Machine Learning. Support Vector Machines. (contains material adapted from talks by Constantin F. Aliferis & Ioannis Tsamardinos, and Martin Law)
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1 Machne Learnng Support Vector Machnes (contans materal adapted from talks by Constantn F. Alfers & Ioanns Tsamardnos, and Martn Law) Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
2 Support Vector Machnes Are bnary classfers Are very popular because They are very effectve classfers n many domans They can tran farly quckly on large data sets They can be used wthout understandng ther underlyng math yet those who want to can geek out on the formulae. 2
3 Man deas: Support Vector Machnes 1. Fnd an optmal hyperplane to splt the data nto two sets: maxmze margn 2. Extend the above defnton for non-lnearly separable problems: have a penalty term for msclassfcatons 3. Map data to hgh dmensonal space where t s easer to classfy wth lnear decson surfaces: reformulate problem so that data s mapped mplctly to ths space
4 Any hyperplane can be wrtten as the set of ponts x satsfyng the equaton below, where w and x are vectors n R d Defnng a hyperplane w x + = 0 The vector w s a normal vector: t s perpendcular to the hyperplane. The parameter b determnes the offset of the hyperplane from the orgn along the normal vector. dst to orgn = b b w Eucldean norm Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
5 A hyperplane n 2 dmensons w! x! + b = 0 b w! Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
6 Some defntons Assume tranng data consstng of vectors of d real values that belong to class 1 or class -1 D = d {( x, y ) x R, y { 1,1}} Fnd a hyperplane to separate the data nto the two classes (assume ths s possble, for the moment). Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
7 Whch hyperplane s best? Class +1 Class -1 Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
8 The margn Defne d+ as the dstance from a hyperplane to the closest postve example. Defne d- as the dstance to the closest negatve example Defne the margn, m as m = d + + d Look for the largest margn! Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
9 An example d + d Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
10 The margn There s some scale for w and for b where the followng equaton holds. d + = d = 1 w Those data ponts that le wthn dstance 1/ w of the hyperplane are called the support vectors. The support vectors defne two planes parallel to the hyperplane separator Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
11 Three hyperplanes to consder w! x! w x + b = 1 + b = 0 Support vectors are ponts on the margns (The margns are the two boundng hyperplanes at dstance 1/ w from the decson lne) 2 w w x + b =1 Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
12 Optmzaton = maxmze margn We wrte the optmzaton problem as maxmze 2 w such that y ( w x + b) 1, { x, y } D remember y s the class label, and y { 1,1} Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
13 Maxmzng margns (not mathy) Gven a guess of w and b we can See f all data ponts are correctly classfed Compute the wdth of the margn Now, we just search the space of w s and b s to fnd the wdest margn hyperplane that correctly classfes the data. How? Gradent descent? Smulated Annealng? Matrx Inverson? Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
14 A lttle more mathy Fnd w and b to maxmze such that 2 margn = w y ( w x + b) 1, { x, y } D Ths s a quadratc functon wth lnear constrants. Quadratc optmzaton problems have known algorthmc solutons. Navely, ths quadratc optmzaton can be solved n O(n^3) tme, where n = the number of data ponts.
15 Maxmzng Stll mathy-er 2 w equvalent to mnmzng w For math convenence we ACTUALLY mnmze 1 w 2 Now, we assocate a Lagrange multpler wth each pont n the data. Ths gves x 2 α L p 1 2 w 2 D = 1 α y ( x w + b) + D = 1 α Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
16 (psst: What s a Lagrange Multpler?) A way of optmzng a functon subject to one (or more) constrant(s) from another functon(s). You ncorporate the orgnal functon and the constrant equatons nto one new equaton and then solve. For more, check out the wkpeda. Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
17 Why use Lagrangan Multplers? We want to put a lne between the sets that maxmze the margn between classes subject to the constrant all ponts are correctly classfed. So The margn maxmzaton s a classc maxmzaton problem Classfyng each pont correctly s a constrant Ths s exactly the setup needed for Lagrangans Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
18 In practce, people don t mnmze ths formula from the prevous slde. Instead they maxmze ts dual, whch wll also gve us what we need and s n a more convenent format for later work. The Lagrangan dual Bryan Pardo, Machne Learnng: EECS 349 Fall 2009 = = ) ( 2 1 D D p b w x y w L α α ) ( j j j D D j D D x x y y L = = = α α α
19 The Lagrangan dual When we maxmze ths, a cool thng happens. Only those ponts defnng the margn have nonzero values for α L D D = 1 α 1 D 2 D = 1 j= 1 α α j y y j ( x x j ) Note also ths formulaton relates two examples to each other va a dot product. Ths wll become meanngful later Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
20 Classfcaton wth SVM For testng, the new data z s classfed as class 1 f and as class -1 f f < 0 f 0 SO our weghts are determned by ths (where S s the number of support vectors) And our decson functon s a normal lnear dscrmnant. Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
21 Classfcaton wth SVM: PART 2 Whle our decson functon s a normal lnear dscrmnant. people usually calculate the class usng the support vectors (those data ponts wth non-0 alpha values that le on the +1 and -1 margns). The new element z s compared to all the support vectors and ts value s determned by where t les n comparson to them. Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
22 What f we have nosy tranng data? Can we combne mnmzng msclassfcatons (wth some forgveness) wth maxmzng the margn? Var 1 w! x! + b = 0 Var 2 Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
23 Man deas: Support Vector Machnes 1. Fnd an optmal hyperplane to splt the data nto two sets: maxmze margn 2. Extend the above defnton for non-lnearly separable problems: have a penalty term for msclassfcatons 3. Map data to hgh dmensonal space where t s easer to classfy wth lnear decson surfaces: reformulate problem so that data s mapped mplctly to ths space
24 Non-Lnearly Separable Data Var 1 w! w! x! + r r w x+ b= 1 b = 0 ξ ξ 1 1 r r w x+ b= 1 Var 2 Allow some nstances to fall wthn the margn, but penalze them Introduce slack varables ξ (one per data pont) y The constrants then become ξ ( w x + b) 1 ( x, y ) D, ξ 0
25 Formulatng the Problem + C w w ξ 2 1 We are now mnmzng ths formula Subject to the constrants Where C determnes the weght to gve msclassfcaton error. 0, ), ( 1 ) ( + D y x b x w y ξ ξ
26 Lnear, Soft-Margn SVMs 1 mn 2 2 w + C ξ y ( w x + b) 1 ξ, x ξ 0 Algorthm tres to mnmze ξ whle maxmzng margn Notce: algorthm does not mnmze the number of msclassfcatons, but the sum of dstances from the margn hyperplanes As C, we get closer to the hard-margn soluton
27 The Lagrange Dual The dual of the soft problem s w s also recovered as The only dfference wth the lnearly separable case s that there s an upper bound C on Once agan, a QP solver can be used to fnd α α
28 Robustness of Soft vs Hard Margn Var 1 Var 1 ξ Var 2 Var 2 Soft Margn SVM Hard Margn SVM
29 Soft vs Hard Margn SVM Soft-Margn always have a soluton Soft-Margn s more robust to outlers Smoother surfaces (n the non-lnear case) Hard-Margn requres no parameters at all
30 Lnear SVMs: Overvew The classfer s a separatng hyperplane. Most mportant tranng ponts are support vectors; they defne the hyperplane. Quadratc optmzaton algorthms can dentfy whch tranng ponts x are support vectors wth non-zero Lagrangan multplers. Both n the dual formulaton of the problem and n the soluton, tranng ponts appear only nsde nner products: L D D = 1 α 1 D 2 D = 1 j= 1 α α j y y j ( x x j )
31 Extenson to Non-lnear Decson Boundary So far, we have only consdered large-margn classfer wth a lnear decson boundary What f the decson boundary s non-lnear? A one-dmensonal decson problem x Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
32 Man deas: Support Vector Machnes 1. Fnd an optmal hyperplane to splt the data nto two sets: maxmze margn 2. Extend the above defnton for non-lnearly separable problems: have a penalty term for msclassfcatons 3. Map data to hgh dmensonal space where t s easer to classfy wth lnear decson surfaces: reformulate problem so that data s mapped mplctly to ths space
33 Hgher dmensonal mappng The dea s to map the vectors to a hgher dmensonal space to gan lnear separaton. x x 2 Mappng x onto {x 2,x} gves lnear separaton. x Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
34 Non-lnear SVMs: Feature spaces General dea: the orgnal feature space can be mapped to some hgher-dmensonal feature space where the tranng set s separable: Φ: x φ(x)
35 What ths does to the math Recall the SVM optmzaton problem wth ts nner product between data ponts If we transform the data nto a new vector space, ths changes the math ever so slghtly to be Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
36 What are Kernel Functons K(x 1, x 2 ) = Φ(x 1 ),Φ(x 2 ) Kernels are functons that return nner products between the mages of data ponts Snce they are nner products, they can be thought of as smlarty functons Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
37 Why Use Kernel Functons K(x 1, x 2 ) = Φ(x 1 ),Φ(x 2 ) You can defne kernels for many thngs that aren t vectors of real values n Eucldean space (e.g. text documents) As long as we can calculate the nner product n the orgnal feature space, we do not need to explctly calculate the mappng Φ It can often be more effcent to compute K drectly, wthout gong through the step of computng Φ Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
38 Ths orgnal formula That math agan becomes ths, when we apply a data transformaton and f we can defne a kernel t becomes ths Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
39 An Example Kernel Suppose s gven as follows An nner product n the feature space s ** NOTE: On ths slde y s not the label, t s a feature vector, just lke x ** So, f we defne the kernel functon as follows, there s no need to carry out f(.) explctly Ths use of kernel functon to avod calculatng explctly s known as the kernel trck Bryan Pardo, Machne Learnng: EECS 349 Fall 2011
40 Examples of Kernel Functons ** NOTE: On ths slde y s not the label, t s a feature vector, just lke x ** Polynomal kernel wth degree d Radal bass functon kernel wth wdth s Closely related to radal bass functon neural networks The feature space s nfnte-dmensonal Sgmod wth parameter k and q It does not satsfy the Mercer condton on all k and q Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
41 Buldng Kernels from Other Kernels Kernels can be composed from other kernels The followng 2 sldes gve some basc rules for buldng complex kernels from smple kernels Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
42 Defntons for the followng slde k 1 (x, x') and k 2 (x, x') are vald kernels on {x,x'} S S s some set (of anythng: emals, mages, ntegers) c > 0 s a constant f ( ) s any functon q( ) s a polynomal wth non-negatve coeffcents φ(x) s a functon from the ->! m k 3 (, )s a vald kernel n! m A s a symmetrc postve semdefnte matrx x = (x a, x b )...essentally, x can be decomposed nto subparts...lke scalars n a vector k a (, ),k b (, ) are vald kernels over ther respectve spaces Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
43 Technques for Kernel Constructon Gven vald kernels k 1 (x, x') and k 2 (x, x'), the followng are also vald kernels k(x, x') = ck 1 (x, x') k(x, x') = f (x)k 1 (x, x') f (x') k(x, x') = q(k 1 (x, x')) k(x, x') = exp(k 1 (x, x')) k(x, x') = k 1 (x, x') + k 2 (x, x') k(x, x') = k 3 (φ(x),φ(x')) k(x, x') = x T Ax' Ths one assumes x,x' are vectors k(x, x') = k a (x a, x' a ) + k b (x b, x' b ) k(x, x') = k a (x a, x' a )k b (x b, x' b ) Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
44 Classfyng wth a Kernel For testng, the new data z s classfed as class 1 f and as class -1 f f < 0 f 0 Orgnal Wth kernel functon NOTE: y s once agan a label from the set {-1, 1} ** Bryan Pardo, Machne Learnng: EECS 349 Fall 2014
45 Strengths and Weaknesses of SVM Strengths Tranng s relatvely easy No local maxmum, unlke n neural networks It scales relatvely well to hgh dmensonal data Tradeoff between classfer complexty and error can be controlled explctly Non-tradtonal data lke strngs and trees can be used as nput to SVM, nstead of feature vectors Weaknesses Tunng SVMs remans a black art: selectng a specfc kernel and parameters s usually done n a try-and-see manner.
46 You as the SVM user You have two man choces to make: 1) What kernel wll you use? Polynomal? Radal Bass Functon? Somethng else? 2) How much slack wll you allow? Depends on how much you trust data collecton and labelng. Bryan Pardo, Machne Learnng: EECS 349 Fall 2009
47 SVM applcatons SVMs were orgnally proposed by Boser, Guyon and Vapnk n 1992 and ganed ncreasng popularty n late 1990s. SVMs are currently among the best performers for a number of classfcaton tasks rangng from text to genomc data. SVMs can be appled to complex data types beyond feature vectors (e.g. graphs, sequences, relatonal data) by desgnng kernel functons for such data. SVM technques have been extended to a number of tasks such as regresson [Vapnk et al. 97], prncpal component analyss [Schölkopf et al. 99], etc. Most popular optmzaton algorthms for SVMs use decomposton to hll-clmb over a subset of α s at a tme, e.g. SMO [Platt 99] and [Joachms 99]
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