Lecture 10 Support Vector Machines. Oct
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1 Lecture 10 Support Vector Machnes Oct
2 Lnear Separators Whch of the lnear separators s optmal?
3 Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron algorthm depends on a concept called margn
4 Intuton of Margn Consder ponts A, B, and C We are qute confdent n our A w x b =0 predcton for A because t s w x b > 0 far from the decson B boundary. In contrast, we are not so confdent n our predcton for w x b < 0 C because a slght change n C the decson boundary may flp the decson. Gven a tranng set, we would lke to make all of our predctons correct and confdent! Ths can be captured by the concept of margn
5 Functonal Margn One possble way to defne margn: We defne ths as the functonal margn of the lnear classfer w.r.t tranng example ( x, y ) The large the value, the better really? What f we rescale (w, b) by a factor α, consder the lnear classfer specfed by (αw, αb) Decson boundary reman the same Yet, functonal margn gets multpled by α We can change the functonal margn of a lnear classfer wthout changng anythng meanngful We need somethng more meanngful
6 What we really want A w x b =0 B C We want the dstances between the examples and the decson boundary to be large ths quantty s what we call geometrc margn But how do we compute the geometrc margn of a data pont w.r.t a partcular lne (parameterzed by w and b)?
7 Some basc facts about lnes w x b = 0 X 1? 1 w x w b
8 Geometrc Margn The geometrc margn of (w, b) w.r.t. x () s the dstance from x () to the decson surface B Ths dstance can be computed as y ( w x b) γ = w C A γ A Gven tranng set S={(x, y ): =1,, }, the geometrc margn of the classfer w.r.t. S s γ = mn γ = 1L ote that the ponts closest to the boundary are called the support ote that the ponts closest to the boundary are called the support vectors n fact these are the only ponts that really matters, other examples are gnorable ( )
9 What we have done so far We have establshed that we want to fnd a lnear decson boundary whose margn s the largest We know how to measure the margn of a lnear decson boundary ow what? We have a new learnng objectve Gven a lnearly separable (wll be relaxed later) tranng set S={(x, y ): =1,, }, we would lke to fnd a lnear classfer (w, b) wth maxmum margn.
10 Maxmum Margn Classfer Ths can be represented as a constraned optmzaton problem. maxγ w, b subject to : () () ( w x b) y () γ, = 1, L, w Ths optmzaton problem s n a nasty form, so we need to do some rewrtng Let γ = γ w, we can rewrte ths as max w, b γ ' w subject to : y ( w x b) γ ', = 1, L,
11 Maxmum Margn Classfer ote that we can arbtrarly rescale w and b to make the functonal margn γ ' large or small So we can rescale them such that =1 max γ ' w, b w subject to : y ( w x γ ' b) γ ', = 1, L, max w, b 1 w subject to : (or equvalently mn y ( w x w, b b) 1, w 2 ) = 1, L, Maxmzng the geometrc margn s equvalent to mnmzng the magntude of w subject to mantanng a functonal margn of at least 1
12 Solvng the Optmzaton Problem mn w, b subject to : y 1 2 w 2 ( w xx b ) 1, = 1, L, Ths results n a quadratc optmzaton problem wth lnear nequalty constrants. t Ths s a well-known class of mathematcal programmng problems for whch several (non-trval) algorthms exst. In practce, we can just regard the QP solver as a black-box box wthout botherng how t works You wll be spared of the excrucatng detals and jump to
13 The soluton We can not gve you a close form soluton that you can drectly yplug n the numbers and compute for an arbtrary data sets But, the soluton can always be wrtten n the followng form w = α y x, s.t. α y = 0 = 1 Ths s the form of w, b can be calculated accordngly usng some addtonal steps The weght vector s a lnear combnaton of all the tranng examples Importantly, many of the α s are zeros These ponts that have non-zero α s are the support vectors = 1
14 A Geometrcal Interpretaton Class 2 α 8 =0.6 α 10=0 α 5 =0 α 7 =0 α 2 =0 α 4 =0 α 9 =0 Class 1 α 6 =1.4 6 α 3 =0 α 1 =0.8
15 A few mportant notes regardng the geometrc nterpretaton gves the decson boundary postve support vectors le on ths lne negatve support vectors le on ths lne We can thnk of a decson boundary now as a tube of certan wdth, no ponts can be nsde the tube Learnng nvolves adjustng the locaton and orentaton t of the tube to fnd the largest fttng tube for the gven tranng set
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