Machine Learning Linear Regression
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1 Machne Learnng Lnear Regresson Lesson 3
2 Lnear Regresson Bascs of Regresson Leas Squares esmaon Polynomal Regresson Bass funcons Regresson model Regularzed Regresson Sascal Regresson Mamum Lkelhood (ML) esmaon Mamum A-Poseror (MAP) esmaon Bayesan Regresson Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( )
3 Inpu: Se of uples: d R s he sample Bascs n Regresson,, D,, R predcon or a value of an unknon funcon over he feaures of Supervsed echnque Goal: creae a funcon y : n one dmenson n d dmensons D : y, θ s he (unknon) se of parameers Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 3 )
4 Graphcal Eample of Regresson *? * Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 4 )
5 Graphcal Eample of Regresson y, * Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 5 )
6 Graphcal Eample of Regresson y *, * Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 6 )
7 arge Lnear regresson: he case of -dmensonal daa 40 Inpu daase D,,,, 0 R (one feaure) Inpu Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 7 )
8 arge Lnear regresson: he case of -dmensonal daa 40 Inpu daase D,,,, 0 R (one feaure) Inpu Predcor: Evaluae lne y, 0 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 8 )
9 arge Lnear regresson: he case of -dmensonal daa 40 Inpu daase so as Inpu D : 0 y D,,,, Learnng: Esmang he regresson coeffcens { 0, } hch are he eghs of he lnear equaon, 0 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 9 )
10 Leas squares lnear f o daa Mos popular esmaon mehod s leas squares. Deermne lnear eghs ha mnmze he sum of squared loss (SSL): J y, 0 Use sandard dfferenal calculus: dfferenae SSL h respec o 0, fnd zeros of each paral dfferenal equaon solve for 0, Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 0 )
11 Dervaves of parameers: J J ) var( ), cov( ˆ 0 ˆ ˆ Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( )
12 More dmensons (d>) R d Inpu has d feaures: d here are d+ lnear eghs for descrbng he regresson funcon y, Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( )
13 Lnear regresson model y, 0 d d 0 Alernave represenaon y, ~ d j j j here ~ 0 d d Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 3 )
14 Sum of Squared Error (SSE) Ho can e quanfy he error? y J, J 0 ~ Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 4 )
15 0 0 0 Error or resdual Predcon Observaon y, y ~, Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 5 ) Sum of Squared Error (SSE) y J ~,
16 Ho can e quanfy he error? ˆ X J X X J d d X d 0 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 6 )
17 Fndng good parameers Wan o fnd parameers hch mnmze he error hnk of a cos surface : error resdual for ha 0 J ˆ arg mn J Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 7 )
18 SSE Mnmzaon Consder a smple problem One feaure (d=), o daa pons (>) o unknons: 0, o equaons: 0 0 Can solve hs sysem drecly (X : ): X ˆ Hoever, mos of he me, > d+ here may be no lnear funcon ha hs all he daa eacly Insead, solve drecly for mnmum of SSE funcon X Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 8 )
19 J Reorderng, e ake he general case X X X J X 0 X X 0 Xˆ X X X X ˆ LS Leas Squares esmaor (X X) - X s called he pseudo-nverse If X s square and ndependen, hs s he nverse X - Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 9 )
20 Even hen (X X) s nverble, mgh be compuaonally epensve f X s huge. rea as an opmzaon problem: ˆ Opmzaon mehods of SSE arg mn Ho o fnd an esmaor? J() s conve n J arg mn J( ) J(, ) X Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 0 )
21 Graden Descen Opmzaon Seepes Descen Funcon decreases mos quckly n he drecon of he negave graden. J() J mn () η: learnng rae (or sep-sze) ne old J Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( )
22 Effec of sep-sze (η) J J (0) (0) Large η : fas convergence bu larger resdual error. May cause oscllaons Small η : slo convergence bu small resdual error. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( )
23 . Graden descen Opmzaon scheme Snce J() s conve, move along negave of graden Inalze: (=0) Updae rule: ( ne) ( ne) ( old) (0) ( old ) X J X Sop: hen some creron s me (e.g. fed # eraons) or hen J (old ) ( old) ŵ (0) Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 3 )
24 . Sochasc Graden descen Prevous scheme as a bach graden descen: all ranng eamples are parcpaed a every sep ( ne) ( old ) X ( old ) ( old ) ( old ) X Sochasc graden descen scheme repeaedly eamnes a sngle eample a every sep: for,, ( ne) ( old) Advanage: ofen ges close o he mnmum much faser han bach mode. I s preferred over bach hen ranng se s large. ( old ) Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 4 )
25 3. eon-raphson Opmzaon scheme Updae rule: ( ne) H J here H: Hessan mar (second dervaves of J()) J H By subsung he prevous e oban he rule: ( ne) ( old) ( old) X J X X X hs s he leas-squares soluon and s eac soluon n one sep X X X X ( old) X X X X X Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 5 )
26 Effecs of SSE (l error) choce Sensvy o oulers cos for hs one daum Heavy penaly for large errors Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 6 )
27 Use he sum of absolue error (SAE) (l error) J 8 L, orgnal daa Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 7 )
28 Use he sum of absolue error (SAE) (l error) J 8 L, orgnal daa 6 L, orgnal daa Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 8 )
29 Use he sum of absolue error (SAE) (l error) J 8 L, orgnal daa L, orgnal daa L, ouler daa L s more robus o oulers. Hoever, he soluon s unsable and he opmzaon problem s harder (use paern search schemes, e.g. SIMPLEX). Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 9 )
30 on-lnear Regresson Consder non-lnear regresson Order polynom al E: hgher-order polynomals Order polynom al 8 Order 3 polynom al Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 30 )
31 J d 0 y k k k 0 0, Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 3 ) Polynomal Regresson
32 Eample: Use 3d polynomal. Sngle feaure, predc arge : D, Add feaures:, D,, Lnear regresson n ne feaures Somemes useful o hnk of feaure ransform y y y [,,, 3 ] Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 3 )
33 Learnng polynomal regresson coeffcens Usng Leas-squares: ˆ LS Usng Graden descen updae rule: ( ne) ( old) ( old) Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 33 )
34 Lnear Bass Funcon Models In general, can use any feaure se e hnk s useful Oher nformaon abou he problem e.g. locaon, age, Polynomal funcons Feaures [,,, 3, ] Oher funcons /, sqr(), *, [,, ] m Regresson remans Lnear = lnear n he parameers Feaures e can make as comple as e an! y m j j j j : R d bass funcons R Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 34 )
35 Eamples of Bass Funcon Polynomal bass funcons hese are global A small change n affec all bass funcons. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 35 )
36 Gaussan bass funcons hese are local A small change n only affec nearby bass funcons. Parameers μ j and s conrol locaon and scale (dh). Relaed o kernel mehods. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 36 )
37 Sgmod bass funcons here Also hese are local: a small change n only affec nearby bass funcons. Parameers μ j and s conrol locaon and scale (slope). Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 37 )
38 Addve models he bass funcons can capure varous properes of he npus (e.g. qualave) hese are called Addve models For eample: e can ry o rae documens based on e descrpors Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 38 )
39 We can ve he addve models graphcally n erms of smple uns and eghs. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 39 )
40 Overfng 0 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 40 )
41 Overfng 3 9 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 4 )
42 Overfng and compley More comple models ll alays f he ranng daa beer Bu hey may overf he ranng daa, learnng comple relaonshps ha are no really presen Smple model Comple model y y Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 4 )
43 Mean squared error ranng versus es error Plo MSE as a funcon of model compley Polynomal order Decreases More comple funcon fs ranng daa beer ranng daa e, es daa Wha abou ne daa? 5 Lo order Error decreases 0 Underfng Hgher order 5 Error ncreases Overfng 0 0 Lo Polynomal order hgh Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 43 )
44 Dealng h Overfng Use more daa Use a unng se Regularzaon Be a Bayesan Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 44 ) 44
45 . Avodng overfng: Cross-valdaon Cross-valdaon allos us o esmae he generalzaon error based on ranng eamples alone Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 45 )
46 Leave-one-ou cross-valdaon reas each ranng eample n urn as a es eample: CV y, ˆ here ˆ are he leas squares esmaes of he parameers hou he h ranng eample. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 46 )
47 Polynomal Regresson eample Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 47 )
48 . Regularzaon Overfng s reduced by mposng a consran on he overall magnude of he parameers. Objecve funcon s modfed J E E λ : regularzaon parameer E () se consrans o lnear eghs D W Daa erm + Regularzaon erm Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 48 )
49 L-Regularzaon or Rdge Regresson he regularzaon erm s quadrac ha penalze large eghs Objecve funcon J M j j W E Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 49 )
50 Regularzaon Dervaon W J 0 ˆ ˆ I 0 ˆ Ideny mar (m m) Rdge regresson or egh decay Snce he squared eghs s compable h he squared error funcon, e ge a nce closed form soluon for he opmal eghs. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 50 )
51 Several regularzed regresson models More general regularzers Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 5 ) m j q j
52 L Regularzaon (LASSO) he regularzaon erm s E Objecve funcon M W j j J Ably o creae sparse models ha are more easly nerpreed Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 5 )
53 Consraned formulaon of he L Regularzaon mn s.. 0 Leas Absolue Selecon and Shrnkage Operaor (LASSO) Several opmzaon echnques for solvng he problem Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 53 )
54 An Opmzaon Scheme: Sequenally added sgn consrans mn X s.. Eamne all possble combnaons of he sgns of he elemens of. L L bshran, R. (996). Regresson shrnkage and selecon va he lasso. Journal of Royal. Sas. Soc B., Vol. 58, o., pages 67-88). Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 54 )
55 L vs. L regularzaon Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 55 )
56 L vs. L regularzaon. L regularzed regresson Quadrac regularzaon has he advanage ha he soluon s n closed form (no appeared n non-quadrac regularzers). L regularzaon shrnks coeffcens oards (bu no o) zero, and oards each oher. L regularzed regresson Lasso represens a conve opmzaon problem solved by quadrac programmng or oher conve opmzaon mehods L regularzaon shrnks coeffcens o zero a dfferen raes; dfferen values of gve models h dfferen subses of feaures. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 56 )
57 Model Lkelhood and Emprcal Rsk o relaed bu dsnc ays o look a a model. Emprcal Rsk: Ho much error does he model have on he ranng daa? Model Lkelhood: Wha s he lkelhood ha a model generaed he observed daa? Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 57 )
58 Sascal ve of lnear regresson Assumpon: (Generave model) Observed oupu = consrucve funcon + sochasc nose y, Whaever e canno capure h our chosen famly of funcons ll be nerpreed as nose Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 58 )
59 Sascal ve of lnear regresson y, If e consder (he) Gaussan nose ~ 0, precson or nverse varance hen e nroduce he sochasc model: ~ y,, Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 59 )
60 Model Lkelhood ranng se: Generave model: Lkelhood funcon: assumng Independenly Idencally dsrbued (d) daa D,,,,, ep,,, y y p p D p,,, Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 60 )
61 Log-lkelhood: L L ln pd ln p, ln ln y, assumng a lnear regresson model of m bass funcons L, ln ln Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 6 )
62 Mamum Lkelhood (ML) esmaon Paral dervave of lnear eghs (): L ln ln, 0 ˆ L ˆ 0 ˆ LS ML ˆ ˆ Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 6 )
63 Mamum Lkelhood (ML) esmaon Paral dervave of nverse varance (β): L ln ln, 0 ˆ L ˆ ML ML ML Mean predcon squared error Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 63 )
64 MAP esmaon of Lnear Regresson MAP dervaon of lnear regresson assumes a pror dsrbuon over lnear eghs : p 0, I α: s a hyperparameer over and s he precson or nverse varance of he dsrbuon. hen, he poseror dsrbuon s obaned as p / ep X,,, p X,, p Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 64 )
65 p Opmze he Bayesan poseror X,,, p X,, p Se he MAP log-lkelhood funcon: L MAP ln p X,, ln p lkelhood + pror Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 65 )
66 p Opmze he Bayesan poseror X,,, p X,, p Se he MAP log-lkelhood funcon: L MAP ln p X,, ln p lkelhood L ML ln p X,, ln p,, ln ln Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 66 )
67 p Opmze he Bayesan poseror X,,, p X,, p Se he MAP log-lkelhood funcon: L MAP ln p X,, ln p pror ln p ln 0, I M M ln ln Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 67 )
68 Opmze he Bayesan poseror Ignorng erms ha do no depend on : L ln p X,, ln p MAP MAP esmaon of lnear eghs: L 0 ˆ I MAP MAP hus regularzed (rdge) L regresson reflecs a 0-mean soropc Gaussan pror on he eghs Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 68 )
69 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 69 ) Deermnsc approaches Sascal approaches Leas squares Mamum Lkelhood (ML) A summary of he Lnear Regresson echnques,, p 0, ~
70 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 70 ) Deermnsc approaches Sascal approaches Leas squares Mamum Lkelhood (ML) L regularzed (rdge) MAP h Gaussan pror m j j A summary of he Lnear Regresson echnques,, p 0, ~ I p 0, ln,, ln p X p
71 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 7 ) Deermnsc approaches Sascal approaches Leas squares Mamum Lkelhood (ML) L regularzed (rdge) MAP h Gaussan pror L regularzed (lasso) MAP h Laplacan pror m j j m j j A summary of he Lnear Regresson echnques,, p 0, ~ I p 0, ln,, ln p X p ~ Ce
72 Herarchcal Bayesan model p a, ~ 0, X ~ 0, I Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 7 )
73 Alernave Regresson models Elasc e Comple Pror for he eghs Penalzaon by eghed L and L norms Weghed Leas Squares: Assgn for each case (, ) a egh v 0 (he hgher he more mporan he case) hen: Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 73 ) ~ Ce V v V V WLS ˆ v v v V 0 0 J
74 Bayesan Lnear Regresson he prevous MLE or MAP dervaon of lnear regresson uses pon esmaes for he egh vecor,. Bayesan modelng esmaes he poseror dsrbuon of eghs afer recevng all observaons. hs allos us o fnd he dsrbuon of he arge of he ne comng npu, and hus make predcon. Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 74 )
75 Generave model: p Le observaons:,,, hen, he jon dsrbuon of observaons ={,, } s: p We rea ha lnear eghs as Gaussan random varables h mean m 0 and covarance mar S 0 p m 0, S 0,, D,, X, p,, I Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 75 )
76 Poseror dsrbuon: produc of o Gaussans 0 0,,,, S m I p X p X p C. M. Bshop, Paern Recognon and Machne Learnng, page 689 Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 76 )
77 p Subsung e ake he general form of he poseror dsrbuon:, X, I m, S, 0 0 here: S If m 0 = 0 and S 0 = α - I hen m S S m S S 0 0 m0 S ai m I I Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 77 )
78 Predcve dsrbuon Assumng a ne npu * e are lookng for makng a predcon of s arge *. hs s equvalen on esmang he poseror dsrbuon: Margnalzng e ake: * * p, * * * *, p, p p d here p m, S m S I ai p * * * *,, Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 78 )
79 Predcve dsrbuon (con.) Accordng o Gaussan properes (B.44) e receve: here: m S p, d p p p,, * * * * * * * *,, p * * * * *,, m p * * * S ai S Machne Learnng 07 Compuer Scence & Engneerng, Unversy of Ioannna ML ( 79 )
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