In the last lecture we consider a learning problem in which
the optimal function belonged to a finite class of functions.
Specifically, for some collection of functions FFwith finite
cardinality |F|≤∞|F|≤∞, we have
min
f
∈
F
R
(
f
)
=
0
⇒
f
*
∈
F
.
min
f
∈
F
R
(
f
)
=
0
⇒
f
*
∈
F
.
(1)
This is almost always not the situation in the real-world
learning problems. Let us suppose we have a finite collection of
candidate functions FF. Furthermore, we do not assume that the
optimal function f*f*, which satisfies
R
(
f
*
)
=
inf
f
R
(
f
)
R
(
f
*
)
=
inf
f
R
(
f
)
(2)
where the infinf is taken over all measurable functions,
is a member of FF. That is, we make few, if any, assumptions about
f*f*. This situation is sometimes termed as Agnostic
Learning. The root of the word agnostic literally means
not known. The term agnostic learning is used to emphasize
the fact that often, perhaps usually, we may have no prior knowledge
about f*f*. The question then arises about how we can reasonably
select an f∈Ff∈F in this setting.
The PAC style bounds discussed in the previous lecture, offer some
help. Since we are selecting a function based on the empirical risk,
the question is how close is R^n(f)R^n(f) to R(f)R(f)∀f∈F∀f∈F. In other words, we wish that the empirical risk is a good
indicator of the true risk for every function in FF. If this is
case, the selection of ff that minimizes the empirical risk
f
n
^
=
arg
min
f
∈
F
n
R
^
n
(
f
)
f
n
^
=
arg
min
f
∈
F
n
R
^
n
(
f
)
(3)
should also yield a small true risk, that is, R(fn^)R(fn^)
should be close to minf∈FR(f)minf∈FR(f). Finally, we can thus
state our desired situation as
P
max
f
∈
F
n
|
R
n
^
(
f
)
-
R
(
f
)
|
>
ϵ
<
δ
,
P
max
f
∈
F
n
|
R
n
^
(
f
)
-
R
(
f
)
|
>
ϵ
<
δ
,
(4)
for small values of ϵϵ and δδ. In other words, with
probability at least 1-δ1-δ, |Rn^(f)-R(f)|>ϵ|Rn^(f)-R(f)|>ϵ, ∀f∈F∀f∈F. In this lecture, we will start
to develop bounds of this form. First we will focus on bounding
P(|Rn^(f)-R(f)|>ϵ)P(|Rn^(f)-R(f)|>ϵ) for one fixed f∈Ff∈F.
To begin, let us recall the definition of empirical risk for
{Xi,Yi}i=1n{Xi,Yi}i=1n be a collection of training data. Then
the empirical risk is defined as
R
^
n
(
f
)
=
1
n
∑
i
=
1
n
ℓ
(
f
(
X
i
)
,
Y
i
)
.
R
^
n
(
f
)
=
1
n
∑
i
=
1
n
ℓ
(
f
(
X
i
)
,
Y
i
)
.
(5)
Note that since the training data {Xi,Yi}i=1n{Xi,Yi}i=1n are
assumed to be i.i.d. pairs, the terms in the sum are
i.i.d random variables.
Let
L
i
=
ℓ
(
f
(
X
i
)
,
Y
i
)
.
L
i
=
ℓ
(
f
(
X
i
)
,
Y
i
)
.
(6)
The collection of losses {Li}i=1n{Li}i=1n is i.i.d
according to some unknown distribution (depending on the unknown
joint distribution of (X,Y) and the loss function). The
expectation of LiLi is E[ℓ(f(Xi),Yi)]=E[ℓ(f(X),Y)]=R(f)E[ℓ(f(Xi),Yi)]=E[ℓ(f(X),Y)]=R(f), the true risk of ff. For now, let's
assume that ff is fixed.
E
[
R
n
^
(
f
)
]
=
1
n
∑
i
=
1
n
E
[
ℓ
(
f
(
X
i
)
,
Y
i
)
]
=
1
n
∑
i
=
1
n
E
[
L
i
]
=
R
(
f
)
E
[
R
n
^
(
f
)
]
=
1
n
∑
i
=
1
n
E
[
ℓ
(
f
(
X
i
)
,
Y
i
)
]
=
1
n
∑
i
=
1
n
E
[
L
i
]
=
R
(
f
)
(7)
We know from the strong law of large numbers that the average (or
empirical mean) Rn^(f)Rn^(f) converges almost surely to the true
mean R(f).R(f). That is, Rn^(f)→R(f)Rn^(f)→R(f) almost surely as n→∞n→∞. The question is how fast.
Concentration inequalities are upper bounds on how fast empirical
means converge to their ensemble counterparts, in probability. The area
of the shaded tail regions in Figure 1 is P(|Rn^(f)-R(f)|>ϵ)P(|Rn^(f)-R(f)|>ϵ). We are interested in finding out how fast this
probability tends to zero as n→∞n→∞.
At this stage, we
recall Markov's Inequality. Let ZZ be a nonnegative random
variable.
E
[
Z
]
=
∫
0
∞
z
p
(
z
)
d
z
=
∫
0
t
z
p
(
z
)
d
z
+
∫
u
∞
z
p
(
z
)
d
z
≥
0
+
t
∫
t
∞
z
p
(
z
)
d
z
=
t
P
(
Z
≥
t
)
⇒
P
(
Z
≥
t
)
≤
E
[
Z
]
t
⇒
P
(
Z
2
≥
t
2
)
≤
E
[
Z
2
]
t
2
E
[
Z
]
=
∫
0
∞
z
p
(
z
)
d
z
=
∫
0
t
z
p
(
z
)
d
z
+
∫
u
∞
z
p
(
z
)
d
z
≥
0
+
t
∫
t
∞
z
p
(
z
)
d
z
=
t
P
(
Z
≥
t
)
⇒
P
(
Z
≥
t
)
≤
E
[
Z
]
t
⇒
P
(
Z
2
≥
t
2
)
≤
E
[
Z
2
]
t
2
(8)
Take
Z
=
|
R
n
(
f
)
^
-
R
(
f
)
|
and
t
=
ϵ
Z
=
|
R
n
(
f
)
^
-
R
(
f
)
|
and
t
=
ϵ
(9)
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
≤
E
[
|
R
n
(
f
)
^
-
R
(
f
)
|
2
]
ϵ
2
≤
var
(
R
^
n
(
f
)
)
ϵ
2
=
∑
i
=
1
n
var
(
L
i
n
)
ϵ
2
=
var
(
ℓ
(
X
)
,
Y
)
n
ϵ
2
=
σ
L
2
n
ϵ
2
.
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
≤
E
[
|
R
n
(
f
)
^
-
R
(
f
)
|
2
]
ϵ
2
≤
var
(
R
^
n
(
f
)
)
ϵ
2
=
∑
i
=
1
n
var
(
L
i
n
)
ϵ
2
=
var
(
ℓ
(
X
)
,
Y
)
n
ϵ
2
=
σ
L
2
n
ϵ
2
.
(10)
So, the probability goes to zero at a rate of at least n-1n-1.
However, it turns out that this is an extremely loose bound. According
to the Central Limit Theorem
R
n
^
(
f
)
=
1
n
∑
i
=
1
n
L
i
→
N
R
(
f
)
,
σ
L
2
n
as
n
→
∞
R
n
^
(
f
)
=
1
n
∑
i
=
1
n
L
i
→
N
R
(
f
)
,
σ
L
2
n
as
n
→
∞
(11)
in distribution. This suggests that for large values of n,
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
≈
O
e
-
n
ϵ
2
2
σ
L
2
.
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
≈
O
e
-
n
ϵ
2
2
σ
L
2
.
(12)
That is, the Gaussian tail probability is tending to zero
exponentially fast.
Note that for any nonnegative random variable ZZ and t>0t>0,
P
(
Z
≥
t
)
=
P
(
e
s
Z
≥
e
s
t
)
≤
E
[
e
s
Z
]
e
s
t
,
∀
s
>
0
by
Markov's
inequality
.
P
(
Z
≥
t
)
=
P
(
e
s
Z
≥
e
s
t
)
≤
E
[
e
s
Z
]
e
s
t
,
∀
s
>
0
by
Markov's
inequality
.
(13)
Chernoff's bound is based on finding the value of ss that
minimizes the upper bound. If ZZ is a sum of independent random
variables. For example, say
Z
=
∑
i
=
1
n
ℓ
(
f
(
X
i
)
,
Y
i
)
-
R
(
f
)
=
n
R
^
n
(
f
)
-
R
(
f
)
Z
=
∑
i
=
1
n
ℓ
(
f
(
X
i
)
,
Y
i
)
-
R
(
f
)
=
n
R
^
n
(
f
)
-
R
(
f
)
(14)
then the bound becomes
P
∑
i
=
1
n
(
L
i
-
E
[
L
i
]
)
≥
t
≤
e
-
s
t
E
[
e
s
∑
i
=
1
n
(
L
i
-
E
[
L
i
]
)
]
≤
e
-
s
t
∏
i
=
1
n
E
[
e
s
(
L
i
-
E
[
L
i
]
)
]
,
from
independence.
P
∑
i
=
1
n
(
L
i
-
E
[
L
i
]
)
≥
t
≤
e
-
s
t
E
[
e
s
∑
i
=
1
n
(
L
i
-
E
[
L
i
]
)
]
≤
e
-
s
t
∏
i
=
1
n
E
[
e
s
(
L
i
-
E
[
L
i
]
)
]
,
from
independence.
(15)
Thus, the problem of finding a tight bound boils down to finding a
good bound for E[ss(Li-E[Li])]E[ss(Li-E[Li])]. Chernoff ('52), first
studied this situation for binary random variables. Then,
Hoeffding ('63) derived a more general result for arbitrary
bounded random variables.
Let Z1,Z2,...,ZnZ1,Z2,...,Zn be independent bounded random variables such that
Zi∈[ai,bi]Zi∈[ai,bi] with probability 1. Let Sn=∑i=1nZiSn=∑i=1nZi. Then for any t>0t>0, we have
P
(
|
S
n
-
E
[
S
n
]
|
≥
t
)
≤
2
e
-
2
t
2
∑
i
=
1
n
(
b
i
-
a
i
)
2
.
P
(
|
S
n
-
E
[
S
n
]
|
≥
t
)
≤
2
e
-
2
t
2
∑
i
=
1
n
(
b
i
-
a
i
)
2
.
(16)
The key to proving Hoeffding's inequality is the following upper
bound: if ZZ is a random variable with E[Z]=0E[Z]=0 and a≤Z≤b,a≤Z≤b, then
E
[
e
s
Z
]
≤
e
s
2
(
b
-
a
)
2
8
.
E
[
e
s
Z
]
≤
e
s
2
(
b
-
a
)
2
8
.
(17)
This
upper bound is derived as follows. By the convexity of the
exponential function,
e
s
z
≤
z
-
a
b
-
a
e
s
b
+
b
-
z
b
-
a
e
s
a
,
for
a
≤
z
≤
b
.
e
s
z
≤
z
-
a
b
-
a
e
s
b
+
b
-
z
b
-
a
e
s
a
,
for
a
≤
z
≤
b
.
(18)
Thus,
E
[
e
s
Z
]
≤
E
Z
-
a
b
-
a
e
s
b
+
E
b
-
Z
b
-
a
e
s
a
=
b
b
-
a
e
s
a
-
a
b
-
a
e
s
b
,
since
E
[
Z
]
=
0
=
(
1
-
θ
+
θ
e
s
(
b
-
a
)
)
e
-
θ
s
(
b
-
a
)
,
where
θ
=
-
a
b
-
a
.
E
[
e
s
Z
]
≤
E
Z
-
a
b
-
a
e
s
b
+
E
b
-
Z
b
-
a
e
s
a
=
b
b
-
a
e
s
a
-
a
b
-
a
e
s
b
,
since
E
[
Z
]
=
0
=
(
1
-
θ
+
θ
e
s
(
b
-
a
)
)
e
-
θ
s
(
b
-
a
)
,
where
θ
=
-
a
b
-
a
.
(19)
Now let
u
=
s
(
b
-
a
)
and
define
φ
(
u
)
≡
-
θ
u
+
log
(
1
-
θ
+
θ
e
u
)
.
u
=
s
(
b
-
a
)
and
define
φ
(
u
)
≡
-
θ
u
+
log
(
1
-
θ
+
θ
e
u
)
.
(20)
Then we have
E
[
e
s
Z
]
≤
(
1
-
θ
+
θ
e
s
(
b
-
a
)
)
e
-
θ
s
(
b
-
a
)
=
e
φ
(
u
)
.
E
[
e
s
Z
]
≤
(
1
-
θ
+
θ
e
s
(
b
-
a
)
)
e
-
θ
s
(
b
-
a
)
=
e
φ
(
u
)
.
(21)
To minimize the upper bound let's express φ(u)φ(u) in a Taylor's series with remainder :
φ
(
u
)
=
φ
(
0
)
+
u
φ
'
(
0
)
+
u
2
2
φ
'
'
(
v
)
for
some
v
∈
[
0
,
u
]
φ
(
u
)
=
φ
(
0
)
+
u
φ
'
(
0
)
+
u
2
2
φ
'
'
(
v
)
for
some
v
∈
[
0
,
u
]
(22)
φ
'
(
u
)
=
-
θ
+
θ
e
u
1
-
θ
+
θ
e
u
⇒
φ
'
(
u
)
=
0
φ
'
'
(
u
)
=
θ
e
u
1
-
θ
+
θ
e
u
-
θ
e
u
(
1
-
θ
+
θ
e
u
)
2
=
θ
e
u
1
-
θ
+
θ
e
u
(
1
-
θ
e
u
1
-
θ
+
θ
e
u
)
=
ρ
(
1
-
ρ
)
.
φ
'
(
u
)
=
-
θ
+
θ
e
u
1
-
θ
+
θ
e
u
⇒
φ
'
(
u
)
=
0
φ
'
'
(
u
)
=
θ
e
u
1
-
θ
+
θ
e
u
-
θ
e
u
(
1
-
θ
+
θ
e
u
)
2
=
θ
e
u
1
-
θ
+
θ
e
u
(
1
-
θ
e
u
1
-
θ
+
θ
e
u
)
=
ρ
(
1
-
ρ
)
.
(23)
Now, φ''(u)φ''(u) is maximized by
ρ
=
θ
e
u
1
-
θ
+
θ
e
u
=
1
2
⇒
φ
'
'
(
u
)
≤
1
4
.
ρ
=
θ
e
u
1
-
θ
+
θ
e
u
=
1
2
⇒
φ
'
'
(
u
)
≤
1
4
.
(24)
So,
φ
(
u
)
≤
u
2
8
=
s
2
(
b
-
a
)
2
8
φ
(
u
)
≤
u
2
8
=
s
2
(
b
-
a
)
2
8
(25)
⇒
E
[
e
s
Z
]
≤
e
s
2
(
b
-
a
)
2
8
.
⇒
E
[
e
s
Z
]
≤
e
s
2
(
b
-
a
)
2
8
.
(26)
Now, we can apply this upper bound to derive Hoeffding's inequality.
P
(
S
n
-
E
[
S
n
]
≥
t
)
≤
e
-
s
t
∏
i
=
1
n
E
[
e
s
(
L
i
-
E
[
L
i
]
)
]
≤
e
-
s
t
∏
i
=
1
n
e
s
2
(
b
i
-
a
i
)
2
8
=
e
-
s
t
e
s
2
∑
i
=
1
n
(
b
i
-
a
i
)
2
8
=
e
-
2
t
2
∑
i
=
1
n
(
b
i
-
a
i
)
2
by
choosing
s
=
4
t
∑
i
=
1
n
(
b
i
-
a
i
)
2
P
(
S
n
-
E
[
S
n
]
≥
t
)
≤
e
-
s
t
∏
i
=
1
n
E
[
e
s
(
L
i
-
E
[
L
i
]
)
]
≤
e
-
s
t
∏
i
=
1
n
e
s
2
(
b
i
-
a
i
)
2
8
=
e
-
s
t
e
s
2
∑
i
=
1
n
(
b
i
-
a
i
)
2
8
=
e
-
2
t
2
∑
i
=
1
n
(
b
i
-
a
i
)
2
by
choosing
s
=
4
t
∑
i
=
1
n
(
b
i
-
a
i
)
2
(27)
Similarly, P(E[Sn]-Sn≥t)≤e-2t2∑i=1n(bi-ai)2P(E[Sn]-Sn≥t)≤e-2t2∑i=1n(bi-ai)2.
This completes the proof of the Hoeffding's theorem.
Let Zi=1f(Xi)≠Yi-R(f),Zi=1f(Xi)≠Yi-R(f), as in the
classification problem. Then for a fixed f, it follows from
Hoeffding's inequality (i.e., Chernoff's bound in this special case)
that
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
=
P
1
n
|
S
n
-
E
[
S
n
]
|
≥
ϵ
=
P
(
|
S
n
-
E
[
S
n
]
|
≥
n
ϵ
)
≤
2
e
-
2
(
n
ϵ
)
2
n
=
2
e
-
2
n
ϵ
2
.
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
=
P
1
n
|
S
n
-
E
[
S
n
]
|
≥
ϵ
=
P
(
|
S
n
-
E
[
S
n
]
|
≥
n
ϵ
)
≤
2
e
-
2
(
n
ϵ
)
2
n
=
2
e
-
2
n
ϵ
2
.
(28)
Now, we want a bound like this to hold uniformly for all f∈Ff∈F.
Assume that FF is a finite collection of models and let |F||F|
denote its cardinality. We would like to bound the
probability that maxf∈F|Rn^(f)-R(f)|≥ϵmaxf∈F|Rn^(f)-R(f)|≥ϵ. Note that the event
max
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
≡
⋃
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
.
max
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
≡
⋃
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
.
(29)
Therefore
P
max
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
=
P
⋃
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
≤
∑
f
∈
F
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
,
the
``
union
of
events
''
bound
≤
2
|
F
|
e
-
2
n
ϵ
2
,
by
Hoeffding's
inequality.
P
max
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
=
P
⋃
f
∈
F
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
≤
∑
f
∈
F
P
(
|
R
n
^
(
f
)
-
R
(
f
)
|
≥
ϵ
)
,
the
``
union
of
events
''
bound
≤
2
|
F
|
e
-
2
n
ϵ
2
,
by
Hoeffding's
inequality.
(30)
Thus, we have shown that with probability at
least 1-2|F|e-2nϵ21-2|F|e-2nϵ2, ∀f∈F∀f∈F
|
R
n
^
(
f
)
-
R
(
f
)
|
<
ϵ
.
|
R
n
^
(
f
)
-
R
(
f
)
|
<
ϵ
.
(31)
And accordingly, we can be reasonably
confident in selecting ff from FF based on the empirical risk
function R^nR^n.
(What does "Endorsed by" mean?)
"A complete course in modern statistical learning theory at the graduate student level."