Florent Chiaroni
email: florent.chiaroni@vedecom.fr

Ghazaleh Khodabandelou
email: ghazaleh.khodabandelou@u-pec.fr

Mohamed-Cherif Rahal

Nicolas Hueber
email: nicolas.hueber@isl.eu

Frédéric Dufaux

Frederic Dufaux
email: frederic.dufaux@l2s.centralesupelec.fr

Generating Relevant Counter-Examples from a Positive Unlabeled Dataset for Image Classification

Keywords: Generative Adversarial Networks (GANs), generative models, semi-supervised learning, partially supervised learning, deep learning

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Introduction

Nowadays, the number of available labeled datasets dedicated to perception applications such as image classification [START_REF] Russakovsky | ImageNet Large Scale Visual Recognition Challenge[END_REF] and semantic scene understanding [START_REF] Cordts | The cityscapes dataset for semantic urban scene understanding[END_REF] has considerably augmented. However, when learning methods trained on these datasets are applied on real data, their performances are likely to deteriorate. Consequently, it is necessary to use a dataset specialized for the given target application. It turns out that it can be easy to get unlabeled data in some applications domains such as autonomous driving. Positive Unlabeled (PU) learning, also called partially supervised classification [START_REF] Liu | Partially supervised classification of text documents[END_REF], enables to use these unlabeled data in combination with labeled samples of our class of interest: the positive class.

The interest is that unlabeled data can contain relevant counter-examples, also called negative examples1 . The difficulty is that unlabeled data can also contain a fraction π P of unlabeled positive examples, as illustrated in Fig. 1. [START_REF] Sansone | Efficient training for positive unlabeled learning[END_REF] enumerates several learning problems which can be addressed in this way such as the challenging information retrieval task. Several PU learning methods exist. While some of them are adapted to time series [de Carvalho Pagliosa and de [START_REF] De Carvalho Pagliosa | Semi-supervised time series classification on positive and unlabeled problems using cross-recurrence quantification analysis[END_REF]] or text classification, for instance by using non-negative matrix factorization [START_REF] Li | Classifying networked text data with positive and unlabeled examples[END_REF], we focus in this work on those that are applied to image classification using deep neural networks. They are generally classified into two categories. The former is censoring PU learning, formalized by [START_REF] Elkan | Learning classifiers from only positive and unlabeled data[END_REF] and recently improved by [START_REF] Northcutt | Learning with Confident Examples: Rank Pruning for Robust Classification with Noisy Labels[END_REF]. The latter is case-control PU learning, introduced by [START_REF] Ward | Presence-only data and the EM algorithm[END_REF], formalized by du Plessis et al. [2014], and then consecutively improved by Du Plessis et al. [2015] and [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF] to reduce the training computational cost and alleviate the overfitting issue. In the context of the proposed approach, we focus our attention in this work on the recently presented GAN-based PU approaches. Thus we classify PU learning approaches into the two following groups suggested by [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF]: one-stage and two-stage PU methods.

One-stage PU methods such as the unbiased PU method (uPU) [START_REF] Du Plessis | Convex formulation for learning from positive and unlabeled data[END_REF] and the non-negative PU method (nnPU) [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF] consist of training a classifier using an unbiased risk directly on the PU dataset. These methods have the advantage to need only one training of the classifier. However, they require dataset prior knowledge and consequently uPU and nnPU need to be combined with an approach estimating the prior knowledge [START_REF] Christoffel | Class-prior Estimation for Learning from Positive and Unlabeled Data[END_REF]. Consequently, they are critically sensitive to slight prior variations per minibatch, as shown experimentally in Section 4.3.1.

Two-stage PU methods prepare during the first stage a Positive Negative (PN) dataset. Some noisy labeled learning strategies consist of detecting automatically the most plausible mislabeled examples [START_REF] Ekambaram | Active cleaning of label noise[END_REF] through an iterative process in order to accelerate the manual PN labeling. Some other methods can prepare automatically a PN dataset without human supervision from end-to-end. For instance, Rank Pruning method (RP) [START_REF] Northcutt | Learning with Confident Examples: Rank Pruning for Robust Classification with Noisy Labels[END_REF] firstly estimates the prior such that it can select only the examples considered as the most confident, in order to substitute the unlabeled samples for the second-stage training of the classifier. RP achieves two-stage state-of-the-art performances without prior knowledge. However, it can only exploit a sub part of the PU dataset during training. This can curb its prediction performances on complex datasets like CIFAR-10. Recently, a new subcategory of two-stage PU methods has appeared:

GAN-based PU methods. They address the PU learning challenge by producing, thanks to an adversarial training [START_REF] Goodfellow | Generative adversarial nets[END_REF], generated samples from a PU dataset during the first step. Then, they are used to train a standard Positive Negative (PN) classifier during the second step.

We discuss in more details these PU methods, namely uPU [START_REF] Du Plessis | Convex formulation for learning from positive and unlabeled data[END_REF], nnPU [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF], RP [START_REF] Northcutt | Learning with Confident Examples: Rank Pruning for Robust Classification with Noisy Labels[END_REF], GenPU [START_REF] Hou | Generative adversarial positive-unlabeled learning[END_REF] and PGAN [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF] in the related work Section 2.

We can nonetheless already make the following remarks, motivating the design of the proposed approach. Unbiased methods [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF], [START_REF] Du Plessis | Convex formulation for learning from positive and unlabeled data[END_REF], and GenPU [START_REF] Hou | Generative adversarial positive-unlabeled learning[END_REF] are by definition sensitive to the prior knowledge in order to deal with a PU dataset.

Conversely, whereas the two-stage censoring methods, such as RP [START_REF] Northcutt | Learning with Confident Examples: Rank Pruning for Robust Classification with Noisy Labels[END_REF], do not require prior information, they suffer from generalization and unsteadiness problems due to their selective process. The PGAN method is the first that does not need prior knowledge nor a selective process, thus preserving a low sensitivity to prior knowledge combined with a training stability. However, as mentioned in the PGAN work, it inherently suffers from first-stage overfitting.

Based on these considerations, we propose in this work a novel GAN-based model, referred to as Divergent-GAN (D-GAN), to overcome the latter issue while preserving the PGAN advantages. To the best of our knowledge, we are the first to propose a GAN-based method to capture exclusively the unlabeled negative samples distribution from a PU dataset without prior knowledge. More specifically, our contribution is the following:

• We propose to incorporate a biased PU learning loss function inside the original GAN [START_REF] Goodfellow | Generative adversarial nets[END_REF] discriminator loss function. The intuition behind it is to have the generative model solving the PU learning problem formulated in the discriminator loss function. In this way, the generator learns the distribution of the examples which are both unlabeled and not positive, namely the negative ones included in the unlabeled dataset.

Consequently, the proposed D-GAN framework compares favorably with PU learning state-of-theart performances on simple MNIST [LeCun et al., 1998] and complex CIFAR-10 [ [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] image datasets. The proposed framework code is available2 .

The remaining of this paper is structured as follow. Section 2 presents previous PU learning approaches. Section 3 describes the proposed method. Section 4 presents the corresponding experimental results. Finally, in Section 5, we draw conclusions and discuss perspectives.

Related work

The PU learning problem consists of trying to distinguish positive samples from negative ones by using a PU dataset. Let X ∈ R m be the input random variable and Y ∈ {0, 1} its associated label. X can be a positive X P , negative X N or unlabeled X U sample which respectively follow the distributions p P = p(X|Y = 0), p N = p(X|Y = 1) and p U = (1 -π P ) • p N + π P • p P . The unknown prior π P ∈ (0, 1) represents the fraction of unlabeled positive examples included in the unlabeled dataset.

Previous works on PU learning [START_REF] Denis | PAC learning from positive statistical queries[END_REF] This bias can limit prediction performances of the learning model.

Several strategies have been proposed to solve this drawback such as unbiased methods [START_REF] Du Plessis | Analysis of learning from positive and unlabeled data[END_REF], [START_REF] Du Plessis | Convex formulation for learning from positive and unlabeled data[END_REF], [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF], pruning method [START_REF] Northcutt | Learning with Confident Examples: Rank Pruning for Robust Classification with Noisy Labels[END_REF],

and more recently GAN based methods [START_REF] Hou | Generative adversarial positive-unlabeled learning[END_REF], [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF]. However, those strategies still present some issues including prior knowledge sensitivity, training unsteadiness, or overfitting problems.

We present in this section different state-of-the-art methods and their respective drawbacks that we aim at overcoming with the proposed GAN-based PU framework.

Unbiased methods

In order to palliate a biased training, the authors of unbiased techniques [START_REF] Du Plessis | Analysis of learning from positive and unlabeled data[END_REF], [START_REF] Du Plessis | Convex formulation for learning from positive and unlabeled data[END_REF], [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF] suggest to avoid the estimator bias by adding some terms in the training loss function. Then, the classifier behaves as if it is trained with a positive negative dataset. The authors firstly used a non convex loss function [START_REF] Du Plessis | Analysis of learning from positive and unlabeled data[END_REF],

which then has been adapted for convex loss functions [START_REF] Du Plessis | Convex formulation for learning from positive and unlabeled data[END_REF] in order to reduce the computational burden. Subsequently, it was proposed to overcome the training overfitting by adding a binary condition (an "if" condition) in the training loss function [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF].

These methods exploit the prior π P in the empirical training loss function. However, we observe that the empirical prior value πP per batch of small size (minibatch) is slightly different to the global dataset prior π P , as its standard deviation depends on the minibatch size, such that:

πP = π P + α, (1)
with α ∼ p α (m), where p α is the probability distribution of the noise α depending on the minibatch size m, as shown in Figure 2. We observe that the worst case scenario is when π P is close to the value 0.5, combined with a small batch size. The cases where π P is higher than 0.5 behave symmetrically to the cases where π P is smaller than 0.5. In our case, we want to train a deep learning model using the stochastic gradient descent (SGD) optimization technique, which is known to be relevant with batches of small size. So the theoretical formulation of unbiased techniques cannot be maintained using SGD with small batch sizes. We will show empirically that in practice, unbiased techniques are highly sensitive to the minibatch size in terms of prediction performances, as they are theoretically sensitive to the prior π P .

It turns out that it is possible to avoid this limitation with two-stage approaches.

Two-stage approaches

Two-stage approaches mainly consist of preparing during the first-stage a positive negative (PN) training dataset which will then be used to directly train a standard classifier during the second stage. One interest of those approaches is that they are not sensitive to the prior knowledge variation. Consequently, they are compatible with the use of minibatches, and thus are suitable when applying SGD optimization.

Pruning approach

Rank Pruning (RP) method [START_REF] Northcutt | Learning with Confident Examples: Rank Pruning for Robust Classification with Noisy Labels[END_REF] is a two-stage technique. It first estimates the prior π P and exploits it to prune the dataset in order to capture only a subset corresponding to the most confident positive and negative samples. Then, during the second stage it considers this subset as a cleanly labeled positive negative dataset to train a classifier. While not requiring prior knowledge in input, RP achieves state-of-the-art results for information retrieval in the One-vs-Rest task on simple datasets such as MNIST. However, by using a pruning strategy, RP can miss some

relevant training examples not included in the selected subset of training. As a consequence, this can limit its generalization, as will be shown experimentally in Table 2, where RP is shown to be relatively unstable when compared to GAN-based approaches in terms of prediction performances.

Using only a training subset is also a weakness on complex datasets like CIFAR-10, where a large training dataset is preferable to obtain better results.

Some approaches have been recently proposed by exploiting GANs benefits, achieving high prediction scores over the same PU learning tasks.

GAN-based approaches

GAN-based PU approaches represent a recent subcategory of two-stage PU methods, as proposed in GenPU [START_REF] Hou | Generative adversarial positive-unlabeled learning[END_REF] and PGAN [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF]. The interest of using GANs is twofold. First, GANs enable relevant data augmentation, as proposed for instance in [START_REF] Zhu | Emotion classification with data augmentation using generative adversarial networks[END_REF] for dealing with unbalanced label distributions for emotion classification. Second, it allows for the use of high-level feature metrics to evaluate generated samples quality, thanks to the adversarial training. This can make capturing a target distribution easier.

In this PU learning context, the generated samples replace the unlabeled ones by learning on the latter as PGAN [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF], or on both unlabeled and positive labeled ones as GenPU [START_REF] Hou | Generative adversarial positive-unlabeled learning[END_REF]. Both methods exploit GANs benefits, but the functioning are different and they are not suitable under the same datasets conditions.

GenPU [START_REF] Hou | Generative adversarial positive-unlabeled learning[END_REF] is based on the original GAN convergence [START_REF] Goodfellow | Generative adversarial nets[END_REF],

such that: GAN framework [START_REF] Goodfellow | Generative adversarial nets[END_REF] to address standard PU learning challenge3 is more computational demanding and not necessary to generate relevant counter-examples. Moreover, using five models amplifies the mode collapse issue, and the corresponding training optimization functions need three additional sets of hyper-parameters combined with prior knowledge. This is impractical in the context of real applications where hyper-parameters may have to be tuned on limited computational resources to adapt the model for a given application dataset.

π P • p G P + (1 -π P ) • p G N - → p U ,
PGAN [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF] is trained to converge towards the unlabeled dataset distribution during the first step. During the second step, it exploits GANs imperfections for capturing the unlabeled distribution, such that the generated distribution at the adversarial equilibrium is still separable from the unlabeled samples distribution by a classifier. It presents a relatively steadier behaviour and better prediction performances than the two-stage baseline RP method on the complex RGB image dataset CIFAR-10 without prior knowledge. However, it is less suitable for relatively simpler datasets like MNIST. The problem is that the generated samples are all considered as negative samples by the classifier. But this is possible only if the generated samples

Table 1: Summary of presented state-of-the-art methods advantages and drawbacks compared to the proposed D-GAN approach. A void cell means that the mentioned criterion is not applicable with the corresponding method.

Proposed Approach

In this section, we first briefly recall the main reasoning which motivates our research work.

Next, we discuss some features of a biased PU risk. We then propose to incorporate this risk into a generic GAN framework in order to guide the generator convergence towards the negative samples distribution, denoted as p N , included inside the unlabeled dataset distribution, denoted as p U .

Motivation

In PU learning, if a classifier associates a given expected label value with positive examples, and in parallel associates a second distinct label value with unlabeled examples, then it is proven that the negative non-labeled examples are exclusively associated with the label of non-labeled examples [START_REF] Denis | PAC learning from positive statistical queries[END_REF]. Concerning GANs, it has been shown that the discriminator learning task influences directly the adversarial generator behaviour [START_REF] Mao | Least squares generative adversarial networks[END_REF]. Based on these considerations, this work aims at incorporating a biased PU risk inside the traditional GAN discriminator cost function.

This compels the discriminator D, to separate negative from positive distributions, which in turn guides the generator G, to exclusively learn the unlabeled counter-examples distribution from a PU dataset. As a matter of the fact, the proposed method is novel in the way it exclusively generates relevant counter-examples without prior knowledge information, while preserving a standard GAN architecture.

Thereafter, we present the biased PU risk that we incorporate in the proposed GAN PU

discriminator training loss function.

Biased PU risk

In what follows, we first explain the expected PU functionality to be incorporated into the GAN discriminator loss function. Biased PU risk setting: Let D : R m → [0, 1] be the decision function which is, later on, considered as the discriminator D, of the proposed framework network.

We have l(ŷ, y) such that l : [0, 1] × [0, 1] → R is the arbitrary cost function with the predicted output ŷ of D for a given example and the corresponding label y as input. D is trained with a PU risk R P U to predict the label value 1 for the unlabeled examples, and the label value 0 for the positive labeled ones such that:

R P U (D) = E x U ∼p U [l(D(x U ), 1)] + E x P ∼p P [l(D(x P ), 0)]. (2)
Given the composition of the distribution p U , we develop: follow the same distribution p P , as mentioned in [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF]. Since expectations are linear, p P is associated to both contradictory labels 0 and 1 as below:

R P U (D) = (1 -π P ) • E x N ∼p N [l(D(x N ), 1)] + π P • E x P ∼p P [l(D(x P ), 1)] + E x P ∼p P [l(D(x P ), 0)]. ( 3
R P U (D) = E x N ∼p N [(1 -π P )l(D(x N ), 1)] + E x P ∼p P [π P l(D(x P ), 1) + l(D(x P ), 0)]. ( 4
)
Positive samples distribution p P is shifted away from the counter-examples dis-tribution p N : When defining the cost-function l as the binary cross-entropy H (Eq. 5) such that l = H, then we can demonstrate that the second term in the Equation 4 is equivalent to associating the positive distribution p P with a unified biased intermediate label value δ. The binary cross-entropy H is defined as:

H(D(X), Y ) = -Y log(D(X)) -(1 -Y )log(1 -D(X)), ( 5
)
where Y is the label value associated with the input X of D. If l = H, then concerning the second term of the Equation 4, we can demonstrate that:

π P H(D(x P ), 1) + 1H(D(x P ), 0) = -π P log(D(x P )) -1log(1 -D(x P )) = -π P log(D(x P )) -(1 + π P -π P )log(1 -D(x P )) =(1 + π P ) • - π P 1 + π P log(D(x P )) -(1 - π P 1 + π P )log(1 -D(x P )) =(1 + π P ) • H D(x P ), π P 1 + π P =(1 + π P ) • H(D(x P ), δ), (6)
with δ = π P /(1 + π P ). Consequently, the PU risk becomes:

R P U (D) = E x N ∼p N [(1 -π P )H(D(x N ), 1)] + E x P ∼p P [(1 + π P )H(D(x P ), δ)]. (7)
Such a PU risk has been previously called biased or constrained in the literature [START_REF] Liu | Partially supervised classification of text documents[END_REF].

The identity between Equations 4 and 7 makes it possible to estimate the restricted interval of possible values for δ without using prior such that if π P ∈ (0, 1) then:

0 < π P < 1 ⇔ 0 < δ < 1 1 + 1 . ( 8
)
In other words, δ ∈ (0, 1/2). This confirms that for any π P value between 0 and 1, labeled and

unlabeled positive examples are associated with a label value δ comprised between 0 and 1/2.

Therefore, when training D with the risk R P U , the D prediction related to the unlabeled positive examples is shifted away from the label value 1. From D prediction output point of view, this risk makes the positive distribution p P diverging from the negative distribution p N . Thus, D is trained to predict the label value 1 exclusively for the counter-examples.

In this way, we have demonstrated in this section that we can consider the discriminator PU training loss function R P U as a PN training loss function, referred to as R P N , by replacing the two opposite labels 0 and 1 associated to positive samples distribution p P by an intermediate label value δ depending on π P , such that we obtain:

R P U (D) = R P N (D), (9)
with:

     R P U (D) = E x U ∼p U [H(D(x U ), 1)] + E x P ∼p P [H(D(x P ), 0)], R P N (D) = E x N ∼p N [(1 -π P )H(D(x N ), 1)] + E x P ∼p P [(1 + π P )H(D(x P ), δ)].
(10)

Proposed generative model

The

L D GAN (G, D) = E x U ∼p U [-logD(x U )] + E z∼pz [-log(1 -D(G(z)))], (11)
where z stands for the input random vector of the generative model G such that G(z) is a generated sample. z follows a uniform or normal distribution. It turns out that the binary cross-entropy formulation (Eq. 5) implies H(D(X), 1) = -log(D(X)) and

H(D(X), 0) = -log(1 -D(X)). L D GAN (G, D) =E x U ∼p U [H(D(x U ), 1)] + E z∼pz [H(D(G(z)), 0)]. ( 12
L D (G, D) =L D GAN (G, D) + E x P ∼p P [H(D(x P ), 0)]. (13)
If we develop the term L D GAN , we then obtain:

L D (G, D) =E x U ∼p U [H(D(x U ), 1)] + E z∼pz [H(D(G(z)), 0)] + E x P ∼p P [H(D(x P ), 0)] =R P U (D) + E z∼pz [H(D(G(z)), 0)]. (14)
In other words, the R P U risk (Eq. 2) is incorporated inside the D-GAN discriminator loss function.

To this extent, D can be trained to only consider the counter-examples as the most real examples by exclusively associating to them the label value 1. This can be considered as applying a constrained optimization.

The generator generates the counter-examples distribution: In contrast, the role of G during the adversarial training is to generate samples considered by D as 1. As suggested by [START_REF] Goodfellow | Generative adversarial nets[END_REF], the training loss function L G of G is such that:

L G (G, D) = E z∼pz [-log(D(G(z)))] = E z∼pz [H(D(G(z)), 1)]. (15
∇ θ D 1 m m i=0 -logD(x (i) U ) -log 1 -D(G(z (i) )) -log 1 -D(x (i) p )
Sample minibatch of m noise samples {z (1) , ..., z (m) } from noise prior p z . Update G by descending its stochastic gradient:

∇ θ G 1 m m i=0 -log D(G(z (i) ))
P } from data distribution p P . Sample minibatch of m noise samples {z (1) , ..., z (m) } from data distribution p z . Update C by descending its stochastic gradient:

∇ θ C 1 2 • m m i=1 l(C(x (i) P ), 1) + l(C(G(z (i) )), 0)

end for

The gradient-based updates can use any standard gradient-based learning rule. We use Adam in our experiments.

As previously discussed, we recall that D exclusively considers the negative examples as 1 thanks to the R P U risk. Thus, if D trainable weights are fixed in the proposed framework, then we propose to reinterpret in L G the label value 1 as D(x N ), as follows:

L G (G, D) = E z∼pz,x N ∼p N [H(D(G(z)), D(x N ))] = E z∼pz,x N ∼p N [-D(x N )log(D(G(z)))],
(16) such that the distance between the generated samples distribution and p N is minimized. Consequently, this justifies the convergence of G in the proposed D-GAN framework towards the negative samples distribution p N , for any π P ∈ (0, 1). G will also generate some examples following a subset of the positive samples distribution. Thus, the D-GAN will tend to behave as the PGAN [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF], which seems to be the best solution in this situation.

Implementation

The next section presents experimental results demonstrating the effectiveness of the proposed approach.

Experimental Results

In this section, we assess the performance of the proposed approach. We first experimentally validate the expected discriminator prediction behaviour when it is applied on a positive unlabeled dataset (Sec. 4.2.1), and study the impact of regularization (Sec. 4.2.2). Then, we show the ability of the generator to generate counter-examples for different types of PU datasets, including two-dimensional points and natural RGB images (Sec. 4.2.3). Finally, we evaluate the proposed model prediction robustness and compare it with state-of-the-art PU learning methods in terms of prior noise (Sec. 4.3.1) and first-stage overfitting (Sec. 4.3.2).

Settings

We detail in this section the settings of the experiments. We have adapted the first-stage discriminator and generator architectures of the proposed GAN based PU framework depending on the dataset on which they are applied, as follows:

• 2D point dataset: In order to deal with 2D point datasets, we have implemented the original GAN [START_REF] Goodfellow | Generative adversarial nets[END_REF] architecture composed of fully connected layers (FullyConnected). The generator and discriminator architectures are summarized in Figure 4.

• MNIST [LeCun et al., 1998]: In order to deal with grayscale images of dimension 28*28 pixels from the MNIST dataset, we use a deep convolutional GAN architecture (DCGAN) 4 such that the generator contains transposed convolutional (DeConv2D) top layers, and the discriminator contains convolutional (Conv2D) bottom layers as illustrated in Figures 5 (a • CIFAR-10 [ [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF]: In order to deal with RGB images of size 32*32 pixels from the CIFAR-10 dataset, we use the same DCGAN and classifier architectures presented in Figures 5 (a),(b) and (c). We only adapt the feature maps size depending on the width (w), the height (h), and the number of channels (ch) of input RGB images.

• celebA [START_REF] Liu | Deep learning face attributes in the wild[END_REF]: In order to deal with RGB images of size 64*64 from the celebA dataset, we use a deeper convolutional GAN architecture6 presented in Figure 6.

Concerning the PU dataset initialization from a standard PN dataset, in all the experiments, except the ones in Sec. 4.3.1, we use the methodology proposed by [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF]

Input: z ∈ R 8 ∼ N (0, 1) FullyConnected (128) ELU FullyConnected (2) Sigmoid Output: 2D point x ∈ R 2 (a) Generator Input: Image x ∈ R 2
FullyConnected (128) ELU

FullyConnected (1) Sigmoid

Output: Scalar y ∈ (0, 1) (b) Discriminator specifically, we set ρ = 0.5 which is the fraction of positive labeled examples of the initial PN dataset that we unlabel such that they are included into the unlabeled dataset. Then, we set π P which is the fraction that represents these unlabeled positive examples among the unlabeled dataset. This method is interesting for testing an approach depending on π P , independently of the selected fraction 1 -ρ of positive labeled samples.

Qualitative analysis

We start by studying qualitatively whether the discriminator behaves as expected in practice.

More precisely, we need to verify whether it exclusively associates the counter-examples distribution with the label value 1, and the positive samples distribution with an intermediate label value between 0 and 1/2.

Empirical Positive Unlabeled risk analysis

We have previously demonstrated (Eq. 6 and Eq. 7) that we can reformulate the discriminator In addition, we have also computed the approximated PN risk RPN using labeled negative and labeled positive samples, instead of labeled positive and unlabeled samples, for several δ values between 0 and 1/2. We can observe that, the global minimum of the PN approximated risk RPN depending on δ, graphically corresponds to the global maximum of the density function corresponding to D output predictions for a positive set. This coincides with the equality presented in Equation 9.

To sum up, this experimentally illustrates that if D is trained with the R P U loss function, then it should predict the label value 1 exclusively for the negative samples, which is the necessary condition to guide the generator during the adversarial training to learn exclusively the counter-examples distribution.

However, this behaviour is only possible if D does not overfit labeled and unlabeled positive samples. In other words, D should be able to discriminate unlabeled positive examples from the unlabeled negative ones. Therefore, in order to generalize the proposed GAN framework to image datasets, we compare in the next section some state-of-the-art regularization techniques commonly used in deep learning models, in order to select the most appropriate one.

Impact of regularizations on the discriminator

Nowadays, Batch Normalization (BN) [START_REF] Ioffe | Batch normalization: Accelerating deep network training by reducing internal covariate shift[END_REF] is considered as a one of the most relevant regularization techniques commonly used in deep neural networks architectures. Its utility for GANs training has been highlighted by [START_REF] Radford | Unsupervised representation learning with deep convolutional generative adversarial networks[END_REF] for the DCGAN architecture in order to stabilize the adversarial training. Other variants like the Wasserstein-GAN [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF] or the Loss-Sensitive GAN [START_REF] Qi | Loss-sensitive generative adversarial networks on lipschitz densities[END_REF] this problem, we could imagine to apply BN on a unified minibatch which contains a fraction of each distribution x P , x U and x F . But the BN effect is greatly influenced by the content of the minibatch on which it is applied. Therefore, the fraction π P of positive examples included in x U will negatively impact the BN outcome.

Compatible normalization techniques: However, BN benefits in a more traditional training are not negligible. Hence, we propose to use an alternative technique in order to replace the BN role in the proposed GAN-based PU framework. Spectral Normalization (SN) [START_REF] Miyato | Spectral normalization for generative adversarial networks[END_REF] is a recent competing technique for GANs training which can stabilize the training of D against input perturbations [START_REF] Farnia | Generalizable adversarial training via spectral normalization[END_REF] by performing a weight normalization. In this way, a training manipulating multiple types of minibatch distributions preserves SN effectiveness. For these reasons, we propose to apply SN instead of BN inside our discriminative model structure.

Dropout alleviates the positive overfitting problem: As mentioned in the previous section, we can only deduce Equation 4if we consider that the positive samples distribution is the same for both labeled and unlabeled ones. In practice, this assumption holds in the case of a large dataset, such that this overfitting problem concerning the positive examples disappears. While some model averaging strategies such as bootstrap aggregating techniques have been previously combined with Support-Vector Machines (SVMs) in order to deal with PU learning [START_REF] Mordelet | A bagging SVM to learn from positive and unlabeled examples[END_REF], the dropout [START_REF] Srivastava | Dropout: a simple way to prevent neural networks from overfitting[END_REF] considerably decreases this overfitting problem. Moreover, the addition of the dropout further helps, such that the dispersion of D predictions is attenuated. This confirms that BN is not compatible with the proposed framework. We conclude that the combination SN + Dropout is the best solution to preserve the distinction between p P and p N for long trainings. This is consistent with the previously discussed arguments. Now that we have validated the discriminator ability to separate positive and negative distributions from a positive unlabeled dataset, we select the most appropriate regularization techniques SN and dropout to train adversarially the discriminator and the generator hereafter. The proposed GAN based PU model ability to generate relevant counter-examples is assessed in the next section.

Generating counter-examples

From a qualitative point of view, and contrary to the PGAN model, the proposed D-GAN paradigm generates items which only follow the counter-examples distribution for diverse data types. This is illustrated in Figure 9 for 2D point datasets and in Figure 10 for image datasets.

In Figure 9, we can observe on the top line that the generated samples exclusively follow the distribution of the counter-examples included in the unlabeled set (i.e. simultaneously not positive and unlabeled). On the bottom line, we can observe that the generator has learned the distribution of confident complements of the positive sample distribution over the uniform distribution of unlabeled sample. In addition, we can also observe that a small area around the positive sample distribution is not captured by the generator. This shows the ability of the proposed generative model to not overfit the positive sample distribution boundary. In order to enable reproducibility, a D-GAN implementation corresponding to Figure 11 is available 7 and is applied on the LS-GAN model [START_REF] Qi | Loss-sensitive generative adversarial networks on lipschitz densities[END_REF]. Our code also includes the method proposed by [START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF] to establish a PU training dataset from a fully labeled dataset with parameters ρ and π P .

We have shown in this section, from a qualitative point of the view, the discriminator ability to separate positive and negative distributions from a positive unlabeled dataset, and the generator ability to learn the counter-examples distribution on various datasets during the first stage. Next, we propose in Sec. 4.3 to quantitatively evaluate the proposed model through an empirical study 7 The code is available in supplementary material.

by focusing on the second-stage classifier C output predictions.

Divergent-GAN for Positive Unlabeled learning

In this section, we evaluate empirically our method on standard PU learning tasks in order to verify its ability to address respective issues of the state-of-the-art methods presented in Section 2.

Concerning these comparative experiments, we use the DCGAN [START_REF] Radford | Unsupervised representation learning with deep convolutional generative adversarial networks[END_REF] architecture.

Robustness to prior noise

Nowadays, SGD remains a useful deep learning regularization technique for large-scale machine learning problems [START_REF] Bottou | Large-scale machine learning with stochastic gradient descent[END_REF]. SGD provides a regularizing effect by using minibatches [START_REF] Wilson | The general inefficiency of batch training for gradient descent learning[END_REF]. However, a smaller batch size implies a higher prior noise per batch. Thus, in this section, we empirically study the proposed model robustness to prior noise using small batch sizes. We choose the prior value π P = 0.5, as the standard deviation of the real prior per minibatch is the highest in this way (see Fig. 2). This eases to observe the prior sensitivity. We reproduce the experiment exp-mnist proposed by nnPU. The PU dataset contains one thousand positive labeled examples, which are even digits. The unlabeled set is composed of the entire initial dataset, thus including also the positive labeled ones. std(π P ) is the standard deviation of the prior per minibatch. uPU and nnPU results have been obtained with the code provided by the authors of the nnPU work. (b) details the prediction scores used to plot the curves in (a).

We reproduce the Even-vs-Odd experiment proposed by [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF] as a function of the batch training size. It consists of learning to discriminate even digits 0, 2, 4, 6, 8 from odd digits 1, 3, 5, 7, 9. We use the same classifier architecture for nnPU, uPU and the D-GAN proposed approach. It is the multilayer perceptron model provided by [START_REF] Kiryo | Positive-Unlabeled Learning with Non-Negative Risk Estimator[END_REF] 8 . We only replace the bottom fully connected layer of the classifier by a convolutional layer, similarly to the generator top layer and discriminator bottom layer in the DCGAN [START_REF] Radford | Unsupervised representation learning with deep convolutional generative adversarial networks[END_REF] structure that we use. This avoids compatibility problems between the generator top convolutional layer output and the bottom classifier layer input. Unwanted artifacts in output of GANs MLP structure are slightly different from unwanted artifacts observed in output of GANs convolutional structures. It turns out that PU approaches using prior such as uPU, nnPU and GenPU make the assumption that the global training dataset prior π P is fixed and known. But in the same PU context, when the minibatch size decreases, the dispersion of π P per minibatch consequently increases. Figure 12 (a)

shows that using small batch training sizes causes critical prediction performances collapse issues for unbiased techniques like nnPU and uPU. On the other hand, our proposed approach without using prior knowledge is drastically less sensitive to this problem: While nnPU and uPU methods become ineffective in terms of test Accuracy (i.e. Accuracy score around 0.5), the D-GAN still provides a prediction test Accuracy of 0.907 for training minibatches of size 16 in D, G and C to address the Even-vs-Odd MNIST superclass classification task, as detailed in Figure 12 (b). We can conclude that the D-GAN outperforms nnPU and uPU in terms of prediction performances such that it can use minibatches to take advantage of SGD. This capacity is also interesting for incremental learning requirements where only small sample sizes may be managed at each new training iteration. Moreover, recent studies show that it is possible to continually train GANs models [START_REF] Lesort | Generative Models from the perspective of Continual Learning[END_REF].

Now that we have shown that the proposed model is robust to prior noise, we continue the comparative tests with the methods which do not use prior knowledge π P in their training cost-functions to address the PU learning task.

One versus Rest challenge

We compare in this section the D-GAN proposed approach with PGAN and RP 9 methods that we consider as baselines for the PU learning task without prior knowledge. D-GAN and in Figures 5 (a Table 2 shows average predictions for the One-vs-Rest task over MNIST and CIFAR-10 datasets.

We use the F1-Score metric for its relevance in such information retrieval and binary classification tasks as highlighted by [START_REF] Liu | Partially supervised classification of text documents[END_REF]: the F1-score measures the positive examples retrieval.

The PU datasets are simulated as proposed by PGAN such that we can evaluate the results as a function of several π P fractions. Concerning the second-stage classifier in these experiments, we have used the convolutional architecture presented in Figure 5 (c). We can observe that the D-GAN globally outperforms PGAN and RP methods in terms of test F1-Score on both MNIST and CIFAR-10 datasets. Moreover, PNGAN results highlight the GAN-based methods data augmentation advantage on complex datasets. This justifies the superior scores obtained by our method compared to RP over the CIFAR-10 dataset.

Reducing the overfitting problem: In addition, we can observe that the proposed model also outperforms PGAN on MNIST by a significant margin. This is due to the fact that, compared To sum up, in Sec. 4.2, we demonstrate that the proposed approach is effective at capturing and observing the counter-examples distribution of our class of interest from only positive and unlabeled data, without using the prior information π P . In addition, comparative experiments in Sec. 4.3 have subsequently highlighted the proposed model ability to address state-of-the-art PU learning issues such as prior sensitivity and first-stage overfitting. It turns out that addressing simultaneously those issues fosters the proposed approach to outperform PU state-of-the-art methods in terms of prediction scores without using prior on both simple and complex image datasets. • D is not sufficiently flexible to separate p P from p N , thus considering these distributions as too similar. Consequently, a unique mutual intermediate label would be predicted by D for both p P and p N . It may be interesting to explore this failure case through distinct distributions sharing common patterns, such as face attributes of celebA dataset [START_REF] Liu | Deep learning face attributes in the wild[END_REF] images;

• The labeled positive set is partially corrupted such that it contains a fraction of negative examples. For reasons of expectation linearity, p N would consequently be associated by D to an intermediate label between 0 and 1. A solution may be to extend the proposed framework to this noisy PU learning challenge by drawing on existing asymmetric noisy labeled learning techniques [START_REF] Chiaroni | Hallucinating a Cleanly Labeled Augmented Dataset from a Noisy Labeled Dataset Using GANs[END_REF]. More specifically, by modifying training loss functions, it is possible to enforce G to learn the distribution corresponding to this given intermediate label predicted by D for p N .

Conclusion

In this work, we proposed to incorporate a constrained PU risk into the GAN discriminator loss function in order to deal with PU learning. In this way, the proposed model generates relevant counter-examples from a PU dataset. It outperforms state-of-the-art PU learning methods by addressing their respective issues. Namely, it addresses the prior knowledge dependence of costsensitive PU methods and the lack of generalization of selective processes. Moreover, it reduces the PGAN first-stage overfitting, while keeping the minimalist standard GAN architecture, such that it is easily adaptable to recent GANs variants.

Nevertheless, although being competitive with the state-of-the-art, the proposed approach may present some failure cases under specific conditions, as outlined in Sec. 4.4. Future research work will aim at addressing these limitations. Furthermore, as adversarial and weakly supervised learning techniques are continuously evolving, we believe that the proposed approach stability, prediction performances, and computational cost still have the potential to be improved. For instance, recent promising GAN training approaches [START_REF] Brock | Large scale GAN training for high fidelity natural image synthesis[END_REF], not mandatorily using BN, may be suitable to extend the proposed approach for higher dimensional image datasets.

Figure 1 :

 1 Figure 1: Illustration of MNIST minibatch training examples for Positive Negative (PN) and Positive Unlabeled (PU) learning. If we consider the class 5 as our class of interest, the positive class, and the class 3 as the negative class representing counter-examples, then (a), (b) and (c) respectively represent positive, negative and unlabeled minibatch examples. While a PN training requires labeled Positive and labeled Negative examples, a PU training only requires labeled Positive and Unlabeled examples. The unlabeled minibatch in (c) is composed by a fraction π P of positive examples, and a fraction (1 -π P ) of negative examples. π P is considered as a prior knowledge which is potentially unknown in real-world applications.

  consider the entire distribution of the unlabeled examples as negative. In this way, all the negative examples, present in the unlabeled dataset, are always considered as negative. However, concerning the positive examples, it implies associating two contradictory labels to the distribution of positive examples in unknown proportions depending on the π P value. Thus, training directly a classifier with positive and unlabeled data provokes a bias in the training estimator, which is not present during a standard positive negative training.

)

  Counter-examples are correctly labeled: Decomposed in this way, the negative examples included in the unlabeled dataset are associated exclusively to the label value 1 for any π P value, such that the negative training examples are all correctly labeled. When there is no overfitting on training positive examples, then one can assume that labeled and unlabeled positive examples

  insight in the proposed D-GAN model can be expressed as follows: D addresses to G the riddle: Show me what IS unlabeled AND NOT positive. It turns out that negative examples included in the unlabeled dataset are both unlabeled and not positive. Consequently, G addresses this riddle by learning to show the negative samples distribution to D. GAN background: We first give a short recall of the original GAN discriminator. It is trained to distinguish real unlabeled samples distribution p U from generated samples distribution p G with the loss function L D GAN defined as:

)

  Towards a GAN biased discriminator loss function: The proposed approach aims at training G to learn the negative samples distribution p N instead of learning the distribution p U . This replaces the standard GAN task "Show me what is unlabeled ", by the task "Show me what is both unlabeled and not positive". We now propose to incorporate the benefits of a biased PU risk (Eq. 2) into the original GAN discriminator loss function (Eq. 11). To this end, we define the D-GAN discriminator loss function L D by adding the term E x P ∼p P [H(D(x P ), 0)] to L D GAN . Consequently, in the proposed D-GAN framework, the training discriminator loss function L D of D becomes:

  end for Classifier training (2 nd step): for number of training iterations do Sample minibatch of m positive labeled examples {x (1) P , ..., x

:

  The corresponding implementation algorithm 1 of the proposed first-stage D-GAN approach enables to adversarially train D and G to respectively minimize loss functions L D and L G . Second-stage: Positive-Generative learning. Once the D-GAN training is completed, the second step can be carried out. It consists of training a classifier C to distinguish fake generated examples x F N = G(z), which are ideally equivalent to the real negative samples, from real positive labeled samples as illustrated in Figure 3.

Figure 3 :

 3 Figure 3: Proposed GAN-based PU approach, x F N represents the generated samples which are similar to real negative samples x N , G is the generative model, D is the discriminator, C is the classifier used to perform the binary Positive-Negative (PN) classification.

  ) and (b). The second-stage classifier 5 presented in Figure 5 (c) also contains convolutional bottom layers.

Figure 4 :

 4 Figure 4: Fully connected GAN model architecture used for two dimensional points datasets. Minibatch size 64, optimizer Adam. We trained the model during 100 epochs on 2D point datasets.

Figure 5 :

 5 Figure 5: Convolutional GAN model architecture used for 28*28 grayscale MNIST and 32*32 RGB CIFAR-10 image datasets. For MNIST we set h=28, w=28, ch=1. For CIFAR-10 we set h=32, w=32, ch=3. Minibatch size: 64, optimizer: Adam, strides of 2 × 2 for the generator Deconv2D and the discriminator Conv2D layers, strides of 1 × 1 for the classifier Conv2D layers. We trained the model during 40 epochs and 1000 epochs respectively on MNIST and CIFAR-10 datasets.

Figure 6 :

 6 Figure 6: Convolutional GAN model architecture used for 64*64 RGB images of celebA dataset. Minibatch size: 64, optimizer: Adam, 2D stride of 2 × 2. We trained the model during 100 epochs on the celebA dataset.

  PU training loss function R P U (Eq. 2) into a PN training loss function R P N (Eq. 10), by replacing the two opposite labels 0 and 1 associated to positive samples distribution p P by an intermediate label value δ depending on π P . It turns out that we can verify the same relation empirically. As illustrated in Figure 7 with 2D point samples following gaussian distributions, if we train the discriminator D, with the multilayer perceptron structure presented in Figure 4 (b), using the PU loss function R P U , then its output predictions for an unlabeled batch sample are partitioned in the vicinity of two different labels. The predictions for positive examples are centered around an intermediate label value corresponding to δ. Conversely, D output predictions for the negative examples are centered around the label value 1.

  for the py(yU) curve, and value for RPU for the py(yU) curve, and value for RPU

Figure 7 :

 7 Figure7: Link between the suggested PN loss function (Eq. 9) and the distribution of the discriminator output predictions for an input PU training minibatch. For this experiment, D is a multi-layer perceptron. D has been trained to distinguish a 2D gaussian distribution from another one by using the risk R P U on a PU dataset. (a) Shows a set of 2D points considered as positive examples. (b) Shows a set of 2D points considered as unlabeled examples. Both curves in (c) and (d) have been normalized to get a better visualization. For (c), p Y (y U ) (in blue), with y U = D(x U ), represents the probability distribution of D predicted outputs for a minibatch of unlabeled samples, with π P = 0.5. RP U (in red) represents the PN risk computed depending on δ with the R P N proposed in Eq. 10 on a minibatch of positive and negative labeled samples, once D is trained with R P U risk (Eq. 2). (d) shows the same curves as in (c) but by giving in input a concatenation of an unlabeled minibatch with a positive labeled minibatch. Unlabeled positive and labeled positive samples provide a unified prediction output distribution.

  generalization technique is also a solution concerning the deep neural networks. Consequently, in the context of the proposed D-GAN training, we propose to introduce dropout in the top fully connected layer of D. We enable it during D training steps, and conversely disable it during G training steps. This improves the evaluation of generated samples which is transmitted from D to G by back-propagation. Hence, dropout alleviates the positive examples overfitting during long D-GAN trainings. This insures to exclusively generate counter-examples. We compare in Figure 8 the ability of D to distinguish positive from negative samples distributions included inside the unlabeled training dataset when D is trained on a PU image dataset without normalization and with BN and SN normalizations. We also consider the cases when they are combined with the dropout regularization. In this experiment, D is trained alone such that it is not adversarially trained with G. This enables to better observe and anticipate the adversarial behaviour of D, and consequently the behaviour of G during the adversarial training. We show in Figure 8 the histograms of D predictions concerning the unabeled training examples. As previously explained in the Section 3.2, if D associates exclusively the label 1 with the distribution p N , then we can observe a mixture of two distributions in the corresponding histograms. The one on the right corresponds to D predictions for unlabeled negative examples. The second one on the left corresponds to D predictions for unlabeled positive examples. It is shifted away from the label 1 and centered around δ. Both distributions cannot be observed with BN. In contrast, SN

Figure 8 :

 8 Figure 8: D predictions on unlabeled training examples. (a), (b), (c), (d), (e), (f) images show the evolution of the histograms of predictions during the training of D. Each horizontal line of pixels represents the histogram of predictions, between 0 and 1 along the horizontal axis, of D on the entire unlabeled training dataset. Clear hot colors represent a high density of prediction. The vertical axis indicates the training iterations from 0 to 50 epochs. Figures (g) and (h) represent the corresponding histograms of predictions after 5 and 25 epochs. Settings are with positive class 8 and negative class 3 of MNIST dataset, with π P = 0.5.

Figure 9 :

 9 Figure 9: Proposed approach applied to two different clusters of 2D points. D and G have a multilayer-perceptron structure with respectively 128 hidden units. From left to right, figures are respectively labeled positive, unlabeled with π P = 0.5, and generated samples. Figures (a), (b), (c) correspond to distributions following circle shapes. Figures (d), (e), (f) correspond to a half circle distribution for positive examples, and a uniform distribution over a defined interval for unlabeled examples.

Figure 10 :Figure 11 :

 1011 Figure 10: Counter-examples generation from Positive Unlabeled image datasets. The two left columns present input positive and unlabeled training samples x P and x U . The right column presents output generated minibatch samples x G . The first row presents results for MNIST classification task 5-vs-3 when π P = 0.5. The second row presents results for CIFAR-10 classification task Car-vs-Airplane when π P = 0.3. The third row presents results for the arbitrary celebA classification task Male-vs-Female when π P = 0.5. Visually, all generated samples follow the counter-examples distribution.

Figure 12 :

 12 Figure12: Prediction test Accuracy on MNIST for the Even-vs-Odd classification task, as a function of the minibatch size. We choose the prior value π P = 0.5, as the standard deviation of the real prior per minibatch is the highest in this way (see Fig.2). This eases to observe the prior sensitivity. We reproduce the experiment exp-mnist proposed by nnPU. The PU dataset contains one thousand positive labeled examples, which are even digits. The unlabeled set is composed of the entire initial dataset, thus including also the positive labeled ones. std(π P ) is the standard deviation of the prior per minibatch. uPU and nnPU results have been obtained with the code provided by the authors of the nnPU work. (b) details the prediction scores used to plot the curves in (a).

  ) and (b). Moreover, D-GAN, PGAN and RP approaches use the same classifier presented in Figure5 (c). We evaluate them on the challenging One-vs-Rest task which consists of trying to distinguish a class from all the other ones. This task is interesting for binary image classification applications where the labeling effort may be exclusively done on the class of interest, the positive class. Another motivation is that One-vs-Rest binary classification brings the tools for multiclass classification[START_REF] Shalev-Shwartz | Understanding machine learning: From theory to algorithms[END_REF].

Figure 13 :

 13 Figure 13: Second-stage Classifier (architecture presented in Figure 5 (c)) test Accuracy evolution as a function of the first-stage GAN epochs. 8-vs-Rest MNIST task, with ρ = 0.5 and π P = 0.5. (a) D-GAN and PGAN are trained without normalization layers. (b) D-GAN and PGAN are respectively trained with SN, SN + dropout, and BN inside the discriminator.

4. 4 .

 4 Limitations of the proposed approach We recall that the counter-examples generator G consists of learning the distribution of examples associated by D to the label 1. For this reason, the proposed approach requires that D associates a mutual intermediate label δ to both labeled and unlabeled positive examples, such that D exclusively associates counter-examples to the label 1, as formalized in Sec. 3.2. These conditions are not met if: • D lacks of generalization such that it overfits positive examples, by associating the labeled ones to the label 0 and the unlabeled ones to the label 1, instead of associating all positive examples to the mutual intermediate label δ. As empirically demonstrated in Sec. 4.2.2, an effective solution is to combine regularization techniques such as dropout with SN;

  with p G P the distribution of positive samples generated by the generator G P , p G N the distribution of the negative samples generated by the generator G N , and p U the distribution of real unlabeled samples. In practice, GenPU is an interesting PU method on simple datasets with few positive labeled samples, and it generates relevant counterexamples. However, its adversarial training of five learning models instead of two as in the original

)

  Algorithm 1 Minibatch SGD training of the D-GAN GAN training (1 st step) for number of training iterations do Sample minibatch of m noise samples {z (1) , ..., z (m) } from noise prior p z . Sample minibatch of m unlabeled examples {x

	(1) U , ..., x Sample minibatch of m positive labeled examples {x (1) (m) U } from data distribution p U . P , ..., x (m) P } from data distribution p P .
	Update D by descending its stochastic gradient:

  . More

	4 The	architecture	is	based	on	this	code:	https://github.com/hwalsuklee/
	tensorflow-generative-model-collections.

  confirmed its interest. As developed in[START_REF] Ioffe | Batch normalization: Accelerating deep network training by reducing internal covariate shift[END_REF], BN addresses issues like vanishing or exploding gradient problems, as well as the risk of getting stuck in a poor local minima, by reducing the internal covariate shift problem of the learning model. A higher learning rate can be used and it can significantly improve the training speed.

	Multiple minibatch manipulation incompatibility. BN regularizes the model, in such a

way that a training example (i.e. single instance) from a given minibatch sample is considered in conjunction with other examples of this minibatch sample. This is the consequence of estimating the mean and variance normalization parameters one time per minibatch, and then applying them

Table 2 :

 2 One-vs-Rest task with two-stage PU methods without prior, as proposed in PGAN[START_REF] Chiaroni | Learning with a generative adversarial network from a positive unlabeled dataset for image classification[END_REF]: From a fully labeled PN dataset, we firstly select a fraction ρ of positive labeled examples that we put in the simulated unlabeled set. Then, we add negative labeled examples in the latter to obtain up to a fraction π P of positive examples in this unlabeled set. Compared to nnPU simulation method, this simulation method has the advantage to simultaneously and independently control the number of positive labeled examples to keep, and the fraction π P for the unlabeled set to simulate. PNGAN expression represents GAN-based methods reference for the ideal case where π P = 0, such that we train during the first stage a GAN exclusively over all the initial cleanly labeled counter-examples set. For each dataset and depending on the fraction π P , we have tested respectively the ten One-vs-Rest task possibilities and display the corresponding average test F1-score predictions. The standard deviation is indicated in parenthesis.

We use the term example to design a single instance (i.e. item, observation) included in a sample set of data following a given distribution.

D-GAN code for RGB images of 64 × 64 pixels is provided in supplementary material.

We use the term standard to refer to the case where we have enough positive labeled examples (at least 100), such that the difficulty is mainly the ability to exploit counter-examples included in the unlabeled set.

The architecture is based on this code: https://github.com/tensorflow/tensorflow/blob/master/ tensorflow/examples/tutorials/mnist/mnist_softmax.py.

The architecture is based on this code: https://github.com/guojunq/lsgan.

The code is available at: https://github.com/kiryor/nnPUlearning.

RP code is available at: https://github.com/cgnorthcutt/rankpruning.

Code availability

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