Usually, supervised learning tasks require a large amount of labeled samples to train their models. Although several kinds of data sources, such as games and e-commerce platforms, can automatically label their samples with clear rules, there is still a large amount of data need to be manually labeled. However, requesting domain experts to label large datasets is very expensive and time-consuming. An alternative choice is to collect large amount of labels from non-experts.
Recently, there are many works [17, 21] interested in using crowdsourcing to make datasets. Many online platforms, like Amazon Mechanical Turk111www.amt.com and CrowdFlower222www.crowdflower.com, provide crowdsourcing services. These platforms split the unlabeled large dataset into small parts and distribute them to a group of registered ordinary workers . Collecting labels from crowdsourcing platform is efficient and cheap, however the non-professional workers usually have low labeling accuracy. To improve the accuracy, the usual practice is to assign each item to different workers, then aggregate the collected redundant labels into the predicted true labels.
Previous works have proposed many label aggregation methods to infer the true label from the observed noisy labels provided by crowds. The most straightforward way to predict the true label is majority voting. It considers each item independently and takes the most frequent class as the true label. This method does not take the reliability of each worker into consideration, and potentially assumes that all the workers are equally good. However, in actually workers have different degrees of reliability depending on their state of mind, expertise and motivation. A variety of advanced methods have been proposed to overcome this problem. These methods make some assumptions about the behavior of workers and design statistical models to generate the observed noisy labels. These assumptions are represented by model parameters and their relation. They reflect the reliability of workers. Whitehill et al.  propose the Generative model of Labels, Abilities, and Difficulties (GLAD) for binary label aggregation. This model can simultaneously infer the true labels, the expertise of workers, and the difficulty of items. Dawid&Skene-EM 
is a label aggregation method based on Expectation Maximization (EM). Many recent works[25, 26, 7, 14] extend this method and improve its performance. We will briefly introduce the existing label aggregation methods in Section 2.
In this paper, we propose a novel label aggregation algorithm which includes a neural network. The neural network is a probabilistic classifier. Given an instance, it can predict the unknown true label, where an instance contains the redundant noisy labels of an item. The challenge is that the corresponding true label of each instance is unknown, the neural network must be trained in an unsupervised way. We need to define a loss function (optimization goal) without any ground truth label. In order to solve this problem we set a guiding model. The training process of our algorithm is to find the consensus between the predictions of the neural network and the guiding model. The loss function is differentiable. Thus our algorithm can be trained using mini-batch stochastic optimization methods, such as SGD, Adam
and RMSProp. The model parameters of the neural network and the guiding model are updated simultaneously.
Compared with existing label aggregation methods, our algorithm is easy to extend, because there are few limitations in designing the neural network and the guiding model. There are many choices for the architecture of our neural network, e.g., MLP, CNN etc. The only limitation of the guiding model is that it should be differentiable. According to our algorithm, we propose two models, one is a binary label aggregation model based on some delicate assumptions about the behavior of workers, the other is designed to aggregate multiclass noisy labels. Our algorithm can be applied to online settings, because it can be trained using mini-batch stochastic optimization methods. However, in order to fairly compare with state-of-the-art methods, our experiments are conducted on fixed datasets. Experiments on four real-world datasets demonstrate that our models achieve superior performance over state-of-the-art methods.
2 Related Work
Dawid and Skene 
proposed the conception of confusion matrix to describe the expertise and the bias of a worker. Based on the confusion matrix, they designed an EM algorithm for label aggregation. In the field of crowdsourcing, the confusion matrix is effective and well known. Raykar et al. used noisy labels to train their classification model. Their two-coin model is a variation of the confusion matrix. BCC 
is the probabilistic graphical model version of Dawid&Skene-EM. It also uses confusion matrix to evaluate workers, and uses Gibbs sampling to perform the parameter estimation. CommunityBCC is an extension of BCC. It divides the workers into worker communities. The workers in the same community have similar confusion matrices. CommunityBCC has better performance than BCC on sparse datasets. BCCWords  also extends BCC. Recently Zhou et al. [25, 26] proposed the minimax entropy estimator and its extensions. In these model, the authors set a separate probabilistic distribution for each worker-item pair. In most cases, minimax entropy and the extensions outperform Dawid&Skene-EM.
There are several other models for label aggregation. These models do not based on confusion matrix. GLAD  is a model that can infer the true labels, the expertise of workers, and the difficulty of items at the same time. It only can used for binary labeling tasks. Liu et al. proposed a model which uses variational inference to approximate the posterior. Zhou and He 
designed a label aggregation approach which is based on tensor augmentation and completion. Li et al. proposed a general crowd targeting framework, it can target good workers for crowdsourcing.
DeepAgg  is a model based on a deep neural network. The model is trained by a seed dataset which contains noisy labels and the corresponding ground truth labels. So it is not an unsupervised approach. DeepAgg has several limitations. It can not aggregate incomplete data, where many annotators only labeled a few items. Recently, Yin et al.  applied Variational Auto-Encoder (VAE)  to label aggregation. Their LAA model contains a classifier and a reconstructor. Both the classifier and the reconstructor are neural networks. LAA is an unsupervised model and works well in most case. We will use it as a baseline in our experiments.
Let’s start by introducing some notations that will be used throughout the paper. Consider that there are items labeled by workers, where each item has possible classes. denotes the label of item given by worker , where . (instance ) denotes the redundant labels for item . is the collection of all the observed labels. Note that each worker may only labels a part of the dataset. If item is not labeled by worker , then the value of is . is the unknown true label of each item . is the collection of all the unknown true labels. For given values of , the goal of label aggregation is to predict the values of .
3.2 Algorithm Framework
In this section, we introduce our novel label aggregation algorithm. This algorithm includes two components: a neural network and a guiding model . The choice of and is very flexible.
can be a multilayer perceptron (MLP), a convolutional neural network (CNN), or any other neural networks. In order to apply stochastic optimization, the loss function should be differentiable respect to the model parameters in. This is the only constraint of . Our algorithm is easy to train using any stochastic optimization method.
In our algorithm, is a label aggregation neural network. Given an instance , it can predict the corresponding unknown true label.
is represented as a probability distribution, where is the input instance (the collected redundant labels for an item) and
denotes the network parameters (e.g., weights, biases, etc.). The network’s output is a C-dimensional vector, the c-th element is the probability that the true label of the input instance is class .
The noisy labels dataset only contains the observed instances , the corresponding true labels
are unknown. Label aggregation is to predict these unknown true labels, so this is an unsupervised learning task. In order to train, we define a model to guide the training. assumes that an instance is generated from some conditional distributions , where means the unknown true label and denotes the model parameters. It also assumes that the true label is generated from a prior distribution . The guiding model potentially defines a posterior distribution:
We do not make any simplifying assumption about and except that the loss function is differentiable respect to .
The optimization goal of this algorithm is to find the consensus between the predictions of and . That means the neural network distribution should be as similar as possible to the posterior distribution
. Thus it is reasonable to use Kullback-Leibler divergence, a widely-used measure of the dissimilarity between two probability distributions, as the loss function:
where is the Kullback-Leibler divergence. We minimize this loss function during the training process. This kind of optimization goal is commonly used in approximate inference and variational inference [1, 20]. Thousands of published papers have shown that this optimization goal is effective. Actually, the data log likelihood can be bounded by
where is the evidence lower bound (ELBO). Minimizing the KL divergence between and is equivalent to maximizing the ELBO. Neural networks are good at fitting probability distributions. Thus, we can efficiently minimize and tightly bound the data log likelihood . That’s why our algorithm framework can work well.
We assume that each collected label is independently generated. The instances in are independent with each other. Plugging and into (2), we have:
Equation (3) cannot be directly used to train and , because the expression of is unknown. The exact expression of the posterior may be intractable. Fortunately, it is not necessary to calculate it in our algorithm, we will further rewrite the loss function. Plugging in (1), we have
According to (3) and (4) the loss function is rewritten as
where the constant is ignored and the loss function is rescaled by . This will not affect the optimization result. The above is the definition of our label aggregation algorithm. The structure of the algorithm is shown in Figure 1.
We are going to solve the following optimization problem
. In our algorithm, we apply mini-batch training that is commonly used in deep learning. The neural network parametersand the guiding model parameters are trained simultaneously. In mini-batch training,
where is a mini-batch sampled from , and denotes the minibatch size. The gradient of is required to update the model parameters. The unobserved variables are discrete variables that take values from to . Therefore, we have
where is the c-th element of the neural network output. According to (7) and (8), the values of and the corresponding stochastic gradient can be easily computed. The advantage of using mini-batch training is that it allows our label aggregation algorithm to be used for large datasets and online setting.
3.3 Label Aggregation Models
In this section, we will introduce two novel label aggregation models based on the aforementioned algorithm framework. In each model, we need to define and .
3.3.1 A Binary Model Based on Worker Ability Assumptions
We take the ability of each worker into consideration and propose a label aggregation model. This model is called NN-WA (NN means neural network and WA means worker ability). In NN-WA, is a MLP. It inputs an instance and outputs a distribution , where denotes the network parameters.
Next, we define a guiding model . As shown in (7) and (8), in order to compute the loss function and its gradient, we need to define and . In NN-WA, for simplicity, we only consider binary labeling tasks (the number of classes ). For each , the ability of each worker is represented by a single parameter . We assume that worker labels each item correctly with the probability
According to this assumption, we have:
We can see that the higher the ability of worker is, the higher the likelihood for him or her to label the item correctly. When , he or she just randomly chooses one class. Therefore our assumption is reasonable. According to (9), the conditional distributions that generated instances are defined as
where is a set of workers who have labeled item . In this model, the prior distribution is fixed during the training process, it has no parameters to be trained. is a multinomial distribution and is estimated by
where the values of the estimators can be calculated by the result of counting the observed labels. Since
is fixed, we introduce a hyperparameterto constrain the Kullback-Leibler divergence term in the loss function (6). We regard this constrained term as a regularizer. Then, using (7) and (8) the mini-batch loss function used in practice is
NN-WA is formally shown in Algorithm 1.
3.3.2 A Multiclass Label Aggregation Model
In this section we design a novel label aggregation model. This model can be applied to aggregate multiclass noisy labels. We call it as NN-MC (MC means multiclass). In NN-MC,
is also a MLP. It is a fully connected neural network with softmax activation function on the last layer.
Now we design a new guiding model for NN-MC. Different from NN-WA, we do not delicately make some assumptions about the ability and the behavior of a worker. We design from another perspective. Every element in an instance is a independent collected label, so we assumes that in , the -th element in an instance is generated by a independent distribution . This distribution is defined as
where is a C-dimensional vector. Then can be defined as
where is the -th element of . Since the softmax function is derivable, can be updated by stochastic optimization methods. Just like NN-WA, the prior in NN-MC is fixed and is estimated by (11). The loss function of this model is also defined as (12). NN-MC is illustrated in Algorithm 2.
. Dawid&Skene-EM is a classic generative model for label aggregation. Minimax Entropy is an extension of Dawid&Skene-EM. This model assumes that the observed labels are generated by a distribution over workers, items, and labels. Bayesian Classifier Combination (BCC) is a Bayesian network using confusion matrix
. Generative model of Labels, Abilities and Difficulties (GLAD) is a binary label aggregation model. It can simultaneously infer the true labels, the expertise of workers, and the difficulty of items. Minimax Conditional Entropy Estimators (MMCE) uses an minimax entropy principle to aggregate noisy labels. Label-Aware Autoencoders (LAA) applies variational auto-encoders to label aggregation.
We use four real-world datasets in our experiments. The detailed information of them is shown in Table 1.
Adult  is a dataset labeled by Mechanical Turk workers to websites. The web pages are classified into four classes according to the amount of adult content on each web page. RTE  is a dataset about recognizing textual entailment. There are workers assign items into classes. The average correct rate of the workers is . Heart dataset is provided by
medical students. The students judge whether the patients have heart disease based on the physical examination results.These physical examination samples and the corresponding diagnostic results are downloaded from the UC Irvine machine learning repository. In order to use Age  in our experiments, its labels have been discretized into 7 bins: .
4.3 True Label Prediction
After training NN-WA and NN-MC, we use maximum likelihood estimation (MLE) to predict the true label of each item. For NN-WA,
where the values of are computed by . The prediction error rates are computed by comparing the predicted true labels with the ground truth labels. Note that the ground truth labels are only used for evaluation. We do not use them in the training stage.
. GLAD, LAA and our two label aggregation models are implemented using TensorFlow333www.tensorflow.org which provides GPU acceleration.
We apply MLP as the label aggregation neural network and use the tangent hyperbolic function (tanh)  as the activation function. Softmax functions are used as the output of the networks. We use RMSprop  as the stochastic optimizer to minimize the loss. In our algorithm, the loss will finally converge to a stable value. As shown in Algorithm 1 and Algorithm 2, we implement MLPs with only layers for NN-WA and NN-MC. Deeper neural networks are also easy to applied in our algorithm. However, limited by the data size, deeper neural networks do not improve the prediction accuracy. We use the likelihood as the criterion to select the hyperparameter. After training the model and predicting the true label, this likelihood is easy to compute. We test multiple values of and select the value that generates the maximal likelihood. In our experiments, .
4.5 Error-Rates of Methods
The prediction error rates of our models and the baselines are illustrated in Table 2, Where GLAD and NN-WA only can aggregate binary noisy labels. The best results are highlighted in bold. NN-WA has the best performance on the Heart dataset. NN-MC outperforms the baselines across the Adult dataset, the Heart dataset and the Age dataset. On the RTE dataset, our models achieve the similar accuracy with GLAD. The results show that our label aggregation algorithm is effective. Our models are more accurate than the baseline methods.
4.6 Further Investigation
4.6.1 Effectiveness of Worker Ability Detection
After training NN-WA, for each class , the ability of each worker can be detected. In order to evaluate the effectiveness of worker ability detection, we take parameters after training on the Heart dataset. However these parameters are not intuitive and not easy to understand. Therefore, for each class , we further use equation (9) to compute the predicted accuracy of each worker. The real accuracy of each worker is easy to compute by counting the collected noisy labels and the ground truth labels. Given class and worker , the corresponding real accuracy is computed by
where denotes the ground truth label of item . The results are illustrated in Figure 2. We can see that the prediction is consistent with the reality. In most case, the predicted accuracy and the real accuracy are quite similar which means NN-WA can effectively detect the ability of workers.
4.6.2 Evaluation of the Trained Parameters in NN-MC
The trained parameters in NN-MC can reflect the reliability of each worker. The values of are hard to understand. Given class and worker , we further use equation (13) to compute the distribution . Table 3 shows the results on the Adult dataset. Due to limited space, we only illustrate the results of two representative workers. We only care about the diagonal values and omit the other values for clarity. The labeling accuracy of each worker is computed by . We can see that worker has higher labeling accuracy, meanwhile, he or she has bigger diagonal values. This is reasonable, according to the definition of , a diagonal value is the probability that the collected label equals the true label and takes value . The results show that, by learning the model parameters, NN-MC can capture the knowledge of the reliability of workers.
We present a novel algorithm which aggregates noisy labels by finding consensus between a neural network and a guiding model. According to the algorithm framework, we design two label aggregation models called NN-WA and NN-MC. In our algorithm, there are very few limitations on the choices of the label aggregation neural network and the guiding model. Therefore, our algorithm is very flexible and easy to extend. The experimental results on four real-world datasets show that our models outperform state-of-the-art label aggregation methods.
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