A neural network approach to ordinal regression

TL;DR

This paper introduces NNRank, a neural network-based method for ordinal regression that outperforms some existing methods and offers advantages like scalability and rapid prediction, suitable for large-scale data tasks.

Contribution

The paper presents NNRank, a neural network approach for ordinal regression that generalizes perceptron methods and performs competitively with Gaussian process and SVM-based methods.

Findings

NNRank outperforms neural network classification on benchmark datasets.

NNRank achieves comparable performance to Gaussian process and SVM ordinal regression methods.

NNRank supports online and batch learning, handling large datasets efficiently.

Abstract

Ordinal regression is an important type of learning, which has properties of both classification and regression. Here we describe a simple and effective approach to adapt a traditional neural network to learn ordinal categories. Our approach is a generalization 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.

Tables

Table 1. 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$\pm$1.8%	16.97$\pm$ 2.3%	0.127$\pm$0.01	0.173$\pm$0.02
Pyrimidines	37.71$\pm$8.1%	41.87$\pm$7.9%	0.450$\pm$0.09	0.508$\pm$0.11
Auto MPG	27.13$\pm$2.0%	28.82$\pm$2.7%	0.281$\pm$0.02	0.307$\pm$0.03
Machine	17.03$\pm$4.2%	17.80$\pm$4.4%	0.186$\pm$0.04	0.192$\pm$0.06
Abalone	21.39$\pm$0.3%	21.74$\pm$ 0.4%	0.226$\pm$0.01	0.232$\pm$0.01
Triazines	52.55$\pm$5.0%	52.84$\pm$5.9%	0.730$\pm$0.06	0.790$\pm$0.09
Boston	26.38$\pm$3.0%	26.62$\pm$2.7%	0.295$\pm$0.03	0.297$\pm$0.03
Diabetes	44.90$\pm$12.5%	43.84$\pm$10.0%	0.546$\pm$0.15	0.592$\pm$0.09

Table 2. 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$\pm$5.0%	54.19$\pm$1.5%	52.91$\pm$2.2%	52.62$\pm$2.7%
Pyrimidines	37.71$\pm$8.1%	41.46$\pm$8.5%	39.79$\pm$7.2%	36.46$\pm$6.5%
Diabetes	44.90$\pm$12.5%	57.31$\pm$12.1%	54.23$\pm$13.8%	54.23$\pm$13.8%
Machine	17.03$\pm$4.2%	17.37$\pm$3.6%	16.53$\pm$3.6%	16.78$\pm$3.9%
Auto MPG	27.13$\pm$2.0%	25.73$\pm$2.2%	23.78$\pm$1.9%	23.75$\pm$1.7%
Boston	26.38$\pm$3.0%	25.56$\pm$2.0%	24.88$\pm$2.0%	24.49$\pm$1.9%
Stocks	12.68$\pm$1.8%	10.81$\pm$1.7%	11.99$\pm$2.3%	12.00$\pm$2.1%
Abalone	21.39$\pm$0.3%	21.58$\pm$0.3%	21.50$\pm$0.2%	21.56$\pm$0.4%

Table 3. 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$\pm$0.07	0.698$\pm$0.03	0.687$\pm$0.02	0.688$\pm$0.03
Pyrimidines	0.450$\pm$0.10	0.450$\pm$0.11	0.427$\pm$0.09	0.392$\pm$0.07
Diabetes	0.546$\pm$0.15	0.746$\pm$0.14	0.662$\pm$0.14	0.665$\pm$0.14
Machine	0.186$\pm$0.04	0.192$\pm$0.04	0.185$\pm$0.04	0.186$\pm$0.04
Auto MPG	0.281$\pm$0.02	0.260$\pm$0.02	0.241$\pm$0.02	0.241$\pm$0.02
Boston	0.295$\pm$0.04	0.267$\pm$0.02	0.260$\pm$0.02	0.259$\pm$0.02
Stocks	0.127$\pm$0.02	0.108$\pm$0.02	0.120$\pm$0.02	0.120$\pm$0.02
Abalone	0.226$\pm$0.01	0.229$\pm$0.01	0.232$\pm$0.01	0.234$\pm$0.01

Full text

1 Introduction

Ordinal regression (or ranking learning) is an important supervised problem of learning a ranking or ordering 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 regression, 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 research in 1990s (Caruana et al., 1996; Herbrich et al., 1998; Cohen et al., 1999). It has attracted the considerable attention in recent years due to its potential 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 protein ranking (Cheng & Baldi, 2006) in Bioinformatics.

A number of machine learning methods have been developed or redesigned to address ordinal regression problem (Rajaram et al., 2003), including perceptron (Crammer & Singer, 2002) and its kernelized generalization (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 vector 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 experts (Paquet et al., 2005), binary classification approach (Frank & Hall, 2001; Li & Lin, 2006) that decomposes 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 (Crammer & 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 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 advantages and disadvantages. Prank (Crammer & Singer, 2002), a perceptron approach that generalizes the binary 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 $O(n^{2})$ ($n$: 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 ordinal 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. Similarly, another state-of-the-art approach, Gaussian process 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 (McCullagh, 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 categories. 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 classification 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 consisting of $n$ data points ($x,y$) , where $x\in R^{d}$ is an input feature vector and $y$ is its ordinal category from a finite 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 predict 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 $t_{k}$ 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 $o=(o_{1},o_{2},...o_{k},...o_{K})$, where $o_{k}$ is closer to 1 and other elements are close to zero, subject to the constraint $\sum_{i=1}^{K}o_{i}=1$.

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 $t_{i}\ (1\leq i\leq k)$ 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 $o=(o_{1},o_{2},...,o_{k},...o_{K})$, where $o_{i}\ (i\leq k)$ is close to 1 and $o_{i}\ (i\geq k)$ is close to 0. $\sum_{i=1}^{K}o_{i}$ 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 approach (Crammer & Singer, 2002). It is also related to the classical cumulative probit model for ordinal regression (McCullagh, 1980), in the sense that we can consider the output probability vector $(o_{1},...o_{k},...o_{K})$ as a cumulative probability distribution on categories $(1,...,k,...,K)$, i.e., $\frac{\sum_{i=1}^{K}o_{i}}{K}$ is the proportion of categories 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) 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 corresponding 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 feedforward 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 hidden 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 softmax $\frac{e^{-z_{i}}}{\sum_{i=1}^{K}e^{-z_{i}}}$(or normalized exponential function) for output nodes, satisfying the constraint that the sum of outputs $\sum_{i=1}^{K}o_{i}$ is 1. $z_{i}$ is the net input to the output node $O_{i}$.

In contrast, each output node $O_{i}$ of our neural network uses a standard sigmoid function $\frac{1}{1+e^{-z_{i}}}$, without including the outputs from other nodes. Output node $O_{i}$ is used to estimate the probability $o_{i}$ that a data point belongs to category $i$ independently, without 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 $O_{i}$ ($i\leq k$) to 1 and $O_{i}$ ($i>k$) to 0. The targets instruct the neural network to adjust weights to produce probability outputs as close as possible to the target vector. It is worth pointing out that using independent sigmoid functions for output nodes does not guaranteed the monotonic relation ($o_{1}>=o_{2}>=...>=o_{K}$), which is not necessary but, desirable for making predictions (Li & Lin, 2006). A more sophisticated approach is to impose the inequality constraints on the outputs to improve the performance.

Training of the neural network for ordinal regression proceeds very similarly as standard neural networks. The cost function for a data point $x$ can be relative entropy or square error between the target vector and the output vector. For relative entropy, the cost function for output nodes is $f_{c}=\sum_{i=1}^{K}{(t_{i}\log o_{i}+(1-t_{i})\log(1-o_{i}))}$. For square error, the error function is $f_{c}=\sum_{i=1}^{K}{(t_{i}-o_{i})^{2}}$. Previous studies (Richard & Lippman, 1991) on neural network cost functions show that relative entropy and square error functions usually yield very similar results. In our experiments, we use square error function and standard back-propagation to train the neural network. The errors are propagated back to output nodes, and from output nodes to hidden nodes, and finally to input nodes.

Since the transfer function $f_{t}$ of output node $O_{i}$ is the independent sigmoid function $\frac{1}{1+e^{-z_{i}}}$, the derivative of $f_{t}$ of output node $O_{i}$ is $\frac{\partial f_{t}}{\partial z_{i}}=\frac{e^{-z_{i}}}{(1+e^{-z_{i}})^{2}}$ = $\frac{1}{1+e^{-z_{i}}}(1-\frac{1}{1+e^{-z_{i}}})$ = $o_{i}(1-o_{i})$. Thus, the net error propagated to output node $O_{i}$ is $\frac{\partial f_{c}}{\partial o_{i}}\frac{\partial f_{t}}{\partial z_{i}}=\frac{t_{i}-o_{i}}{o_{i}(1-o_{i})}\times o_{i}(1-o_{i})=t_{i}-o_{i}$ for relative entropy cost function, $\frac{\partial f_{c}}{\partial o_{i}}\frac{\partial f_{t}}{\partial z_{i}}=-2(t_{i}-o_{i})\times o_{i}(1-o_{i})=-2o_{i}(t_{i}-o_{i})(1-o_{i})$ for square error cost function. The net errors are propagated through neural networks to adjust weights using 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 $O_{1},O_{2},...,O_{K}$. 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 $O_{k}$ 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 regression (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 & Ghahramani, 2005a) discretized the real-value targets into five equal intervals, corresponding to five ordinal categories. The authors randomly split each dataset into training/test datasets and repeated the partition 20 times independently. We use the exactly same partitions as in (Chu & Ghahramnai, 2005a) to train and test our method.

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 optimized on the training data, we train five models on the training data with the optimal parameters, starting from different initial weights. The ensemble of five trained models are then used to estimate the generalized 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 ($k^{\prime}$) 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 compute 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 standard 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 ordinal regression neural network consistently achieves the better performance than the standard classification 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 methods (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 methods are quoted from (Chu & Ghahramani, 2005a). Table 2 reports the zero-one error on the eight datasets. NNRank achieves the best results on Diabetes, Triazines, 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 regression. Our neural network approach can be considered 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 classification. 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 regression, our method has several advantages of neural networks. 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 large datasets iteratively, while training is more complex than support vector machines and Gaussian processes. 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 Bioinformatics. We are currently applying the method to rank proteins according to their structural relevance 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 neural network (or multi-layer perceptron) approach for ordinal regression. One direction is to design a transfer 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 implementations of the multi-threshold multi-layer perceptron approach for ordinal regression are possible. Since machine learning ranking is a fundamental problem that has wide applications in many diverse domains such as web page ranking, information retrieval, image retrieval, collaborative filtering, bioinformatics and so on, we believe the further exploration of the neural network (or multi-layer perceptron) approach for ranking and ordinal regression is worthwhile.