Machine Learning. Support Vector Machines. Eric Xing. Lecture 4, August 12, Reading: Eric CMU,
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1 Machne Learnng Support Vector Machnes Erc Xng Lecture 4 August Readng: Erc CMU
2 Erc CMU What s a good Decson Boundar? Wh e a have such boundares? Irregular dstrbuton Ibalanced tranng szes outlners
3 Erc CMU Classfcaton and Margn Paraeterzng decson boundar Let denote a vector orthogonal to the decson boundar and b denote a scalar "offset" ter then e can rte the decson boundar as: + b 0 Class 2 Class d - d +
4 Erc CMU Classfcaton and Margn Paraeterzng decson boundar Let denote a vector orthogonal to the decson boundar and b denote a scalar "offset" ter then e can rte the decson boundar as: + b 0 Margn +b/ > +c/ for all n class 2 +b/ < c/ for all n class Class Class 2 d - d + Or ore copactl: +b / >c/ he argn beteen to ponts d + d +
5 Erc CMU Mau Margn Classfcaton he argn s: * * 2c Here s our Mau Margn Classfcaton proble: a s.t 2c + b / c /
6 Mau Margn Classfcaton con'd. he optzaton proble: a s.t b c + b / c / But note that the agntude of c erel scales and b and does not change the classfcaton boundar at all! h? So e nstead ork on ths cleaner proble: a s.t b he soluton to ths leads to the faous Support Vector Machnes - -- beleved b an to be the best "off-the-shelf" supervsed learnng algorth + b Erc CMU
7 Support vector achne A conve quadratc prograng proble th lnear constrans: a b s.t + b he attaned argn s no gven b Onl a fe of the classfcaton constrants are relevant support vectors d + d - Constraned optzaton We can drectl solve ths usng coercal quadratc prograng QP code But e ant to take a ore careful nvestgaton of Lagrange dualt and the soluton of the above n ts dual for. deeper nsght: support vectors kernels ore effcent algorth Erc CMU
8 Erc CMU Dgresson to Lagrangan Dualt he Pral Proble Pral: he generalzed Lagrangan: the 's ι 0 and β's are called the Lagarangan ultplers Lea: A re-rtten Pral: l h k g f s.t. n l k h g f β β L o/ s constrant pral satsfes f a f β β L 0 a n β β L 0
9 Erc CMU Lagrangan Dualt cont. Recall the Pral Proble: he Dual Proble: n a β 0 L β a β 0 n L β heore eak dualt: d * a β 0 n L β n a β 0 L β p * heore strong dualt: Iff there est a saddle pont of L β e have * d p *
10 Erc CMU he KK condtons If there ests soe saddle pont of L then the saddle pont satsfes the follong "Karush-Kuhn-ucker" KK condtons: heore: If * * and β* satsf the KK condton then t s also a soluton to the pral and the dual probles. g g l k β β β L L
11 Erc CMU Solvng optal argn classfer Recall our opt proble: a s.t b + b hs s equvalent to n b 2 s.t Wrte the Lagrangan: + b 0 * L b 2 Recall that * can be reforulated as No e solve ts dual proble: [ + ] b n b a L b 0 a n L b 0 b
12 Erc CMU *** he Dual Proble We nze L th respect to and b frst: Note that * ples: Plug *** back to L and usng ** e have: n a b b L 0 b 0 L b b 0 L * b 2 L ** [ ] + b b 2 L
13 Erc CMU he Dual proble cont. No e have the follong dual opt proble: a J 2 s.t. 0 k 0. hs s agan a quadratc prograng proble. A global au of can alas be found. But hat's the bg deal?? Note to thngs:. can be recovered b 2. he "kernel" See net More later
14 Erc CMU I. Support vectors Note the KK condton --- onl a fe 's can be nonzero!! g Class Call the tranng data ponts hose 's are nonzero the support vectors SV Class
15 Erc CMU Support vector achnes Once e have the Lagrange ultplers { } e can reconstruct the paraeter vector as a eghted cobnaton of the tranng eaples: SV For testng th a ne data z Copute z + b SV z + b and classf z as class f the su s postve and class 2 otherse Note: need not be fored eplctl
16 Erc CMU Interpretaton of support vector achnes he optal s a lnear cobnaton of a sall nuber of data ponts. hs sparse representaton can be veed as data copresson as n the constructon of knn classfer o copute the eghts { } and to use support vector achnes e need to specf onl the nner products or kernel beteen the eaples We ake decsons b coparng each ne eaple z th onl the support vectors: * sgn SV z + b
17 Erc CMU II. he Kernel rck Is ths data lnearl-separable? Ho about a quadratc appng φ?
18 Erc CMU II. he Kernel rck Recall the SVM optzaton proble he data ponts onl appear as nner product As long as e can calculate the nner product n the feature space e do not need the appng eplctl Man coon geoetrc operatons angles dstances can be epressed b nner products Defne the kernel functon K b 2 a J 0. 0 s.t. C K φ φ
19 Erc CMU II. he Kernel rck Coputaton depends on feature space Bad f ts denson s uch larger than nput space a 2 K s.t. 0 k 0. Where K φ t φ * z sgn K SV z + b
20 Erc CMU ransforng the Data Input space φ. Coputaton n the feature space can be costl because t s hgh densonal he feature space s tpcall nfnte-densonal! he kernel trck coes to rescue φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ Feature space Note: feature space s of hgher denson than the nput space n practce
21 Erc CMU An Eaple for feature appng and kernels Consder an nput [ 2 ] Suppose φ. s gven as follos An nner product n the feature space s So f e defne the kernel functon as follos there s no need to carr out φ. eplctl φ ' ' 2 2 φ φ 2 ' ' K +
22 Erc CMU More eaples of kernel functons Lnear kernel e've seen t K ' ' Polnoal kernel e ust sa an eaple ' p K ' + here p 2 3 o get the feature vectors e concatenate all pth order polnoal ters of the coponents of eghted appropratel Radal bass kernel K ' ep ' 2 In ths case the feature space conssts of functons and results n a nonparaetrc classfer. 2
23 Erc CMU he essence of kernel Feature appng but thout pang a cost E.g. polnoal kernel Ho an densons e ve got n the ne space? Ho an operatons t takes to copute K? Kernel desgn an prncple? Kz can be thought of as a slart functon beteen and z hs ntuton can be ell reflected n the follong Gaussan functon Slarl one can easl coe up th other K n the sae sprt Is ths necessarl lead to a legal kernel? n the above partcular case K s a legal one do ou kno ho an denson φ s?
24 Erc CMU Kernel atr Suppose for no that K s ndeed a vald kernel correspondng to soe feature appng φ then for e can copute an atr here hs s called a kernel atr! No f a kernel functon s ndeed a vald kernel and ts eleents are dot-product n the transfored feature space t ust satsf: Setr KK proof Postve sedefnte proof? Mercer s theore
25 SVM eaples Erc CMU
26 Erc CMU Eaples for Non Lnear SVMs Gaussan Kernel
27 Eaple Kernel s a bag of ords Defne φ as a count of ever n-gra up to nk n. hs s huge space 26 k What are e easurng b φ t φ? Can e copute the sae quantt on nput space? Effcent lnear dnac progra! Kernel s a easure of slart Must be postve se-defnte Erc CMU
28 Non-lnearl Separable Probles Class 2 Class We allo error ξ n classfcaton; t s based on the output of the dscrnant functon +b ξ approates the nuber of sclassfed saples Erc CMU
29 Erc CMU Soft Margn Hperplane No e have a slghtl dfferent opt proble: ξ are slack varables n optzaton Note that ξ 0 f there s no error for ξ s an upper bound of the nuber of errors C : tradeoff paraeter beteen error and argn s.t b + 0 ξ ξ + b C 2 ξ n
30 Erc CMU Hnge Loss Reeber Rdge regresson Mn [squared loss + λ t ] Ho about SVM? regularzaton Loss: hnge loss
31 Erc CMU he Optzaton Proble he dual of ths ne constraned optzaton proble s hs s ver slar to the optzaton proble n the lnear separable case ecept that there s an upper bound C on no Once agan a QP solver can be used to fnd 2 a J 0. 0 s.t. C
32 Erc CMU he SMO algorth Consder solvng the unconstraned opt proble: We ve alread seen several opt algorths!??? Coordnate ascend:
33 Coordnate ascend Erc CMU
34 Erc CMU Sequental nal optzaton Constraned optzaton: Queston: can e do coordnate along one drecton at a te.e. hold all [-] fed and update? 2 a J 0. 0 s.t. C
35 Erc CMU he SMO algorth Repeat tll convergence. Select soe par and to update net usng a heurstc that tres to pck the to that ll allo us to ake the bggest progress toards the global au. 2. Re-optze J th respect to and hle holdng all the other k 's k ; fed. Wll ths procedure converge?
36 Erc CMU Convergence of SMO Let s hold 3 fed and reopt J.r.t. and 2 2 a J. 0 s.t. k C 0 KK:
37 Erc CMU Convergence of SMO he constrants: he obectve: Constraned opt:
38 Cross-valdaton error of SVM he leave-one-out cross-valdaton error does not depend on the densonalt of the feature space but onl on the # of support vectors! Leave - one - out CV error # support ve ctors # of tranng eaples Erc CMU
39 Erc CMU Suar Ma-argn decson boundar Constraned conve optzaton Dualt he K condtons and the support vectors Non-separable case and slack varables he SMO algorth
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