lecture02 - Yinyu Ye, MS&E, Stanford...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 1 MoreModeling,FormulationsandApplications YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 2 Formulation4:SupportingVectorMachine Supposewehavetwo-class discriminationdata .Weassignthefirstclasswith 1 andthesecondwith- 1 .Apowerful discriminationmethod isthe Supporting VectorMachine(SVM) . Letthedatapoint i begivenby a i R d , i =1 ,...,n .Withthisdataset,we havesome y i =1 (inthefirstclass)andtherest y i =- 1 (inthesecondclass). Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 3 * * * * * * * * * * * * * * * * * * * * * Figure1:LinearSupportVectorMachine Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 4 SupportingVectorMachinecontinued Wewishtofinda hyper-plane in R d toseparate a i swithredfrom a j swithgreen. Mathematically,wewishtofind x R d and R suchthat a T i x + 1 { i : y i =1 } and a T i x + - 1 { i : y i =- 1 } , thatis, y i ( a T i x + )- 1 i. Thehyperp-lanewouldbe { y : x T y + =0 } . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 5 SupportingVectorMachinecontinued Ifa cleanseparation ispossible,wecanformulatetheproblemasamaximization problem: minimize i ( y i ( a T i x + )- 1) subjectto y i ( a T i x + )- 1 i. Or mmnimize i i subjectto y i ( a T i x + )- 1 - i , i, i , i. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 6 Formulation5:AdwordsAllocation Given n different Adwords that m companies arebiddingtoputtheir advertisementsonthepagewhentheAdwordisclicked. A order , i ,onasetofAdwordsincludes...
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lecture02 - Yinyu Ye, MS&E, Stanford...

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