Information Technology Reference
In-Depth Information
16.2.1 The Pointwise Approach
m
We denote all the
m
documents associated with query
q
as
x
={
x
j
}
j
=
1
, and their
m
j
relevance degrees as
y
1
. Note that in the subset ranking framework, there
is no assumption of sampling of each individual documents. Instead, it is
(
x
,
y
)
(which is a representation for the corresponding query) that is regarded as a random
variable sampled from the space
={
y
j
}
=
m
m
X
×
Y
according to an unknown probability
distribution
P
.
Suppose the pointwise loss function is
L(f
;
x
j
,y
j
)
. Then the expected risk can
be represented as follows,
m
1
m
=
;
R(f )
L(f
x
j
,y
j
)P (d
x
,d
y
).
(16.9)
X
m
×
Y
m
j
=
1
Intuitively, the expected risk means the
average
loss that a ranking model
f
would make for all the documents associated with a random query
q
. Since it is
almost impossible to compute the expected risk, in practice, the empirical risk on
the training set is used as an estimate of the expected risk.
n
m
L
f
.
1
n
1
m
x
(i)
j
,y
(i)
j
R(f )
=
;
(16.10)
i
=
j
=
1
1
16.2.2 The Pairwise Approach
For the pairwise approach, once again, we denote all the
m
documents associated
with query
q
as
x
m
j
m
j
={
x
j
}
1
, and denote the relevance degrees as
y
={
y
j
}
1
.We
=
=
m
m
regard
(
x
,
y
)
as a random variable sampled from the space
X
×
Y
according to
an unknown probability distribution
P
.
Suppose the pairwise loss function is
L(f
;
x
v
,x
v
,y
u,v
)
. For any two different
documents
x
u
and
x
v
, we denote
y
u,v
=
2
·
I
{
y
u
y
v
}
−
1. Accordingly, the expected
risk can be represented as follows,
m
m
2
m(m
−
R(f )
=
L(f
;
x
u
,x
v
,y
u,v
)P (d
x
,d
y
).
(16.11)
1
)
X
m
×
Y
m
u
=
1
v
=
u
+
1
Intuitively, the expected risk means the
average
loss that a ranking model
f
would make for all the document pairs associated with a random query
q
. Since it
is almost impossible to compute the expected risk, in practice, the empirical risk on
the training set is used as an estimate of the expected risk. In particular, given the
training data
(
x
(i)
,
y
(i)
)
n
i
{
}
1
,the
empirical risk
can be defined as follows,
=
n
m
m
L
f
u,v
.
1
n
2
m(m
R(f )
x
(i)
u
,x
(i)
v
,y
(i)
=
;
(16.12)
−
1
)
i
=
1
u
=
1
v
=
u
+
1
