Information Technology Reference

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the form of
b
k
−
1
≤

b
k
, while the implicit constraint uses redundant training exam-

ples to guarantee the ordinal relationship between thresholds.

2.4.3 Ordinal Regression with Threshold-Based Loss Functions

In [
18
], different loss functions for ordinal regression are compared. Basically two

types of threshold-based loss functions are investigated, i.e., immediate-threshold

loss and all-threshold loss. Here, the thresholds refer to
b
k
(k

=

1
,...,K

−

1
)
which

separate different ordered categories.

Suppose the scoring function is
f(x)
, and
φ
is a margin penalty function.
φ
can

be the hinge, exponential, logistic, or square function. Then the immediate-threshold

loss is defined as follows:

L(f
;
x
j
,y
j
)
=
φ
f(x
j
)
−
b
y
j
−
1
+
φ
b
y
j
−
f(x
j
)
,

(2.14)

where for each labeled example
(x
j
,y
j
)
, only the two thresholds defining the “cor-

rect” segment
(b
y
j
−
1
,b
y
j
)
are considered. In other words, the immediate-threshold

loss is ignorant of whether multiple thresholds are crossed.

The all-threshold loss is defined as below, which is a sum of all threshold-

violation penalties.

K

φ
s(k,y
j
)
b
k
−
f(x
j
)
,

L(f
;
x
j
,y
j
)
=

(2.15)

k

=

1

where

−

1
,k<
j
,

s(k,y
j
)

=

(2.16)

+

1
,k

≥

y
j
.

Note that the slope of the above loss function increases each time a threshold is

crossed. As a result, solutions are encouraged that minimize the number of thresh-

olds that are crossed.

The aforementioned two loss functions are tested on the MovieLens dataset.
3
The

experimental results show that the all-threshold loss function can lead to a better

ranking performance than multi-class classification and simple regression methods,

as well as the method minimizing the immediate-threshold loss function.

2.5 Discussions

In this section, we first discuss the relationship between the pointwise approach and

some early learning methods in information retrieval, such as relevance feedback.

Then, we discuss the limitations of the pointwise approach.

3
http://www.grouplens.org/node/73
.