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offprint
offprint
206 Feilong CAO, Xing XING
Throughout this paper, if ξ is a random variable, i.e., a real valued on a
probability space Z, we will use E(ξ) to denote the expected value and σ
2
(ξ)
to denote its variance. Thus,
E(ξ)=
Z
ξdρ, σ
2
(ξ)=E(ξ
2
) − (E(ξ))
2
.
2Mainresults
Our target of error analysis is to understand how sgn(f
z
) approximate the Bayes
rule as m become large.
An important result on the relation between the misclassification error and
generalization error for a convex loss function is due to [4,5,18]. Here, we
consider the hinge loss
φ(t)=(1− t)
+
=max{1 −t, 0}.
From [14] and [18], we obtain
R(sgn(f) − R(f
c
) E (f) − E (f
c
). (4)
Thus, the excess misclassification error R(sgn(f
z
)) − R(f
c
) can be bounded
by the excess generalization error E (f) − E (f
c
), and the following error
decomposition holds (see [15]):
E (π(
f
z
)) − E (f
c
)+
λf
z
2
Λ
D(
λ)+{E (π(
f
z
)) − E
z
(π(
f
z
)) + E
z
(
f
Λ,
λ
) − E (
f
Λ,
λ
)}, (5)
where
f
Λ,λ
=arg min
f∈H
Λ
{E (f)+λf
2
Λ
},
D(λ)= inf
f∈H
Λ
{E (f) − E (f
c
)+λf
2
Λ
}.
The regularization error for a regularizing function f
Λ,λ
is defined [11] as
D(λ)=E (f
Λ,λ
) − E (f
c
)+λf
Λ,λ
2
Λ
.
The information we need about the capacity K is measured by covering
numbers. The covering numbers play an essential role in sample error estimates.
Definition 3 [6] Let F be a subset of a metric space. For any ε>0, the
covering number N(F ,ε) is defined to be the minimal integer l ∈ N such that
there exist l balls with radius ε covering F .
The function sets in our situation are balls of the multi-kernel space in the
forms of
B
R
= {f ∈ H
Λ
: f
Λ
R}.
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