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Support Vector Learning for Ordinal Regression
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Support Vector Learning for Ordinal Regression
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Support
Vector Learning for Ordinal Regression
Ralf Herbrich, Thore Graepel, Klaus Obermayer
Technical University of Berlin
Department of Computer Science
Franklinstr.
28/29
10587
Berlin
ralfhlgraepel2loby@cs.tu-berlin.de
Abstract
We investigate the problem of predicting variables
of
or-
dinal scale. This task is referred to as
ordinal regression
and is complementary to the standard machine learning
tasks of classification and metric regression. In contrast
to statistical models we present a distribution independent
formulation
of
the problem together with uniform bounds
of
the risk functional. The approach presented is based
on a mapping from objects to scalar utility values. Sim-
ilar to Support Vector methods we derive a new learning
algorithm for the task of ordinal regression based on large
margin rank boundaries. We give experimental results
for
an information retrieval task: learning the order of doc-
uments w.r.t. an initial query. Experimental results indi-
cate that the presented algorithm outperforms more naive
approaches to ordinal regression such as Support Vector
classification and Support Vector regression
in
the case of
more than two ranks.
1
Introduction
Problems of ordinal regression arise
in
many fields, e.g.,
in
information retrieval (Herbrich et al. 1998),
in
econo-
metric models (Tangian and Gruber 1995), and
in
clas-
sical statistics (McCullagh 1980; Anderson 1984). They
can be related to the standard machine learning paradigm
as follows:
Given an i.i.d. sample
S
=
{(~i,yi)}f=~
-
Piy
and
a set
3t
of
mappings
h
from
X
to
Y,
a learning proce-
dure selects one mapping
he
such that
-
using a prede-
fined loss
1
:
Y
x
Y
R
-
the risk functional
R(
he)
is
minim'ized. Typically,
in
machine learning the risk func-
tional
R(h)
under consideration is the expectation value
of the
loss
I(y,
h(x)),
i.e., the loss at each point
(x,
y)
weighted by its (unknown) probability
Pxy
(x,
y). Us-
ing the principle of Empirical Risk Minimization (ERM),
one chooses that function
he
which minimizes the mean
of the
loss
R,,,(he)
given the sample
S.
Two main sce-
narios were considered
in
the past:
(i)
If
Y
is a finite
unordered set (nominal scale), the task is referred to as
classijication.
Since
Y
is unordered, the
0
-
1
loss,
i.e.,
lo-l(y,y)=Oiffy=y,andlo-l(yly)
=
liffy#jl,is
adequate to capture the
loss
at each point
(x,
y).
(ii)
If
Y
is a metric space, e.g., the set of real numbers, the task
is referred to as
regression estimation.
In this case the
loss function can take into account the full metric struc-
ture (see Smola (1998) for a detailed discussion of
loss
functions for regression).
In ordinal regression, we consider a problem which
shares properties of both classification
(i)
and metric re-
gression (ii). Like
in
(i)
Y
is a finite set and like
in
(ii)
there exists an ordering among the elements of
Y.
A
variable of the above type exhibits an
ordinal scale
and
can be thought of as the result of coarse measurement
of a continuous variable (Anderson 1984). The ordinal
scale leads to problems
in
defining an appropriate
loss
function for
our
task (see McCullagh 1980). In Section
2
we present a distribution independent model for ordi-
nal regression, which is based on a
loss
function that acts
on pairs of ranks. We give explicit uniform convergence
Artificial Neural Networks,
7
-
10 September 1999, Conference Publication No.
470
0
IEE
1999
97
xlliu0226
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