Heterogeneous Domain Adaptation with Positive and Unlabeled Data

The common idea of existing HUDA techniques is to learn feature transformers to transform the source and target data into a homogeneous feature space to minimize the distribution distance of the two domains on that feature space. Due to this difficulty of dealing with two heterogeneous feature spaces, most HUDA techniques require some supplementary information to bridge the two domains. For example, KCCA [5] and HHTL [6] require paired (common) samples between the two domains, which are often difficult to obtain in real cases. DSFT [7] and OT(optimal transport)-based method [8] require common features between the two domains as in this paper¹¹1While DSFT and this paper consider the situation where the source and target each have domain-specific features, OT-based method considers the situation where only the target has domain-specific features.. CL-SCL [14] and FSR [15] require semantically equivalent word pairs and meta-features, respectively. There are some studies that do not explicitly require supplementary information, such as GLG [10] and SFER [9] although they implicitly assume that the two domains have sufficiently similar features. All of the methods listed here require both positive and negative examples in the source domain. A comparison of this paper with these methods is summarized in Table I.

PU learning trains a binary classification from positive and unlabeled examples without labeled negative examples [11]. Early PU learning methods adopt the two-step technique that first identifies reliable negative examples and then conducts supervised learning [16, 17]. Some studies propose PU learning methods by considering unlabeled data as negative data having label noise [18, 19]. Another promising branch of PU learning employs the framework of cost-sensitive learning, such as uPU [20] and nnPU [21]. Recently, Hu et al. [13] proposed the state-of-the-art PU learning method, Predictive Adversarial Networks (PAN) based on the revised architecture of GAN. Besides performance, PAN has the advantage of not requiring class prior probability.

As in our study, there are several recent studies that address domain adaptation in the context of PU learning. For example, Sonntag et al. [22] considers the domain adaptation scenario where the source domain has labeled data of all classes, and the target domain has unlabeled data and a few positive labeled data in the homogeneous feature space. The setting differs from ours which is more difficult, in that the target domain has positive labeled data and the source domain has both positive and negative data, and in the homogeneity of the feature space. The scenario where both target and source domains have unlabeled and positive labeled data is also treated for a link prediction task in the homogeneous setting [23] and heterogeneous setting [24], respectively. Their settings are also different from ours where the source domain has only positive labeled data and the target domain has only unlabeled data. In addition, there are studies that address open set domain adaptation (OSDA) by considering it as PU learning [25, 26]. OSDA performs domain adaptation while also rejects target classes that are not present in the source domain as unknown. In their settings, the source domain has a positive (known) labeled data and the target domain has unlabeled data including positive and negative (unknown) data. However, they consider the homogeneous feature space unlike our study. Thus far, no study has examined the PU learning task in the HUDA setting.

We define a problem addressed in this paper, PU-HUDA. We suppose that the source domain and target domain share some common features as in DSFT [7] (although our experimental results show that our proposed method works even when the number of common features is sufficiently small). Let $X_{s}=[S_{c};S_{s}]\in\mathbb{R}^{n_{s}\times c}\times\mathbb{R}^{n_{s}\times s}$ and $X_{t}=[T_{c};T_{t}]\in\mathbb{R}^{n_{t}\times c}\times\mathbb{R}^{n_{t}\times t}$ be the source data matrix with $n_{s}$ data samples and target data matrix with $n_{t}$ data samples, respectively. Here, $S_{c}$ and $T_{c}$ are the data matrices in the common feature space with $c$ dimension in the source and target domain, respectively. $S_{s}$ and $T_{t}$ are the data matrices in the source and target specific feature spaces with $s$ and $t$ dimensions, respectively. We suppose that all data in the source domain is labeled as positive while all data in the target domain is unlabeled (including both positive and negative examples). We remark that this setting is different from the conventional HUDA setting where the source domain has both positive and negative labeled data. The main tasks of the conventional HUDA and PU-HUDA are the same, i.e., to predict the binary labels of the target data. These settings are visualized in Fig. 1.

The simplest baseline for solving PU-HUDA is to use only common features $S_{c}$ and $T_{c}$. Existing PU learning methods can be directly applied to the common features. However, since existing PU learning methods cannot use domain-specific features, this baseline method provides only limited performance. In solving PU-HUDA, PU learning that can take advantage of domain specific features in the heterogeneous setting is essential.

One naive solution to tackle PU-HUDA is to directly combine the existing HUDA and PU learning methods. In this naive combination approach, we first train feature transformers that transform $X_{s}$ and $X_{t}$ into $\hat{X}_{s}$ and $\hat{X}_{t}$ in a new homogeneous feature space by using HUDA methods, such as DSFT, GLG, and SFER. Then, we can apply an existing PU learning method to the unlabeled data $\hat{X}_{t}$ and the positive labeled data $\hat{X}_{s}$.

However, this naive combination approach can be ineffective due to the following the reason. HUDA methods learn feature transformers that reduce the distribution divergence between $\hat{X}_{s}$ and $\hat{X}_{t}$ under some metric, such as MMD. However, recall that $\hat{X}_{s}$ includes only positive data while $\hat{X}_{t}$ includes both positive and negative data. Due to this gap, the mapping that reduces the distribution divergence between $\hat{X}_{s}$ and $\hat{X}_{t}$ does not necessarily generate a feature space that gives high discriminative performance on the target data, as described in Fig. 2 (top). Indeed, we experimentally confirmed that this naive combination of HUDA and PU learning did not work in Sec. V. In the next section, we propose a new method to overcome this difficulty.

Specifically, we regard PU-HUDA as a task to identify likely positive examples from the target data and reduce the distribution divergence between the whole source examples and the likely positive target examples. To solve this task, we exploit adversarial training and thus introduce a discriminator. Hence, our proposed method mainly consists of the following three models as described in Fig. 3 (Details of Fig. 3 are explained in Sec. IV-C).

(1) Feature Transformer $F:\mathbb{R}^{c+t}\rightarrow\mathbb{R}^{s}$ transforms the target feature space to that of the source. The target data $X_{t}$ is transformed into the data $\hat{T}_{s}=F(X_{t})$ in the source-specific feature space via $F$. Thus, we get $\hat{X}_{t}=[T_{c};\hat{T}_{s}]$ and $X_{s}$ in the same feature space (see Fig. 3). Here, we employed an asymmetric transformation [4] (i.e., only target data is transformed) because it is difficult to train feature transformers for each of the source and target data, which have different label distributions and feature spaces, and therefore it is easy to fall into non-optimal solutions.

(2) Classifier $C:\mathbb{R}^{c+s}\rightarrow\mathbb{R}^{2}$ identifies positive examples in the transformed unlabeled target data (classifies positive and negative target examples). We represent the output confidence scores of $C$ as $C(\cdot)=(C(\cdot)_{0},C(\cdot)_{1})$, where $C(\cdot)_{1}$ and $C(\cdot)_{0}$ represent the probabilities that the input samples are predicted to be positive and negative, respectively.

(3) Discriminator $D:\mathbb{R}^{c+s}\rightarrow\mathbb{R}^{2}$ determines whether the likely positive examples are from the target or source domain (negative examples are predicted as target). The same notation as $C$ applies to $D$ although positive and negative mean source and target for $D$, respectively.

Since $D$ is trained to discriminate the source and target examples, $F$ and $C$ are trained as follows:

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  $F$ is trained so that the source data and the transformed target data predicted by $C$ to be positive cannot be distinguished by $D$.

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  $C$ is trained to identify likely positive examples from the target data transformed by $F$ so that they cannot be distinguished from the source data that are all positive by $D$. Therefore, $C$ can classify positive and negative examples.

$C$ and $F$ provide feedback to each other. That is, if $C$ can predict the correct positive examples, $F$ can reduce the distribution divergence between the truly positive target data and the source data, and conversely if $F$ can transform the target positive data into the data close to the source data, $C$ can predict the likely positive examples more indistinguishable from the source data. Therefore, by alternately training them with $D$, we can obtain good $C$ and $F$, which are used for binary classification inference on the target data.

We next consider the optimization ways to train $C$, $D$, and $F$.

We first consider focusing on only positive target data while ignoring negative target data and reducing the distribution divergence between the target positive data and the source data in the transformed feature space.

The standard objective function used in domain adaptation methods using adversarial learning [27, 28, 29] to optimize $D$ and $F$ is the following cross-entropy loss:

where $\mathcal{P}_{s}$ and $\mathcal{P}_{t}$ are data distributions of the source and target data, $\hat{x}_{t}=[t_{c};F(x_{t})]$, and $x_{t}=[t_{c};t_{t}]$. By maximizing Eq. IV-B, $D$ is trained to recognize source and target domains, and by minimizing Eq. IV-B, $F$ is trained to fool $D$, i.e., to reduce the divergence between the source and target data.

Since optimizing only Eq. IV-B causes the label-gap problem described in Sec.III-B, we incorporate the outputs of $C$ into the second term to focus on the positive target data and train $C$ simultaneously. That is, we optimize the following objective function to train $C$, $F$, and $D$:

By weighting the second term with $C(\cdot)_{1}$, the training of $D$ and $F$ focuses on the target data that $C$ predicts are likely positive. The denominator of the second term prevents $C$ from outputting 0 for any input. Therefore, by the weighted second term, $C$, $F$, and $D$ are expected to be trained as described in Sec. IV-A. We call this naive optimization approach weighted adversarial domain adaptation (WADA). At first glance, WADA seems to work. However, WADA has the following two problems.

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  Since WADA places no restrictions on $C$ incorrectly predicting positive examples to be negative, $C$ will predict as positive only those samples that are sufficiently likely to be positive.

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  Since the second term in Eq. IV-B tends to be smaller for negative examples, $F$ will overfit on positive examples, failing to transform negative target data into the features that $C$ can easily classify.

To address the problems in WADA, we propose predictive adversarial domain adaptation (PADA). PADA treats positive and negative examples equally by using an objective function based on KL divergence for confidence scores inspired by PAN [13], thus solving the two problems of WADA. The objective function of PADA is as follows:

where $\hat{x}_{t}^{i}=[t_{c}^{i};F(x_{t}^{i})]$ and $x_{t}^{i}=[t_{c}^{i};t_{t}^{i}]$. Each $x_{*}^{i}$ is a data sample from $X_{*}$. $\tilde{C}(\cdot)$ denotes the opposite output of $C(\cdot)$, i.e., $\tilde{C}(\cdot)=(C(\cdot)_{1},C(\cdot)_{0})$.

The first and second terms: cross-entropy for $D$, corresponding to the empirical version of Eq. IV-B. Optimizing these terms has the following effects on $D$ and $F$.

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  $D$ can identify the source and target data to some extent. Here, all the target data is regarded as negative for $D$ to give $D$ the ability to recognize the target data.

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  Since $F$ is trained adversarial to $D$, $F$ can transform the whole target data into the data close to the source data to some extent.

The third and fourth terms: KL divergence between the outputs of $C$ and $D$. These terms help to alleviate the label-gap problem caused by using only the first and second terms and the problems in WADA. Since the fourth term only symmetrizes the gradient by the third term and improves performance [13], only the effects of the third term are discussed below.

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  $C$ is trained to output similar scores to those of $D$ for the transformed target data. That is, $C$ tries to give a high score of positivity to the target example that $D$ has difficulty distinguishing from source examples, i.e., the positive target example. $C$ also gives a low score of positivity to the target example that is easy to distinguish from source examples by $D$, i.e., the negative target example. Therefore, $C$ obtains an ability to classify target positive and negative examples.

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  $D$ aims to discriminate source examples and positive target examples. $D$ is trained to output the opposite scores from those of $C$ by the third term. If $C$ identifies the likely positive target example, $D$ tries to predict it as target (negative). Conversely, if $C$ identifies the likely negative target example, $D$ tries to predict it as source (positive). The latter effect is not desirable but is canceled with the effect of the second term.

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  $F$ is trained so that $D$ outputs similar scores to those of $C$, i.e., the likely positive (resp. negative) target examples predicted by $C$ are indistinguishable (resp. distinguishable) from the source examples by $D$. In other words, $F$ tries to make $D$ predict the likely positive (resp. negative) target example predicted by $C$ as source (resp. target). This allows $F$ to transform the positive target data into the data close to the source data while keeping the negative target data away from the source data. The latter is an effect not obtained with WADA.

As a result, we obtain a label-discriminative feature space in the target domain and a good classifier that works on that feature space. Algorithm 1 describes the learning algorithm of PADA using stochastic gradient descent.

We finally mention the role of common features in PADA. Common features prevent PADA falls into a trivial solution where $C$ predicts all target data to be negative. That is, common features are useful for finding some examples in the target data that are difficult for $D$ to distinguish from the source data, especially in the early stage of learning. It is clear that PADA will not fall into the other trivial solution where $C$ predicts all target data to be positive because $D$ is learned with the target data as negative by the second term of Eq. IV-C. See the next subsection for how to use common features more explicitly to facilitate the learning of $C$.

In the early stage of learning PADA, since $F$ is not fully trained, the features $\hat{T}_{s}$ created by $F$ may make the whole features $\hat{X}_{t}=[T_{c};\hat{T}_{s}]$ useless. As a result, $C$ might fail to identify likely positive examples from $\hat{X}_{t}$, which leads to the unstable learning of $C$ and $F$. Therefore, in this section, we consider guiding the learning of $C$ by a base classifier that can be trained stably using only common features without $F$.

First, we apply an existing PU learning method to the common features of the source data and target data, $S_{c}$ and $T_{c}$. Let $C_{0}$ be the base classifier obtained from the existing PU learning method using $S_{c}$ and $T_{c}$ as the positive and unlabeled data, respectively. If the common features contain some useful information for classification, $C_{0}$ should attain some degree of classification accuracy. Therefore, we can use $C_{0}$ to guide the learning of $C$. By using $C_{0}$, the objective of PADA is modified as follows:

where $\eta$ is a balancing parameter. Note that $C_{0}$ takes only common features as inputs. The second and third terms are added to the original objective of PADA. Since the third term is a symmetrization term, we discuss only the second term. In the second term, $C_{0}$ gives soft labels²²2Soft labels are usually used in the context of distillation [30], which transfers the knowledge of large models to small models. to the unlabeled data $\hat{X}_{t}$, and $C$ is trained using them as teacher labels. By this term, the learning of $C$ will be accelerated when $D$ and $F$ are not yet learned enough, which also leads to accelerated learning of $D$ and $F$ as a result.

Once the learning of $C$ is finished, $C$ is expected to perform better than $C_{0}$. Therefore, it is also expected that training PADA again using this $C$ as a base classifier $C_{0}$ will yield a better classifier. With this insight, we adopt repeating this soft-labeling process until the accuracy of the finally produced classifier saturates. We note that, in the second and later rounds, $C_{0}$ can take both common and target-specific features as inputs.

In this section, we experimentally confirm that PADA transforms the positive target data into data close to the source data while keeping the negative target data away from the source data using the Movielens-Netflix dataset.

We compare PADA with three baselines: (1) Common, which uses only the raw common features of the source and target data, (2) DSFT, a naive combination of HUDA and PU learning methods, and (3) WADA. To see the distribution distance between the source data and the positive (resp. negative) target data in the resulting feature space, we define a metric $Acc_{d}(s,t_{p})$ (resp. $Acc_{d}(s,t_{n})$), the test accuracy of discrimination by a test discriminator that discriminates the source data and the positive (resp. negative) target data. The test discriminator is trained on the resulting feature space after finishing the learning of each method or only on the common feature space for Common. Note that the test discriminator is different from the discriminator trained for PADA and WADA. The true labels of the target data are used to train the test discriminator. If $Acc_{d}(s,t_{p})$ (resp. $Acc_{d}(s,t_{n})$) is close to 0.5, the distribution divergence between the source data and the positive (resp. negative) target data is considered small. Therefore, small $Acc_{d}(s,t_{p})$ and large $Acc_{d}(s,t_{n})$ are desirable in PU-HUDA.

The results are presented in Table VI. As expected, we can see that $Acc_{d}(s,t_{p})$ of PADA is smaller than those of the other methods, while $Acc_{d}(s,t_{n})$ of PADA is larger than it. This means that the positive target data are transformed into data close to the source data while keeping the distance between the negative target data and the source data. $Acc_{d}(s,t_{p})$ of Common is smaller than that of PADA, but $Acc_{d}(s,t_{n})$ of Common is also small. As a result, the accuracy of the existing PU learning methods using only common features is low. Even for DSFT, the difference between $Acc_{d}(s,t_{p})$ and $Acc_{d}(s,t_{n})$ is small, which is consistent with the problem of the naive combination approaches described in Sec. III-B. We also note that although the difference between $Acc_{d}(s,t_{p})$ and $Acc_{d}(s,t_{n})$ of WADA is large, its $Acc_{d}(s,t_{p})$ is also large. Thus, if WADA is trained to make $Acc_{d}(s,t_{p})$ smaller, $Acc_{d}(s,t_{n})$ is expected to be even closer to $Acc_{d}(s,t_{p})$, which is an undesirable result as described in Sec. IV-B.