# Lecture 21 Bounds on the generalization error of voting classiﬁers. 18.465 ```Lecture 21
Bounds on the generalization error of voting classiﬁers.
18.465
We continue to prove the lemma from Lecture 20:
�d
Lemma 21.1. Let Fd = convd H = { i=1 λi hi , hi ∈ H} and ﬁx δ ∈ (0, 1]. Then
�
��
� ��
dV log nδ
t
Eϕδ − ϕ&macr;δ
≥ 1 − e−t .
≤K
+
P ∀f ∈ Fd , √
n
n
Eϕδ
Proof. We showed that
log D(ϕδ (yFd ) , ε/2, dx,y ) ≤ KdV log
2
.
εδ
By the result of Lecture 16,
n
1�
k
Eϕδ (yf (x)) −
ϕδ (yi f (xi )) ≤ √
n i=1
n
�
√
Eϕδ
�
log
1/2
D(ϕδ (yFd (x)) , ε)dε +
0
tEϕδ
n
with probability at least 1 − e−t . We have
� √Eϕδ
� √Eϕδ �
k
k
2
1/2
√
log D(ϕδ (yFd (x)) , ε)dε ≤ √
dV log dε
εδ
n 0
n 0
�
� √
k 2 δ Eϕδ /2 √
1
=√
dV log dx
x
nδ 0
�
k 2√
δ�
2
≤√
dV 2
Eϕδ log √
2
nδ
δ Eϕδ
= x, ε = 2x
. Without loss of generality, assume Eϕδ ≥ 1/n.
� δ
Otherwise, we’re doing better than in Lemma: √EE ≤ logn n ⇒ E ≤ logn n . Hence,
�
√
� √Eϕδ
k
dV Eϕδ
2 n
1/2
√
log D(ϕδ (yFd (x)) , ε)dε ≤ K
log
n
δ
n 0
�
dV Eϕδ
n
≤K
log
n
δ
where we have made a change of variables
2
εδ
So, with probability at least 1 − e−t ,
n
1�
Eϕδ (yf (x)) −
ϕδ (yi f (xi )) ≤ K
n i=1
�
dV Eϕδ (yf (x))
n
log +
n
δ
�
tEϕδ (yf (x))
n
which concludes the proof.
�
The above lemma gives a result for a ﬁxed d ≥ 1 and δ ∈ (0, 1]. To obtain a uniform result, it’s enough
to consider δ ∈ Δ = {2−k , k ≥ 1} and d ∈ {1, 2, . . .}. For a ﬁxed δ and d, use the Lemma above with tδ,d
deﬁned by e−tδ,d = e−t d26πδ2 . Then
�
P ∀f ∈ Fd , . . . +
and
⎛
P⎝
�
d,δ
∀f ∈ Fd , . . . +
�
tδ,d
n
�
�
≥ 1 − e−tδ,d = 1 − e−t
6δ
d2 π 2
�⎞
�
tδ,d ⎠
6δ
≥1−
e−t 2 2 = 1 − e−t .
n
d π
δ,d
53
Lecture 21
Bounds on the generalization error of voting classiﬁers.
Since tδ,d = t + log
18.465
d2 π 2
6δ ,
�
⎞
⎛�
2 2
t + log d 6δπ
dV log nδ
Eϕδ − ϕ&macr;δ
⎠
∀f ∈ Fd , √
+
≤K⎝
n
n
Eϕδ
��
�
� �
dV log nδ
d2 π 2
t
≤K
+ log
+
n
n
6δ
��
� �
n
dV
log
t
δ
≤ K�
+
n
n
�
2 2
dV log n
δ
since log d 6δπ , the penalty for union-bound, is much smaller than
.
n
Recall the bound on the misclassiﬁcation error
n
1 �
P (yf (x) ≤ 0) ≤
I(yi f (xi ) ≤ δ) +
n i=1
If
�
�
n
1�
Eϕδ (yf (x)) −
ϕδ (yi f (xi )) .
n i=1
�n
Eϕδ − n1 i=1 ϕδ
√
≤ ε,
Eϕδ
then
n
Eϕδ − ε
�
Eϕδ −
1�
ϕδ ≤ 0.
n i=1
Hence,
�
�
�� �
n
�
ε
ε 2 1�
Eϕδ ≤ + �
+
ϕδ
2
2
n i=1
Eϕδ ≤ 2
� ε �2
2
n
+2
1�
ϕδ .
n i=1
The bound becomes
⎛
⎞
n
⎜ 1 �
dV
n t⎟
⎜
⎟
P (yf (x) ≤ 0) ≤ K ⎜
I(yi f (xi ) ≤ δ) +
log + ⎟
n
δ
n
⎝ n i=1
⎠
� �� �
(*)
where K is a rough constant.
(*) not satisfactory because in boosting the bound should get better when the number of functions grows.
We prove a better bound in the next lecture.
54
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