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class05 (1) - Support Vector Machines For Classification...

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Unformatted text preview: Support Vector Machines For Classification 9.520 Class 06, 24 February 2003 Ryan Rifkin Plan • Regularization derivation of SVMs • Geometric derivation of SVMs • Optimality, Duality and Large Scale SVMs • SVMs and RLSC: Compare and Contrast The Regularization Setting (Again) We are given ‘ examples ( x 1 , y 1 ) , . . . , ( x ‘ , y ‘ ), with x i ∈ IR n and y i ∈ {- 1 , 1 } for all i . As mentioned last class, we can find a classification function by solving a regularized learning problem: min f ∈H 1 ‘ ‘ X i =1 V ( y i , f ( x i )) + λ || f || 2 K . Note that in this class we are specifically consider binary classification . The Hinge Loss The classical SVM arises by considering the specific loss function V ( f ( x ) , y ) ≡ (1- yf ( x )) + , where ( k ) + ≡ max( k, 0) . The Hinge Loss-3-2-1 1 2 3 0.5 1 1.5 2 2.5 3 3.5 4 y * f(x) Hinge Loss Substituting In The Hinge Loss With the hinge loss, our regularization problem becomes min f ∈H 1 ‘ ‘ X i =1 (1- y i f ( x i )) + + λ || f || 2 K . Slack Variables This problem is non-differentiable (because of the “kink” in V ), so we introduce slack variables ξ i , to make the problem easier to work with: min f ∈H 1 ‘ ∑ ‘ i =1 ξ i + λ || f || 2 K subject to : y i f ( x i ) ≥ 1- ξ i i = 1 , . . . , ‘ ξ i ≥ i = 1 , . . . , ‘ Applying The Representer Theorem Substituting in: f * ( x ) = ‘ X i =1 c i K ( x , x i ) , we arrive at a constrained quadratic programming problem: min c ∈ IR ‘ 1 ‘ ∑ ‘ i =1 ξ i + λ c T K c subject to : y i ∑ ‘ j =1 c j K ( x i , x j ) ≥ 1- ξ i i = 1 , . . . , ‘ ξ i ≥ i = 1 , . . . , ‘ Adding A Bias Term If we add an unregularized bias term b , which presents some theoretical difficulties to be discussed later, we arrive at the “primal” SVM: min c ∈ IR ‘ ,ξ ∈ IR ‘ 1 ‘ ∑ ‘ i =1 ξ i + λ c T K c subject to : y i ( ∑ ‘ j =1 c j K ( x i , x j ) + b ) ≥ 1- ξ i i = 1 , . . . , ‘ ξ i ≥ i = 1 , . . . , ‘ Forming the Lagrangian For reasons that will be clear in a few slides, we derive the Wolfe dual quadratic program using Lagrange multiplier techniques: L ( c , ξ, b, α, ζ ) = 1 ‘ ‘ X i =1 ξ i + λ c T K c- ‘ X i =1 α i y i ‘ X j =1 c j K ( x i , x j ) + b - 1 + ξ i - ‘ X i =1 ζ i ξ i We want to minimize L with respect to c , b , and ξ , and maximize L with respect to α and ζ , subject to the con- straints of the primal problem and nonnegativity constraints on α and ζ . Eliminating b and ξ ∂L ∂b = 0 = ⇒ ‘ X i =1 α i y i = 0 ∂L ∂ξ i = 0 = ⇒ 1 ‘- α i- ζ i = 0 = ⇒ ≤ α i ≤ 1 ‘ We write a reduced Lagrangian in terms of the remaining variables: L R ( c , α ) = λ c T K c- ‘ X i =1 α i ( y i ‘ X j =1 c j K ( x i , x j )- 1) Eliminating c Assuming the K matrix is invertible, ∂L R ∂ c = 0 = ⇒ 2 λK c- KY α = 0 = ⇒ c i = α i y i 2 λ Where Y is a diagonal matrix whose i ’th diagonal element is y i ; Y α is a vector whose i ’th element is α i y i . The Dual Program...
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This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.

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class05 (1) - Support Vector Machines For Classification...

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