# adversarial_partial_multilabel_learning_with_label_disambiguation__50c156a1.pdf

Adversarial Partial Multi-Label Learning with Label Disambiguation

Yan Yan, 1 Yuhong Guo2

1 School of Computer Science and Engineering, Northwestern Polytechnical University, China 2 School of Computer Science, Carleton University, Canada yanyan.nwpu@gmail.com, yuhong.guo@carleton.ca

Partial multi-label learning (PML), which tackles the problem of learning multi-label prediction models from instances with overcomplete noisy annotations, has recently started gaining attention from the research community. In this paper, we propose a novel adversarial learning model, PML-GAN, under a generalized encoder-decoder framework for partial multilabel learning. The PML-GAN model uses a disambiguation network to identify irrelevant labels and uses a multi-label prediction network to map the training instances to their disambiguated label vectors, while deploying a generative adversarial network as an inverse mapping from label vectors to data samples in the input feature space. The learning of the overall model corresponds to a minimax adversarial game, which enhances the correspondence of input features with the output labels in a bi-directional mapping. Extensive experiments are conducted on both synthetic and real-world partial multi-label datasets, while the proposed model demonstrates the state-of-the-art performance.

Introduction In partial multi-label learning (PML), each training instance is assigned multiple candidate labels which are only partially relevant; that is, some irrelevant noise labels are assigned together with the ground-truth labels. As it is typically difficult and costly to precisely annotate instances for multi-label data (Xie and Huang 2018), the task of PML naturally arises in many real-world scenarios with crowdsource annotations. In such a scenario, in order to collect the complete set of positive labels for each data instance, one can gather all labels provided by multiple annotators to form the candidate label set, which is usually overcomplete and contains additional noisy labels beyond all the true labels, leading to the PML problem. Figure 1 presents such example of an overcompletely annotated training image for object recognition, where the candidate labels provided by crowdsource annotators cover all the ground truth labels (in black color) and some irrelevant noise labels (in red color). PML is much more challenging than standard multi-label learning as the true labels are hidden among irrelevant labels and the number of true labels is unknown. The goal of PML is to learn a

Work conducted while visiting Carleton University. Copyright c 2021, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

good multi-label prediction model from such a partial label training set, and hence reduce the annotation cost. An intuitive strategy of PML is to treat all candidate labels as relevant ground truth, thus any off-the-shelf multi-label classification method can be adapted to induce an expected multi-label predictor (Zhang and Zhou 2014). This strategy, though simple, cannot work well since taking the noise labels as part of the true labels will mislead the multi-label training and induce inferior prediction models. The PML work in (Xie and Huang 2018) assumes that each candidate label has a confidence score of being a true label, and learns the confidence scores and the classifier in an alternative manner by minimizing a confidence weighted ranking loss. Although this work yields some reasonable results, the estimation of label confidence scores is error-prone, especially when noise labels dominate, which can seriously impair the classifier s performance. The recent work in (Xie and Huang 2020) proposes to perform ground-truth label recovery and noise label identification simultaneously by exploring the label correlations and the relationships between the noise labels and feature representations. Another recent work in (Fang and Zhang 2019) presents a two-stage PML method. It estimates the confidence values of the candidate labels using iterative label propagation and then chooses the highly confident candidate labels as credible labels to induce a multi-label prediction model. This work however suffers from the cumulative errors induced in propagation, which can impact the label confidence estimation and consequently impair the prediction. In this paper, we propose a novel adversarial learning model, PML-GAN, under a generalized encoder-decoder framework to tackle the partial multi-label learning problem. The PML-GAN model comprises four component networks: a disambiguation network that predicts the probability of each candidate label being an additive noise for a training instance; a prediction network that predicts the disambiguated true labels of each instance from its input features; a generation network that generates samples in the feature space given latent vectors in the label space; and a discrimination network that separates the generated samples from the real data. The prediction network and disambiguation network together form an encoder that maps data samples in the input feature space to the disambiguated label vectors, while the generation network and discrimination network form a

The Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI-21)

Figure 1: Example of an annotated image under the partial multi-label learning (PML) setting.

generative adversarial network (GAN) as an inverse decoding mapping from vectors in the multi-label space to samples in the input feature space. The learning of the overall model corresponds to a minimax adversarial game, which enhances the correspondence of input features with the output labels through the bi-directional encoder-decoder mapping mechanism, and consequently boosts multi-label prediction performance. To the best of our knowledge, this is the first work that exploits a generative adversarial model based bi-directional mapping mechanism for PML. We conduct extensive experiments on multiple multi-label datasets under partial multi-label learning setting. The empirical results show the proposed PML-GAN yields the state-of-theart PML performance.

Related Work Multi-label learning is a prevalent classification problem in many real world domains, where each instance can be assigned into multiple classes simultaneously. Many multilabel learning methods developed in the literature exploit label correlations at different degrees to produce multi-label classifiers (Zhang and Zhou 2014), including the first order methods (Zhang et al. 2018), second order methods (Li, Zhao, and Guo 2014), and high-order methods (Burkhardt and Kramer 2018). Nevertheless, standard multi-label learning methods all assume each training instance is annotated with a complete set of ground truth labels, which can be impractical in many domains, where the annotations are obtained through crowdsourcing. With the union of annotations produced by multiple noisy labelers under the crowdsourcing setting, the partial multi-label learning (PML) problem arises naturally in real world scenarios, where the set of labels assigned to each training instance not only contain the ground truth labels, but also some additional irrelevant labels. PML is more challenging than standard multi-label learning. The previous PML work in (Xie and Huang 2018) proposes two methods, PML-FP and PML-LC, to estimate the label confidence values and optimize the relevance ordering of labels by exploring the structural information in both feature and label spaces. However, due to the inherent property of alternative optimization, in these methods, the estimation error of labeling confidence values can negatively impact the coupled multi-label predictor. The work in (Sun et al. 2019) denoises the observed label matrix based on low-rank

and sparse matrix decomposition. The recent work in (Xie and Huang 2020) proposes to learn the multi-label classifier and noisy label identifier by exploiting the label correlations as well as exploring the feature-induced noise model with the observed noise-corrupted label matrix. The work in (Xu, Liu, and Geng 2020) attempts to recover the label distributions by exploiting the topological information from the feature space and label correlations from the label space, and then induces a predictive model by fitting the recovered label distributions. Another work in (Chen et al. 2020) proposes to tackle multi-view PML problem using graph-based disambiguation. In another recent work (Fang and Zhang 2019), the authors propose to address PML problem using a twostage strategy. It first estimates the label confidence value of each candidate label with iterative label propagation, and then performs multi-label learning over selected credible labels based on the confidence values by using pairwise label ranking (PARTICLE-VLS) or maximum a posteriori reasoning (PARTICLE-MAP). The work in (Wang et al. 2019) also presents a two-stage PML method that estimates the label confidence matrix in the first stage. However, in these two-stage methods, the confidence label estimation errors can consequently degrade the multi-label learning performance without correction interaction, especially when there are many noise labels. Studies on weak learning, partial label learning, and noisy label learning have some connections with PML, but address different problems. Weak label learning tackles the problem of multi-label learning with incomplete labels (Sun, Zhang, and Zhou 2010; Wei et al. 2018), where some ground truth labels are missed out from the annotations. Partial label learning (PLL) tackles multi-class classification under the setting where for each training instance there is one groundtruth label among the given candidate label set (Cour, Sapp, and Taskar 2011; Liu and Dietterich 2012; Zhang and Yu 2015; Yu and Zhang 2016; Chen, Patel, and Chellappa 2018). PLL methods cannot be directly applied on the more challenging PML problems, as under PML one has unknown numbers of ground truth labels among the candidate label set for each training instance. Noisy label learning (NLL) tackles multi-class classification problems where some groundtruth labels are replaced by noise labels (Thekumparampil et al. 2018; Lee et al. 2018; Zhang and Sabuncu 2018; Han et al. 2018; Kaneko, Ushiku, and Harada 2019; Lee et al. 2019). The off-the-shelf NLL methods cannot be directly applied on the more challenging PML problems due to the difference of problem settings. Generative adversarial networks (GANs) (Goodfellow et al. 2014), which perform minimax adversarial training over a generation network and a discrimination network, are one of the most popular generative models since its introduction. During the past years, a vast range of GAN-based adversarial learning methods have been developed to address different tasks, including semi-supervised learning (Kumar, Sattigeri, and Fletcher 2017; Lecouat et al. 2018), unsupervised learning (Jakab et al. 2018), and learning with noisy labels (Thekumparampil et al. 2018). The proposed work in this paper however is the first one that exploits generative adversarial models for partial multi-label learning.

Figure 2: The proposed PML-GAN model. It has four component networks: generator G, disambiguator e D, predictor F, and discriminator D.

Proposed Approach In this section, we present the proposed adversarial partial multi-label learning model, PML-GAN, under the following setting. Assume we have a training set S = (X, Y ) = {(xi, yi)} N i=1, where xi Rd denotes the input feature vector for the i-th instance, and yi {0, 1}L is the corresponding annotated label indicator vector. There are multiple 1 values in each yi, which indicate either the ground truth labels or the additional mis-annotated noise labels. We aim to learn a good multi-label prediction model from this partially labeled training set. The proposed PML-GAN model is illustrated in Figure 2, which comprises four component networks: disambiguation network e D, prediction network F, generation network G and discrimination network D. The four components coordinate with and enhance each other under an encoder-decoder learning framework, which forms inverse mappings between the instance vectors in the input feature space and the continuous label vectors in the output class label space, aiming to facilitate the identification of a suitable prediction function. The prediction network and the disambiguation network also naturally enhance each other from a mutual learning perspective. Below we present these model components, the learning objective and the training algorithm in details.

Prediction with Disambiguated Labels Comparing to standard multi-label learning, the main difficulty of PML is that the annotated labels {yi} in the training data contain additive noise labels. The main challenge lies in identifying the ground truth labels z i from each annotated candidate label vectors yi; that is dropping the additional 1s from each candidate label vector yi. We propose to tackle this challenge by using a disambiguation network e D : Ωx Ω (Ω denotes the corresponding domain space), which predicts the irrelevant labels for a given instance. Hence the true label indicator vector z i can be recovered as z i = Re LU(yi i) in the ideal case, where i 0 denotes the output of the disambiguation network e D(xi), and Re LU( ) = max( , 0) denotes the commonly used rectified linear unit activation function. Re LU is used here to ensure the disambiguation effort is only counted on the candidate labels. Then we can learn a prediction network F : Ωx Ωz, i.e., a multi-label classifier, to predict the dis-

ambiguated ground truth labels for each instance. Although the label indicator vectors in the training data are provided as discrete values, it is difficult for either the disambiguation network or the prediction network to directly produce discrete output values. Instead, by using a sigmoid activation function on the last layer of each network, e D(x) and F(x) can predict the probability of each class label being the additive irrelevant label and the ground truth label respectively. With the disambiguation network and prediction network, we can perform partial multi-label learning by minimizing the classification loss on training data S:

min F, e D Lc(X, Y ; F, e D) = X

(xi,yi) S ℓc(F(xi), zi) (1)

s.t. zi = Re LU(yi i), i = e D(xi), (xi, yi) S

where zi denotes the disambiguated label confidence vector with continuous values in [0, 1], which can be viewed as a relaxation of a true label indicator vector, while ℓc( , ) denotes the cross-entropy loss between the predicted probability of each label and its confidence of being a ground-truth label. We expect that the disambiguation network and the prediction network can coordinate with each other to mutually minimize this disambiguated classification loss.

Inverse Mapping with GANs The prediction network can be viewed as an encoder that maps data samples in the input feature space to the disambiguated label vectors. To enhance the label disambiguation and hence improve multi-label classification, we propose to conduct an inverse decoding mapping from label vectors ˆz [0, 1]L in the continuous label vector space to samples in the input feature space. In particular, we propose to deploy a generative adversarial network (GAN) model to transform continuous label vectors in the label space into samples in the input feature space. The GAN model comprises a generation network G and a discrimination network D. Given a label vector ˆz sampled from a prior distribution P(ˆz), which can be viewed as a low-dimensional representation vector, one can generate a sample ˆx using the generation network, x = G(ˆz). A two-class discriminator D is used to discriminate the generated samples from the real samples in S. The training of the GAN model is a minimax optimization problem over an adversarial loss function:

min G max D Ladv(G, D, S) = Exi S[log D(xi)]+ Eˆz P (ˆz)[log(1 D(G(ˆz)))]

where the discriminator D tries to maximally distinguish the generated samples G(ˆz) from the real data samples in S, and the generator G tries to generate samples that are similar to the real data as much as possible such that the discriminator cannot tell the difference. In theory, the samples generated by the adversarially trained generator G can have an identical distribution with the real data S (Goodfellow et al. 2014). To further ensure the generator G can provide an inverse mapping function from low-dimensional vectors in the label space to samples

Algorithm 1 Minibatch based stochastic gradient descent training algorithm for PML-GAN

Input: training set S; trade-off parameter β; k # of update steps for the discriminator. for # of training iterations do Sample a minibatch {(xi, yi)}m i=1 from training set S. Sample n label vectors {ˆzi}n i=1 from a prior P(ˆz). Update the network parameters of G, e D, F by descending with their stochastic gradients:

1 n Pn i=1 β[log(1 D(G(ˆzi)))] +

1 m Pm i=1 h ℓc F(xi), Re LU(yi e D(xi)) +

ℓg G(Re LU(yi e D(xi))), xi i

for r=1:k do Sample n label vectors {ˆzi}n i=1 from a prior P(ˆz). Update the parameters of the discrimination network by ascending with its stochastic gradient:

i=1 log D(xi) + 1

i=1 log(1 D(G(ˆz))) i

end for end for

in the feature space, we further propose to decode the disambiguated training label vectors into the training samples S with G by deploying a generation loss:

Lg(G, S) = X

(xi,yi) S ℓg(G(zi), xi), (3)

with zi = Re LU(yi e D(xi)), (4)

where ℓg( , ) measures the generation loss on each training instance, which can be a least squares function. This generation loss can enhance the label disambiguation and improve multi-label learning.

Learning with PML-GANs By integrating the classification loss in Eq.(1), the adversarial loss in Eq.(2), and the generation loss in Eq.(3) together, we obtain the following minimax optimization problem for the proposed PML-GAN model:

min G, e D,F max D E(xi,yi) S ℓc(F(xi), zi) + ℓg(G(zi), xi) +

β Exi S[log D(xi)] + Eˆz P (ˆz)[log(1 D(G(ˆz)))] (5)

s.t. zi = Re LU(yi e D(xi)), (xi, yi) S

where β is a trade-off hyperparameter that controls the relative importance of the adversarial loss; the objective function can be denoted as L(G, e D, F, D). The learning of the overall model corresponds to a minimax adversarial game, which enhances the bi-directional mapping between the feature and label vector spaces, and consequently boosts multilabel prediction performance.

Figure 3: Dependence graph of PML-GAN.

We perform training using a minibatch based stochastic gradient descent algorithm. In each iteration of the training, the minimization over G, e D, F and the maximization over D are conducted alternatively. The overall training algorithm is presented in Algorithm 1.

Theoretical Results In the proposed PML-GAN model, given the generator G, the discriminator D is conditionally independent from the predictor F and the disambiguator e D. Between G, F and e D, G and F are conditionally independent from each other given e D. Their independence relationship can be illustrated using the undirected dependence graph in Figure 3. Based on these conditional independence relationships, we have the following optimality results.

Proposition 1. For any G, e D, and F, the optimal discriminator D is given by

D G, e D,F (x) = D G(x) = p S(x)/ p S(x) + pg(x) (6)

where p S( ) and pg( ) denote the probability distributions of real and generated data respectively. Proof. Due to the conditional independence relationship between D and {F, e D}, the optimal discriminator D only depends on the generator G. Given fixed G, the optimal discriminator can be derived in the same way as in the standard GANs (Goodfellow et al. 2014, Proposition 1). Proposition 2. Assume the model has sufficient capacity. Let C(G, e D, F) = max D L(G, e D, F, D). Given fixed e D, the minimum of C(G, e D, F) is lower bounded by E(xi,yi) S H Re LU(yi e D(xi)) β log 4, which can be achieved when F(xi) = Re LU(yi e D(xi)), G(F(xi)) = xi, and pg = p S. Here H( ) denotes an entropy function. This proposition suggests that F and G should be inverse mapping functions for each other in the ideal optimal case.

Experiments We conducted extensive experiments to validate the empirical performance of the proposed PML-GAN model. In this section, we report our experimental setting and results.

Experimental Setting Datasets. We conducted experiments on twelve multilabel classification datasets. Three of them have existing partial multi-label learning settings (mirflickr, music style and music emotion (Fang and Zhang 2019)). For each of the

Dataset #Inst. #Feats #Classes avg.#CLs Dataset #Inst. #Feats #Classes avg.#CLs music emotion 6833 98 11 5.29 music style 6839 98 10 6.04 mirflickr 10433 100 7 3.35 image 2000 294 5 2,3,4 scene 2407 294 6 3,4,5 yeast 2417 103 14 9,10,11,12 enron 1702 1001 15 8,9,10,11,12,13 corel5k 5000 499 15 8,9,10,11,12,13 eurlex dc 8636 100 15 8,9,10,11,12,13 eurlex sm 12679 100 15 8,9,10,11,12,13 delicious 14000 500 15 8,9,10,11,12,13 tmc2007 28596 49060 15 8,9,10,11,12,13

Table 1: Information of the experimental data sets. The number of instances, features and classes are recorded. The avg.#CLs column lists the average number of candidate labels in each PML set.

other nine datasets (Zhang and Zhou 2014), we transformed it into a PML dataset by randomly adding irrelevant labels into the candidate label set of each training instance. By adding different numbers of irrelevant labels, for each dataset we can create multiple PML variants with different average numbers of candidate labels. Following the setting of (Xie and Huang 2018), we also filtered out the rare labels and kept at most 15 classes in each dataset. The detailed characteristics of the processed datasets are given in Table 1.

Comparison Methods. We compared our proposed method with five state-of-the-art PML methods and one baseline multi-label learning method. We adopted a simple but effective neural network based multi-label learning method, ML-RBF (Zhang 2009), as a baseline method, which performs PML by treating all the candidate labels as ground-truth labels. Then we used five recently developed PML methods for comparison, including the PML-LC and PML-FP methods from (Xie and Huang 2018), the PARTICLE-VLS and PARTICLE-MAP methods from (Fang and Zhang 2019), and the PML-NI from (Xie and Huang 2020).

Implementation. The proposed PML-GAN model has four component networks, all of which are designed as multilayer perceptrons with Leaky Re Lu activation for the middle layers. The disambiguator, predictor, and discriminator are all three-layer networks with sigmoid activation in the output layer, while the generator is a five layer network with Tanh activation in the output layer. We used the Adam (Kingma and Ba 2014) optimizer in our implementation. The mini-batch size, m, is set to 64. The hyperparameters k (the number of steps for discriminator update) and n (the number of label vectors sampled) in Algorithm 1 are set to 1 and 210 respectively. The hyperparameter β is chosen from {0.001, 0.01, 0.1, 1, 10} based on the classification loss value Lc in the training objective function; that is, the β value that leads to the smallest training Lc loss will be chosen. This is a heuristic parameter selection method we create specifically for PML.

Comparison Results

We compared the proposed PML-GAN method with the six comparison methods on the twelve datasets. For each dataset, we randomly select 80% of the data for training and use the remaining 20% for testing. We repeat each experiment 10 times with different random partitions of the

datasets. The comparison test results in terms of four commonly used evaluation metrics (Hamming loss, ranking loss, one error and average precision) (Zhang and Zhou 2014) are reported in Table 2. The results are the means and standard deviations over the 10 repeated runs. We can see that the methods specially developed for PML problems all outperform the baseline multi-label neural network classifier, ML-RBF, in most cases. But it is difficult to beat the baseline competitor on all the datasets with different evaluation metrics. Among the total 48 cases over 12 datasets and 4 evaluation metrics, PML-NI, PARTICLE-VLS, PARTICLEMAP, PML-LC and PML-FP outperform ML-RBF in 39, 42, 45, 35 and 40 cases respectively. By contrast, the proposed PML-GAN method outperforms ML-RBF consistently across all the 48 cases with remarkable performance gains. Even comparing with all the other five PML methods, PML-GAN produced the best results in 40 out of the total 48 cases. Moreover, the performance gains yield by PML-GAN over all the other methods are quite notable in many cases. For example, in terms of average precision, PML-GAN outperforms the best alternative comparison method by 4.6%, 4.8%, and 3.4% on eurlex dc, scene and image respectively. These results clearly demonstrate the effectiveness of the proposed PML-GAN model. The results reported in Table 2 and discussed above are produced on each dataset with a selected average number of candidate labels. As shown in Table 1, we have multiple PML variants with different numbers of candidate labels for nine of the datasets in the list. In total this provides us 49 PML datasets. We hence also conducted experiments on each of these 49 variant datasets, by comparing the proposed PML-GAN with each of the other methods in terms of the 4 evaluation metrics. In total there are 196 comparison cases for each pair of methods. For the comparison of PML-GAN vs other method in each case, we conducted pairwise t-test at significance level of p < 0.05. The win/tie/loss counts in all cases are reported in Table 3. We can see that overall the proposed PML-GAN significantly outperforms PMLNI, PARTICLE-VLS, PARTICLE-MAP, PML-LC, PMLFP, and ML-RBF in 80.6%, 75%, 77%, 81.1%, 82.6%, and 90.8% of the cases respectively. This again validates the efficacy of the proposed method for PML.

Ablation Study

As shown in Eq.(5), the objective of PML-GAN contains three parts: classification loss, generation loss and adversarial loss. The generation loss and adversarial loss are in-

Data set avg.#C.Ls PML-GAN PML-NI PARTICLE -VLS PARTICLE -MAP PML-LC PML-FP ML-RBF

Hamming loss (the smaller, the better) music emotion 5.29 .200 .004 .212 .003 .212 .004 .215 .004 .236 .003 .245 .004 .779 .004 music style 6.04 .115 .002 .116 .004 .121 .003 .175 .005 .126 .004 .126 .004 .856 .001 mirflickr 3.35 .170 .003 .167 .003 .178 .035 .189 .081 .202 .057 .202 .057 .748 .002 image 3 .202 .006 .210 .009 .234 .065 .269 .096 .264 .072 .267 .063 .754 .003 scene 4 .132 .007 .175 .003 .184 .037 .174 .035 .178 .029 .187 .038 .820 .001 yeast

.213 .008 .232 .004 .226 .004 .220 .008 .226 .008 .219 .009 .694 .003 enron .186 .003 .235 .005 .197 .032 .190 .036 .206 .027 .206 .027 .813 .004 corel5k .118 .001 .135 .003 .189 .012 .269 .027 .151 .008 .152 .008 .886 .001 eurlex dc .044 .001 .067 .001 .061 .001 .064 .004 .096 .001 .071 .001 .933 .001 eurlex sm .083 .002 .091 .008 .067 .001 .076 .002 .119 .006 .122 .002 .885 .001 delicious .249 .002 .260 .002 .260 .003 .290 .005 .290 .004 .290 .004 .712 .002 tmc2007 .084 .001 .089 .001 .090 .003 .110 .003 .103 .002 .103 .002 .857 .001 Ranking loss (the smaller, the better) music emotion 5.29 .242 .007 .251 .007 .263 .008 .240 .007 .267 .009 .275 .010 .365 .010 music style 6.04 .145 .006 .140 .009 .163 .007 .147 .005 .215 .005 .150 .005 .242 .006 mirflickr 3.35 .124 .014 .124 .004 .227 .029 .129 .108 .160 .029 .143 .028 .195 .015 image 3 .191 .010 .217 .008 .239 .077 .250 .085 .291 .134 .217 .120 .251 .019 scene 4 .123 .009 .213 .010 .177 .049 .167 .060 .192 .032 .238 .056 .188 .014 yeast

.194 .011 .222 .005 .203 .007 .208 .012 .219 .011 .203 .008 .270 .007 enron .182 .012 .236 .013 .240 .078 .182 .029 .239 .048 .239 .047 .244 .010 corel5k .295 .011 .392 .009 .367 .032 .311 .008 .366 .035 .398 .025 .404 .082 eurlex dc .067 .005 .126 .010 .150 .004 .085 .004 .137 .008 .131 .001 .135 .003 eurlex sm .122 .007 .246 .037 .129 .007 .127 .009 .282 .007 .182 .008 .183 .003 delicious .258 .004 .287 .002 .314 .005 .276 .004 .277 .005 .276 .005 .316 .003 tmc2007 .070 .001 .077 .001 .096 .008 .095 .007 .082 .005 .080 .005 .153 .002 One error (the smaller, the better) music emotion 5.29 .450 .028 .500 .014 .473 .016 .475 .018 .556 .028 .540 .027 .587 .019 music style 6.04 .347 .016 .355 .016 .374 .005 .399 .019 .409 .013 .408 .013 .385 .006 mirflickr 3.35 .236 .059 .307 .020 .165 .150 .229 .306 .300 .129 .298 .121 .338 .002 image 3 .342 .014 .401 .028 .369 .134 .387 .147 .542 .191 .549 .174 .398 .034 scene 4 .321 .022 .413 .018 .340 .078 .349 .082 .497 .089 .523 .118 .428 .022 yeast

.245 .017 .290 .009 .248 .019 .252 .018 .257 .017 .263 .027 .408 .023 enron .307 .035 .498 .024 .411 .101 .351 .040 .494 .039 .498 .038 .495 .019 corel5k .685 .015 .792 .016 .835 .025 .721 .035 .784 .029 .787 .024 .809 .015 eurlex dc .307 .013 .521 .015 .390 .016 .374 .014 .707 .014 .518 .011 .342 .008 eurlex sm .339 .013 .516 .019 .350 .014 .360 .015 .506 .031 .542 .018 .340 .005 delicious .368 .009 .415 .007 .366 .015 .414 .018 .401 .015 .399 .013 .450 .009 tmc2007 .202 .007 .214 .008 .194 .029 .267 .018 .235 .019 .236 .019 .388 .006 Average precision (the larger, the better) music emotion 5.29 .621 .013 .598 .007 .605 .012 .612 .009 .574 .013 .568 .014 .506 .012 music style 6.04 .732 .010 .729 .012 .715 .009 .709 .009 .702 .008 .703 .008 .646 .010 mirflickr 3.35 .777 .027 .787 .008 .678 .027 .791 .202 .736 .043 .758 .039 .676 .048 image 3 .775 .010 .740 .013 .741 .090 .729 .086 .644 .131 .725 .119 .723 .021 scene 4 .801 .012 .688 .011 .750 .074 .753 .064 .689 .047 .710 .079 .728 .015 yeast

.732 .014 .701 .004 .724 .010 .714 .010 .721 .012 .728 .010 .634 .008 enron .665 .019 .580 .009 .595 .099 .661 .047 .556 .041 .575 .041 .560 .009 corel5k .441 .012 .345 .010 .377 .025 .415 .008 .345 .027 .384 .021 .334 .008 eurlex dc .797 .009 .704 .022 .692 .013 .751 .008 .693 .019 .716 .014 .710 .000 eurlex sm .720 .009 .558 .023 .705 .009 .683 .011 .438 .016 .679 .011 .656 .000 delicious .630 .006 .597 .003 .596 .007 .601 .008 .607 .007 .608 .006 .576 .004 tmc2007 .821 .002 .807 .003 .799 .013 .759 .013 .793 .012 .794 .012 .662 .003

Table 2: Comparison results of in terms of Hamming loss, ranking loss, one error and average precision. The best results are presented in bold font. The average number of candidate labels is presented under the column avg.#C.Ls .

Evaluation Metric PML-GAN vs PML-NI PARTICLE-VLS PARTICLE-MAP PML-LC PML-FP ML-RBF Hamming loss 36/11/2 39/6/4 38/9/2 40/7/2 40/3/6 45/4/0 Ranking loss 38/11/0 38/9/2 38/8/3 38/8/3 40/5/4 44/3/2 One error 44/5/0 33/12/4 39/7/3 41/8/0 40/9/0 43/6/0 Average precision 40/8/1 37/8/4 36/10/3 40/6/3 42/4/3 46/3/0 Total 158/35/3 147/35/14 151/34/11 159/29/8 162/21/13 178/16/2

Table 3: Win/tie/loss counts of pairwise t-test (with p < 0.05 ) between PML-GAN and each comparison method over all dataset variants with different numbers of candidate labels.

Data set PML-GAN CLS-GEN CLS-GAN CLS-ML PML-GAN CLS-GEN CLS-GAN CLS-ML Hamming loss (the smaller, the better) Ranking loss (the smaller, the better) music emotion .200 .004 .203 .003 .202 .004 .207 .004 .242 .007 .249 .007 .244 .007 .250 .010 music style .115 .002 .118 .003 .117 .004 .121 .001 .145 .006 .147 .007 .149 .007 .155 .004 mirflickr .170 .003 .173 .004 .174 .005 .177 .004 .124 .014 .131 .019 .133 .021 .136 .019 image .202 .006 .206 .005 .204 .008 .220 .006 .191 .010 .195 .010 .196 .016 .201 .010 scene .132 .007 .140 .010 .138 .005 .148 .008 .123 .009 .130 .007 .137 .010 .140 .020 yeast .213 .008 .219 .007 .216 .003 .222 .006 .194 .011 .199 .008 .195 .005 .203 .006 enron .186 .003 .273 .015 .277 .014 .281 .012 .182 .012 .185 .009 .188 .009 .189 .009 corel5k .118 .001 .118 .001 .120 .003 .122 .003 .295 .011 .304 .013 .299 .007 .306 .015 eurlex dc .044 .001 .050 .001 .047 .001 .054 .001 .067 .005 .068 .004 .068 .005 .071 .008 eurlex sm .083 .002 .085 .001 .086 .001 .088 .002 .122 .007 .125 .003 .125 .005 .127 .004 delicious .249 .002 .251 .003 .252 .001 .255 .002 .258 .004 .261 .007 .259 .005 .269 .003 tmc2007 .084 .001 .086 .001 .086 .002 .091 .001 .070 .001 .073 .002 .072 .001 .075 .003 Average precision (the larger, the better) One error (the smaller, the better) music emotion .621 .013 .608 .013 .612 .014 .605 .013 .450 .028 .465 .019 .469 .029 .478 .028 music style .732 .010 .725 .012 .726 .011 .720 .004 .347 .016 .359 .021 .356 .018 .367 .007 mirflickr .777 .027 .765 .037 .761 .036 .754 .036 .336 .059 .384 .084 .399 .067 .417 .086 image .775 .010 .766 .009 .766 .018 .758 .011 .342 .014 .359 .016 .359 .029 .364 .021 scene .801 .012 .793 .009 .783 .013 .780 .021 .321 .022 .330 .017 .349 .020 .350 .027 yeast .732 .014 .723 .012 .730 .008 .715 .009 .245 .017 .262 .025 .247 .012 .264 .018 enron .665 .019 .658 .017 .648 .028 .645 .021 .307 .035 .328 .024 .347 .049 .350 .035 corel5k .441 .012 .431 .015 .439 .009 .428 .017 .685 .015 .705 .023 .690 .018 .707 .020 eurlex dc .797 .009 .790 .009 .792 .009 .779 .015 .307 .013 .310 .015 .312 .015 .315 .021 eurlex sm .720 .009 .713 .005 .713 .008 .711 .005 .339 .013 .351 .009 .350 .011 .356 .010 delicious .630 .006 .627 .009 .626 .005 .620 .002 .369 .009 .375 .015 .381 .012 .386 .003 tmc2007 .821 .002 .817 .003 .818 .004 .815 .004 .202 .007 .205 .004 .206 .007 .210 .007

Table 4: Comparison results of PML-GAN and its three ablation variants.

tegrated to assist the predictor training. To investigate and validate the contribution of the generation loss and adversarial loss, we conducted an ablation study by comparing PML-GAN with three of its ablation variants: (1) CLS-GEN, which drops the adversarial loss; (2) CLS-GAN, which drops the generation loss; and (3) CLS-ML, which only uses the classification loss by dropping both the adversarial loss and generation loss. The comparison results are reported in Table 4. We can see that comparing to the full model, all three variants produced inferior results in general. Among the three variants, both CLS-GEN and CLS-GAN outperform CLS-ML in most cases. This suggests that both the generation loss and the adversarial loss are critical terms for the proposed model. Moreover, even the baseline variant CLSML still produces some reasonable PML results. This sug-

gests the integration of our proposed prediction network and disambiguation network is also effective.

Conclusion In this paper, we proposed a novel adversarial model for PML. The proposed model comprises four component networks, which form an encoder-decoder framework to improve noise label disambiguation and boost multi-label learning performance. The training problem forms a minimax adversarial optimization, which is solved using an alternative min-max procedure with minibatch stochastic gradient descent. We conducted extensive experiments on multiple PML datasets. The results show that the proposed model outperforms all the comparison methods and achieves the state-of-the-art PML performance.

Acknowledgments This research was supported in part by the NSERC discovery grant, the Canada Research Chairs program, and the China Scholarship Council.

A Appendix A.1 Proof for Proposition 2 Proof. This proposition suggests that F and G should be inverse mapping functions for each other in the ideal optimal case. Based on the solution for the optimal discriminator D in Proposition 1, we have: Ladv(G, D , S)

= Ex p S[log D G(x)] + Eˆz P (ˆz)[log(1 D G(G(ˆz)))]

= Ex p S[log D G(x)] + Ex pg[log(1 D G(x))]

log p S(x) p S(x) + pg(x)

log pg(x) p S(x) + pg(x)

Hence, C(G, e D, F) = max D L(G, e D, F, D)

E(xi,yi) S ℓc(F(xi), Re LU(yi e D(xi))) +

E(xi,yi) S ℓg(G(Re LU(yi e D(xi))), xi) +

β Ex p S[log p S(x) p S(x)+pg(x)] + Ex pg[log pg(x) p S(x)+pg(x)]

Note given fixed e D, F is conditionally independent from G and D. Hence the minimization of C(G, e D, F) over F can be independently conducted from the minimization over G. Let zi = Re LU(yi e D(xi)). With the cross-entropy loss function ℓc( , ), we have:

min F C(G, e D, F)

min F E(xi,yi) S ℓc F(xi), Re LU(yi e D(xi))

min F E(xi,yi) S h z i log F(xi) (1 zi) log(1 F(xi)) i

min F E(xi,yi) S H(zi) + KL(zi F(xi))

E(xi,yi) S H(zi) (7) where H( ) denotes the entropy over a binomial distribution vector and KL( ) denotes the KL-divergence between two sets of binomial distributions. Assume sufficient capacity for F, in the ideal case the minimum can be reached when the predictor obtains the same distributions as the zi; that is F (xi) = zi = Re LU(yi e D(xi)), (xi, yi) S (8) Next let s consider the minimization problem over G. Note G is involved in both the generation loss and adversarial loss. If we could find solutions that lead to minimals in both losses separately, we can guarantee a minimal in the united loss. Based on (Goodfellow et al. 2014, Theorem 1), the adversarial loss part in C(G, e D, F) can be rewritten as βLadv

= β Ex p S[log p S(x) p S(x) + pg(x)] + Ex pg[log pg(x) p S(x) + pg(x)]

= β KL(p S, p S + pg

2 ) log 2 + KL(pg, p S + pg

= β KL(p S, p S + pg

2 ) + KL(pg, p S + pg

β log 4 (9)

where the minimal can be achieved when p S = pg which leads to zero KL-divergence values. The generation loss part (with least squares loss function) in C(G, e D, F) can be rewritten as

E(xi,yi) S ℓg(G(Re LU(yi e D(xi))), xi)

= E(xi,yi) S G(Re LU(yi e D(xi))) xi 2

where the minimal 0 can only be achieved when

G(Re LU(yi e D(xi))) = xi, (xi, yi) S (11)

The optimal condition above can be satisfied simultaneously together with the condition pg = p S. Together with the condition in (8), these conditions lead to a lower bound of C(G, e D, F) and the proposition is proved.

A.2 Network Architecture Information for the PML-GAN Model

The proposed PML-GAN model has four component networks, and all of them are designed as multilayer perceptrons with Leaky Re Lu activation function for the middle layers. The disambiguator, predictor, and discriminator are three-layer networks with sigmoid activation in the output layer, while the generator is a five layer network with Tanh activation in the output layer. Batch normalization is also deployed in the middle three layers of the generation network. The detailed input and output dimension information for each layer of the networks is given in Table 5.

Generator G Input Output BN Act. ˆz 512 LRe LU 512 1024 LRe LU 1024 256 LRe LU 256 128 LRe LU 128 dim Tanh Discriminator D data 512 LRe LU 512 256 LRe LU 256 1 Sigmoid Disambiguator e D x 512 LRe LU 512 256 LRe LU 256 class num Sigmoid Predictor F x 512 LRe LU 512 256 LRe LU 256 class num Sigmoid

Table 5: The network architecture of PML-GAN. BN: Batch normalization; LRe LU: Leaky rectified linear unit; Act.: Activation function; dim: Feature dimension of training samples x; class num: the number of class labels.

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