Classification : Logistic regression. Generative classification model.

Transcription

1 CS 75 Mache Lear Lecture 8 Classfcato : Lostc reresso. Geeratve classfcato model. Mlos Hausrecht mlos@cs.ptt.edu 539 Seott Square 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 mme: E Error Frst step: e eed to devse a model of the fucto CS 75 Mache Lear

2 scrmat fuctos Oe a to represet a classfer s b us 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

3 scrmat fuctos CS 75 Mache Lear scrmat fuctos CS 75 Mache Lear

4 efe decso boudar scrmat fuctos CS 75 Mache Lear Quadratc decso boudar 3 ecso boudar CS 75 Mache Lear

5 Lostc reresso model efes a lear decso boudar scrmat fuctos: here / + e f - s a lostc fucto Iput vector f d Lostc fucto d CS 75 Mache Lear Lostc fucto fucto + e Is 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

6 Lostc reresso model scrmat fuctos: Values of dscrmat fuctos var [] Probablstc terpretato f p p Iput vector d d CS 75 Mache Lear Lostc reresso We lear a probablstc fucto f : X [] here f descrbes the probablt of class ve f p Note that: p p rasformato to bar class values: If p / the choose Else choose CS 75 Mache Lear

7 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

8 CS 75 Mache Lear Lelhood of outputs Let he Fd ehts that mame 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 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 l

9 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 tale 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

10 Ole alorthm. Eample. CS 75 Mache Lear Ole alorthm. Eample. CS 75 Mache Lear

11 CS 75 Mache Lear Ole alorthm. Eample. CS 75 Mache Lear ervato of the radet Lo lelhood ervatves of the lolelhood lo lo l µ µ + f l [ ] j j l + lo lo µ µ [ ] + + lo lo µ µ ervatve of a lostc fucto + j j

12 Geeratve approach to classfcato Idea:. Represet ad lear the dstrbuto p. Use t to defe probablstc dscrmat fuctos E.. o p p pcal model p p p p Class-codtoal dstrbutos destes bar classfcato: to class-codtoal dstrbutos p p p Prors o classes - probablt of class bar classfcato: Beroull dstrbuto p + p CS 75 Mache Lear Geeratve approach to classfcato Eample: Class-codtoal dstrbutos multvarate ormal dstrbutos N µ Σ for p µ Σ ~ ~ N µ Σ for Multvarate ormal ~ N µ Σ π d / Σ / ep µ Σ µ Prors o classes class Beroull dstrbuto p θ θ θ ~ Beroull {} CS 75 Mache Lear

13 Lear of parameters of the model est estmato statstcs We see eamples e do ot o the parameters of Gaussas class-codtoal destes p µ Σ π ML estmate of parameters of a multvarate ormal N µ Σ for a set of eamples of Optme lo-lelhood: l µ Σ lo p µ Σ µ ˆ Ho about class prors? d / Σ / ep Σˆ CS 75 Mache Lear µ Σ µ ˆ µ µ ˆ Geeratve model CS 75 Mache Lear

14 Gaussa class-codtoal destes. CS 75 Mache Lear Ma class decso Bascall e eed to des dscrmat fuctos o possble choces: Lelhood of data choose the class Gaussa that eplas the put data better lelhood of the data p µ Σ > p µ Σ the else Posteror of a class choose the class th better posteror probablt p > p the else p p µ Σ p µ Σ p p + p µ Σ CS 75 Mache Lear p

15 Gaussas: Quadratc decso boudar Cotours of class-codtoal destes CS 75 Mache Lear Gaussas: Quadratc decso boudar 3 ecso boudar CS 75 Mache Lear

16 Gaussas: Lear decso boudar Whe covaraces are the same ~ N µ Σ ~ N µ Σ CS 75 Mache Lear Gaussas: Lear decso boudar Cotours of class-codtoal destes CS 75 Mache Lear

17 Gaussas: lear decso boudar ecso boudar CS 75 Mache Lear