1 Introduction
In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance Mannor and Meir (2001). It has been proved that ensemble methods can boost weak learners whose accuracies are slightly better than random guessing into arbitrarily accurate strong learners Bai et al. (2014); Zhang et al. (2016). When it could not be possible to directly design a strong complicated learning system, ensemble methods would be a possible solution. In this paper, we are inspired by ensemble techniques to combine several weak loss functions in order to design a stronger ensemble loss function for text classification.
In this paper we will focus on multiclass classification where the class to predict is encoded as a vector
with the onehot encoding of the target label, and the output of a classifier
is a vector of probability estimates of each label given input sample
and training parameters . Then, a loss function is a positive function that measures the error of estimation Steinwart and Christmann (2008). Different loss functions have different properties, and some wellknown loss functions are shown in Table 1. Different loss functions lead to different Optimum Bayes Estimators having their own unique characteristics. So, in each environment, picking a specific loss function will affect performance significantly Xiao et al. (2017); Zhao et al. (2010).Name of loss function  

ZeroOne Xiao et al. (2017)  
Hinge Loss MasnadiShirazi and Vasconcelos (2009); Steinwart (2002)  
Smoothed Hinge Zhao et al. (2010)  
Square Loss  
Correntropy Loss Liu et al. (2007, 2006)  
CrossEntropy Loss MasnadiShirazi et al. (2010)  
Absolute Loss 
In this paper, we propose an approach for combining loss functions which performs substantially better especially when facing annotation noise. The framework is designed as an extension to regular neural networks, where the loss function is replaced with an ensemble of loss functions, and the ensemble weights are learned as part of the gradient propagation process. We implement and evaluate our proposed algorithm on several text classification datasets.
The paper is structured as follows. An overview of several loss functions for classification is briefly introduced in Section 2. The proposed framework and the proposed algorithm are explained in Section 3. Section 4 contains experimental results on classifying several text datasets. The paper is concluded in Section 5.
2 Background
A typical machine learning problem can be reduced to an expected loss function minimization problem Bartlett et al. (2006); Painsky and Rosset (2016)
. rosasco2004loss studied the impact of choosing different loss functions from the viewpoint of statistical learning theory. In this section, several wellknown loss functions are briefly introduced, followed by a review of ensemble methods.
In the literature, loss functions are divided into marginbased and distancebased categories. Marginbased loss functions are often used for classification purposes Steinwart and Christmann (2008); Khan et al. (2013); Chen et al. (2017). Since we evaluate our work on classification of text datasets, in this paper we focus on marginbased loss functions.
A marginbased loss function is defined as a penalty function based in a margin
. In any given application, some marginbased loss functions might have several disadvantages and advantages and we could not certainly tell which loss function is preferable in general. For example, consider the ZeroOne loss function which penalizes all the misclassified samples with the constant value of 1 and the correctly classified samples with no loss. This loss function would result in a robust classifier when facing outliers but it would have a terrible performance in an application with margin focus
Zhao et al. (2010).A loss function is margin enforcing if minimization of the expected loss function leads to a classifier enhancing the margin MasnadiShirazi and Vasconcelos (2009)
. Learning a classifier with an acceptable margin would increase generalization. Enhancing the margin would be possible if the loss function returns a small amount of loss for the correct samples close to the classification hyperplane. For example, ZeroOne does not penalize correct samples at all and therefore it does not enhance the margin, while Hinge Loss is a margin enhancing loss function.
The general idea of ensemble techniques is to combine different expert ideas aiming at boosting the accuracy based on enhanced decision making. Predominantly, the underlying idea is that the decision made by a committee of experts is more reliable than the decision of one expert alone Bai et al. (2014); Mannor and Meir (2001). Ensemble techniques as a framework have been applied to a variety of real problems and better results have been achieved in comparison to using a single expert.
Having considered the importance of the loss function in learning algorithms, in order to reach a better learning system, we are inspired by ensemble techniques to design an ensemble loss function. The weight applied to each weak loss function is tuned through the gradient propagation optimization of a neural network working on a text classification dataset.
Other works Shi et al. (2015); BenTaieb et al. (2016)
have combined two loss functions where the weights are specified as a hyperparameter set prior to the learning process (e.g. during a finetuning process with crossvalidation). In this paper, we combine more than two functions and the hyperparameter is not set apriory but it is learned during the training process.
3 Proposed Approach
Let be a sample where is the input and is the onehot encoding of the label ( is the number of classes). Let
be the parameters of a neural network classifier with a top softmax layer so that the probability estimates are
. Let denote weak loss functions. In addition to finding the optimal , the goal is to find the best weights , , to combine weak loss functions in order to generate a better applicationtailored loss function. We need to add a further constraint to avoid yielding near zero values for all weights. The proposed ensemble loss function is defined as below.(1) 
The optimization problem could be defined as follows, given training samples.
(2)  
s.t. 
To make the optimization algorithm simpler, we use instead of , so the second constraint can be omitted. We then incorporate the constraint as a regularization term based on the concept of Augmented Lagrangian. The modified objective function using Augmented Lagrangian is presented as follows.
(3)  
Note that the amount of must be significantly greater that Nocedal and Wright (2006) . The first and the second terms of the objective function cause values to approach zero but the third term satisfies .
Figure 1 illustrates the framework of the proposed approach with the dashed box representing the contribution of this paper. In the training phase, the weight of each weak loss function is trained through the gradient propagation optimization method. The accuracy of the model is calculated in a test phase not shown in the figure.
4 Experimental Results
We have applied the proposed ensemble loss function to several text datasets. Table 2 provides a brief description of the datasets. To reach a better ensemble loss function we choose three loss functions with different approaches in facing with outliers, as weak loss functions: Correntropy Loss which does not assign a high weight to samples with big errors, Hinge Loss which penalizes linearly and Crossentropy Loss function which highly penalizes the samples whose predictions are far from the targets. We compared results with 3 loss functions which are widely used in neural networks: Crossentropy, Square Loss, and Hinge Loss.
We picked near zero and in (3).
Name of Datasets  Description 

20newsgroup  This data set is a collection of 20,000 messages,collected from 20 different netnews newsgroups. 
Moviereviews in corpus  The NLTK corpus moviereviews data set has the reviews, and they are labeled already as positive or negative. 
EmailClassification (TREC)  It is a collection of sample emails (i.e. a text corpus). In this corpus, each email has already been labeled as Spam or Ham. 
Reuters21578  The data was originally collected and labeled by Carnegie Group, Inc. and Reuters, Ltd. in the course of developing the CONSTRUE text categorization system 
Since this work is a proof of concept, the neural networks of each application are simply a softmax of the linear combination of input features plus bias:
where the input features are the word frequencies in the input text. Thus, in our notation is composed of and
. We use Python and its TensorFlow package for implementing the proposed approach. The results are shown in Table
3. The table compares the results of using individual loss functions and the ensemble loss.Dataset  Crossentropy  Hinge  Square  Ensemble 

20newsgroups  0.80  0.69  0.82  0.85 
Moviereview  0.83  0.81  0.85  0.83 
EmailClassification (TREC)  0.88  0.78  0.96  0.97 
Reuters  0.79  0.79  0.81  0.81 
Dataset  Crossentropy  Hinge  Square  Ensemble 

20newsgroups  0.79  0.67  0.69  0.83 
Moviereviews  0.75  0.74  0.73  0.78 
EmailClassification (TREC)  0.86  0.57  0.82  0.96 
Reuters  0.76  0.69  0.71  0.73 
We have also compared the robustness of the proposed loss function with the use of individual loss functions. In particular, we add label noise by randomly modifying the target label in the training samples, and keep the evaluation set intact. We conducted experiments with 10% and 30% of noise, where e.g. 30% of noise means randomly changing 30% of the labels in the training data. Tables 4 and 5 show the results, with the best results shown in boldface. We can observe that, in virtually all of the experiments, the ensemble loss is at least as good as the individual losses, and in only two cases the loss is (slightly) worse. And, in general, the ensemble loss performed comparatively better as we increased the label noise.
5 Conclusion
This paper proposed a new loss function based on ensemble methods. This work focused on text classification tasks and can be considered as an initial attempt to explore the use of ensemble loss functions. The proposed loss function shows an improvement when compared with the use of wellknown individual loss functions. Furthermore, the approach is more robust against the presence of label noise. Moreover, according to our experiments, the gradient descent method quickly converged.
Dataset  Crossentropy  Hinge  Square  Ensemble 

20newsgroups  0.57  0.64  0.55  0.82 
moviereview  0.55  0.54  0.55  0.6 
EmailClassification (TREC)  0.80  0.46  0.81  0.93 
Reuters  0.64  0.54  0.53  0.68 
We have used a very simple neural architecture in this work but in principle this method could be used for systems that use any neural networks. In future work we will explore the integration of more complex neural networks such as those using convolutions and recurrent networks. We also plan to study the application of this method to other tasks such as sequence labeling (e.g. for NER and PoS tagging). Another possible extension could focus on handling sparseness by adding a regularization term.
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