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