A neural network approach to ordinal regression
A Neural Network Approach to Ordinal Regression
Jianlin Cheng jcheng@cs.ucf.edu
School of Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL 32816, USA
Abstract
Ordinal regression is an important type of
learning, which has properties of both clas-
sification and regression. Here we describe
a simple and effective approach to adapt a
traditional neural network to learn ordinal
categories. Our approach is a generaliza-
tion of the perceptron method for ordinal
regression. On several benchmark datasets,
our method (NNRank) outperforms a neural
network classification method. Compared
with the ordinal regression methods using
Gaussian processes and support vector
machines, NNRank achieves comparable
performance. Moreover, NNRank has the
advantages of traditional neural networks:
learning in both online and batch modes,
handling very large training datasets, and
making rapid predictions. These features
make NNRank a useful and complementary
tool for large-scale data processing tasks
such as information retrieval, web page
ranking, collaborative filtering, and protein
ranking in Bioinformatics.
1. Introduction
Ordinal regression (or ranking learning) is an impor-
tant supervised problem of learning a ranking or or-
dering on instances, which has the property of both
classification and metric regression. The learning task
of ordinal regression is to assign data points into a set
of finite ordered categories. For example, a teacher
rates students’ performance using A, B, C, D, and E
(A > B > C > D > E) (Chu & Ghahramani, 2005a).
Ordinal regression is different from classification due
to the order of categories. In contrast to metric re-
gression, the response variables (categories) in ordinal
regression is discrete and finite.
The research of ordinal regression dated back to the
ordinal statistics methods in 1980s (McCullagh, 1980;
McCullagh & Nelder, 1983) and machine learning re-
search in 1990s (Caruana et al., 1996; Herbrich et al.,
1998; Cohen et al., 1999). It has attracted the con-
siderable attention in recent years due to its poten-
tial applications in many data-intensive domains such
as information retrieval (Herbrich et al., 1998), web
page ranking (Joachims, 2002), collaborative filtering
(Goldberg et al., 1992; Basilico & Hofmann, 2004; Yu
et al., 2006), image retrieval (Wu et al., 2003), and pro-
tein ranking (Cheng & Baldi, 2006) in Bioinformatics.
A number of machine learning methods have been de-
veloped or redesigned to address ordinal regression
problem (Rajaram et al., 2003), including perceptron
(Crammer & Singer, 2002) and its kernelized gener-
alization (Basilico & Hofmann, 2004), neural network
with gradient descent (Caruana et al., 1996; Burges
et al., 2005), Gaussian process (Chu & Ghahramani,
2005b; Chu & Ghahramani, 2005a; Schwaighofer et
al., 2005), large margin classifier (or support vec-
tor machine) (Herbrich et al., 1999; Herbrich et al.,
2000; Joachims, 2002; Shashua & Levin, 2003; Chu
& Keerthi, 2005; Aiolli & Sperduti, 2004; Chu &
Keerthi, 2007), k-partite classifier (Agarwal & Roth,
2005), boosting algorithm (Freund et al., 2003; Dekel
et al., 2002), constraint classification (Har-Peled et al.,
2002), regression trees (Kramer et al., 2001), Naive
Bayes (Zhang et al., 2005), Bayesian hierarchical ex-
perts (Paquet et al., 2005), binary classification ap-
proach (Frank & Hall, 2001; Li & Lin, 2006) that de-
composes the original ordinal regression problem into
a set of binary classifications, and the optimization of
nonsmooth cost functions (Burges et al., 2006).
Most of these methods can be roughly classified into
two categories: pairwise constraint approach (Herbrich
et al., 2000; Joachims, 2002; Dekel et al., 2004; Burges
et al., 2005) and multi-threshold approach (Cram-
mer & Singer, 2002; Shashua & Levin, 2003; Chu &
Ghahramani, 2005a). The former is to convert the full
ranking relation into pairwise order constraints. The
latter tries to learn multiple thresholds to divide data
A Neural Network Approach to Ordinal Regression
into ordinal categories. Multi-threshold approaches
also can be unified under the general, extended binary
classification framework (Li & Lin, 2006).
The ordinal regression methods have different advan-
tages and disadvantages. Prank (Crammer & Singer,
2002), a perceptron approach that generalizes the bi-
nary perceptron algorithm to the ordinal multi-class
situation, is a fast online algorithm. However, like a
standard perceptron method, its accuracy suffers when
dealing with non-linear data, while a quadratic kernel
version of Prank greatly relieves this problem. One
class of accurate large-margin classifier approaches
(Herbrich et al., 2000; Joachims, 2002) convert the
ordinal relations into O(n2) (n: the number of data
points) pairwise ranking constraints for the structural
risk minimization (Vapnik, 1995; Schoelkopf & Smola,
2002). Thus, it can not be applied to medium size
datasets (> 10,000 data points), without discarding
some pairwise preference relations. It may also overfit
noise due to incomparable pairs.
The other class of powerful large-margin classifier
methods (Shashua & Levin, 2003; Chu & Keerthi,
2005) generalize the support vector formulation for or-
dinal regression by finding K − 1 thresholds on the
real line that divide data into K ordered categories.
The size of this optimization problem is linear in the
number of training examples. However, like support
vector machine used for classification, the prediction
speed is slow when the solution is not sparse, which
makes it not appropriate for time-critical tasks. Simi-
larly, another state-of-the-art approach, Gaussian pro-
cess method (Chu & Ghahramani, 2005a), also has the
difficulty of handling large training datasets and the
problem of slow prediction speed in some situations.
Here we describe a new neural network approach for
ordinal regression that has the advantages of neural
network learning: learning in both online and batch
mode, training on very large dataset (Burges et al.,
2005), handling non-linear data, good performance,
and rapid prediction. Our method can be considered
a generalization of the perceptron learning (Crammer
& Singer, 2002) into multi-layer perceptrons (neural
network) for ordinal regression. Our method is also
related to the classic generalized linear models (e.g.,
cumulative logit model) for ordinal regression (Mc-
Cullagh, 1980). Unlike the neural network method
(Burges et al., 2005) trained on pairs of examples
to learn pairwise order relations, our method works
on individual data points and uses multiple output
nodes to estimate the probabilities of ordinal cate-
gories. Thus, our method falls into the category of
multi-threshold approach. The learning of our method
proceeds similarly as traditional neural networks using
back-propagation (Rumelhart et al., 1986).
On the same benchmark datasets, our method yields
the performance better than the standard classifica-
tion neural networks and comparable to the state-of-
the-art methods using support vector machines and
Gaussian processes. In addition, our method can learn
on very large datasets and make rapid predictions.
2. Method
2.1. Formulation
Let D represent an ordinal regression dataset consist-
ing of n data points (x, y) , where x ∈ Rd is an input
feature vector and y is its ordinal category from a fi-
nite set Y . Without loss of generality, we assume that
Y = 1, 2, ...,K with ”<” as order relation.
For a standard classification neural network without
considering the order of categories, the goal is to pre-
dict the probability of a data point x belonging to
one category k (y = k). The input is x and the
target of encoding the category k is a vector t =
(0, ..., 0, 1, 0, ..., 0), where only the element tk is set to
1 and all others to 0. The goal is to learn a function
to map input vector x to a probability distribution
vector o = (o1, o2, ...ok, ...oK), where ok is closer to 1
and other elements are close to zero, subject to the
constraint
i=1 oi = 1.
In contrast, like the perceptron approach (Crammer &
Singer, 2002), our neural network approach considers
the order of the categories. If a data point x belongs
to category k, it is classified automatically into lower-
order categories (1, 2, ..., k − 1) as well. So the target
vector of x is t = (1, 1, .., 1, 0, 0, 0), where ti (1 ≤ i ≤ k)
is set to 1 and other elements zeros. Thus, the goal
is to learn a function to map the input vector x to
a probability vector o = (o1, o2, ..., ok, ...oK), where
oi (i ≤ k) is close to 1 and oi (i ≥ k) is close to 0.∑K
i=1 oi is the estimate of number of categories (i.e.
k) that x belongs to, instead of 1. The formulation
of the target vector is similar to the perceptron ap-
proach (Crammer & Singer, 2002). It is also related
to the classical cumulative probit model for ordinal re-
gression (McCullagh, 1980), in the sense that we can
consider the output probability vector (o1, ...ok, ...oK)
as a cumulative probability distribution on categories
(1, ..., k, ..., K), i.e.,
is the proportion of cate-
gories that x belongs to, starting from category 1.
The target encoding scheme of our method is related to
but, different from multi-label learning (Bishop, 1996)
and multiple label learning (Jin & Ghahramani, 2003)
A Neural Network Approach to Ordinal Regression
because our method imposes an order on the labels (or
categories).
2.2. Learning
Under the formulation, we can use the almost exactly
same neural network machinery for ordinal regression.
We construct a multi-layer neural network to learn
ordinal relations from D. The neural network has d
inputs corresponding to the number of dimensions of
input feature vector x and K output nodes correspond-
ing to K ordinal categories. There can be one or more
hidden layers. Without loss of generality, we use one
hidden layer to construct a standard two-layer feedfor-
ward neural network. Like a standard neural network
for classification, input nodes are fully connected with
hidden nodes, which in turn are fully connected with
output nodes. Likewise, the transfer function of hid-
den nodes can be linear function, sigmoid function,
and tanh function that is used in our experiment. The
only difference from traditional neural network lies in
the output layer. Traditional neural networks use soft-
max e
(or normalized exponential function) for
output nodes, satisfying the constraint that the sum of
outputs
i=1 oi is 1. zi is the net input to the output
node Oi.
In contrast, each output node Oi of our neural net-
work uses a standard sigmoid function 1
1+e−zi
, with-
out including the outputs from other nodes. Output
node Oi is used to estimate the probability oi that a
data point belongs to category i independently, with-
out subjecting to normalization as traditional neural
networks do. Thus, for a data point x of category
k, the target vector is (1, , 1, .., 1, 0, 0, 0), in which the
first k elements is 1 and others 0. This sets the target
value of output nodes Oi (i ≤ k) to 1 and Oi (i > k)
to 0. The targets instruct the neural network to ad-
just weights to produce probability outputs as close
as possible to the target vector. It is worth pointing
out that using independent sigmoid functions for out-
put nodes does not guaranteed the monotonic relation
(o1 >= o2 >= ... >= oK), which is not necessary but,
desirable for making predictions (Li & Lin, 2006). A
more sophisticated approach is to impose the inequal-
ity constraints on the outputs to improve the perfor-
mance.
Training of the neural network for ordinal regres-
sion proceeds very similarly as standard neural net-
works. The cost function for a data point x can
be relative entropy or square error between the tar-
get vector and the output vector. For relative en-
tropy, the cost function for output nodes is fc =∑K
i=1 (ti log oi + (1− ti) log(1− oi)). For square er-
ror, the error function is fc =
i=1 (ti − oi)
2. Pre-
vious studies (Richard & Lippman, 1991) on neural
network cost functions show that relative entropy and
square error functions usually yield very similar re-
sults. In our experiments, we use square error function
and standard back-propagation to train the neural net-
work. The errors are propagated back to output nodes,
and from output nodes to hidden nodes, and finally to
input nodes.
Since the transfer function ft of output node Oi is
the independent sigmoid function 1
1+e−zi
, the deriva-
tive of ft of output node Oi is
(1+e−zi )2
1+e−zi
(1 − 1
1+e−zi
) = oi(1 − oi). Thus, the net error
propagated to output node Oi is
= ti−oi
oi(1−oi)
oi(1 − oi) = ti − oi for relative entropy cost function,
= −2(ti−oi)×oi(1−oi) = −2oi(ti−oi)(1−oi)
for square error cost function. The net errors are prop-
agated through neural networks to adjust weights us-
ing gradient descent as traditional neural networks do.
Despite the small difference in the transfer function
and the computation of its derivative, the training of
our method is the same as traditional neural networks.
The network can be trained on data in the online
mode where weights are updated per example, or in
the batch mode where weights are updated per bunch
of examples.
2.3. Prediction
In the test phase, to make a prediction, our method
scans output nodes in the order O1, O2, ..., OK . It
stops when the output of a node is smaller than the
predefined threshold T (e.g., 0.5) or no nodes left. The
index k of the last node Ok whose output is bigger than
T is the predicted category of the data point.
3. Experiments and Results
3.1. Benchmark Data and Evaluation Metric
We use eight standard datasets for ordinal regres-
sion (Chu & Ghahramani, 2005a) to benchmark our
method. The eight datasets (Diabetes, Pyrimidines,
Triazines, Machine CUP, Auto MPG, Boston, Stocks
Domain, and Abalone) are originally used for metric
regression. Chu and Ghahramani (Chu & Ghahra-
mani, 2005a) discretized the real-value targets into
five equal intervals, corresponding to five ordinal cat-
egories. The authors randomly split each dataset into
training/test datasets and repeated the partition 20
times independently. We use the exactly same parti-
tions as in (Chu & Ghahramnai, 2005a) to train and
test our method.
A Neural Network Approach to Ordinal Regression
We use the online mode to train neural networks. The
parameters to tune are the number of hidden units, the
number of epochs, and the learning rate. We create
a grid for these three parameters, where the hidden
unit number is in the range [1..15], the epoch number
in the set (50, 200, 500, 1000), and the initial learning
rate in the range [0.01..0.5]. During the training, the
learning rate is halved if training errors continuously
go up for a pre-defined number (40, 60, 80, or 100) of
epochs. For experiments on each data split, the neural
network parameters are fully optimized on the training
data without using any test data.
For each experiment, after the parameters are opti-
mized on the training data, we train five models on
the training data with the optimal parameters, start-
ing from different initial weights. The ensemble of five
trained models are then used to estimate the general-
ized performance on the test data. That is, the average
output of five neural network models is used to make
predictions.
We evaluate our method using zero-one error and mean
absolute error as in (Chu & Ghahramani, 2005a).
Zero-one error is the percentage of wrong assignments
of ordinal categories. Mean absolute error is the root
mean square difference between assigned categories
(k′) and true categories (k) of all data points. For
each dataset, the training and evaluation process is
repeated 20 times on 20 data splits. Thus, we com-
pute the average error and the standard deviation of
the two metrics as in (Chu & Ghahramani, 2005a).
3.2. Comparison with Neural Network
Classification
We first compare our method (NNRank) with a stan-
dard neural network classification method (NNClass).
We implement both NNRank and NNClass using
C++. NNRank and NNClass share most code with
minor difference in the transfer function of output
nodes and its derivative computation as described in
Section 2.2.
As Table 1 shows, NNRank outperforms NNClass in
all but one case in terms of both the mean-zero error
and the mean absolute error. And on some datasets
the improvement of NNRank over NNClass is sizable.
For instance, on the Stock and Pyrimidines datasets,
the mean zero-one error of NNRank is about 4% less
than NNClass; on four datasets (Stock, Pyrimidines,
Triazines, and Diabetes) the mean absolute error is
reduced by about .05. The results show that the or-
dinal regression neural network consistently achieves
the better performance than the standard classifica-
tion neural network. To futher verify the effectiveness
of the neural network ordinal regression approach, we
are currently evaluating NNRank and NNclass on very
large ordinal regression datasets in the bioinformatics
domain (work in progress).
3.3. Comparison with Gaussian Processes and
Support Vector Machines
To further evaluate the performance of our method, we
compare NNRank with two Gaussian process meth-
ods (GP-MAP and GP-EP) (Chu & Ghahramani,
2005a) and a support vector machine method (SVM)
(Shashua & Levin, 2003) implemented in (Chu &
Ghahramani, 2005a). The results of the three meth-
ods are quoted from (Chu & Ghahramani, 2005a). Ta-
ble 2 reports the zero-one error on the eight datasets.
NNRank achieves the best results on Diabetes, Tri-
azines, and Abalone, GP-EP on Pyrimidines, Auto
MPG, and Boston, GP-MAP on Machine, and SVM
on Stocks.
Table 3 reports the mean absolute error on the eight
datasets. NNRank yields the best results on Diabetes
and Abalone, GP-EP on Pyrimidines, Auto MPG, and
Boston, GP-MAP on Triazines and Machine, SVM on
Stocks.
In summary, on the eight datasets, the performance
of NNRank is comparable to the three state-of-the-art
methods for ordinal regression.
4. Discussion and Future Work
We have described a simple yet novel approach to
adapt traditional neural networks for ordinal regres-
sion. Our neural network approach can be consid-
ered a generalization of one-layer perceptron approach
(Crammer & Singer, 2002) into multi-layer. On the
standard benchmark of ordinal regression, our method
outperforms standard neural networks used for classi-
fication. Furthermore, on the same benchmark, our
method achieves the similar performance as the two
state-of-the-art methods (support vector machines and
Gaussian processes) for ordinal regression.
Compared with existing methods for ordinal regres-
sion, our method has several advantages of neural net-
works. First, like the perceptron approach (Crammer
& Singer, 2002), our method can learn in both batch
and online mode. The online learning ability makes
our method a good tool for adaptive learning in the
real-time. The multi-layer structure of neural network
and the non-linear transfer function give our method
the stronger fitting ability than perceptron methods.
Second, the neural network can be trained on very
A Neural Network Approach to Ordinal Regression
large datasets iteratively, while training is more com-
plex than support vector machines and Gaussian pro-
cesses. Since the training process of our method is the
same as traditional neural networks, average neural
network users can use this method for their tasks.
Third, neural network method can make rapid
prediction once models are trained. The ability of
learning on very large dataset and predicting in
time makes our method a useful and competitive
tool for ordinal regression tasks, particularly for
time-critical and large-scale ranking problems in
information retrieval, web page ranking, collaborative
filtering, and the emerging fields of Bioinformat-
ics. We are currently applying the method to
rank proteins according to their structural rele-
vance with respect to a query protein (Cheng &
Baldi, 2006). To facilitate the application of this
new approach, we make both NNRank and NNClass
to accept a general input format and freely available at
http://www.eecs.ucf.edu/∼jcheng/cheng software.html.
There are some directions to further improve the neu-
ral network (or multi-layer perceptron) approach for
ordinal regression. One direction is to design a trans-
fer function to ensure the monotonic decrease of the
outputs of the neural network; the other direction
is to derive the general error bounds of the method
under the binary classification framework (Li & Lin,
2006). Furthermore, the other flavors of implemen-
tations of the multi-threshold multi-layer perceptron
approach for ordinal regression are possible. Since ma-
chine learning ranking is a fundamental problem that
has wide applications in many diverse domains such
as web page ranking, information retrieval, image re-
trieval, collaborative filtering, bioinformatics and so
on, we believe the further exploration of the neural net-
work (or multi-layer perceptron) approach for ranking
and ordinal regression is worthwhile.
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A Neural Network Approach to Ordinal Regression
Table 1. The results of NNRank and NNClass on the eight datasets. The results are the average error over 20 trials along
with the standard deviation.
Mean zero-one error Mean absolute error
Dataset NNRank NNClass NNRank NNClass
Stocks 12.68±1.8% 16.97± 2.3% 0.127±0.01 0.173±0.02
Pyrimidines 37.71±8.1% 41.87±7.9% 0.450±0.09 0.508±0.11
Auto MPG 27.13±2.0% 28.82±2.7% 0.281±0.02 0.307±0.03
Machine 17.03±4.2% 17.80±4.4% 0.186±0.04 0.192±0.06
Abalone 21.39±0.3% 21.74± 0.4% 0.226±0.01 0.232±0.01
Triazines 52.55±5.0% 52.84±5.9% 0.730±0.06 0.790±0.09
Boston 26.38±3.0% 26.62±2.7% 0.295±0.03 0.297±0.03
Diabetes 44.90±12.5% 43.84±10.0% 0.546±0.15 0.592±0.09
Table 2. Zero-one error of NNRank, SVM, GP-MAP, and GP-EP on the eight datasets. SVM denotes the support vector
machine method (Shashua & Levin, 2003; Chu & Ghahramani, 2005a). GP-MAP and GP-EP are two Gaussian process
methods using Laplace approximation (MacKay, 1992) and expectation propagation (Minka, 2001) respectively (Chu &
Ghahramani, 2005a). The results are the average error over 20 trials along with the standard deviation. We use boldface
to denote the best results.
Data NNRank SVM GP-MAP GP-EP
Triazines 52.55±5.0% 54.19±1.5% 52.91±2.2% 52.62±2.7%
Pyrimidines 37.71±8.1% 41.46±8.5% 39.79±7.2% 36.46±6.5%
Diabetes 44.90±12.5% 57.31±12.1% 54.23±13.8% 54.23±13.8%
Machine 17.03±4.2% 17.37±3.6% 16.53±3.6% 16.78±3.9%
Auto MPG 27.13±2.0% 25.73±2.2% 23.78±1.9% 23.75±1.7%
Boston 26.38±3.0% 25.56±2.0% 24.88±2.0% 24.49±1.9%
Stocks 12.68±1.8% 10.81±1.7% 11.99±2.3% 12.00±2.1%
Abalone 21.39±0.3% 21.58±0.3% 21.50±0.2% 21.56±0.4%
Table 3. Mean absolute error of NNRank, SVM, GP-MAP, and GP-EP on the eight datasets. SVM denotes the support
vector machine method (Shashua & Levin, 2003; Chu & Ghahramani, 2005a). GP-MAP and GP-EP are two Gaussian
process methods using Laplace approximation and expectation propagation respectively (Chu & Ghahramani, 2005a).
The results are the average error over 20 trials along with the standard deviation. We use boldface to denote the best
results.
Data NNRank SVM GP-MAP GP-EP
Triazines 0.730±0.07 0.698±0.03 0.687±0.02 0.688±0.03
Pyrimidines 0.450±0.10 0.450±0.11 0.427±0.09 0.392±0.07
Diabetes 0.546±0.15 0.746±0.14 0.662±0.14 0.665±0.14
Machine 0.186±0.04 0.192±0.04 0.185±0.04 0.186±0.04
Auto MPG 0.281±0.02 0.260±0.02 0.241±0.02 0.241±0.02
Boston 0.295±0.04 0.267±0.02 0.260±0.02 0.259±0.02
Stocks 0.127±0.02 0.108±0.02 0.120±0.02 0.120±0.02
Abalone 0.226±0.01 0.229±0.01 0.232±0.01 0.234±0.01
