Ensemble Multi-source Domain Adaptation with Pseudolabels

Ensemble Multi-source Domain Adaptation with Pseudolabels

Abstract

Given multiple source datasets with labels, how can we train a target model with no labeled data? Multi-source domain adaptation (MSDA) aims to train a model using multiple source datasets different from a target dataset in the absence of target data labels. MSDA is a crucial problem applicable to many practical cases where labels for the target data are unavailable due to privacy issues. Existing MSDA frameworks are limited since they align data without considering conditional distributions of each domain. They also miss a lot of target label information by not considering the target label at all and relying on only one feature extractor. In this paper, we propose Ensemble Multi-source Domain Adaptation with Pseudolabels (EnMDAP), a novel method for multi-source domain adaptation. EnMDAP exploits label-wise moment matching to align conditional distributions , using pseudolabels for the unavailable target labels, and introduces ensemble learning theme by using multiple feature extractors for accurate domain adaptation. Extensive experiments show that EnMDAP provides the state-of-the-art performance for multi-source domain adaptation tasks in both of image domains and text domains.

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1 Introduction

Given multiple source datasets with labels, how can we train a target model with no labeled data? A large training data are essential for training deep neural networks. Collecting abundant data is unfortunately an obstacle in practice; even if enough data are obtained, manually labeling those data is prohibitively expensive. Using other available or much cheaper datasets would be a solution for these limitations; however, indiscriminate usage of other datasets often brings severe generalization error due to the presence of dataset shifts (TorralbaE11). Unsupervised domain adaptation (UDA) tackles these problems where no labeled data from the target domain are available, but labeled data from other source domains are provided. Finding out domain-invariant features has been the focus of UDA since it allows knowledge transfer from the labeled source dataset to the unlabeled target dataset. There have been many efforts to transfer knowledge from a single source domain to a target one. Most recent frameworks minimize the distance between two domains by deep neural networks and distance-based techniques such as discrepancy regularizers (LongCWJ15; LongZWJ16; LongZWJ17), adversarial networks (GaninUAGLLML16; TzengHSD17), and generative networks (LiuBK17; ZhuPIE17; HoffmanTPZISED18).

While the above-mentioned approaches consider one single source, we address multi-source domain adaptation (MSDA), which is very crucial and more practical in real-world applications as well as more challenging. MSDA is able to bring significant performance enhancement by virtue of accessibility to multiple datasets as long as multiple domain shift problems are resolved. Previous works have extensively presented both theoretical analysis (Ben-DavidBCKPV10; MansourMR08; CrammerKW08; HoffmanMZ18; ZhaoZWMCG18; ZellingerMS20) and models (ZhaoZWMCG18; XuCZYL18; PengBXHSW19) for MSDA. MDAN (ZhaoZWMCG18) and DCTN (XuCZYL18) build adversarial networks for each source domain to generate features domain-invariant enough to confound domain classifiers. However, these approaches do not encompass the shifts among source domains, counting only shifts between source and target domain. M3SDA (PengBXHSW19) adopts moment matching strategy but makes the unrealistic assumption that matching the marginal probability would guarantee the alignment of the conditional probability . Most of these methods also do not fully exploit the knowledge of target domain, imputing to the inaccessibility to the labels. Furthermore, all these methods leverage one single feature extractor, which possibly misses important information regarding label classification.

In this paper, we propose EnMDAP, a novel MSDA framework which mitigates the limitations of these methods of not explicitly considering conditional probability , and relying on only one feature extractor. The model architecture is illustrated in Figure 1. EnMDAP aligns the conditional probability by utilizing label-wise moment matching. We employ pseudolabels for the inaccessible target labels to maximize the usage of the target data. Moreover, integrating the features from multiple feature extractors gives abundant information about labels to the extracted features.Extensive experiments show the superiority of our proposed methods.

Our contributions are summarized as follows: {itemize*}

Method. We propose EnMDAP, a novel approach for MSDA that effectively obtains domain-invariant features from multiple domains by matching conditional probability , not marginal one, utilizing pseudolabels for inaccessible target labels to fully deploy target data, and using multiple feature extractors. It allows domain-invariant features to be extracted, capturing the intrinsic differences of different labels.

Analysis. We theoretically prove that minimizing the label-wise moment matching loss is relevant to bounding the target error.

Experiments. We conduct extensive experiments on image and text datasets. We show that 1) EnMDAP provides the state-of-the-art accuracy, and 2) each of our main ideas significantly contributes to the superior performance.

2 Related Work

Single-source Domain Adaptation. Given a labeled source dataset and an unlabeled target dataset, single-source domain adaptation aims to train a model that performs well on the target domain. The challenge of single-source domain adaptation is to reduce the discrepancy between the two domains and to obtain appropriate domain-invariant features. Various discrepancy measures such as Maximum Mean Discrepancy (MMD) (TzengHZSD14; LongCWJ15; LongZWJ16; LongZWJ17; GhifaryKZBL16) and KL divergence (ZhuangCLPH15) have been used as regularizers. Inspired from the insight that the domain-invariant features should exclude the clues about its domain, constructing adversarial networks against domain classifiers has shown superior performance. LiuBK17 and HoffmanTPZISED18 deploy GAN to transform data across the source and target domain, while GaninUAGLLML16 and TzengHSD17 leverage the adversarial networks to extract common features of the two domains. Unlike these works, we focus on multiple source domains.

Multi-source Domain Adaptation. Single-source domain adaptation should not be naively employed for multiple source domains due to the shifts between source domains. Many previous works have tackled MSDA problems theoretically. MansourMR08 establish distribution weighted combining rule that the weighted combination of source hypotheses is a good approximation for the target hypothesis. The rule is further extended to a stochastic case with joint distribution over the input and the output space in HoffmanMZ18. CrammerKW08 propose the general theory of how to sift appropriate samples out of multi-source data using expected loss. Efforts to find out transferable knowledge from multiple sources from the causal viewpoint are made in ZhangGS15. There have been salient studies on the learning bounds for MSDA. Ben-DavidBCKPV10 found the generalization bounds based on -divergence, which are further tightened by ZhaoZWMCG18. Frameworks for MSDA have been presented as well. ZhaoZWMCG18 propose learning algorithms based on the generalization bounds for MSDA. DCTN (XuCZYL18) resolves domain and category shifts between source and target domains via adversarial networks. M3SDA PengBXHSW19 associates all the domains into a common distribution by aligning the moments of the feature distributions of multiple domains. However, all these methods do not consider multimode structures (PeiCLW18) that differently labeled data follow distinct distributions, even if they are drawn from the same domain. Also, the domain-invariant features in these methods contain the label information for only one label classifier which lead these methods to miss a large amount of label information. Different from these methods, our frameworks fully count the multimodal structures handling the data distributions in a label-wise manner and minimize the label information loss considering multiple label classifiers.

Moment Matching. Domain adaptation has deployed the moment matching strategy to minimize the discrepancy between source and target domains. MMD regularizer (TzengHZSD14; LongCWJ15; LongZWJ16; LongZWJ17; GhifaryKZBL16) can be interpreted as the first-order moment while SunFS16 address second-order moments of source and target distributions. ZellingerGLNS17 investigate the effect of higher-order moment matching. M3SDA (PengBXHSW19) demonstrates that moment matching yields remarkable performance also with multiple sources. While previous works have focused on matching the moments of marginal distributions for single-source adaptation, we handle conditional distributions in multi-source scenarios.

3 Proposed Method

Figure 1: EnMDAP for =2. EnMDAP consists of pairs of feature extractor and label classifier, one extractor classifier, and one final label classifier. Colors and symbols of the markers indicate domains and class labels of the data, respectively.

In this section, we describe our proposed method, EnMDAP. We first formulate the problem definition in Section 3.1. Then, we describe our main ideas in Section 3.2. Section 3.3 elaborates how to match label-wise moment with pseudolabels and Section 3.4 extends the approach by adding the concept of ensemble learning. Figure 1 shows the overview of EnMDAP.

3.1 Problem Definition

Given a set of labeled datasets from source domains and an unlabeled dataset from a target domain , we aim to construct a model that minimizes test error on . We formulate source domain as a tuple of the data distribution on data space and the labeling function : . Source dataset drawn with the distribution is denoted as . Likewise, the target domain and the target dataset are denoted as and , respectively. We narrow our focus down to homogeneous settings in classification tasks: all domains share the same data space and label set .

3.2 Overview

We propose EnMDAP based on the following observations: 1) existing methods focus on aligning the marginal distributions not the conditional ones , 2) knowledge of the target data is not fully employed as no target label is given, and 3) there exists a large amount of label information loss since domain-invariant features are extracted for only one single label classifier. Thus, we design EnMDAP aiming to solve the limitations. Designing such method entails the following challenges:

  1. Matching conditional distributions. How can we align the conditional distribution, , of multiple domains not the marginal one, ?

  2. Exploitation of the target data. How can we fully exploit the knowledge of the target data despite the absence of the target labels?

  3. Maximally utilizing feature information. How can we maximally utilize the information that the domain-invariant features contain?

We propose the following main ideas to address the challenges:

  1. Label-wise moment matching (Section 3.3). We match the label-wise moments of the domain-invariant features so that the features with the same labels have similar distributions regardless of their original domains.

  2. Pseudolabels (Section 3.3). We use pseudolabels as alternatives to the target labels.

  3. Ensemble of feature representations (Section 3.4). We learn to extract ensemble of features from multiple feature extractors, each of which involves distinct domain-invariant features for its own label classifier.

3.3 Label-wise Moment Matching with pseudolabels

We describe how EnMDAP matches conditional distributions of the features from multiple distinct domains. In EnMDAP, a feature extractor and a label classifier lead the features to be domain-invariant and label-informative at the same time. The feature extractor extracts features from data, and the label classifier receives the features and predicts the labels for the data. We train the two components, and , according to the losses for label-wise moment matching and label classification, which make the features domain-invariant and label-informative, respectively.

Label-wise Moment Matching. To achieve the alignment of domain-invariant features, we define a label-wise moment matching loss as follows:

(1)

where is a hyperparameter indicating the maximum order of moments considered by the loss, and are two distinct domains amongst the domains, and is the number of data labeled as in . We introduce pseudolabels for the target data, which are determined by the outputs of the model currently being trained, to manage the absence of the ground truths for the target data. In other words, we leverage to give the pseudolabel to the target data . Drawing the pseudolabels using the incomplete model, however, brings mis-labeling issue which impedes further training. To alleviate this problem, we set a threshold and assign the pseudolabels to the target data only when the prediction confidence is greater than the threshold. The target examples with low confidence are not pseudolabeled and not counted in label-wise moment matching.

By minimizing , the feature extractor aligns data from multiple domains by bringing consistency in distributions of the features with the same labels. The data with distinct labels are aligned independently, taking account of the multimode structures that differently labeled data follow different distributions.

Label Classification. The label classifier gets the features projected by as inputs and makes the label predictions. The label classification loss is defined as follows:

(2)

where is the softmax cross-entropy loss. Minimizing separates the features with different labels so that each of them gets label-distinguishable.

3.4 Ensemble of Feature Representations

In this section, we introduce ensemble learning for further enhancement. Features extracted with the strategies elaborated in the previous section contain the label information for a single label classifier. However, each label classifier leverages only limited label characteristics, and thus the conventional scheme to adopt only one pair of feature extractor and label classifier captures only a small part of the label information. Our idea is to leverage an ensemble of multiple pairs of feature extractor and label classifier in order to make the features to be more label-informative.

We train multiple pairs of feature extractor and label classifier in parallel following the label-wise moment matching approach explained in Section 3.3. Let denote the number of the feature extractors in the overall model. We denote the (feature extractor, label classifier) pairs as and the resultant features as where is the output of the feature extractor . After obtaining different feature mapping modules, we concatenate the features into one vector . The final label classifier takes the concatenated feature as input, and predicts the label of the feature.

Naively exploiting multiple feature extractors, however, does not guarantee the diversity of the features since it resorts to the randomness. Thus, we introduce a new model component, extractor classifier, which separates the features from different extractors. The extractor classifier gets the features generated by a feature extractor as inputs and predicts which feature extractor has generated the features. For example, if , the extractor classifier attempts to predict whether the input feature is extracted by the extractor or . By training the extractor classifier and multiple feature extractors at once, we explicitly diversify the features obtained from different extractors. We train the extractor classifier utilizing the feature diversifying loss, :

(3)

where is the number of feature extractors.

3.5 EnMDAP: Ensemble Multi-Source Domain Adaptation with pseudolabels

Our final model EnMDAP consists of pairs of feature extractor and label classifier, , one extractor classifier , and one final label classifier . We first train the entire model except the final label classifier with the loss :

(4)

where is the label classification loss of the classifier , is the label-wise moment matching loss of the feature extractor , and and are the hyperparameters. Then, the final label classifier is trained with respect to the label classification loss using the concatenated features from multiple feature extractors.

4 Analysis

We present a theoretical insight regarding the validity of the label-wise moment matching loss. For simplicity, we tackle only binary classification tasks. The error rate of a hypothesis on a domain is denoted as where is the labeling function on the domain . We first introduce -th order label-wise moment divergence.

Definition 1.

Let and be two domains over an input space where is the dimension of the inputs. Let be the set of the labels, and and be the data distribution given that the label is , i.e. and for the data distribution and on the domains and , respectively. Then, the -th order label-wise moment divergence of the two domains and over is defined as

(5)

where is the set of the tuples of the nonnegative integers, which add up to , and are the probability that arbitrary data from and to be labeled as respectively, and the data is expressed as .∎

The ultimate goal of MSDA is to find a hypothesis with the minimum target error. We nevertheless train the model with respect to the source data since ground truths for the target are unavailable. Let datasets be drawn from labeled source domains respectively. We denote -th source dataset as . The empirical error of hypothesis in -th source domain estimated with is formulated as . Given a weight vector such that , the weighted empirical source error is formulated as . We extend the theorems in Ben-DavidBCKPV10; PengBXHSW19 and derive a bound for the target error , for trained with source data, in terms of -th order label-wise moment divergence.

Theorem 1.

Let be a hypothesis space of VC dimension , be the number of samples from source domain , be the total number of samples from source domains , and with . Let us define a hypothesis that minimizes the weighted empirical source error, and a hypothesis that minimizes the true target error. Then, for any and , there exist integers and constants such that

(6)

with probability at least , where and .∎

Proof.

See the appendix. ∎

Speculating that all datasets are balanced against the annotations, i.e., for any , is expressed as the sum of the estimates of with . The theorem provides an insight that label-wise moment matching allows the model trained with source data to have performance comparable to the optimal one on the target domain.

5 Experiments

We conduct experiments to answer the following questions of EnMDAP. {itemize*}

Accuracy (Section 5.2). How well does EnMDAP perform in classification tasks?

Ablation Study (Section 5.3). How much does each component of EnMDAP contribute to performance improvement?

Effects of Degree of Ensemble (Section 5.4). How does the performance change as the number of the pairs of the feature extractor and the label classifier increases?

5.1 Experimental Settings

Datasets. We use three kinds of datasets, Digits-Five, Office-Caltech101, and Amazon Reviews2. Digits-Five consists of five datasets for digit recognition: MNIST3 (LeCunBBH98), MNIST-M4 (GaninL15), SVHN5 (NetzerWCBWN11), SynthDigits6 (GaninL15), and USPS7 (HastieFT01). We set one of them as a target domain and the rest as source domains. Following the conventions in prior works (XuCZYL18; PengBXHSW19), we randomly sample 25000 instances from the source training set and 9000 instances from the target training set to train the model except for USPS for which the whole training set is used. Office-Caltech10 is the dataset for image classification with 10 categories that Office31 dataset and Caltech dataset have in common. It involves four different domains: Amazon, Caltech, DSLR, and Webcam. We double the number of data by data augmentation and exploit all the original data and augmented data as training data and test data respectively.Amazon Reviews dataset contains customers’ reviews on 4 product categories: Books, DVDs, Electronics, and Kitchen appliances. The instances are encoded into 5000-dimensional vectors and are labeled as being either positive or negative depending on their sentiments. We set each of the four categories as a target and the rest as sources. For all the domains, 2000 instances are sampled for training, and the rest of the data are used for the test. Details about the datasets are summarized in appendix.

Competitors. We use 3 MSDA algorithms, DCTN (XuCZYL18), M3SDA (PengBXHSW19), and M3SDA- (PengBXHSW19), with state-of-the-art performances as baselines. All the frameworks share the same architecture for the feature extractor, the domain classifier, and the label classifier for consistency. For Digits-Five, we use convolutional neural networks based on LeNet5 (LeCunBBH98). For Office-Caltech10, ResNet50 (HeZRS16) pretrained on ImageNet is used as the backbone architecture. For Amazon Reviews, the feature extractor is composed of three fully-connected layers each with 1000, 500, and 100 output units, and a single fully-connected layer with 100 input units and 2 output units is adopted for both of the extractor and label classifiers. With Digits-Five, LeNet5 (LeCunBBH98) and ResNet14 (HeZRS16) without any adaptation are additionally investigated in two different manners: Source Combined and Single Best. In Source Combined, multiple source datasets are simply combined and fed into a model. In Single Best, we train the model with each source dataset independently, and report the result of the best performing one. Likewise, ResNet50 and MLP consisting of 4 fully-connected layers with 1000, 500, 100, and 2 units are investigated without adaptation for Office-Caltech10 and Amazon Reviews respectively.

Training Details. We train our models for Digits-Five with Adam optimizer (KingmaB14) with , , and the learning rate of for 100 epochs. All images are scaled to and the mini batch size is set to . We set the hyperparameters , , and . For the experiments with Office-Caltech10, all the modules comprising our model are trained following SGD with the learning rate , except that the optimizers for feature extractors have the learning rate . We scale all the images to and set the mini batch size to . All the hyperparameters are kept the same as in the experiments with Digits-Five. For Amazon Reviews, we train the models for epochs using Adam optimizer with , , and the learning rate of . We set , , and the mini batch size to . For every experiment, the confidence threshoold is set to .

Method M+S+D+U/T T+S+D+U/M T+M+D+U/S T+M+S+U/D T+M+S+D/U Average
LeNet5 (SC) 97.580.18 61.721.38 75.150.76 80.290.66 81.581.51 79.270.90
ResNet14 (SC) 98.220.26 63.530.84 79.081.63 92.850.48 94.510.31 85.640.70
LeNet5 (SB) 97.090.14 51.101.87 76.750.57 79.920.50 83.280.92 77.630.80
ResNet14 (SB) 97.071.03 49.481.30 81.400.70 91.790.53 91.542.68 82.331.25
DCTN 99.280.06 71.991.58 78.341.10 91.550.65 98.430.23 87.920.72
M3SDA 98.750.05 67.770.71 81.750.61 88.510.29 97.170.22 86.790.38
M3SDA- 98.990.03 72.470.19 81.400.28 89.510.37 97.400.19 87.950.21
EnMDAP (n=2) 99.310.04 83.950.90 86.930.39 93.150.17 98.490.08 92.370.31
(a) Digits-Five
Method C+D+W/A A+D+W/C A+C+W/D A+C+D/W Average
ResNet50 (SC) 95.470.25 91.590.51 99.360.78 99.260.37 96.420.48
ResNet50 (SB) 95.030.48 89.050.88 99.870.28 98.240.61 95.550.56
DCTN 95.050.24 90.600.71 100.00.00 99.460.62 96.280.39
M3SDA 95.140.31 93.590.40 99.490.53 99.860.19 97.020.36
M3SDA- 94.360.26 91.700.71 99.750.35 99.390.15 96.300.37
EnMDAP (n=2) 95.740.29 93.910.28 99.870.28 99.860.19 97.350.26
(b) Office-Caltech10
Method D+E+K/B B+E+K/D B+D+K/E B+D+E/K Average
MLP (SC) 79.760.70 82.180.59 84.420.27 87.230.51 83.400.52
MLP (SB) 79.000.92 80.380.61 84.760.45 87.460.36 82.900.58
DCTN 78.920.56 81.221.01 83.561.52 86.470.71 82.540.95
M3SDA 78.970.79 80.510.99 83.630.68 85.990.85 82.270.83
M3SDA- 80.260.43 81.800.72 85.020.34 86.990.56 83.520.51
EnMDAP (n=2) 81.140.29 83.130.45 86.470.35 88.530.33 84.820.35
(c) Amazon Reviews
Table 1: Classification accuracy on Digits-Five, Office-Caltech10, and Amazon Reviews with and without domain adaptation. The letters before and after the slash represent source domains and a target domain respectively. In Digits-Five, T, M, S, D, and U stands for MNIST, MNIST-M, SVHN, SynthDigits, and USPS respectively. In Office-Caltech10 and Amazon Reviews, we indicate each domain using the first letter of its name. SC and SB indicate Source Combined and Single Best respectively. Note that EnMDAP shows the best performance.

5.2 Performance Evaluation

We evaluate the performance of EnMDAP with against the competitors. We repeat experiments for each setting five times and report the mean and the standard deviation. The results are summarized in Tables 1. Note that EnMDAP provides the best accuracy in all the datasets, showing their consistent superiority in both image datasets (Digits-Five, Office-Caltech10) and text dataset (Amazon Reviews). The enhancement is especially remarkable when MNIST-M is the target domain in Digits-Five, improving the accuracy by compared to the state-of-the-art methods.

5.3 Ablation Study

We perform an ablation study on Digits-Five to identify what exactly enhances the performance of EnMDAP. We compare EnMDAP with 3 of its variants: MDAP-L, MDAP, and EnMDAP-R. MDAP-L has the same strategies as M3SDA, aligning moments regardless of the labels of the data. MDAP trains the model without ensemble learning theme. EnMDAP-R exploits ensemble learning strategy but relies on randomness without extractor classifier and feature diversifying loss.

The results are shown in Table 2. By comparing MDAP-L and MDAP, we observe that considering labels in moment matching plays a significant role in extracting domain-invariant features.The remarkable performance gap between MDAP and EnMDAP with verifies the effectiveness of ensemble learning. On the other hand, the performance of EnMDAP-R and EnMDAP have little difference. It indicates that two feature extractors trained independently without any diversifying techniques are unlikely to be correlated even though it resorts to randomness.

5.4 Effects of Ensemble

We vary , the number of pairs of feature extractor and label classifier, and repeat the performance evaluation on Digits-Five. The results are summarized in Table 2. While ensemble of two pairs gives much better performance than the model with one single pair, using more than two pairs rarely brings further improvement. This result demonstrates that two pairs of feature extractor and label classifier are able to cover most information without losing important label information in Digits-Five. It is notable that increasing sometimes brings small performance degradation. As more feature extractors are adopted to obtain final features, the complexity of final features increases. It is harder for the final label classifiers to manage the features with high complexity compared to the simple ones. This deteriorates the performance when we exploit more than two feature extractors.

Method M+S+D+U/T T+S+D+U/M T+M+D+U/S T+M+S+U/D T+M+S+D/U Average
MDAP-L 98.750.05 67.770.71 81.750.61 88.510.29 97.170.22 86.790.38
MDAP 99.140.06 79.320.73 84.770.39 91.910.05 98.490.16 90.730.28
EnMDAP-R (n=2) 99.340.05 83.240.81 86.960.34 92.880.15 98.560.17 92.200.30
EnMDAP (n=2) 99.310.04 83.950.90 86.930.39 93.150.17 98.490.08 92.370.31
EnMDAP (n=3) 99.310.05 82.780.67 87.100.29 92.850.24 98.480.09 92.100.27
EnMDAP (n=4) 99.300.07 82.740.55 86.650.41 92.860.15 98.500.08 92.010.25
Table 2: Experiments with EnMDAP and its variants.

6 Conclusion

We propose EnMDAP, a novel framework for the multi-source domain adaptation problem. EnMDAP overcomes the problems in the existing methods of not directly addressing conditional distributions of data , not fully exploiting knowledge of target data, and missing large amount of label information. EnMDAP aligns data from multiple source domains and the target domain considering the data labels, and exploits pseudolabels for unlabeled target data. EnMDAP further enhances the performance by introducing multiple feature extractors. Our framework exhibits superior performance on both image and text classification tasks. Considering labels in moment matching and adding ensemble learning theme is shown to bring remarkable performance enhancement through ablation study. Future works include extending our approach to other tasks such as regression, which may require modification in the pseudolabeling method.

References

Appendix A Appendix

a.1 Proof for Theorem 1

We prove Theorem 1 in the paper by extending the proof in the existing studies (Ben-DavidBCKPV10; PengBXHSW19). We first define -th order label-wise moment divergence , and disagreement ratio of the two hypotheses on the domain .

Definition 1.

Let and be two domains over an input space where is the dimension of the inputs. Let be the set of the labels, and and be the data distributions given that the label is , i.e. and for the data distribution and on the domains and , respectively. Then, the -th order label-wise moment divergence of the two domains and over is defined as

(7)

where is the set of the tuples of the nonnegative integers, which add up to , and are the probability that arbitrary data from and to be labeled as respectively, and the data is expressed as .∎

Definition 2.

Let be a domain over an input space with the data distribution . Then, we define the disagreement ratio of the two hypotheses on the domain as

(8)

Theorem 2.

(Stone-Weierstrass Theorem (stone37)) Let be a compact subset of and be a continuous function. Then, for every , there exists a polynomial, , such that

(9)

Theorem 2 indicates that continuous functions on a compact subset of are approximated with polynomials. We next formulate the discrepancy of the two domains using the disagreement ratio and bound it with the label-wise moment divergence.

Lemma 1.

Let and be two domains over an input space , where n is the dimension of the inputs. Then, for any hypotheses and any , there exist and a constant such that

(10)

Proof.

Let the domains and have the data distribution of and , respectively, over an input space , which is a compact subset of , where is the dimension of the inputs. For brevity, we denote as . Then,

(11)

For any hypotheses , the indicator function is Lebesgue integrable on , i.e. is a function. Since a set of continuous functions is dense in , for every , there exists a continuous function defined on such that

(12)

for every , and the fixed and that drive Equation 5 to the supremum. Accordingly,

(13)

By integrating every term in the inequality over , the inequality,

(14)

follows. Likewise, the same inequality on the domain with instead of holds. By subtracting the two inequalities and reformulating it, the inequality,

(15)

is induced. By substituting the inequality in Equation 9 to the Equation 5,

(16)

By the Theorem 2, there exists a polynomial such that

(17)

and the polynomial is expressed as

(18)

where is the order of the polynomial, is the set of the tuples of the nonnegative integers, which add up to , is the coefficient of each term of the polynomial, and . By applying Equation 11 to the Equation 10 and substituting the expression in Equation 12,

(19)

where and are the probability that an arbitrary data is labeled as class in domain and , respectively, and and are the data distribution given that the data is labeled as class on domain and , respectively. For ,

(20)

for . ∎

Let datasets be drawn from labeled source domains respectively. We denote -ith source dataset as . The empirical error of hypothesis in -th source domain estimated with is formulated as . Given a positive weight vector such that and , the weighted empirical source error is formulated as .

Lemma 2.

For source domains , let be the number of samples from source domain , be the total number of samples from source domains, and with . Let be the weighted true source error which is the weighted sum of . Then,

(21)
Proof.

It has been proven in Ben-DavidBCKPV10. ∎

We now turn our focus back to the Theorem 1 in the paper and complete the proof.

Theorem 1.

Let be a hypothesis space of VC dimension , be the number of samples from source domain , be the total number of samples from source domains , and with . Let us define a hypothesis that minimizes the weighted empirical source error, and a hypothesis that minimizes the true target error. Then, for any and , there exist integers and constants such that

(22)

with probability at least , where and .∎

Proof.
(23)

We define for every for the following equations. We also note that the 1-triangular inequality (CrammerKW08) holds for binary classification tasks, i.e., for any hypothesis and domain . Then,

(24)

for the ground truth labeling function on the domain and two hypotheses . Applying the definition and the inequality to Equation 17,

(25)

By the definition of , for . Additionally, according to Lemma 1, for any , there exists an integer and a constant such that

(26)

By applying these relations,

(27)

By Lemma 2 and the standard uniform convergence bound for hypothesis classes of finite VC dimension (Ben-DavidBCKPV10),