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
