1 Introduction
In many reallife applications, we are confronted with the task of building a binary classification model from a number of positive data and plenty of unlabeled data without extra information on the negative data. For example, it is common in disease gene identification [1] that only known disease genes and unknown genes are available, because the reliable nondisease genes are difficult to obtain. Similar scenarios occur in deceptive review detection [2], web data mining [3]
, inlierbased outlier detection
[4], etc. Such a task is certainly beyond the scope of the standard supervised machine learning, and where positiveunlabeled (PU) learning comes in handy.
A straightforward approach for PU learning is to employ a twostep strategy: First reliable negative data are identified from the unlabeled data by some heuristic techniques
[5, 6, 7, 8], then the classifier can be trained by traditional supervised learning or expectationmaximizationlike semisupervised learning algorithms
[9, 10]. Furthermore, the two steps can be iteratively executed so that more negative data can be accurately identified [11]. Most of the twostep strategy based methods assume that the positive and negative data distributions can be well separated with almost nonoverlapping supports, which is difficult to satisfy in complex practical problems. Recently, applications of generative adversarial networks (GAN) in PU learning have received growing attention [12, 13], where the generative models learn to generate fake positive and negative samples (or only negative samples), and the classifier is trained by using the fake samples. Experiments show that GAN can improve the performance of PU learning when the size of positive labeled data is extremely small, but some strong assumptions of data distributions, including the data separability, are still required for the GAN based methods.Another widely used approach is to train the classifier by minimizing a weighted loss function, where unlabeled data are interpreted as negative samples with noisy labels, and the weights can be constant hyperparameters
[14, 15] or modeled as a continuous weight function according to the estimated mislabeling probabilities [16, 17]. In [18], a universal framework for classification with noisy labels are developed under the data separability assumption, and the PU learning can be efficiently performed by the presented rank pruning algorithm as a special case within this framework.One solution to the PU learning problem with general data distributions is given by [19, 20]
, where an unbiased estimator for the misclassification risk of supervised learning is derived for PU data, and the classifier can be trained through minimizing the estimate. However, the direct minimization of the estimated risk easily leads to severe overfitting. In order to address this difficulty, a nonnegative risk estimator is presented in
[21], which is biased but more robust to statistical noise. The main limitation of this approach is that the class prior, i.e., the proportion of positive data (including labeled and unlabeled) to the whole data, is needed. In practical applications, the class prior can be estimated by some class prior estimation methods [22, 23, 24, 25], but the classification performance could be badly affected by an inaccurate estimate.In this paper, we propose a novel PU learning framework called discriminative adversarial networks (DAN). The key idea of DAN is to approximate the ideal Bayes classifier via reducing the distribution distance between the labeled positive data and those identified by the classifier from the whole dataset. DAN measures and minimizes the distance through a minimax game between the classifier and another discriminative model by analogy to the wellknown generative adversarial networks (GAN) [26], and provides a more efficient way to recover the positive and negative data distributions from unlabeled data than GAN based PU learning methods. Moreover, it can effectively avoid the phenomenon of mode collapse, from which GAN easily suffers. Both theoretical analysis and experimental results show that the proposed framework can achieve high classification accuracy in general cases without the class prior or the common assumption of data separability in PU learning. The paper is organized as follows. The next section is devoted to problem statement. Section 3 presents DAN and its detailed mathematical formulation. Section 4 compares our approach and some related works, and experiment results are provided in Section 5. Finally, further discussion and some future research directions of DAN are given in Section 6.
2 Problem statement
Let be independent samples drawn from an underlying distribution density with labels , where only the first samples are labeled as positive, i.e., for , and the labels of the other samples are unavailable. We further assume that the empirical distribution of the positive data in is consistent with the ground truth . The goal of PU learning is to learn a binary classification model from the positive dataset and the unlabeled dataset , which can predict the label of a new instance .
It is wellknown that the optimal classifier in the sense of minimum misclassification probability can be given by with being the conditional probability of the positive label. In the case of positivenegative (PN) learning, where all training samples are labeled, can be effectively approximated by minimizing some empirical misclassification risk (e.g., crossentropy loss). But such an approximation is difficult for PU learning due to the absence of labeled negative training data.
Remark 1.
We only consider here the singletrainingset scenario of PU learning with . Another common scenario in the literature is called casecontrol [27], where samples in and are drawn from and independently, and the method proposed in this paper can be naturally extended to this scenario (see Section C in Supplementary Information).
3 Discriminative adversarial networks
3.1 Motivation
Unlike some popular PU learning methods [19, 21, 12], the class prior of the positive class is not
assumed to be known in this paper, and only distributions of positive data and the whole dataset are available. According to the Bayes’ theorem, the two distributions are connected via
as , where(1) 
represents the positive data distribution reconstructed by a function . Furthermore, by replacing with the empirical distribution of , Eq. (1) can be rewritten as
(2) 
where denotes the Dirac function and is the mean value of over . Hence, we can generate samples via resampling with probability proportional to .
The above analysis suggests that
of can be trained via minimizing the distance between and . However, It is worth pointing out that for according to (1). So, we can only get the value of up to a proportional constant even if holds exactly. The scale invariance of has been thoroughly discussed in the research of mixture proportion estimation, and some theoretical conclusions can be seen in [28, 29]. Here, we make the following assumption so that is identifiable for given and :(3) 
i.e., at least one sample can be predicted to be positive with probability one, which comprises many practical cases. Under this assumption, we can obtain
(4) 
if .
3.2 Method
Inspired by the remarkable success of generative adversarial networks (GAN) [30], here we represent as a deep neural network, and define a second deep discriminative model , which maps a sample to the probability that came from rather than . Then the distance between and can be measured and minimized through the following game between and :
(5)  
Intuitively, as illustrated in Fig. 1, intends to separate the samples uniformly drawn from and those obtained by resampling from with weights given by , whereas is trained to correctly identify positive samples in so as to fool . Under some technical assumptions, it can be shown that for a fixed ,
(6) 
at the limit of infinite data size (see Proposition 2 in Supplementary Information), where denotes the JensenShannon divergence. Hence, in the ideal situation, the training procedure converges to the equilibrium point where and cannot distinguish the two distributions with , and can be obtained after the normalization described in (4).
The adversarial training method described in above can provide satisfying performance when is small. But for highdimensional PU learning tasks, the training procedure defined by (5) also suffers from mode collapse like training of GAN, i.e., tends to predict for a part of positive samples especially when the positive data distribution has multiple modes. In order to address this problem, we introduce a penalty factor
(7) 
and change the learning objective as
(8) 
so that is still satisfied by the optimal solution, where is the average of over , and is a small constant to avoid singularity. The numerator of penalizes small values of for , and can effectively prevent the phenomenon of model collapse because as for some . Furthermore, the normalization constraint (3) of model can be automatically satisfied by solving (8) since for . The denominator of is designed according to our experimental experience (see Section B in Supplementary Information for some other choices), which increases the gap between for in and , and can improve the classification performance. More detailed analysis of and is given in Section A of Supplementary Information.
The learning framework developed in this section is similar to GAN, and is based on a zerosum game between two discriminators instead of a generator and a discriminator. Thus, we call this framework discriminative adversarial networks (DAN).
3.3 Implementation
The detailed DAN learning algorithm adopted in this paper is summarized by Algorithm 1, where are both deep networks with weights denoted by
, and the Sigmoid output neurons (or the other bounded output neurons) can be used so that
for all . For applications in big data scenarios, all mean values involved in the objective function are approximated by minibatches in each iteration. Notice that is updated only by using the gradient of in Step (9) because is independent of . For , the value of is usually positive under the condition that performs better than random guess, which is usually satisfied in training process. But when the model is badly initialized with , updating according to the gradient of may yield the divergence of the algorithm. So we implement the update of as shown in (10) for numerical stability.(9)  
(10) 
(11) 
4 Related work
An important idea of DAN is to approximate by matching and , which has in fact been investigated in literature (see, e.g., [31, 32, 33, 34, 24]). However, the direct approximation based on (1) involves the probability density estimation and is difficult for highdimensional applications. In [34, 24], by modeling the ratio between and as a linear combination of basis functions, this problem is transformed into a quadratic programming problem. But the approximation results cannot meet the requirement for classification, and are only applicable to estimation of the class prior of . One main contribution of our approach compared to the previous works is that we find a general and effective way to optimize the model of by adversarial training.
It is also interesting to compare DAN to GenPU, a GAN based PU learning method [12], since they share the similar adversarial training architecture. In DAN, the discriminative model plays the role of the generative model in GAN by approximating positive data distribution in an implicit way, and can be efficiently trained together with . In contrast, GenPU is much more timeconsuming and easily suffers from mode collapse as stated in [12] due to that it contains three generators and two discriminators. (Notice that the penalty factor cannot be applied to GenPU for the probability densities of samples given by generators are unknown.) Furthermore, the consistency of the GenPU needs the assumptions that class prior is given and there is no overlapping between positive and negative data distributions, which are not necessary for DAN.
5 Experiments
In this section, we conduct a series of PU learning experiments on both synthetic and realworld datasets to evaluate the performance of DAN. The detailed settings of datasets and algorithms are provided in Section D of Supplementary Information, and the software code for DAN is also available.^{1}^{1}1The software code will be publicly available after the blind review process.
We first visualize the learning results of DAN on four twodimensional toy examples in Fig. 2, from which we can observe that an accurate classification boundary can be deduced from the conditional class probability approximated by DAN even if the positive and negative data cannot be well separated.
Next, we conduct experiments on three benchmark datasets taken from the UCI Machine Learning Repository [35, 36], and the performance of DAN is compared to that of some recently developed PU learning methods, including the unbiased risk estimator based uPU and nnPU [19, 21], the generative model based GenPU [12], and the rank pruning (RP) proposed in [18].^{2}^{2}2The software codes are downloaded from https://github.com/kiryor/nnPUlearning, https://qibinzhao.github.io/index.html and https://github.com/cgnorthcutt/rankpruning. Considering that uPU and nnPU require the class prior , we implement uPU and nnPU under two different conditions: (a) The exact value of is known, and (b) is estimated by KM2 proposed in [29], which is one of the stateoftheart class prior estimation algorithms. For GenPU, the hyperparameters of the algorithm are determined by greedy grid search (see Section D.5 in Supplementary Information). The classification results are summarized in Table 1
. It can be seen that DAN outperforms the other methods with high accuracies and low variances on almost all the datasets. Only the nnPU obtains a higher accuracy on the dataset of Grid Stability with “unstable vs stable” when the exact value of
is given, and its accuracy decreases significantly with estimated . In addition, RP interprets unlabeled data as noisy negative data and can get an accurate classifier when the proportion of positive data is small in unlabeled data. But in the opposite case where the proportion is too large, RP performs even worse than random guess. ( and in Page Blocks with ’2,3,4,5’ vs ’1’ and Grid Stability with ’unstable’ vs ’stable’.)Dataset  DAN  nnPU  nnPU(KM2)  uPU  uPU(KM2)  GenPU  RP 

Page Blocks  
Page Blocks  
Grid Stability  
Grid Stability  
Avila  
Avila 
Classification accuracies (%) of compared methods on UCI datasets. The accuracies are evaluated on test sets, and the mean and standard deviation values are computed from
independent runs. Definitions of labels (’Positive’ vs ’Negative’) are as follows: Page Blocks: ’1’ vs ’2,3,4,5’. Page Blocks: ’2,3,4,5’ vs ’1’. Grid Stability: ’stable’ vs ’unstable’. Grid Stability: ’unstable’ vs ’stable’. Avila: ’A’ vs the rest. Avila: ’A, F’ vs the rest. Labeled positive data are randomly selected from the training data with and .Finally, all the methods are compared on two image datasets: FashionMNIST and CIFAR10,^{3}^{3}3Datasets are downloaded from https://github.com/zalandoresearch/fashionmnist and https://www.cs.toronto.edu/~kriz/cifar.html. and the classification results are collected in Table 2, where the superior performance of DAN is also evident. Here uPU performs much worse than nnPU due to the overfitting problem [21] (see Fig. 4 in Supplementary Information). Moreover, the performance of GenPU is also not satisfying because of the mode collapse of generators as shown in Fig. 3. In contrast, different modes of positive and negative data can be successfully sampled from the distributions defined by and in DAN.
Dataset  DAN  nnPU  nnPU(KM2)  uPU  uPU(KM2)  GenPU  RP 

FashionMNIST  
FashionMNIST  
CIFAR10  
CIFAR10 
6 Discussion
The framework of DAN can be viewed as a mixture of discriminative learning and generative learning: A discriminative model is trained by minimizing a loss function defined by the distribution distance as a generative model. Due to the existence of unlabeled data, it is very difficult, if not impossible, to perform the PU learning in a pure discriminative manner. Even uPU and nnPU, which are developed based on the estimator of the discriminative loss, still need to model positive and negative data distributions for the approximation of class prior. But DAN demonstrates that, in PU learning, the classifier can be trained directly without solving the problem of probability density estimation as an intermediate step. It is interesting to extend this idea to more general semisupervised learning problems, such as PNU learning, where some data are labeled as positive or negative while most data are unlabeled, and DAN has the potential to address such classification challenges especially in application scenarios where labeled positive and negative data cannot cover all modes of datasets.
It is also worthy to note that DAN is a very flexible framework, and the performance can be expected to be further improved by utilizing many advanced GAN techniques developed in recent years. For example, by analogy to WGAN and MMDGAN, we can simply establish DAN models based on the Wasserstein metric and maximum mean discrepancy between distributions. Another research direction in future is to investigate robust DAN for semisupervised learning with noisy labels.
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Appendix A Theoretical analysis of DAN learning
In this section, we analyze the properties of (8) and its optimal solution under the following assumptions.
Assumption 1.
have enough capacity and both and tends to infinity with being fixed.
Assumption 2.
The marginal distribution for all .
Assumption 3.
There exists a measurable set so that for all .
Proposition 1.
defined by (7) satisfies: (i) for . (ii) . (iii) as for some .
Proof.
The proof of (i) is trivial, and (ii) and (iii) are direct conclusions of the following inequality:
(12)  
∎
Proposition 2.
For a given ,
(13) 
and the maximum is achieved when
(14) 
Proof.
Proposition 3.
Proof.
Appendix B Penalty factors
Besides the penalty factor given in (7), we also considered the following factors:
(19)  
(20)  
(21) 
where
(22)  
denotes the mutual information between sample and its label defined by , and denotes the average of over in (22). All the above choices of the penalty factor can lead to consistency of learning. We choose defined by (7) because it achieves the best performance in our experiments.
Appendix C Casecontrol scenario
Under the scenario of casecontrol, the empirical approximation (2) of becomes
(23) 
where is the mean value of over . Therefore, the method and theory presented in this paper can be extended to the casecontrol scenario by defining
(24)  
Appendix D Experiment details
In FashionMNIST, CIFAR10 and Avila, datasets have been separated into training and test sets. For the two UCI datasets, we adopt the train_test_split function in scikitlearn to get test sets.
d.1 Toy examples
The former three toy examples in our experiments are generated by functions of make_circles, make_moons, make_blobs in the package of scikitlearn, where the centers of blobs are , , ,
. The dataset of the fourth example are given by a Gaussian mixture model with centers
, , , , , , , . The other details are shown in Table 3.Dataset  parameters 

Concentric circles  factor= noise= 
Half moons  noise= 
Blobs  cluster_std= 
Gaussian mixture model  covariance matrix= 
d.2 UCI datasets
We first clarify the UCI datasets used in our experiments in Table 4. Then, we give the detailed experimental settings of each experiment in Table 5.
Dataset  size of test set  

Page Blocks  
Grid Stability  
Avila 
Experiment  setting  Data amount  

Page Blocks  ’2,3,4,5’ vs ’1’  = =  
Page Blocks  ’1’ vs ’2,3,4,5’  = =  
Grid Stability  ’stable’ vs ’unstable’  = =  
Grid Stability  ’unstable’ vs ’stable’  = =  
Avila  ’A’ vs The rest  = =  
Avila  ’A,F’ vs The rest  = = 
d.3 FashionMNIST and CIFAR10
The details of the experiments are shown in Table 6
. Classification errors of nnPU, uPU and DAN on CIFAR10 test data with different numbers of epochs are plotted in Fig.
4.Experiment  Setting  Data amount  KM2  

FashionMNIST  ’1,4,7’ vs ’0,2,3,5,6,8,9’  = =  
FashionMNIST  ’0,2,3,5,6,8,9’ vs ’1,4,7’  = =  
CIFAR10  ’0,1,8,9’ vs ’2,3,4,5,6,7’  = =  
CIFAR10  ’2,3,4,5,6,7’ vs ’0,1,8,9’  = = 
d.4 Other details
We choose Adam as the optimizer for DAN in our experiments, and the hyperparameters in Adam are . The architectures for models in DAN are shown in Table 7. The epoch number of DAN for image datasets is , and for the other datasets. Moreover, the hyperparameter .
Dataset  Network  Model  Initial learning rate 

Toy examples  D 
layers MLP with ReLU 

layers MLP with ReLU  
UCI datasets  D  layers MLP with ReLU  
layers MLP with ReLU  
FashionMNIST  D  layers CNN with ReLU  
layers CNN with ReLU  
CIFAR10  D  layers CNN with ReLU  
layers CNN with ReLU 
d.5 Choice of hyperparameters of GenPU
GenPU contains four hyperparameters: , , , . Although the parameters are coupled for given in [12], our experience shows that the better performance can be achieved by selecting the four parameters independently. Table 8 shows the best hyperparameters which lead to the largest classification accuracies on test sets. They are selected in by greedy grid search.
Dataset  FashionMNIST  CIFAR10  Page Blocks  Grid Stability  Avila 

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