fucto f : X [] here f descrbes the probablt of class ve f p Note that: p p rasformato to dscrete class values: If p / the choose Else choose CS 75 Mache Lear
9 Lear decso boudar Lostc reresso model defes a lear decso boudar Wh? Aser: Compare to dscrmat fuctos. ecso boudar: For the boudar t must hold: o lo o lo lo ep + ep lo lo ep + ep CS 75 Mache Lear Lostc reresso model. ecso boudar LR defes a lear decso boudar Eample: classes blue ad red pots ecso boudar CS 75 Mache Lear
10 CS 75 Mache Lear Lelhood of outputs Let he Fd ehts that mamze the lelhood of outputs Appl the lo-lelhood trc he optmal ehts are the same for both the lelhood ad the lo-lelhood Lostc reresso: parameter lear. l lo lo µ µ µ µ P L µ µ z p µ lo lo µ µ + > < CS 75 Mache Lear Lostc reresso: parameter lear Lo lelhood ervatves of the lolelhood Gradet descet: lo lo l µ µ + f l ] [ l α Nolear ehts!! + f ] [ α j j z l
11 Lostc reresso. Ole radet descet O-le compoet of the lolelhood J lo µ + lo µ ole O-le lear update for eht J ole α [ J ] ole th update for the lostc reresso ad < > + α [ f ] CS 75 Mache Lear Ole lostc reresso alorthm Ole-lostc-reresso umber of teratos talze ehts Kd for :: umber of teratos do select a data pot < > from set α / update ehts parallel + α [ f ] ed for retur ehts CS 75 Mache Lear
12 Ole alorthm. Eample. CS 75 Mache Lear Ole alorthm. Eample. CS 75 Mache Lear
13 Ole alorthm. Eample. CS 75 Mache Lear
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