Information Technology Reference
In-Depth Information
Note that in some previous work, LambdaRank is regarded as a kind of direct
optimization method (see Sect. 4.2). However, according to our understanding, it
is not such a method because the integral of the lambda function as defined above
(which will correspond to the implicit loss function) is not highly related to NDCG.
Therefore, in our opinion, it is more appropriate to say that LambdaRank considers
the evaluation measures to set its pair weight than to say it directly optimizes the
evaluation measures.
3.3.8 Robust Sparse Ranker
In [
31
], ranking is reduced to a weighted pairwise classification, whose weights are
defined with the gain and discount functions in NDCG. In this way, one can also
introduce positional information to the pairwise approach. Specifically, in [
31
], the
following weighted classification loss is minimized:
·
I
{
y
u
>y
v
}
·
c(x
u
,x
v
)
,
L(c
;
x
,
y
)
=
ω(y
u
,y
v
,c)
(3.32)
u<v
1
where
ω(y
u
,y
v
,c)
denotes the im-
portance weight for the ordered pair
(x
u
,x
v
)
, and
c
is a classifier:
c(x
u
,x
v
)
=
Z
m
|
(G(y
u
)
−
G(y
v
))(η(π(u))
−
η(π(v)))
|
=
1if
c
prefers
x
v
to
x
u
and 0 otherwise.
On the one hand, given any distribution
P
on the training data
{
x
,
y
}
, the expected
loss of the classifier
c
is given by
E
x
,
y
L(c
x
,
y
)
.
L(c
;
P)
=
;
(3.33)
Then the regret of
c
on
P
is defined as follows:
L(c
∗
;
;
=
;
−
r(c
P)
L(c
P)
min
c
∗
P).
(3.34)
On the other hand, given the evaluation measure NDCG, we can also define its
expectation with respect to the ranking model
h
as follows:
E
x
,
y
NDCG
(h
x
,
y
)
.
G(h
;
P)
=
;
(3.35)
Correspondingly, we can define the regret of
h
on
P
as
G(h
∗
;
r(h
;
P)
=
max
h
∗
P)
−
G(h
;
P).
(3.36)
Then if the ranking model
h
determines the position of document
x
u
according
to
u
=
v
c(x
u
,x
v
)
, the following theoretical result has been proven in [
31
]:
r(h
;
P)<r(c
;
P
),
(3.37)
