Linear Regression & Least Squares!

Transcription

1 Lner Regresson & Lest Squres Al Borj UWM CS 790 Slde credt: Aykut Erdem

2 Ths&week Lner&regresson&prolem&& ' con0nuous&outputs& ' smple&model Introduce&key&concepts:&& ' loss&func0ons& ' generlz0on& ' op0mz0on& ' model&complety& &regulrz0on Sldes'dpted'from'Rchrd'Zemel,'Ern'Hlpern,'Zv:Br'Joseph,'Ar>'Sngh,'Brns'Poczos,'J.P.'Lews,' Erk'Sudderth 2

3 Clssfc0on Input:&X& ' Rel&vlued,&vectors&over&rel.& ' Dscrete&vlues&(0,,2, )& ' Other&structures&(e.g.,&strngs,&grphs,&etc.) Output:&Y& ' Dscrete&(0,,2,...) Sports% Scence% News% Anemc%cell% Helthy%cell% X''Document' Y''Topc' X''Cell'Imge' Y''Dgnoss' 3

4 Regresson Input:&X& ' Rel&vlued,&vectors&over&rel.& ' Dscrete&vlues&(0,,2, )& ' Other&structures&(e.g.,&strngs,&grphs,&etc.) Output:&Y& ' Rel&vlued,&vectors&over&rel. Stock%Mrket%% Predcon% Y''?' X''Fe0'' 4

5 Choosng&&resturnt In&everydy&lfe&we&need&to&mke&decsons& y&tkng&nto&ccount&lots&of&fctors& The&ques0on&s&wht&weght&we&put&on& ech&of&these&fctors&(how&mportnt&re& they&wth&respect&to&the&others).& Assume&we&would&lke&to&uld&& recommender&system&sed&on&n& ndvduls &preferences& If&we&hve&mny& oserv0ons&we& my&e&le&to& recover&the&weghts Revews (out of 5 strs) Dstnce Cusne (out of 0) ? 5

6 Some%other%emples Weght%+%heght% cholesterol%level% Age%+%gender%6me%% 6me%spent%n%front%of%the%TV% Pst%choces%of%%user% 'NeHl%score'% Profle%of%%jo% (user,%mchne,%6me) Memory%usge%of% %sumked%process. 6

7 Emple:%Polynoml%Curve%FQng The%green%curve%s%the%true%func6on% (whch%s%not%%polynoml)% %not%known% The%dt%ponts%re%unform%n%%ut%hve% nose%n%t.% t() f() + Am:%ft%%curve%to%these%ponts% Key%ques6ons:% own from Bshop %How%do%we%prmetrze%the%model%(the%curve)?% %%%% %%Wht%loss%(ojec6ve)%func6on%should%we%use%to%judge%ft?%% %%%% %%How%do%we%op6mze%ft%to%unseen%test%dt%(generlz6on)?%% 7

8 \D%regresson 8

9 One\dmensonl%regresson Fnd%%lne%tht%represent%the% est %lner%rel6onshp: 9

10 One\dmensonl%regresson Prolem:%the%dt%does%not% go%through%%lne% e 0

11 One\dmensonl%regresson Prolem:%the%dt%does%not% go%through%%lne% e Fnd%the%lne%tht%mnmzes% the%sum:% ( 2 )

12 Prolem:%the%dt%does%not% go%through%%lne% Fnd%the%lne%tht%mnmzes% the%sum:% We%re%lookng%for%%%%%%tht% mnmzes% One\dmensonl%regresson 2 ˆ e 2 ) ( ) ( e 2 ) (

13 Mtr%not6on Usng%the%followng%not6ons & : % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%nd n # " & : % n # " 3

14 Usng%the%followng%not6ons %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%nd We%cn%rewrte%the%error%func6on%usng%lner% lger%s: Mtr%not6on 4 " # % & n : " # % & n : 2 2 ) ( ) ( ) ( ) ( ) ( e e T

15 Emple:%Boston%House%Przes Es6mte%medn%house% prce%n%%neghorhood% sed%on%neghorhood% st6s6cs%% Look%t%frst%(of%3)% Krutes:%per%cpt% crme%rte%% Use%ths%to%predct%house% prces%n%other% neghorhoods hkps://rchve.cs.uc.edu/ml/dtsets/housng 5

16 Represent%the%dt Dt%descred%s%prs%D%%(( (),t () ),%( (2),t (2) ),...,%( (N),t (N) ))%% \ %s%the%nput%feture%(per%cpt%crme%rte)% \ t%s%the%trget%output%(medn%house%prce)% Here%t%s%con6nuous,%so%ths%s%%regresson%prolem%% Could%tke%frst%300%emples%s%trnng%set,%remnng%206% s%test%set% \ Use%the%trnng%emples%to%construct%hypothess,%or% func6on%ppromtor,%tht%mps%%to%predcted%y% \ Evlute%hypothess%on%test%set 6

17 Nose A%smple%model%typclly%does%not%ectly%ft%the%dt% % lck%of%ft%cn%e%consdered%nose Sources%of%nose% %Imprecson%n%dt%Krutes%(nput%nose)% %%% %% %%Errors%n%dt%trgets%(mslelng)%% %%% %% %%Add6onl%Krutes%not%tken%nto%ccount%y%dt% Krutes,%ffect%trget%vlues%(ltent%vrles)%% %%% %% %%Model%my%e%too%smple%to%ccount%for%dt% trgets 7

18 Lest\Squres%Regresson Stndrd%loss/cost/ojec6ve% func6on%mesures%the% squred%error%n%the% predc6on%of%t()%from%. N J(w) " [t (n) (w + w (n) )] 2 0 n from Bshop The%loss%for%the%red% hypothess%s%the%sum%of%the% squred%ver6cl%errors. 8

19 Op6mzng%the%Ojec6ve One%strghHorwrd%method:%n6lze%w%rndomly,% repetedly%updte%sed%on%grdent%descent%n%j% w w " #J #w Here%λ%s%the%lernng%rte% rnng rte For%%sngle%trnng%cse,%ths%gves%the%LMS%updte% rule: w w + 2(t (n) " y( (n) )) (n) Note:%s%error%pproches%zero,%so%does%updte error pproches zero, so does upd 9

20 Effect%of%step\sze%λ Lrge%λ%>%Fst%convergence%ut%lrger%resdul%error% %%%%%%%%%%%%%%%%%%%Also%possle%oscll6ons% Smll%λ%>%Slow%convergence%ut%smll%resdul%error 20

21 Op6mzng%Across%Trnng%Set Two%wys%to%generlze%ths%for%ll%emples%n%trnng%set:%. Stochs6c/onlne%updtes% updte%the%prmeters% for%ech%trnng%cse%n%turn,%ccordng%to%ts%own% grdents%% 2. Btch%updtes:%sum%or%verge%updtes%cross%every% emple%,%then%chnge%the%prmeter% vlues% w w + 2 N # n Underlyng%ssump6on:%smple%s%ndependent%nd% ssumpton: smple s ndepend den6clly%dstruted%(..d.) (t (n) " y( (n) )) (n) 2

22 Non\ter6ve%Lest\squres%Regresson An%ltern6ve%op6mz6on%pproch%s%non\ ter6ve:%tke%derv6ves,%set%to%zero,%nd%solve% for%prmeters. N dj(w) 2 "[t (n) (w + w (n) )] 0 dw 0 0 N n w (" t (n) w (n) ) / N t w 0 n w " n " n (t (n) t )( (n) ) ( (n) ) 2 22

23 Mul6\dmensonl%lner%regresson Usng%%model%wth%m%prmeters m m j j j 23

24 Mul6\dmensonl%lner%regresson Usng%%model%wth%m%prmeters m m j j j 2 24

25 Mul6\dmensonl%lner%regresson Usng%%model%wth%m%prmeters m m j j j 2 25

26 Mul6\dmensonl%lner%regresson Usng%%model%wth%m%prmeters nd%n%mesurements j j j m m , 2, ) ( ) ( A " # % & ' m j j j n m j j j e

27 Mul6\dmensonl%lner%regresson Usng%%model%wth%m%prmeters nd%n%mesurements j j j m m , 2, ) ( ) ( ) ( A " # % & ' e e m j j j n m j j j

28 :A 28 " # % & " # % & " # % & m m n n m n :... : :.. :,,,, A

29 :A 29 " # % & " # % & " # % & " # % & )... ( : )... ( :... : :.. :,,,,,,,, m m n n n m m n m n n m n A

30 :A 30 " # % & " # % & " # % & " # % & )... ( : )... ( :... : :.. :,,,,,,,, m m n n n m m n m n n m n A prmeter'

31 :A 3 " # % & " # % & " # % & " # % & )... ( : )... ( :... : :.. :,,,,,,,, m m n n n m m n m n n m n A mesurement'n prmeter'

32 Emple:%Boston%House%Przes% %revsted One%method%of%etendng%the% model%s%to%consder%other% nput%dmensons y() w + w + w In%the%Boston%housng% emple,%we%cn%look%t%the% numer%of%rooms%nput% feture% We%cn%use%grdent%descent% to%solve%for%ech%coeffcent,% or%use%lner%lger% %solve% system%of%equ6ons 2 32

33 Lner%Regresson Imgne%now%wnt%to%predct%the%medn%house%prce% from%these%mul6\dmensonl%oserv6ons%% Ech%house%s%%dt%pont%n,%wth%oserv6ons% ndeed%y%j:% (n) ( (n),..., d (n) ) Smple%predctor%s%nlogue%of%lner%clssfer,% producng%rel\vlued%y%for%nput%%wth%prmeters%w% (effec6vely%fng% 0 %%): 0 y w 0 + d j w j j w T 33

34 Mul6\dmensonl%lner%regresson e() A 2 2 ( A) T ( A) T A T A T A T T A + T. A%mnmum%occurs%when%%.%The%frst%derv6ve%s%zero,%% 2.%The%second%derv6ve%s%pos6ve.% Mul6dmensonl%cse:%% \ st %derv6ve%of%%func6on%f()%s%the%grdent,% f()%(%row%vector)% \ 2 nd %derv6ve,%the%hessn,%s%%mtr%tht%we%wll%denote%s%h f' (). e()2a T A 2A T. H e ()2A T A. 34

35 Mnmzng% e() mn mnmzes e( ) f

36 Mnmzng% e() mn mnmzes e( ) f e() mn

37 Mnmzng% e() e() s flt t mn mn mnmzes e( ) f e() mn

38 Mnmzng% e() e() s flt t mn e( ) 0 mn mn mnmzes e( ) f e() mn

39 Mnmzng% e() e() s flt t mn e( ) 0 mn mn mnmzes e( ) f e() does not go down round mn e() mn

40 Mnmzng% e() e() s flt t mn e( ) 0 mn mn mnmzes e( ) f e() does not go down round mn H e ( mn ) s postve e() sem - defnte mn

41 Recp:%Pos6ve%sem\defnte A s postve sem - defnte T A 0, for ll In -D In 2-D 4

42 Mnmzng% e( ) A 2 A T Aˆ A T ˆ mnmzes e( ) f 2A T A s postve sem - defnte

43 Mnmzng% e( ) A 2 A T Aˆ A T ˆ mnmzes e( ) f 2A T A s postve sem - defnte Alwys%true

44 Mnmzng% e( ) A 2 A T Aˆ A T The%norml'equton ˆ mnmzes e( ) f 2A T A s postve sem - defnte Alwys%true

45 Geometrc%nterpret6on 45

46 Geometrc%nterpret6on %s%%vector%n%r n 46

47 Geometrc%nterpret6on %s%%vector%n'r n % The%columns%of%A%defne%%vector%spce%rnge(A) 2 47

48 Geometrc%nterpret6on %s%%vector%n%r n % The%columns%of%A%defne%%vector%spce%rnge(A)% A%s%n%rtrry%vector%n%rnge(A) A 2 48

49 Geometrc%nterpret6on %s%%vector%n%r n % The%columns%of%A%defne%%vector%spce%rnge(A)% A%s%n%rtrry%vector%n%rnge(A) A A 2 49

50 Geometrc%nterpret6on %%%%%%s%the%orthogonl%projec6on%of%%onto%rnge(a) Aˆ A T ( ) T T Aˆ 0 A Aˆ A Aˆ ˆ ˆ Aˆ 2 50

51 The norml equton: A T Aˆ A T

52 The norml equton: A T Aˆ A T T T Estence:%%%%%%%%%%%%%%%%%%%%%%%%hs%lwys%%soluton A Aˆ A

53 The norml equton: A T Aˆ A T T T Estence:%%%%%%%%%%%%%%%%%%%%%%%%hs%lwys%%soluton% A Aˆ A Unqueness:%the%soluton%s%unque%f%the%columns%of% A%re%lnerly%ndependent%

54 The norml equton: A T Aˆ A T T T Estence:%%%%%%%%%%%%%%%%%%%%%%%%hs%lwys%%soluton% A Aˆ A Unqueness:%the%soluton%s%unque%f%the%columns%of% A%re%lnerly%ndependent% Aˆ 2

55 Lner%models % %%It%s%mthem6clly%esy%to%ft%lner%models%to%dt. %We%cn%lern%%lot%out%model\fQng%n%ths%rel6vely% smple%cse.%% % %%There%re%mny%wys%to%mke%lner%models%more%powerful%whle%retnng% ther%nce%mthem6cl%proper6es:% %By%usng%non\lner,%non\dp6ve%ss%func6ons,%we%cn%get%generlzed% lner%models%tht%lern%non\lner%mppngs%from%nput%to%output%ut%re% lner%n%ther%prmeters% %only%the%lner%prt%of%the%model%lerns.%% %%% %%%%%% %By%usng%kernel%methods%we%cn%hndle%epnsons%of%the%rw%dt%tht%use% %huge%numer%of%non\lner,%non\dp6ve%ss%func6ons.% % %By%usng%lrge%mrgn%kernel%methods%we%cn%vod%overfQng%even%when% we%use%huge%numers%of%ss%func6ons.% %But%lner%methods%wll%not%solve%most%AI%prolems.%% % %They%hve%fundmentl%lmt6ons.% 55

56 Some%types%of%ss%func6ons%n%\D { Sgmods Gussns Polynomls ( ) ( φ j () ep { ( µ } µj ) φ j) 2 j () σ s 2s 2 σ() +ep( ). 56

57 Two%types%of%lner%model%tht%re%equvlent%wth% y(, w) y(, w) s w w 0 0 respect%to%lernng + + w w φ + w 2 ( ) + 2 w 2 + φ... 2 w ( ) + T... w T Φ( ) The%frst%model%hs%the%sme%numer%of%dp6ve%coeffcents%s% the%dmensonlty%of%the%dt%+.%% The%second%model%hs%the%sme%numer%of%dp6ve% coeffcents%s%the%numer%of%ss%func6ons%+.%% Once%we%hve%replced%the%dt%y%the%outputs%of%the%ss% func6ons,%fqng%the%second%model%s%ectly%the%sme%prolem% s%fqng%the%frst%model%(unless%we%use%the%kernel%trck)% So%we ll%just%focus%on%the%frst%model 57

58 Generl%lner%regresson%prolem Usng%our%new%not6ons%for%the%ss%func6on%lner% regresson%cn%e%wrken%s n y w () j j j 0 where%%%%%%%%%cn%e%ether% j () j %for%mul6vrte%regresson%or% one%of%the%nonlner%ss%we%defned% er Once%gn%we%cn%use% lest%squres %to%fnd%the%op6ml% solu6on. 58

59 LMS%for%the%generl%lner%regresson%prolem regresson prolem Our gol s to mnmze the followng loss functon: J(w) (y w j j ( )) Movng to vector nottons we get: j 2 y n j 0 w j j () w vector of dmenson k+ ( ) vector of dmenson k+ y scler J(w) (y w T ( )) 2 We tke the dervtve w.r.t w w (y w T ( )) 2 2 (y w T ( )) Equtng to 0 we get 2 (y w T ( )) ( ) T ( ) T 0 y ( ) T w T ( ) ( ) T 59

60 LMS%for%the%generl%lner%regresson%prolem We tke the dervtve w.r.t w w (y w T ( )) 2 2 (y w T ( )) ( ) T Equtng to 0 we get 2 (y w T ( )) ( ) T 0 Defne: y ( ) T w T ( ) 0 ( ) ( ) m ( ) 0 ( 2 ) ( 2 ) m ( 2 ) 0 ( n ) ( n ) m ( n ) ( ) T J(w) (y w T ( )) 2 Then dervng w we get: w ( T ) T y 60

61 LMS%for%the%generl%lner%regresson%prolem Dervng w we get: w ( T ) T y J(w) (y w T ( )) 2 k+ entres vector n entres vector n y k+ mtr Ths soluton s lso known s psuedo nverse 6

62 FQng%%polynoml Now%we%use%one%of%these% ss%func6ons:%n%m th %order %polynoml%func6on% own from Bshop We%cn%use%the%sme%pproches%to%op6mze%the% vlues%of%the%weghts%on%ech%coeffcent:%nly6c,% nd%ter6ve 62

63 0 th %order%polynoml 63

64 st %order%polynoml 64

65 3 rd %order%polynoml 65

66 9 th %order%polynoml 66

67 Root%Men%Squre%(RMS)%Error E(w) 2 N n {y( n, w) t n } 2 t 0 M 0 t 0 M E RMS 2E(w )/N 0 0 The%dvson%y%N%llows%us%to%compre% dfferent%szes%of%dt%sets%on%n%equl% foo6ng,%nd%the%squre%root%ensures% tht%erms%s%mesured%on%the%sme% scle%(nd%n%the%sme%unts)%s%the% trget%vrle%t% t 0 M 3 0 t 0 M

68 Root%Men%Squre%(RMS)%Error Trnng Test ERMS M 6 9 Root>Men>Squre'(RMS)'Error:' E(w) 2 NX (t n ( n ) T w) 2 t w 2 2 n Theoverf)ngprolem 68

69 Root%Men%Squre%(RMS)%Error Tle of the coeffcents w for polynomls of vrous order. Oserve how the typcl mgntude of the coeffcents ncreses drmtclly s the order of the polynoml ncreses. M 0 M M 6 M 9 w w w w w w w w w w Theoverf)ngprolem 69

70 Incresng%the%sze%of%trnng%dt N 5 N 00 t t M 9 0 For%%gven%model%complety,%the%over\fQng%prolem%ecome%less%severe% s%the%sze%of%the%dt%set%ncreses.% Another%wy%to%sy%ths%s%tht%the%lrger%the%dt%set,%the%more%comple%(n% other%words%more%flele)%the%model%tht%we%cn%fford%to%ft%to%the%dt.% 70

71 \D%regresson%llustrtes%key%concepts Dt%fts% %s%lner%model%est%(model%selec6on)?% %%% %% %%Smplest%models%do%not%cpture%ll%the%mportnt%% %%% %% %%More%comple%model%my%overft%the%trnng%dt%(ft% not%only%the%sgnl%ut%lso%the%nose%n%the%dt),% especlly%f%not%enough%dt%to%constrn%model%% One%method%of%ssessng%ft:%test%generlz6on%%model s% lty%to%predct%the%held%out%dt%% Op6mz6on%s%essen6l:%stochs6c%nd%tch%ter6ve% pproches;%nly6c%when%vlle 7

72 Regulrzed%Lest%Squres A%technque%to%control%the%overfQng%phenomenon% Add%%penlty%term%to%the%error%func6on%n%order%to% dscourge%the%coeffcents%from%rechng%lrge% vlues Rdge regresson Ẽ(w) 2 N n w whch's'mnmzed'y' {y( n, w) t n } 2 + λ 2 w 2 w 2 w T w w w w 2 M mportnce of the regulrzton term com 72

73 The%effect%of%regulrz6on ln λ 8 ln λ 0 t t M 9 73

74 The%effect%of%regulrz6on ERMS 0.5 Trnng Test ln λ ln λ 8 ln λ 0 w w w w w w w w w w ln λ The%correspondng%coeffcents%from%the%fKed%polynomls,%showng% tht%regulrz6on%hs%the%desred%effect%of%reducng%the%mgntude% of%the%coeffcents. 74

75 A%more%generl%regulrzer 2 N {t n w T φ( n )} 2 + λ 2 n M j w j q q 0.5 q q 2 q 4 75