Support vector machines
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1 CS 75 Mache Learg Lecture Support vector maches Mlos Hauskrecht 539 Seott Square CS 75 Mache Learg Outle Outle: Algorthms for lear decso boudary Support vector maches Mamum marg hyperplae. Support vectors. Support vector maches. Etesos to the o-separable case. Kerel fuctos. CS 75 Mache Learg
2 Learly separable classes here s a hyperplae that separates trag staces th o error Hyperplae: + = Class (+ + > Class (- + < CS 75 Mache Learg Logstc regresso Separatg hyperplae: + = y >.5? d d We ca use gradet methods or Neto Rhapso for sgmodal stchg fuctos ad lear the eghts Recall that e lear the lear decso boudary CS 75 Mache Learg
3 Perceptro algorthm Perceptro algorthm: Smple teratve procedure for modfyg the eghts of the lear model Italze eghts Loop through eamples, y the dataset D. Compute y ˆ =. If y yˆ = the + 3. If y yˆ = + the Utl all eamples are classfed correctly Propertes: guarateed covergece CS 75 Mache Learg Solvg va LP Lear program soluto: Fds eghts that satsfy the follog costrats: + For all, such that y = + + For all, such that = y ogether: y ( + Property: f there s a hyperplae separatg the eamples, the lear program fds the soluto CS 75 Mache Learg
4 Optmal separatg hyperplae here are multple hyperplaes that separate the data pots Whch oe to choose? Mamum marg choce: mamzes dstace d + d + here d + s the shortest dstace of a postve eample from the hyperplae (smlarly for egatve eamples d Marg dstace d d + CS 75 Mache Learg Mamum marg hyperplae For the mamum marg hyperplae oly eamples o the marg matter (oly these affect the dstaces hese are called support vectors CS 75 Mache Learg
5 Fdg mamum marg hyperplaes Assume that eamples the trag set are, y such that y { +, } Assume that all data satsfy: + for y = + + for y = he equaltes ca be combed as: y ( + for all Equaltes defe to hyperplaes: + = + = CS 75 Mache Learg Fdg the mamum marg hyperplae Dstace of a pot th label from the hyperplae: d ( = ( + / - ormal to the hyperplae.. L - Eucldea orm L Dstace of a pot th label -: d ' = ( ' + / Dstace of a pot th label y: L ρ,, y = y( + / L CS 75 Mache Learg
6 Fdg the mamum marg hyperplae Geometrcal marg: ρ,, y = y( + / L For pots satsfyg: y ( + = he dstace s L Wdth of the marg: d + + d = L CS 75 Mache Learg Mamum marg hyperplae We at to mamze d We do t by mmzg + + d = L, - varables = L / / But e also eed to eforce the costrats o pots: [ y ( + ] CS 75 Mache Learg
7 Mamum marg hyperplae Soluto: Icorporate costrats to the optmzato Optmzato problem (Lagraga [ y ( + ] = α - Lagrage multplers J (,, α = / α Mmze th respect to, (prmal varables Mamze th respect to α (dual varables Lagrage multplers eforce the satsfacto of costrats [ y ( + ] > If α Else α > Actve costrat CS 75 Mache Learg Ma marg hyperplae soluto Set dervatves to (Karush-Kuh-ucker (KK codtos J (,, α = α y = J (,, α = = α y = = No e eed to solve for Lagrage parameters (Wolfe dual J ( α = = α, = α α y Quadratc optmzato problem: soluto αˆ for all y Subect to costrats α for all, ad α = y = mamze CS 75 Mache Learg
8 Mamum hyperplae soluto he resultg parameter vector ŵ ca be epressed as: ˆ = ˆ α αˆ s the soluto of the dual problem = y he parameter s obtaed through Karush-Kuh-ucker codtos αˆ y ( ˆ + = [ ] Soluto propertes αˆ = for all pots that are ot o the marg ŵ s a lear combato of support vectors oly he decso boudary: ˆ + = αˆ y + = SV CS 75 Mache Learg he decso boudary: he decso: Support vector maches ˆ ˆ = α y SV + yˆ = sg αˆ y SV + + CS 75 Mache Learg
9 he decso boudary: ˆ he decso: Support vector maches ˆ = α y SV + yˆ = sg αˆ y + SV (!!: Decso o a e requres to compute the er product betee the eamples Smlarly, the optmzato depeds o J ( α = α α α y y =, = CS 75 Mache Learg + Eteso to a learly o-separable case Idea: Allo some fleblty o crossg the separatg hyperplae CS 75 Mache Learg
10 Eteso to the learly o-separable case Rela costrats th varables + ξ + + ξ for for = + Error occurs f ξ, ξ s the upper boud o the = umber of errors Itroduce a pealty for the errors mmze Subect to costrats / + C ξ = ξ y y = C set by a user, larger C leads to a larger pealty for a error CS 75 Mache Learg Eteso to learly o-separable case Lagrage multpler form (prmal problem Dual form after, are epressed ( ξ s cacel out J ( α = α α α y y = he parameter, = [ y ( + + ξ ], α = / + C ξ α = = J (, µ ξ Subect to: α C for all, ad α y = = Soluto: ˆ = αˆ y = he dfferece from the separable case: α C s obtaed through KK codtos = CS 75 Mache Learg
11 he decso boudary: ˆ he decso: Support vector maches ˆ = α y SV + yˆ = sg αˆ y + SV (!!: Decso o a e requres to compute the er product betee the eamples Smlarly, the optmzato depeds o J ( α = α α α y y =, = CS 75 Mache Learg + Nolear case he lear case requres to compute ( he o-lear case ca be hadled by usg a set of features. Essetally e map put vectors to (larger feature vectors φ( It s possble to use SVM formalsm o feature vectors Kerel fucto φ( φ( ' Crucal dea: If e choose the kerel fucto sely e ca compute lear separato the feature space mplctly such that e keep orkg the orgal put space!!!! K, ' = φ( φ( ' CS 75 Mache Learg
12 Kerel fucto eample Assume = [ ad a feature mappg that maps the put, ] to a quadratc feature set φ( = [,,,,,] Kerel fucto for the feature space: K ', = φ( ' φ( = ' + ' + ' ' + ' + ' + = ' + ' + = ( + ' he computato of the lear separato the hgher dmesoal space s performed mplctly the orgal put space CS 75 Mache Learg Nolear eteso Kerel trck Replace the er product th a kerel A ell chose kerel leads to effcet computato CS 75 Mache Learg
13 Kerel fucto eample Lear separator the feature space No-lear separator the put space CS 75 Mache Learg Polyomal kerel Kerel fuctos Lear kerel K, ' = ' [ ] ' k K, ' = + Radal bass kerel K, ' = ep ' CS 75 Mache Learg
14 Kerels SVM researchers have proposed kerels for comparso of varety of obects: Strgs rees Graphs Cool thg: SVM algorthm ca be o appled to classfy a varety of obects CS 75 Mache Learg
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