1 Introduction
Supervised Learning consists in learning a model on a sample of data drawn from an unknown distribution. It assumes that the data distribution is conserved at test time. This assumption may not be hold for a large wide of industrial applications (Candela et al., 2009; Kull & Flach, 2014) since several factors may alter the data generative process: different preprocessing, sample rejection, conditions of data collection… Domain Adaptation (DA) (Patel et al., 2015; Pan et al., 2010) is a typical case of such an issue. By transferring knowledge learned on source domain where labels are available, Domain Adaptation aims to learn a better model on the target domain. Prior work on Domain Adaptation mainly differs on the assumption made on the change in data distribution between the source and the target domains.
A first family of approaches, known as Importance Sampling (Candela et al., 2009), consists in reweighting the importance of each sample in order to incorporate during learning the fact that the joint distribution may change across domains. The optimal reweighting is obtained computing the ratio between target and source joint distributions. This requires the availability of labels on the target domain, which is often unfeasible. To overcome this issue, two exclusive assumptions are commonly made depending on a prior knowledge of the distributional shift nature. The first, called Covariate Shift or Sample Selection Bias (Sugiyama et al., 2008; Wen et al., 2014; Zadrozny et al., 2003; Bickel et al., 2007; Huang et al., 2007) assumes changes in while the conditional distribution is conserved across domains. Under this assumption the reweighting depends only on . This results in a wellposed problem since samples of are available in both source and target domains. The second, called Target Shift (Storkey, 2009) or Endogeneous Stratified Sampling (Manski & Lerman, 1977) assumes changes in while the conditional distribution is conserved across domains. Under this assumption, the reweighting depends only on . The major drawback of this assumption is the lack of labeled data in the target domain: the reweighting is performed using estimated labels in the target domain. Challenging cases for Importance Sampling methods are the Conditional Shift (when changes) and Concept Drift (when changes).
For addressing such challenging cases of distributional shift, Invariant Representation methods aim to learn a representation that makes source and target data indistinguishable (Baktashmotlagh et al., 2013). Such a representation is essentially learned by matching the distribution between the source and target domains using adversarial learning (Ganin & Lempitsky, 2014; Ganin et al., 2016; Tzeng et al., 2014). One of the major drawbacks of such approaches is that they do not naturally handle the case of Target Shift. To address this issue, (Manders et al., 2018; Yan et al., 2017) generalizes the approach of (Ganin & Lempitsky, 2014) by matching of and estimating label distributional shift during learning.
Claims
As suggested in (Courty et al., 2017), there is no clear reason why learning a distribution such that is conserved across domains will help for solving the task using the intermediate state . Likewise, methods based on class ratio estimation between source and target domains implicitly make the assumption that it exists a nonlinear mapping such that is conserved across domains. Furthermore, those methods make a second assumption that the conservation of across domains will help to solve the task using the intermediate state . This assumption reveals the underlying causal scenario (Gong et al., 2016) and there is no clear reason why this assumption necessarily holds for real world data.
Contribution
We believe the hypothesis of Hidden Covariate Shift (Kull & Flach, 2014) is a rather minimal assumption for performing Domain Adaptation in challenging distributional shift settings. Hidden Covariate Shift is a particular case of Concept Drift (where may change across domains (Gama et al., 2014)) where there exists a non linear function such that is conserved across domains. We believe that learning such a function is the underlying objective of any Domain Adaptation method. Using Maximum Mean Discrepancy (), we theoretically show that this assumption can be expressed as a null maximum mean discrepancy involving the joint distribution and the density ratio of between target and source domains. We must however estimate the joint distribution on the target domain. Our formulation naturally handles challenging distributional shift cases (Concept Drift and Target Shift). We show on several experiments with both synthetic and real world data that our formulation is robust to these cases.
2 Proposed Approach
2.1 Introduction
2.1.1 Notations
We denote a random variable of features
(a realization ) and a random variable labels (a realization ) which respectively take their values in a feature space and a label space. On such spaces, we introduce two probability distributions:
(source distribution) and (target distribution) on . We introduce the source domain and the target domain . The expectation over the source domain is noted and the target domain . We call a Hidden Covariate Representation, a function such that . For the purpose of notation, we identify with as hidden covariate representation. For a given set , the set denotes the set of measurable and bounded functions from to .2.1.2 Distribution matching with Maximum Mean Discrepancy
Invariant Representation methods for learning crossdomain representations rely on the quantification on how data distributions from the source and the target domains differ. Formally, for two given distributions and , such methods introduce a proxy which quantifies how close and are. In the present work, we suggest to use the Maximum Mean Discrepancy measure (Gretton et al., 2007, 2012) denoted . Such measure is based on the following property:
(1) 
where is the set of measurable and bounded functions of . We can derive a proxy of this property called Maximum Mean Discrepancy:
(2) 
2.1.3 Main contributions
In this section, we derive an optimization procedure from the Hidden Covariate Shift assumption. More specifically, we show that a given representation is a Hidden Covariate Representation if in the target domain and a normalized joint distribution in the source domain are equal. The normalization consists in a reweighting by a factor . This implies the introduction of three different losses which we detail further:

For a given representation , we need to estimate the density ratio (step in Figure 2). The loss proxy for learning for a is denoted and called Hidden Weight Loss.

Since we address the context of Unsupervised Domain Adaptation, labeled samples in the target domain are not available. We estimate them through a target labeler (step in Figure 2). We show that learning is equivalent to a Covariate Shift Adaptation in the representation space. The loss proxy for learning for given and is noted called Hidden Covariate Loss.

Assuming and labeled samples are available in target domain, it is possible to learn such that is an hidden covariate representation by distribution matching (step in Figure 2). The loss proxy for learning for given and is noted called Reweighted Distribution Matching Loss.
2.2 Formulation
2.2.1 From hidden covariate shift assumption to a distribution matching problem
Let denote an Hidden Covariate Representation. Therefore, by definition, for all ,
(3) 
The fact that equation 3 holds for all is equivalent to:
(4) 
which holds for all . Since equation 4 holds for all , we can add a dependency with in ^{1}^{1}1The choice of a relevant may differ with a given value of . Considering the case where it exists two such that and . The function which exhibits may be different of the function which exhibits . Thus, equation 4 is equivalent to:
(5) 
where is the set of functions such that is measurable bounded and is measurable. Then, it is equivalent to:
(6) 
which consists in taking the expectation over the variable for . Noting that where , we obtain:
(7) 
The application of the transfer theorem with leads to:
(8) 
where the equality holds for all . To summarize, the Hidden Covariate Shift assumption is equivalent to the equality between two families of distributions and . We have shown this equality between two distribution families is equivalent to a null discrepancy between the join distribution in the target domain and a reweighted version in the source domain. To compute this discrepancy, we must evaluate the density ratio . Such a task can be challenging in high dimension and we may want to keep the representation dimension of reasonably low. Additionally, we need to estimate the label in the target domain.
2.2.2 Losses
Hidden Covariate Loss
For a given Hidden Covariate Representation , and verify the covariate shift assumption introduced in (Sugiyama et al., 2007)
. Authors have shown that Domain Adaptation problem can be solved by instance reweighting in the loss function of label estimation. In our context, the reweighting is done in the representation space
. Thus, the label estimation in the target domain is obtained by minimizing the following loss:(9) 
where and .
Hidden Weight Loss
Reweighted Distribution Matching Loss
Assuming that an estimation of both and are available, from 2.2.1 an estimation is obtained by minimizing the following loss:
(11) 
where .
2.3 Learning
2.3.1 Unbiased estimation of the losses with RKHS
Since the supremum on is highly intractable, the work of (Gretton et al., 2007) suggests to use Reproducing Kernel Hilbert Space (RKHS). For a given unit ball of RKHS associated with a kernel , the supremum has a close form:
(12) 
Furthermore, authors have derived an unbiased empirical estimate of given samples drawn from and samples drawn from :
(13) 
In our specific context, for a given and and noting that () and , the Reweighted Distribution Matching Loss and the Hidden Weight Loss can be expressed as follows:
(14)  
(15) 
Following the unbiased estimation suggested in
(Gretton et al., 2007), considering and associated with estimated labels , we derive the loss estimators with :(16) 
(17) 
Details on kernel estimation are given in Appendix B
2.3.2 Optimization procedure
Using the previous results, the learning of a Hidden Covariate Representation consists in solving the following optimization problem :
(18)  
such that  (19)  
(20) 
Since the model can collapse to a state where and are independent in the source and in the target domains (which ensures that ) we add the loss of the supervised task in target domain as a regularization term:
(21) 
2.3.3 Model selection with hidden reversed validation
Most of Domain Adaptation methods depend on a large set of hyperparameters. A major difficulty of Domain Adaptation is the lack of labeled data in the target domain. (Zhong et al., 2010) suggest to use the Reverse Validation to select the best hyperparameters. This method consists in training the domain adaptation procedure in order to infer labels on the target domain. Then, the same Domain Adaptation method is used in the reversed source/target situation using estimated labels in the target domain. The best model is finally chosen by comparing these new labels with the groundtruth. Assuming and is a Covariate Shift situation, Domain Adaptation only needs to learn instance weight . It is straightforward to show that the second adaptation only consists in learning , by minimizing where are estimated labels in the target domain and the density ratio learned during the first adaptation. In our context, we aim to learn a function such that and is a situation of Covariate Shift. We suggest to apply Hidden Reverse Validation which consists in learning with our proposed approach and validate the model with reverse validation in by only learning a new by minimizing . Hidden Reverse Validation has the major advantage to reduce dramatically computation time since the second adaptation is only an instance reweighted supervised problem.
3 Experiments
3.1 Baselines
The literature on Domain Adaptation is large and vastly different methods can be employed (learning invariant features with or without label conditioning). Moreover, several discrepancy measures may be used: divergence measure (e.g. KLdivergence, Mutual Information), integral divergence measure () or Optimal Transport based measure (Wasserstein distance^{2}^{2}2which is also an integral divergence measure). For the sake of simplicity and to ensure a fair comparison with methods of the literature, we focus on differences between invariance assumptions. Moreover, the same discrepancy measure is used (i.e. with the same kernels). Furthermore, we detail how the discrepancy loss of distribution matching is modified for each baseline:
Based on invariance
Based on invariance
It learns to match . Thus, depends only of the labels . The distribution matching loss is . This invariance covers namely (Yan et al., 2017; Manders et al., 2018; Chen et al., 2018; Zhang et al., 2013).
We will flag our method as based on invariance.
3.2 A toy experiment on synthetic data
We generate synthetic two dimensional data of a binary classification problem which verifies a challenging case of distributional shift (equations 22, 23, 24 bellow and Figure 3 (Top)). In that context, we observe the situation of Target Shift since . Besides, it suffers from Concept Drift with an inverted dependency with between source and target domains. Furthermore with but the distribution of is normal in the source domain and uniform in the target domain thus the data verifies the assumption of Hidden Covariate Shift. The model we considered is a linear model where features which lie in two dimensional space are projected to which lie in a one dimensional space. Then a sigmoid layer is applied on to determine the label. The estimation of
is done with a two layers neural network with
nonlinearity and 10 dimensions of hidden states.We report in Figure 3 the distribution of and the estimation of for four snapshots equally separated during learning. We observe that after the pretraining step (top left figure), representations from the target domain are not well separated and seem to have a gaussian component. The main reason is the model uses to infer . During learning (from top left, to top right, to bottom left and to bottom right), we observe that the distribution of
from the target domain switches from a gaussian distribution to a uniform distribution (which is the distribution of
): the model has learned to forget the feature to infer on the target domain. Furthermore, we can observe the estimation of which integrates both label ratio (to the left of the linear separation, density ratio is much higher than to the right) and the fact that representation from the target domain has a higher density close to the separator than representation from the source domain (since representation in the target domain are harder to separate).(22)  
(23)  
(24) 
3.3 Amazon Reviews dataset
Task  No DA  

ED  72.1  76.1  74.8  75.4 / 76.1 
EB  71.8  74.4  71.8 / 73.3  73.2 / 74.2 
EK  83.8  86.1 / 87.3  85.8 / 86.8  85.4 
DE  73.6  82.2  79.8 / 81.0  80.2 / 81.0 
DK  77.3  83.0  83.0  82.2 / 83.4 
DB  78.6  78.8 / 79.4  77.0  77.9 / 78.6 
BD  78.7  80.3  79.7 / 80.0  78.4 / 78.6 
BE  72.9  80.4 / 80.6  79.2 / 79.6  78.5 / 79.0 
BK  76.1  83.7  81.8  82.8 
KE  83.2  84.4  82.0 / 83.1  83.6 
KD  73.9  80.4 / 80.6  79.2 / 79.6  78.5 / 79.0 
KB  72.4  80.5 / 82.7  79.8 / 81.0  80.2 / 81.0 
AVG  76.2  80.9 / 81.2  79.5 / 80.1  79.7 / 80.2 
In our experiment on realworld data, we used the Amazon Reviews dataset (Blitzer et al., 2007). This dataset contains reviews collected on the Amazon website about products (Book (B), DVD (D), Kitchen (K), Electronics (E)) such that reviews with a rating higher than 4 are considered as positive and considered as negative otherwise. Authors introduced a preprocessed version where relevant unigrams and bigrams are selected. It allows to evaluate the model on 12 tasks of Domain Adaptation ().
In order to compare with literature preprocessing choices, for each task, we selected the 5000 most frequent features encoded as a bag of words (Ganin & Lempitsky, 2014). The model we used during experiments is a one layer neural networks with 50 dimensions and a non linearity for learning
, a two hidden layer neural network with 10 dimensions for hidden representation for learning
, andare sigmoid layers. We used a slow optimizer RMSProp with a learning rate of
for learning and for learning and , a large batch of 128 samples and and and are updated 5 times at each update of. Training is stopped after 30 epochs. Those choices are motivated by numerical stability observed during experiments. The hyperparameter
(equation 21) was selected by Hidden Reversed Validation among repeating the experiment for 3 different but fixed seeds for replicability. Thus, performance on each task is selected among 9 runs. For transparency, we report two results separated with ’’: left is performance of the model selected by Hidden Reverse Validation and right is best performance observed among the 9 runs. This allows to observe the variability of methods. We bold best model comparing methods based on versus since methods based on invariance has a clear advantage since the hypothesis is verified for the standard benchmark and the Concept Drift experiment. We report in Table 1 the accuracy of our approach compared to baselines on the standard Amazon Reviews (standard benchmark).3.3.1 Concept Drift
We filtered original datasets for obtaining a Concept Drift situation. For a couple of datasets , we build the source domain by overrepresenting positive reviews from and negative reviews from . The target domain is balanced: it has the same number of negative and positive reviews from each domain. More formally, we sample initial datasets such that for the source domain , while conserving and . For the target domain, we took a nonoverlapping sample with the source dataset such that and . The Concept Drift situation occurs since determining whether the sample is drawn from or helps to infer in the source domain but do not help in the target domain. We report results in Table 2 as the accuracy of the conventional test set of .
No DA  Task  

(E, D)  71.1  78.1  76.4  75.8 
(E, B)  70.3  75.3 / 77.3  74.0 / 76.1  73.0 / 75.7 
(E, K)  83.2  86.6  84.9 / 85.4  85.1 / 85.2 
(D, E)  76.7  83.0  80.3 / 82.7  82.1 
(D, K)  75.5  85.2  83.9 / 85.4  82.7 / 83.1 
(D, B)  75.4  78.0  75.2 / 76.4  76.3 / 78.0 
(B, D)  75.0  79.3  76.5 / 77.9  78.0 / 78.1 
(B, E)  76.7  82.3  81.4 / 81.8  81.0 
(B, K)  77.7  85.3  82.5 / 85.2  83.4 / 84.7 
(K, E)  81.6  82.1 / 83.1  80.3 / 82.1  82.4 / 83.2 
(K, D)  73.7  76.0 / 76.6  74.2 / 76.7  75.6 / 76.3 
(K, B)  71.7  76.9  74.6 / 76.4  74.0 / 75.7 
AVG  75.7  80.7 / 80.9  78.7 / 80.2  79.1 / 79.9 
3.3.2 Target shift
Task  No DA  

ED  71.2  71.6 / 73.0  72.6  73.8 
EB  70.9  72.2  70.6 / 71.7  72.3 
EK  85.1  81.1 / 81.5  83.7 / 85.1  84.9 / 85.9 
DE  74.2  79.0 / 79.5  83.2  80.8 
DK  76.4  79.0 / 79.4  81.3  78.1 / 81.6 
DB  78.7  75.9 / 76.9  77.6  77.6 
BD  78.2  78.1  78.3 / 78.6  79.1 
BE  71.1  76.9 / 77.3  78.8 / 80.4  77.2 
BK  76.9  81.0  82.6  82.6 
KE  82.2  81.6  83.4  80.4 
KD  73.1  72.9 / 74.4  71.8 / 72.0  76.1 
KB  73.2  71.7 / 73.1  70.2 / 75.1  73.9 
AVG  76.0  76.8 / 77.3  77.8 / 78.6  78.1 / 78.4 
69.4  71.3  72.8 

69.5  66.7 / 67.8  69.0 / 69.9 
74.8 / 75.0  82.9  84.0 
71.1/74.8  77.1 / 79.1  79.3 
73.4  80.4  78.5 / 78.6 
73.1 / 73.3  75.9  75.5 
72.8  76.8 / 77.7  77.3 / 78.2 
69.8 / 72.5  73.0 / 77.4  75.6 / 76.4 
75.3  79.9 / 80.5  75.4 / 79.2 
74.3  81.4 / 82.3  83.2 
70.9  72.7 / 73.4  73.8 / 74.1 
72.1  70.9 / 71.5  73.3 
72.2 / 72.8 
75.8 / 76.7  76.5 / 77.0 
We filtered original dataset for obtaining a Target Shift situation. It consists in rejecting randomly samples in order to obtain a target set with a desired amount of Target Shift. We investigate two cases of target shift: Soft Target Shift with and Hard Target Shift with . We report in Table 3 results of the experiment.
3.3.3 Analysis
We note our approach learns since it improves the model without Domain Adaptation of for the standard benchmark (see Table 1). In the situation of Concept Drift, our approach still improves the model without Domain Adaptation (). In these both cases, our approach remains significantly under the baseline based on invariance (respectively and ). This drop of performance is explained by the fact that invariance baseline has a major advantage since it has a prior knowledge on the target distribution by knowing . Nevertheless, in the context of Target Shift, baseline based on invariance fails to learn in such situation: the model degrades badly compared to the model without Domain Adaptation in the context of Hard Target Shift (). Our approach still allows to learn in such situation by improving performances with respect to the model without Domain Adaptation (smooth: , hard: ). For the standard benchmark, the Concept Drift and Target Shift situation, our approach is (globally) marginally better than the baseline based on invariance (from for the standard benchmark to for Hard Target Shift). This may be explained by the fact our approach can adapt itself to by learning to set making it a more generic formulation.
4 Related work
Addressing Target Shift is known as a difficult problem. (Manders et al., 2018) and (Yan et al., 2017) has shown it is possible to adapt respectively the adversarial framework of (Ganin & Lempitsky, 2014) and MMD based methods (Gretton et al., 2007) for learning invariant representation by reweighting at each training step the domain discrepancy measure with an estimated class distribution in the target domain. In the context of Optimal Transport based discrepancy measure (Arjovsky et al., 2017; Shen et al., 2018), (Chen et al., 2018) suggests to learn adversarially the class ratio incorporating it into the supremum over the dual critic function of the Wasserstein measure. Conditional Shift (where may change while keeping constant) is also a current assumption for extending invariant representation methods in challenging context of distributional shift. They traditionally use a localscale transformation for learning such (Zhang et al., 2013). (Gong et al., 2016) suggests a componentwise version of localscale transformations for avoiding noisy features which can not be well matched. Those methods are naturally extended in their original work to the context of both target shift and conditional shift using class ratio estimation.
Previous methods are essentially based on marginals or representations conditioned to label matching. However, Joint Domain Adaptation methods have recently received a lot of attention and seem wellsuited for tackling challenging distributional shift cases. (Courty et al., 2017; Damodaran et al., 2018) have shown it is possible to learn a non linear function minimizing an Optimal Transport cost between the joint distribution of on the source and the estimated joint distribution . Furthermore, they successfully adapted it to the context of target shift (Redko et al., 2018). (Long et al., 2016, 2018)
is the first, to our knowledge, to perform MMD on joint distributions. They suggest to join internal states of a neural network and to perform MMD in the tensor product of reproducing kernel Hilbert spaces associated to each layer. They claim their formulation handles harder distributional shift cases than traditional invariant representation methods since it weights each layer of the network with respect to others in the kernel mean embedding space. Our work mainly differs from the work
(Long et al., 2016, 2018) by proposing a constrained reweighting structure in the kernel mean embedding space. This structural constraint is derived from the hypothesis of Hidden Covariate Shift assumption.5 Discussion and Future Work
In the present work, we have explored a method for performing Unsupervised Domain Adaptation based on the Hidden Covariate Shift assumption. To our knowledge, proposed approach is novel and differentiates to current approaches by incorporating during learning the implicit assumption of Domain Adaptation: learning such that is conserved. We adapted the reverse validation method for model selection in our specific case suggesting Hidden Reverse Validation. Furthermore, our approach has the interest to take best of two traditional approaches, namely Covariate Shift and Domain Invariant Representation. We have shown the viability of our formulation in context of Target Shift or Concept Drift on both synthetic and realworld data. We reported performances comparable to stateoftheart approaches on Amazon Review dataset. Besides, we have observed that Hidden Reverse Validation may not always reflect the performance of selected model at test time on the target set. This may result to the fact we need to estimate a density ratio in high dimension which may be highly noisy.
In a future work, we want to exhibit situation where our formulation has a clear advantage with respect to stateoftheart approaches. We believe that invariance is a weaker constraint that invariance. One the one hand, this allows to learn representations with domain specific information although such information is not used during inference. This can be a desiderata in a context of MultiTask Learning or Transfer Learning. On the other hand, this weaker constraint leads to introduce an overparametrized formulation. Although it is not theoretically justified to look for invariance for learning with the intermediate state
, this has the advantage to regularize the model with a receivable heuristic. This intuition was confirmed during experiment when some learnings collapse to learn exactly the ratio of labels distribution (i.e.
). Therefore, future work will focus on introducing regularization compatible with the hypothesis of Hidden Covariate Shift. Finally, we want to extend our formulation to deep neural network where the network learns a sequence of representation such that at each layer is conserved across domains.Acknowledgments
This work was funded by Sidetrade and ANRT (France).
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Appendix A Optimization procedure details
Appendix B Training details: Kernels
We used the following kernels for computing Maximum Mean Discrepancy:

Liner kernels,

Gaussian kernels:
with

Quadratic kernels:
with
During learning, we need to compute kernel in joint space . Since is discrete, we suggest to cast the variable into a continuous variable .
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