Information Technology Reference
In-Depth Information
The MP loss is defined as follows:
=
f(x
u
)
y
u,v
f(x
v
)
−
β
.
L
MP
(f
;
x
u
,x
v
,y
u,v
)
−
(3.21)
The HMP loss is defined as follows:
0
,
u,v
(f (x
u
)
−
f(x
v
))
≥
0
,
;
=
L
HMP
(f
x
u
,x
v
,y
u,v
)
(3.22)
β
,
|
−
−
y
u,v
|
(f (x
u
)
f(x
v
))
otherwise
.
The SVR loss is defined as follows:
0
,
|
y
u,v
−
(f (x
u
)
−
f(x
v
))
|
<ε,
L
SVR
(f
;
x
u
,x
v
,y
u,v
)
=
(3.23)
β
,
|
(f (x
u
)
−
f(x
v
))
−
y
u,v
−
ε
|
otherwise
.
The differences between the three loss functions lie in the different conditions
that they penalize a mis-ranked pair (but not to what degree). For example, for the
MP loss, not only the mis-ranked pairs but also the correctly-ranked pairs will be
penalized if their magnitude of the predicted preference is too large; for the HMP
loss, only the miss-ranked pairs are penalized; for the SVR loss, only if the magni-
tude of the predicted preference is different from the labeled preferences to a certain
degree, the pair (no matter correctly or mis-ranked) will be penalized.
Then a
L
2
regularization term is introduced to these loss functions, and the loss
functions are optimized using kernel methods. Experimental results show that the
magnitude-preserving loss functions can lead to better ranking performances than
the original pairwise ranking algorithms, such as RankBoost [
18
].
3.3.3 IR-SVM
According to the second problem of the pairwise approach as mentioned above, the
difference in the numbers of document pairs of different queries is usually signifi-
cantly larger than the difference in the number of documents. This phenomenon has
been observed in some previous studies [
9
,
27
].
In this case, the pairwise loss function will be dominated by the queries with
a large number of document pairs, and as a result the pairwise loss function will
become inconsistent with the query-level evaluation measures. To tackle the prob-
lem, Cao et al. [
9
] propose introducing query-level normalization to the pairwise
loss function. That is, the pairwise loss for a query will be normalized by the total
number of document pairs associated with that query. In this way, the normalized
pairwise losses with regards to different queries will become comparable to each
other in their magnitude, no matter how many document pairs they are originally
associated with. With this kind of query-level normalization, Ranking SVM will
become a new algorithm, referred to as IR-SVM [
9
]. Specifically, given
n
training
queries
1
, their associated document pairs
(x
(i
u
,x
(i
v
)
, and the corresponding
n
i
{
q
i
}
=
