lecture02

# lecture02 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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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 α ∈ R suchthat a T i x + α ≥ 1 ∀{ i :¯ y i =1 } and a T i x + α ≤- 1 ∀{ i :¯ y i =- 1 } , thatis, ¯ y i ( a T i x + α )- 1 ≥ ∀ i. Thehyperp-lanewouldbe { y : x T y + α =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 + α )- 1) subjectto ¯ y i ( a T i x + α )- 1 ≥ ∀ i. Or mmnimize ∑ i δ i subjectto ¯ y i ( a T i x + α )- 1 ≥- δ i , ∀ i, δ 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 MS&E211 Lecture...

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