Solved–Week 9 –Solution

Description

Question 1 Corrected (For Assessment)

Show that, for the maximum margin classifier, the correct value of β0 is
β0=− maxi:yi=−1(β∗)T xi + mini:yi=1 (β∗)T xi ,
where β∗ = P 2
n λi∗yixi is the optimal value for β.
i=1
Question 2

Work through labs 9.6.2 and 9.6.5 in the text book. This should give you a feeling for how support vector classifiers work in R.

Question 3

Consider the support vector classifier with the Lagrangian

1 n n n
X X X
L(β, β0, ξ, λ, µ) = βTβ+C ξi − λi yi(xiT β + β0) − 1 + ξi − µiξi
2
i=1 i=1 i=1
Using the KKT equations, show that the optimal β can be written as
n
β = X
λiyixi.
i=1
Show that λ solves
n 1 n n
X X X
max λ y y λ λ xT x
i − 2
λ i j i j i j
i=1 i=1 j=1
Subject to:

0 ≤ λi ≤ C, i = 1, . . . , n

n
X
λiyi = 0.

i=1

Argue that

λi = 0 ⇒ yi(βT xi + β0) ≥ 1

λi = C ⇒ yi(βT xi + β0) ≤ 1

0 < λi < C ⇒ yi(βT xi + β0) = 1.

1