Supervised learning: Linear regression Logistic regression

Transcription

1 CS 57 Itroducto to AI Lecture 4 Supervsed learg: Lear regresso Logstc regresso Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 57 Itro to AI

2 Data: D { D D.. D D Supervsed learg d a set of eamples s a put vector of sze d s the desred output gve b a teacher Obectve: lear the mappg s.t. f for all.. } f Y Regresso: Y s cotuous Eample: eargs product orders compa stock prce Classfcato: Y s dscrete Eample: hadrtte dgt bar form dgt label : X CS 57 Itro to AI

3 Supervsed learg Net: To basc models of f : used supervsed learg X Y Lear regresso: Regresso here Y s R Logstc regresso Classfcato th classes CS 57 Itro to AI

4 Lear regresso Fucto f : X Y s a lear combato of put compoets f k - parameters eghts d d d Bas term f Iput vector d d CS 57 Itro to AI

5 Lear regresso Shorter vector defto of the model Iclude bas costat the put vector f d CS 57 Itro to AI d d k - parameters eghts Iput vector d d T f

6 Lear regresso. Error. Data: Fucto: D f We ould lke to have f for all.. Error fucto measures ho much our predctos devate from the desred asers Mea-squared error J.. f Learg: We at to fd the eghts mmzg the error! CS 57 Itro to AI

7 Lear regresso. Eample dmesoal put CS 57 Itro to AI

8 Lear regresso. Eample. dmesoal put CS 57 Itro to AI

9 CS 57 Itro to AI Lear regresso. Optmzato. We at the eghts mmzg the error For the optmal set of parameters dervatves of the error th respect to each parameter must be Vector of dervatves:.... T f J T J J grad d d J

10 CS 57 Itro to AI Lear regresso. Optmzato. For the optmal set of parameters dervatves of the error th respect to each parameter must be defes a set of equatos ] [ J k k ] [ k k J.... ] [ k k f J ] [ k k J grad J

11 CS 57 Itro to AI Solvg lear regresso B rearragg the terms e get a sstem of lear equatos th d+ ukos d d J d d A b d d d d

12 CS 57 Itro to AI Solvg lear regresso The optmal set of eghts satsfes: Leads to a sstem of lear equatos SLE th d+ ukos of the form Solutos to SLE: e.g. matr verso f the matr s sgular T J d d A b b A

13 Gradet descet soluto There are other as to solve the eght optmzato problem the lear regresso model J Error f.. A smple techque: Gradet descet Idea: Adust eghts the drecto that mproves the Error The gradet tells us hat s the rght drecto Error - a learg rate scales the gradet chages CS 57 Itro to AI

14 Gradet descet method Desced usg the gradet formato Error Error * * Drecto of the descet Chage the value of accordg to the gradet Error CS 57 Itro to AI

15 Gradet descet method Error Error * * Ne value of the parameter Error * * For all - a learg rate scales the gradet chages CS 57 Itro to AI

16 Gradet descet method Iteratvel coverge to the optmum of the Error fucto Error 3 CS 57 Itro to AI

17 Ole regresso algorthm The error fucto defed for the hole dataset D J Error f.. Istead of the error for all data pots e use error for each eample D Jole Error f Chage regresso eghts after ever eample accordg to the gradet: Error vector form: Error - Learg rate that depeds o the umber of updates CS 57 Itro to AI

18 Gradet for o-le learg T Lear model f O-le error Jole Error f O-le algorthm: sequece of ole updates -th update for the lear model: D Vector form: Error f -th eght: Error f Aealed learg rate: - Graduall rescales chages eghts CS 57 Itro to AI

19 Adaptve models T Lear model f O-le error Jole Error f O-le algorthm: Sequece of ole updates oe eample at the tme Useful for cotuous data streams Adaptve models: the uderlg model s ot statoar ad ca chage over tme Eample: seasoal chages O-le algorthm ca be made adaptve b keepg the learg at some costat value c CS 57 Itro to AI

20 Ole regresso algorthm Ole-lear-regresso D umber of teratos Italze eghts for =:: umber of teratos do select a data pot D from D ed for retur eghts set / d update eght vector f Advatages: ver eas to mplemet cotuous data streams CS 57 Itro to AI

21 O-le learg. Eample CS 57 Itro to AI

22 Etesos of smple lear model Replace puts to lear uts th feature bass fuctos to model oleartes f m - a arbtrar fucto of f d m m The same techques as before to lear the eghts CS 75 Mache Learg

23 Etesos of the lear model Models lear the parameters e at to ft f Bass fuctos eamples: k k k a hgher order polomal oe-dmesoal put 3 3 Multdmesoal quadratc Other tpes of bass fuctos s cos m... m - parameters... - feature or bass fuctos m CS 75 Mache Learg

24 CS 75 Mache Learg Etesos of the lear case Error fucto.. / f J φ.. f J Leads to a sstem of m lear equatos m φ Assume: m m Ca be solved eactl lke the lear case

25 Eample. Regresso th polomals. Regresso th polomals of degree m Data pots: pars of Feature fuctos: m feature fuctos m Fucto to lear: m f m m m m CS 75 Mache Learg

26 CS 75 Mache Learg Eample: Regresso th polomals of degree m O le update for <> par f f Eample. Regresso th polomals. m m f

27 Learg th feature fuctos Fucto to lear: f O le gradet update for the <> par f k f Gradet updates are of the same form as the lear regresso models CS 75 Mache Learg

28 Multdmesoal addtve model eample CS 75 Mache Learg

29 Multdmesoal addtve model eample CS 75 Mache Learg

30 To classes Y Bar classfcato {} Our goal s to lear to classf correctl to tpes of eamples Class labeled as Class labeled as We ould lke to lear f Zero-oe error loss fucto : X { } f Error f Error e ould lke to mmze: E Error Frst step: e eed to devse a model of the fucto CS 75 Mache Learg

31 Dscrmat fuctos Oe a to represet a classfer s b usg Dscrmat fuctos Works for bar ad mult-a classfcato Idea: For ever class = k defe a fucto mappg X Whe the decso o put should be made choose the class th the hghest value of g g So hat happes th the put space? Assume a bar case. CS 75 Mache Learg

32 Dscrmat fuctos Eample: To classes -D.5.5 g g CS 75 Mache Learg

33 Dscrmat fuctos Dscrmat fuctos g ad g defe the decso boudar g g g g g g g g g g CS 75 Mache Learg

34 Quadratc decso boudar 3 Decso boudar g g g g g g CS 75 Mache Learg

35 Logstc regresso model Defes a lear decso boudar Dscrmat fuctos: T g g here g z / e z T T f g g T g g - s a logstc fucto Iput vector d z Logstc fucto f d CS 75 Mache Learg

36 fucto Logstc fucto g z z e Is also referred to as a sgmod fucto Replaces the threshold fucto th smooth stchg takes a real umber ad outputs the umber the terval [] CS 75 Mache Learg

37 Dscrmat fuctos: T g g Logstc regresso model T g g Values of dscrmat fuctos var [] Probablstc terpretato T f p g g z p Iput vector d d CS 75 Mache Learg

38 Logstc regresso We lear a probablstc fucto f : X [] here f descrbes the probablt of class gve f Note that: T g p p p Trasformato to bar class values: If p / the choose Else choose CS 75 Mache Learg

39 Logstc regresso model. Decso boudar Logstc Regresso defes a lear decso boudar Eample: classes blue ad red pots Decso boudar CS 75 Mache Learg

40 CS 75 Mache Learg Lkelhood of outputs Let The Fd eghts that mamze the lkelhood of outputs Appl the log-lkelhood trck The optmal eghts are the same for both the lkelhood ad the log-lkelhood Logstc regresso: parameter learg D l log log P D L T g z g p log log D

41 CS 75 Mache Learg Logstc regresso: parameter learg Log lkelhood Dervatves of the loglkelhood Gradet descet: Ole update for k-th eample log log D l T f g D l ] [ k D l k k k Nolear eghts!! k k k f k ] [ z g D l

42 Logstc regresso. Ole gradet descet O-le compoet of the loglkelhood J ole D log log O-le learg update for eght J D ole k k k k [ J D ] k ole k Ole update for the logstc regresso for k-th eample D k k k k[ k k k f k ] k CS 75 Mache Learg

43 Ole logstc regresso algorthm Ole-logstc-regresso D umber of teratos talze eghts for =:: umber of teratos do select a data pot D from D ed for retur eghts d set / update eghts parallel [ f ] CS 75 Mache Learg

44 Ole algorthm. Eample. CS 75 Mache Learg

45 Ole algorthm. Eample. CS 75 Mache Learg

46 Ole algorthm. Eample. CS 75 Mache Learg

47 CS 75 Mache Learg Apped: Dervato of the gradet Log lkelhood Dervatves of the loglkelhood log log D l T f g D l z z D l log log z z g z g z z g z g z log log z g z g z z g Dervatve of a logstc fucto z g z g z g z