Information Technology Reference
In-Depth Information
;
Suppose the pointwise loss function is
L(f
x
j
,y
j
)
. Then the expected risk is
defined as follows:
D
q
dx
j
,dy
j
P
Q
(dq).
=
;
R(f )
L(f
x
j
,y
j
)
(16.15)
Q
X
×
Y
Intuitively, the expected risk means the loss that a ranking model
f
would make
for a random document associated with a random query. As both the distributions
P
Q
and
D
q
are unknown, the average of the loss over a set of training queries
n
i
{
q
i
}
1
(i.i.d. observations according to
P
Q
) and their associated training documents
=
m
(i)
j
{
(x
j
,y
j
)
}
(i.i.d. observations according to
D
q
) is used to estimate the above
=
1
expected risk,
m
(i)
n
L
f
;
x
(i)
j
.
1
n
1
m
(i)
R(f )
=
,y
(i)
j
(16.16)
i
=
1
j
=
1
16.3.2 The Pairwise Approach
Similar to the document ranking framework, there are also two views on the pairwise
approach in the two-layer ranking framework.
(1) The U-statistics View
With the U-statistics view, one assumes i.i.d. distribu-
tion of the documents and their ground-truth labels with respect to a query. Given
two documents associated with query
q
and their ground truth labels,
(x
u
,y
u
)
and
(x
v
,y
v
)
, we denote
y
u,v
=
2
·
I
}
−
1. Then the expected risk can be defined as
{
y
u
y
v
R(f )
=
X
×
Y
)
2
L(f
;
x
u
,x
v
,y
u,v
)
D
q
(dx
u
,dy
u
)
D
q
(dx
v
,dy
v
)P
(dq).
Q
Q
(
(16.17)
Intuitively, the expected risk means the loss that a ranking model
f
would make
for two random documents associated with a random query
q
. As both the distri-
butions
P
and
D
q
are unknown, the following empirical risk is used to estimate
Q
R(f )
:
m
(i)
m
(i)
n
L
f
u,v
.
1
n
2
m
(i)
(m
(i)
R(f )
x
(i)
u
,x
(i)
v
,y
(i)
=
;
(16.18)
−
1
)
i
=
u
=
v
=
u
+
1
1
1
(2) The Average View
The average view assumes the i.i.d. distribution of docu-
ment pairs. More specifically, with the average view, each document pair
(x
u
,x
v
)
is given a ground-truth label
y
u,v
∈
Y
={−
1
,
1
}
, where
y
u,v
=
1 indicates that doc-
ument
x
u
is more relevant than
x
v
and
y
u,v
=−
1 otherwise. Then
(x
u
,x
v
,y
u,v
)
is
