EMTL: A Generative Domain Adaptation Approach
Authors: Jianfeng Zhang
Illyyne Saffar
Aladin Virmaux
Balázs Kégl
Venue:  (2021)
Keywords: unsupervised domain adaptation
EM
generative model
density estimation
deep learning
transfer learning

Abstract: We propose an unsupervised domain adaptation approach based on generative models. We show that when the source probability density function can be learned, one-step Expectation–Maximization iteration plus an additional marginal density function constraint will produce a proper mediator probability density function to bridge the gap between the source and target domains. The breakthrough is based on modern generative models (autoregressive mixture density nets) that are competitive to discriminative models on moderate-dimensional classification problems. By decoupling the source density estimation from the adaption steps, we can design a domain adaptation approach where the source data is locked away after being processed only once, opening the door to transfer when data security or privacy concerns impede the use of traditional domain adaptation. We demonstrate that our approach can achieve state-of-the-art performance on synthetic and real data sets, without accessing the source data at the adaptation phase.

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EMTL: A GENERATIVE DOMAIN ADAPTATION APPROACH
Anonymous authors Paper under double-blind review
ABSTRACT
We propose an unsupervised domain adaptation approach based on generative models. We show that when the source probability density function can be learned, one-step Expectation-Maximizaae
iteration plus an additional marginal density function constraint will produce proper mediator probability density function to bridge the gap between the source and target domains. The breakthrough is based on modern generative models (autoregressive mixture density nets) that are competitive to discriminative models on problems. By decoupling the source density estimation from the adaption steps, we can design domain adaptation approach where the source data is locked away after being processed only once, opening the door to transfer when data security privacy concerns impede the use of traditional domain adaptation. We demon- strate that our approach can achieve state-of of-the-art performance on synthetic and real data sets, without accessing the source data at the adaptation phase.
classification
INTRODUCTION
In the classical supervised learning paradigm, we assume that the training and test data come from the same distribution. In practice, this assumption often does hold. When the pipeline includes massive data labeling, models are routinely retrained after each data collecion campaign. However, data labeling costs often make retraining impractical. Without labeled data, it is still possible to train the model by using a training set which is relevant but not identically distributed to the test set. Due to the distribution shift between the training and test sets, the performance usually cannot be guaranteed.
Domain adaptation (DA) is machine learning subdomain that aims at learning a model from biased training data. It explores the relationship between source (labeled training data) and target (test data) domains to find the mapping function and fix the bias, so that the model learned on the source data can be applied in target domain. Usually some target data is needed during the training phase to calibrate the model. In unsupervised domain adaptation (UDA) only unlabeled target data is needed during training phase. UDA is an appealing learning paradigm since obtaining unlabeled data is usually easy in a lot of applications. UDA allows the model to be deployed in various target domains with different shifts using a single labeled source data set.
Due to these appealing operational features, UDA has became a prominent research field with var- ious approaches. Kouw & Loog (2019) and Zhuang al. (2020) surveyed the latest progress on UDA and found that most of the approaches are based on discriminative models, either by reweight- ing the source instances to approximate the target distribution or learning a feature mapping function to reduce the statistical distance between the source and target domains. After calibrating a discrim- inative model is trained on the adjusted source data and used in target domain. In this workflow, the adaptation algorithm usually have to access the source and target data simultaneously. However, accessing the source data during the adaptation phase is not possible when the source data is sensi- tive (for example because of security or privacy issues). In particular, in our application workflow an industrial company is selling devices to various service companies which cannot share their cus- tomer data with each other. The industrial company may contract with of the service companies to access their data during an R&D phase, but this data will not be available when the industrial company sells the device (and the predictive model) other service companies.

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In this paper we propose EMTL, a generative UDA algorithm for binary classification that does not have to access the source data during the adaptation phase. We use density estimation to estimate the joint source probability function p'(x. y) and the marginal target probability function p"(x) and use them for domain adaption. To solve the data security issue, EMTI. decouples source density estimation from the adaptation steps. In this way, after the source preprocessing delete the source data. Our approach is motivated by the theory on domain adaptation (Ben-David al., 2010) which claims that the error of a hypothesis h on the target domain can be bounded by three items: the eтror on the source domain, the distance between source and target distributions, and the expected difference in labeling functions. This theorem motivated us define mediator density function p" (xx,) y) i) whose conditional probability
can put away or
is equal to the conditional probability of the source and ii) whose marginal density on x equal to the marginal density of the target. We can then construct Bayes optimal classifier on the target domain under the assumption of covariate shift (the distribution y/ x is the same in the source and target domains).
Our approach became practical with the recent advances in (autoregressive) neural density estima- tion (Uria et al., 2013). We learn p= y) from p"(x, y) and p¹ px() to bridge the gap between the source and target domains. We regard the label on the target data as latent variable and show that p"(x y = i) be learned perfectly for é E {0, 1). then a one-step Expectation
Expectation-Maxiiizatiinn Maximization (and this is why our algorithm named EMTL) iteration will produce a density function y p= y) with the following properties on the target data: i) minimizing the Kullback-Leibles divergence between p" ("( x.) and ("a) x,): ii) maximizing the log-likelihood Σ log p (x,). Then, by adding an additional marginal constraint on p px") to make it close to px(x, on the target data explicitly,
we obtain the final objective function for EMTL. Although this analysis assumes a simple covariate shift, we will experimentally show that EMTL can go beyond this assumption and work well in other distribution shifts.
We conduct experiments on synthetic and real data to demonstrate the effectiveness of EMTL. First, we construct a simple two-dimensional data set to visualize the performance of EMTI. Second, we use UCI benchmark data sets and the Amazon reviews data set to show that EMTI is competitive with state-of-the-an art UDA algorithms, without accessing the source data at the adaptation phase. To our best knowledge, EMTL is the first work using density estimation for unsupervised domain adaptation. Unlike other existing generative approaches (Kingma et al., 2014; Karbalayghareh et al., 2018: Sankaranarayanan et al., 2018), EMTL can decouple the source density estimation process from the adaption phase and thus it can be used in situations where the source data is not available at the adaptation phase due security or privacy reasons.
RELATED WORK
Zhuang et al. (2020), Kouw & Loog (2019) and Pan & Yang (2009) categorize DA approaches into mnsance-based and feature-based techniques. Instance-based approaches reweight labeled source samples according to the ratio of between the source and the target densities. Importance weighting methods reweight source samples to reduce the divergence between the source and target densities (Huang et al., 2007; Gretton et al., 2007; Sugiyama et al., 2007). In contrast, class importance weighting methods reweight source samples make the source and target label distribution the same (Azizzadenesheli et al., 2019; Lipton et al., 2018; Zhang et al., 2013). Feature-based approaches learn new representation for the source and the target by minimizing the divergence between the source and target distributions. Subspace mapping methods assume that there is common subspace between the source and target (Fernando al., 2013; Gong et al., 2012). Courty et al. (2017) proposed to use optimal transport to constrain the learning process of the transformation function. Other methods aim at learning a representation which is domain-invariant among domains (Gong et al., 2016; Pan et al., 2010).
Besides these shallow models, deep learning has also been widely applied in domain adaptation (Tzeng et al., 2017; Ganin et al., 2016; Long et al., 2015). DANN (Ganin et al., 2016) learns a representation using a neural network which is discriminative for the source task while cannot distinguish the source and target domains from each other. Kingma et al. (2014) and Belhaj et al. (2018) proposed a variational inference based semi-supervised learning approach by regarding the missing label as latent variable and then performing posterior inference.
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3 NOTATION AND PROBLEM DEFINITION
We consider the unsupervised domain adaptation problem in a binary classification setting (the setup trivial to extend multi-class classification). Let p(x, y) be joint density function defined on X' x yi where x E I&P is the feature vector and E {0, 1] is the label. We denote the conditional probability p'y 11 x) by g(x). A hypothesis or model is a function h X' + [0, 1]. We define the error of h as the expected disagreement between hxx) and (xx, i..,
<(h) = Fx h(xx) ((x)l-
We use superscripts and t to distinguish the source and target domains, that is, p² y) and p' y) are the joint density functions in the source and target domains respectively. In general, we assume that p" (x. v) + p (x.
Let D [(xt, bi and UL' = {x]] be i...d. data sets generated from the source distribution px(x, y) and the marginal target distribution p" p'(x). respectively, where n² and n² are source and target sample sizes. The objective of unsupervised domain adaptation is to learn model h by using labeled D and unlabeled Lf", which achieves lowest error in target domain.
4 GENERATIVE APPROACH
Ben-David et al. (2010) proved that the error of a hypothesis in the target domain (h) can be bounded by the sum of error in source domain ch()) the distribution distance between the two domains, and the expected L distance between two conditional probabilities.
Theorem (Ben-David et al. (2010), Theorem 1) For a hypothesis h,
(h) S (h) di (1 (x) (x)) min (x o' (xx), Ex-op' lg (x)) -x
(x)))
where d1 d2(p" (x"(x), (x)) 2sup (B)) Pr((B) is the twice the total variation distance of two BEB
domain distributions and of (x) and (x) are the source and target probabilities of y 1| x, re- spectively.
In the covariate shift setting, we assume that the conditional probability p(y) x) is invariant between the source and the target domains. Thus in the right hand side of Eq. (2), the third component will be zero, which means that the target error is bounded by the source error plus the distance between two domains. Many current unsupervised domain adaptation solutions work on how to reduce the distance between the two domain densities. sample the source data to mimic the target data distribution, and feature- feature-mapping bapedd approaches do that by learning transformation function o(x) for the source data. However, both approaches need to access source and target data simultaneously.
aeportance-sampligg based approaches manage to re-
In this paper, we propose domain adaptation approach based generative models. First, learn all multivariate densities using RNADE (Uria et al., 2013), an autoregressive version of Bishop (1994)'s mixture density nets. We found RNADE excellent in learning mediun-dimennsonal densi- ties, and in a certain sense it is RNADE that made our approach feasible. Second, we introduce a mediator joint density function p"'(x, 3) that bridges the gap between p"(x., y) and p' y). Since the source distribution information is stored in the learned generative model after training, we do not need access source data in the adaptation phase.
4.1 DENSITY FUNCTION
Due to recent developments in neural density estimation, we estimate moderrate-dimensionnal densities efficiently. this paper, we use real-valued autoregressive density estimator (RNADE) of Uria al. (2013). RNADE is autoregressive version of mixture density nets of Bishop (1994) which fights the curse of dimensionality by estimating conditional densities, and provides explicit likelihood by using mixtures of Gaussians.
To estimate p(x), let x = [2, Z2, ... xp) be p dimensional random vector. RNADE decom- poses the joint density function using the chain rule and models each p(r; xxe) with mixture of
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Gaussians whose parameters depend on observed x Formally,
p(x) = II p(x x
II 2=1
Σ Mes
(3)
where X<i [x1,''.. - and d is the number of Gaussian components. The weights aj, means Mj. and variances o, are modeled by a single neural net whose architecture makes sure that the parameter (xxx) depends only X-ç- The neural net is trained maximize the likelihood of the training data. We denote the RNADE model by the function f(x;w) where w represents all the parameters (neural net weights) in RNADE, and use approximate p(x). The conditional density p(x 1g) can be estimated in the same way by just selecting x as the training data. In following sections, we denote the maximum likelihood parameters of p² (x ly = 0), p p"(x by 1), and p' (x) by w.o. Ws1, and wi, respectively. We further denote the proportion of class 0 in the source domain by 75o #(y-=0} The full parameter vector
[wko: W41: TsO] of p* (x, y) and p² p²(x) is denoted by 0g.
4.2 THE MEDIATOR DISTRIBUTION
By Eq. (2), the target error can be bounded by the source error plus the distance between the two marginal distributions plus the expected difference in ply 1|x) between two domains. This motivated us to construct mediator distribution pr v) (Figure 1) which has two properties:
it has the same conditional distribution as the source: p='(yl x) = p²(ylx), and
it has the same marginal distribution as the target: p=((x) = p'(x).
n (x)xx=p'(x)
pa [x, y) p"(yl (=( x)=
(x, y)
p p'(x,y)
Figure 1: The mediator has the same conditional probability as the source and the same marginal probability as target. According Theorem 1, we will have e (h) (h) for any hypotheses h since the last two terms are zero.
In the covariate shift setting, perfectly: i) the first property forces p(yl x) to be the same in source and mediator distribu- tions, and in the covariate shift setting have p²(y) x) p²(y) x), then this property makes ""| x) p'(y) x); ii) the second property makes the marginal distributions of the mediator and the target the same, which leads d₁ (x"(x) (x))
then solve the unsupervised domain adaptation problem
Under these two conditions, for any model h, we will have ehhh) < (h) since the last two terms of Eq. (2) will be zero. Furthermore, given the mediator distribution p" (x, 1/) it easy learn the best model (Bayes classifier)
h(x) p ly 1) ( (v= (v=
(4)
p (x)
which achieves the tightest bound for the target error. summary, by introducing the mediator dis- tribution px y). we can bound the target error by the mediator error. In the following sections, we will introduce how to learn p" (x, y) from p"(x, 8) and p' (x) using the expectation-mant maximizaiiinnn (EM) algorithm combined with marginal constraint term.
EMTL
If we regard the missing label y as latent variable that generates observed x in the target domain, we can use the EM algorithm to infer y. We consider that the target density p(x; 0) is a mixture with two components p(x ly i; 0) where i E {0, 1}. When 0 converges to its limit 0* in EM, we can recover the joint density function p(x, y: 8"). We denote this joint density function by p" (x, y). However, this p (x, y) may be far away from the ground truth p² (x y). The mismatch comes from two facts: i) EM can easily converge to a bad local minimum because of a bad initialization, and ii) EM tends find inner structure (e.g., clusters) of the data but this structure may be irrelevant
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to the true label. The local minimum problem is due parameter initialization, and the structure- label mismatching problem comes from not having a-priori information of the label. When we have fully known source distribution p'(x. y), these two issues can be solved by selecting proper initialization plus constraint on marginal distribution.
The first observation is that in lot of cases and it is better than random guess. We use this intuition to make the source model p"(x, y) as the initial guess of pmh y). Following section 4.1, we use RNADE model p" (y) and denote parameters of pd (x, y) by 0m = [wmo, Wm1 Fmol- Initializing p" (x, y) by using p*(x, y) means set om, the initial state of Om in the EM algorithm, to 0g. The next EM iterations can be seen as a way to fine-tune Om using the target data. In the next sections we will formally analyze this intuitive algorithm.
directly use the source model in the target domain
5.1 ANALYSIS !
First we link the EM algorithm with initial 0 0, to Theorem 1. In each iteration, EM alternates between two steps: E step defines Q function as Q(110()) x,00) log p(0; y) and M step do the maximization g(t-1) arg max ((0)0()). After the first EM iteration, we have
1
(()) arg max (8) ) arg max
Σ
E wl log p(x), y).
(5)
T2"
Suppose 0g is learned perfectly from source data, which means that we can replace p(x, yr ) by p²(x. y). Thus the expectation operation Eq. (5) can be written as
E F. xt 0.)
=
Σ (0,1)
p(us = 51x; x1xx3 0&)E
E ,'u jE(0,1)
(6)
for any random variable ç. This expectation links the source distribution with the target. We rewrite the full expectation expression of Eq. (5) as
E yydxx xt.0 log (ogp(x), 01:0)
Σ p' jl xx) logp(x;;, yi 0;
JE(0.1)
= DDi (x: I 0)) + log p(x; 0) - Hp x!).
where (ys/xx) x²) the conditional entropy on probability p². This equation shows that the ex- pected log-likelihood can be decomposed into the sum of three items. the first item is the negative KL-divergence between the two conditional distributions (3(y x²)) and p(y,) x"; f); the second item the target likeliicl log p(x) (0); the last item is the negative entropy of the source conditional distribution, which irrelevant to parameter so can be ignored during the optimization.
Therefore, by setting 100 Og and maximizing the Q function in the first EM iteration, we will get a px"(x, 3/) which minimizes the KI.-divergence between pd (y) x) with p²(y) x) and maximizes log pmh (x). Minimizing the KL-divergence reduces the third term of Eq. (2) and maximizing the log-likelihood forces p" (x) to move towards p² (xx implicitly, which reduces the second item of
Eq. (2). This suggests that the Bayes classifier nT ("(( x) can be proper classifier for target domain.
5.2 MARGINAL CONSTRAINT
In the previous section, we implicitly reduce the distance between pr (x) and (x(x by maximizing the log-likelihood of p(x; 0) on the target data. To further control the target error bound Eq. (2), we explicitly add a marginal constraint for p y) by minimizing the distance between the two marginal distributions. Rather than calculating di (n² (x), y (x)) directly, we use the KL--divereence to measure the distance between two distributions since we can explicitly calculate the p (x!) and p² (xt) by using our density estimators. Furthermore, according to Pinsker's inequality (Tsybakov, 2008), we have
di di ( (x)) 22D.I. (x))),
(8)
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thus minimizing the KL-divergence also controls d1 (x""(x), (x)). Since only have samples x! from the target domain, we use an empirical version of the KL-divergence. The marginal constraint is defined as
n2'
p' ('())
$ = -V2x x ( f(x):w)) log
[(x]:u))) & p(x;; 0)
M(0) = v2x
Σ
p'
(x+)
log
(9)
im (x!)
where P p/ Ep and f f/ are normalized discrete distributions on the target samples.
5.3 OBJECTIVE FUNCTION EMTL
By putting the Q and M functions together, get the objective function
0 arg min - Q(0 ( + 10)
(10)
of our generative domain adaptation approach, where 0 = 0g and is non-negative hyperpa- rameter that controls the trade-off of the two terms.
In real-life scenarios, both pxx) and plyl x) can be different in the source and target domains so the covariate shift assumption may be violated. To go beyond this assumption, we need relax the constraint on p"(yl x) p² (yl x) which is used justifying ((0)(()). As we will show in Section 6, by setting a large and doing more iterations, EMTL will reduce the weight on the Q function and allow us escape from covariate shift constraints. We summarize the process of EMTL in Algorithm 1.
Algorithm 1: EMTL Algorithm
Result: EMTL classifier (y = 1/x)
Initialize 0₂ [wso: Wa1, Tool and using D and Li", respectively;
Initialize ( by 0, and = 1;
while S n itr do ((i) arg ming Q((000 + (0);
t=t + 1;
end
nd (x, y) p(xxx, vi 000 ):
pm
(y 1| x)
pr ly=1) (y=1)
(1-1(0) )f (x(w(())) m0 (x(wwl) ) + f(xcw))))
(x)
EXPERIMENTS
In this section, present experiments on both synthetic (Section 6.1) and real-life data (Section 6.2) to validate the effectiveness of EMTL.
6.1 EXPERIMENTS ON SYNTHETIC DATA SET
We study the performance of EMTL under conditional shift where (x A (x(xly) using variant of inter-twinning moons example (Ganin et al., 2016). In the source domain we generate an upper moon (class 0) and lower moon (class 1) with 1000 points each class. In the target domain, we first generate 2000 samples in the source then rotate the data by 40 to make the target distribution of x ly different from the source. Figure (left) shows the source and target distributions. In this experiments,
set the number of Gaussian components to 10 and the hidden layer dimension to 30 the RNADE model.
We set 1 and 200 to illustrate how large 1) helps the model to escape from covariate shift constraint. Figure 2 (upper right) shows the prediction results in the target data using 7 = 1. When n_itr 0, the EMTL classifier is the source Bayes classifier. In the upper moon, the model mis- classifies the middle and the tail parts as class 1. This is because according to the source distribu- tion, these areas closer to class 1. The same misclassification occurs in lower moon. As itr

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n n_ttr=0
n_itr=1
n. n_itr=10
n. 1tr=t50
wedktes cast Prockted
Uniisblled Target
1.1
-2.5
22000
-2.5
0.0
2.5
-25 0.0 2.5
-25
0.5
2.5
-2.5
0.0
2.5
Figure 2: Inter-twining moons example. (Left) Samples from the source and target distributions where there is 40 rotation in target; (Right) EMTL result the target test data under different iterations and r7s. Small results in local optima. Larger 7) allows the objective function to escape from the p"(yl x) p (yl x) constraint which is wrong in this case.
increases, the misclassification reduces slightly, because the objective function focuses more on op- timizing the Q function thus keeping p(y x) stable in each iteration. As contrast, in Figure 2 (bottom right), when setting 77 200, the first iteration reduces the misclassification significantly and finally the error converges to zero. By setting large 7). the conclusion of this example is two- fold: i) the pyy) x) p' ('yyl x) constraint will be relieved thus resulting in a better adaptation result, and ii) one-step iteration will increase the performance significantly thus suggesting that we do not need too many iterations. According to ii), our following experiments the n_itr is fixed as 1. We show more experimental results using different r7s in Appendix A.1 and Figure 3.
6.2 EXPERIMENTS ON REAL-LIFE DATA SETS
In this section, we validate EMTL on real-life data sets by comparing its performance with two standard supervised learning and three domain adaptation algorithms. The validation is conducted on three UCI data sets and the Amazon reviews data set. First, we create two benchmarks: the source RF/SVM is the model trained only using source data (as a baseline) and the target RF/SVM the model trained only using labeled target data (as an upper bound). A random forest (RF) classifier used on the UCI data sets and support vector machine (SVM) is used on the Amazon reviews data set. The three DA algorithms are kernel mean matching (KMM, Huang et al. (2007)), subspace alignment (SA, Fernando et al. (2013)) and domain adversarial neural network (DANN, Ganin et al. (2016)). For the UCI data sets, both KMM and SA are based RF and for Amazon reviews data set SVM is used. In KMM., we us an RBF kernel with the kernel width set as the median distance among the data. In DANN, À fixed as 0.1. In EMTL, we set the number of components to 5 and the hidden layer size to 10 for RNADE model and to 1. For each transfer task, five-fold cross validation (CV) is conducted. In each CV fold, we randomly select 90% source samples and 90% target samples respectively to train the model. We average the output of the five models and calculate the 95% confidence interval of the mean. For the UCI tasks, ROC AUC score is the used metric since we are dealing with imbalanced classification tasks. For Amazon reviews tasks accuracy is the used metric. Table and 2 summarize the experimental results. Numbers marked in bold indicate the top performing DA algorithms (more than one bold means they not significantly different).
UCI data sets. Three UCI data sets (Abalone, Adult, and Bank Marketing) are used in our experi- ments (Dua & Graff, 2017; Moro et al., 2014). We preprocess the data first: i) only select numerical features; ii) add uniform noise to smooth the data from integer to real for Adult and Bank data sets. Since the original goal in these data sets is not transfer learning, we use a variant biased sampling approach proposed by Gretton et al. (2009) and Bifet & Gavaldà (2009) to create different domains for each data set. More precisely, for each data set we train a RF classifier to find the most important feature, then sort the data along this feature and split the data in the middle. We regard the first 50% (denoted by A) and second 50% (denoted by B) as the two domains. When doing domain

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Table 1: Experimental results UCI data sets. AUC(%) is used metric.
Task
Source
Target RF
KMM
DANN
EMTL
Abalone A +
67.1 1.1
72.7 0.5
66.5 1 2.2
67.8 1 0.6
67.5 0.4
65.7 1 2.8
Abalone B + A
67.5 + 1.2
81.2 0.4
59.4 4.6
68.5 + 2.1
69.5 0.7
70.8 1 0.7
Adult A
84.4 1.44:00 0.2
84.8 0.2
83.4
82.8 0.2
84.7
84.8 1 0.3
Adult B
82.1 0.1
8.1
81.3
81.0
82.8
82.7 + 0.4
Bank A +
70.1 0.3
8.1
69.3
70.4
70.8 0.5
70.5 1.7
Bank B + A
76.7 + 0.7
0.6
74.8 + 0.5
76,6 0.4
78,4 0.2
79.3 + 0.8
Table 2: Experimental result on Amazon reviews data set. Accuracy(%) is used metric.
Task
Source SVM
Target SVM
KMM
SA
DANN
EMTL
B +
80.0 + 0.0
79.9
79.7
79.9 0.1
79.9 + 0.0
79.5
B +
70.3 + 0.1
0.2
0.2
0.3
0.2
B + K
75.7 0.1
0.1
76.3
0.1
7..7
76.0
D
0.0
0.1
0.1
0.1
7..4
75.7
0.1
7.12
74.6
74.4
7..5
0.2
D + K
0.1
0.0
0.1
0.1
0.3
7.21
E + B
0.1
0.1
0.1
0.1
70.5
0.3
E + D
0.0
73.1
7.3
7.11
7.11
72.7 0.2
E + K
85.8
0.0
0.8
0.1
8.2
85.3
71.5
0.1
0.2
0.0
7.1
71.6 0.1
K -
70.6 0.0
7.2
0.3
0.1
7.16
7.26
K B
83.9 : 0.0
84.3
84.2
8.1
8.10
83,9 = 0.2
adaptation, we use 75% of the target domain samples to train the model and use other 25% target domain samples test data. Finally, and target data sets respectively. Table 3 Appendix A.2 summarizes the features of the data sets we created for the experiments. Table shows the results the test data for UCI data sets. We find that the performance of EMTL is not significantly different from DANN all tasks (remember that our goal was not the beat the state of the art but match it, without accessing the source data the adaptation phase). On the two Adult tasks and Bank - A, although the average score of EMTL less than that of Target RF, the differences are small.
normal quantile transformation normalize the source
Amazon reviews. This data set (Ganin al., 2016) includes four products, books (B), DVD (D), electronics (E) and kitchen (K) reviews from the Amazon website. Each product (or domain) has 2000 labeled reviews and about 4000 unlabeled reviews. Each review is encoded by a 5000- dimensional feature vector and binary label (if it is labeled): O if its ranking is lower than three stars, and otherwise. We create twelve transfer learning tasks using these four domains. As RNADE is not designed for ultra high dimensional cases, overcome this constraint by reducing the number features from 5000 to 5 using a feed forward Neuronal Network (FNN). More pre- cisely, for each task we train 2-hidden layer FNN the source data. Then, cut the last layer and we use the trained network to encode both source and target to dimensions. Table 2 shows the results the data for Amazon reviews data set. We notice that EMTL is slightly better than DANN in most of the tasks and still comparable with both KMM and SA.
CONCLUSIONS AND FUTURE WORK
In this paper, we have presented proach EMTL. Thanks to the excellent performance of autoregressive mixture density models (e.g., RNADE) on medium-dimenniionll problems, EMTL is competitive to state-of-the-art solutions. The advantage of EMTL is to decouple the source density estimation phase from the model adaptation phase: we do not need to access the source data when adapting the model the target domain. This property allows our solution to be deployed in applications where the source data is not available after preprocessing. In our future work, we aim to extend EMTL to more general cases, including high-dimensional as well more complex data (e.g., time series).
ponssit-estimation-aaass unsupervised domain adaptation ap-
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Under review as a conference paper at ICLR 2021
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Marouan Belhaj, Pavlos Protopapas, and Weiwei Pan. Deep variational transfer: Transfer learning through cemi-supervised deep generative models. arXiv preprint arXiv:1812.03323, 2018.
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Christopher M. Bishop. Mixture density networks. Technical report, 1994.
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Alexandre B Tsybakov. Introduction to nonparametric estimation. Springer Science & Business Media, 2008
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Kun Zhang, Bernhard Schölkopf, Krikamol Muandet, and Zhikun Wang. Domain adaptation under target and conditional shift. In International Conference on Machine Learning, pp. 819-827, 2013.
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A APPENDIX
A.1 INTER-TWINNING MOONS EXAMPLE
We test three settings and compare the corresponded AUC and accuracy in Appendix Figure 3. We find that as n itr increase, the AUC and accuracy will increase too. In each fixed itr, larger 7) always has higher AUC and accuracy.
AUC
Accuracy
1.0
1.0
0.9
0.9
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n= 100.0
n 200.0
200.0
0.7
0.7
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3
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Figure 3: inter-twinning moons example, as 7j increase, both AUC and accuracy will increase.
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Under review as a conference paper at ICLR 2021
A.2 UCI EXPERIMENTS
We summarize the size and class ratio information of UCI data sets in Appendix Table 3.
Table 3: UCI data sets
Task
Source
Target
Dimension
Class (/1(uurrce)
Abalone A ++ B
2,088
1,566
523
24% vs. 76%
Abalone B -> A 2,089
1,566
522
76% 24%
Adult A
16,279
12,211
4,071
75% 25%
Adult + A 16,282
12,209
4,070
77% vs. 23%
Bank A + B
22,537
17,005
5,669
97% vs. 03%
Bank B 7 A 22,674
16,902
5,635
80% vs. 20%
Parameter settings in UCI data sets. We enumerate the parameter settings on UCI experiment here.
Random forest models with 100 trees are used as the classifier.
For DANN, set the feature extractor, the label predictor, and the domain classifier as two-layer neural networks with hidden layer dimension 20. The learning rate is fixed as 0.001.
For EMTL, fix the learning rate 0.1 except for the task Abalone B - A (where set to 0.001) as it did not converge. As mentioned in section 6.1, we only do one EM iteration.
Parameter settings in Amazon reviews dataset. We enumerate the parameter settings choice of Amazon reviews experiment here.
SVM has been chosen over RF because it showed better results in the case of Amazon reviews experimentation
We run a grid search to find the best C parameter for SVM over one task (from books to dvd) the best result C 4.64E 04 is then used for all tasks and for source svm, target svm, KMM and SA solutions.
For DANN, we set the feature extractor, the label predictor, and domain classifier as one-layer neural networks with hidden layer dimension 50. The learning rate is fixed as 0.001.
FNN composed of hidden layers of dimensions 10 and (the encoding dimension). we added Gaussian Noise, Dropout, Activity Regularization layers in order to generalize better and guarantee better encoding on target data.
For EMTL, fix the learning rate 0.001 and only do one EM iteration.
Note that the presented result of Amazon reviews data set in Table have been rounded to digit. This explains why the 95% confidence interval the mean is sometimes equal to 0.0 and why some values not in bold.
11


Reviews:
[{"review_id": "w7zf12ZF9vZ", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Update after author response: I appreciate the clarifications, but given the lack of comparisons or discussion to related prior methods (at least Liang et al 2020 or some alternative equivalent), I cannot recommend acceptance at this point. The authors did not submit a paper revision as well. I think the idea seems promising, so don't take this as a critique of the research direction.\n\n#########################################################################\n\nSummary:\n\nThis paper tackles the problem of unsupervised domain adaptation, where there may be privacy constraints on the source, so we have access to a source model but not the source data. Their approach first learns a generative model p(x, y | theta_s) for the source. They use this to pseudolabel every target example p(y^t_i | x^t_i, theta_s), and then fit a new generative model to these examples. They iterate this process multiple times to get their final model p(x, y | theta), from which they can predict p(y | x, theta).\n\n#########################################################################\n\nReasons for score:\n\nI think this is a cool idea, but the paper in its current form seems incomplete. One of the goals is to preserve privacy of the source dataset, but it seems unlikely that a generative model p(x, y | theta_s) learned on the source preserves privacy. In particular, we could effectively “generate” the source and get access to private information. Is there any prior work that suggests that this is not the case? This seems to be a key point, since barring privacy concerns their method doesn’t seem to do better than baselines. The conceptual explanation for the method seems unclear and incomplete (more below). The method could also be better tied into /compared with the existing privacy preserving domain adaptation literature, and the self-training literature. Overall, this is a promising and interesting idea, and with more work would be good to publish.\n\n#########################################################################\n\nPros:\n\n- I like the idea of using a generative model for x and y, and repeatedly applying that by pseudolabeling the target and self-training.\n\n- Their synthetic experiments show some promise, and on some of the UCI datasets they seem to do well.\n\n\n#########################################################################\n\nCons:\n\n- No explanation given for why a generative model p(x, y | theta_s) learned on the source preserves privacy, when having a generative model would allow us to generate samples and potentially get sensitive information. This seems to be a key point of the paper, since experimental results are not better than baselines, but the argument is that those don’t preserve privacy.\n\n- There are many other related alternatives. One prominent alternative is self-training or pseudolabeling based approaches, which train a classifier on the source, and use it to pseudolabel the target, then training a regularized classifier on the pseudolabeled data. This, and related methods such as entropy minimization, has been used in the context of domain adaptation (Shu et al, Kumar et al), also in the context of privacy preserving domain adaptation (Liang et al). These methods are highly scalable as well, and all the above papers use multi-layer neural networks and high dimensional datasets. Would be good to compare with at least some existing baseline.\n\n- The conceptual / mathematical explanation seems unclear and incomplete. If there is only covariate shift, then P_m and P_t should be identical. This is because P_t and P_m have the same distribution over x. P_m has the same y | x distribution as P_s, but in covariate shift this is the same as P_t. So it’s unclear what P_m is doing. Section 5.1 argues that the EM training objective enforces P_m to have similar x distribution to P_t and similar y | x distribution to P_s. However, there isn’t a proper explanation of why this objective is a good thing. There seems to be an attempt to explain this in Section 4.2, but in the case of no covariate shift, the Bayes classifier on the source = Bayes classifier on the target, so it suffices to simply train a model on the source.\n\n#########################################################################\n\nQuestions and things to improve:\n\nPlease address questions and comments listed in cons.\n\n#########################################################################\n\nReferences mentioned:\n\nA DIRT-T Approach to Unsupervised Domain Adaptation. Rui Shu, Hung H. Bui, Hirokazu Narui, Stefano Ermon. ICLR 2018.\n\nUnderstanding Self-Training for Gradual Domain Adaptation. Ananya Kumar, Tengyu Ma, Percy Liang. ICML 2020.\n\nDo We Really Need to Access the Source Data? Source Hypothesis Transfer for Unsupervised Domain Adaptation. Jian Liang, Dapeng Hu, Jiashi Feng. ICML 2020.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "tQzg5ouEnKX", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "In this paper, the authors propose generative domain adaptation approach called EMTL. The key idea is to model a mediator distribution which can approximate the true target joint distribution. Specifically, the authors apply an E-M strategy to infer the model parameters. Experimental studies are done on both synthetic and real-world datasets. \n\nThe paper is well-organized and easy to follow. My major concern is on the significance of the paper, which is not significantly novel especially compared with the recent DNN-based domain adaptation methods. Moreover, there are some technical flaws, which need to be further clarified. Here are the detailed comments:\n\n(1)\tOne motivation of the paper is the sensitivity of the source data. Due to security or privacy issues, source data may not be accessible. While it is a practical and nice point of motivation, the paper misses one important research line on federated learning that is specially proposed for privacy issues. It is necessary to discuss with some federated learning related works. \n\n(2)\tRegarding the access of the source data, the proposed method still requires the source data to do source density estimation. In this sense, I am not convinced by the claim on the privacy preservation. From Algorithm 1, it can be clearly seen that D^s is still used for the initialization of \\theta_s. \n\n(3)\tThe authors highlight no access of source data in the adaptation phase. Could you elaborate on what is the adaptation phase? Based on my understanding, the whole Algorithm 1 is for adaptation, but it still needs source data. Is it better to claim that the proposed method only requires source model parameters trained previously? \n\n(4)\tThe related work section can be further improved, by discussing more on both subspace-based and deep-learning based domain adaptation methods. \n\n(5)\tThe key idea is based on Theorem 1, and aims to build a mediator distribution to approximate the target joint distribution. Most of existing domain adaptation methods share the same idea although they are not generative models, please highlight the main advantage of proposed generative model over existing subspace-based and deep-learning based methods. \n\n(6)\tRegarding Eq. (5), why \\theta_s is used in the subscription of \\mathbb{E}? It should be \\thetha_m^(0), right? Moreover, how to obtain y_i for each target data point? \n\n(7)\tFor Eq. (5), does it only hold for the first iteration where the source parameters are used as the initialization? For the following iterations, are you still using p^s(y_i = j | x_i^t)? or using p^m(y|x)? If the latter is used, does it mean y_i is updated in each iteration? \n\n(8)\tThe proposed synthetic dataset is very naïve. It is more convincing to test on more complex datasets with higher dimensionality data.  For the real-world datasets, there are a lot of benchmark datasets for domain adaptation, e.g., office 31, office-caltech 10, and office-home etc. It is more convincing to test on these well-known datasets. More importantly, please compare with more state-of-the-art baselines, on both subspace-based and deep learning based. Even on the reported datasets, the improvements of EMTL over SA and DANN (these 2 are not state-of-the-art) are marginal. \n\nUpdate: Thanks for the authors' response. After reading the response and the other reviewers' comments, I think the paper needs to be further improved, and thus I will keep my score.\n\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "hSmZ-PufiD", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "--------Updates after rebuttal-----------\n\nSince the author did not propose an updated paper and new experiments. I keep my original score.\n\n---------------------------------------------------\n\n\n\nSummary:\nThis paper proposed a generative domain adaptation (DA) approach under covariate shift. Different from previous domain discriminator methods, this paper introduced a mediator distribution and adopted an autoregressive approach (RNADE) to estimate the distribution density. Empirical results on simple datasets (UCI and Amazon) verified its practical benefits.\n\n------------------------------------------------------\nOverall review \n\nPros:\n\n[1] As far as I know, this is the first paper that used the autoregressive approach in DA. \n\n[2] The proposed adaptation algorithm does not require accessing the source data at the adaptation phase, which has some practical potential.\n\n[3] The high-level idea seems logical and correct (But some technical details seem problematic.)\n\nCons:\n\n[1] The motivation of the proposed approach is unclear: it seems a simple plug-in approach with RNADE in DA. A thorough analysis is lacking.\n\n[2] The empirical significance of the paper is rather limited: the paper did not effectively show its practical utility.\n\n[3]  Some technical details are difficult to follow or flawful. \n\nBased on these reasons, I recommend rejection.\n\n--------------------------------------------------\nDetailed explanations\n\n[1] Motivation\n\nI am rather confused about the motivation of the proposed approach. As for the generative model, the particular reason to use RNADE is unclear. Is the discriminator unable to solve the source-target separation issue? An alternative approach is to train a model on source only and apply the unlabelled target for fine-tuning. (see recent paper [1]). Discussion on the benefits of these settings is highly expected. \n\n[2] Experiments\n\nSince it is an empirical paper, I am most concerned about the empirical results.\n\n[a] The current results are rather limited. The author only evaluated on UCI and Amazon review dataset. Both are simple datasets and linear models can achieve good results.\n\n[b] The compared baselines are NOT SOTA. DANN is the standard baseline.\n\n[3] Technical details\n\n[a] The RNADE is a high time complexity approach for high dimensional data. I would like to see an empirical and theoretical discussion on the high-dimensional dataset (such as the image)\n\n[b] The notation in the EM algorithm is rather confusing and difficult to follow. \nBesides, Eq (7) is the log-MLE approach, then it can be naturally decomposed in three terms. Eq (9) is not correct, $p(x)$ should be a continuous function (not discrete). Using the empirical counterpart to estimate the KL divergence is problematic in the high dimensional dataset. \n\n[c] Sec 5.3 “As we will show in Section 6, by setting a large $\\eta$ and doing more iterations, EMTL will reduce the weight on the Q function and allow us to escape from covariate shift constraints”. This discovery is really important and interesting. I think it deserves a better justification.\n\n--------------------------------------------\nSuggestions \n\nI suggest extensive empirical results for showing the effectiveness of the proposed approach.\n\n\n[1] Do We Really Need to Access the Source Data? Source Hypothesis Transfer for Unsupervised Domain Adaptation. ICML 2020\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "97XgRYd7qC", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a novel method for Unsupervised Domain Adaptation (UDA) when the source domain's privacy should be preserved. The authors propose EMTL, which is a generative method using multivariate densities using RNADE (Uria et al., 2013) and a mediator joint density function bridging both source and target domains. EMTL achieves comparable performances to those of DANN (Ganin et al., 2016) on a single dataset.\n\n**Pros**\n\n- Unique motivation for UDA and privacy-preserving.\n- Well formulated method using RNADE and a mediator density function. In the adaptation phase, the source domain data can be deleted.\n\n**Cons**\n\n- There is a closely related paper for privacy-preserving UDA (Song et al., 2020) before the deadline of ICLR 2021. The method by Song et al. utilized a framework of federated learning and encryption. Thus the approaches are different from each other, but the motivation for privacy-preserving is close. The authors should compare them quantitatively.\n\nSong et al. Privacy-Preserving Unsupervised Domain Adaptation in Federated Setting. IEEE Access, Vol. 8, pp.143233-143240, 2020.\n\n- Although the adaptation phase does not require the source domain data, a probabilistic function $p^m(y|x)$ should be available. The reviewer just concerns if model inversion attacks, such as (Fredrikson et al., 2015), violate the source domain's privacy.\n\nFredrikson et al. Model Inversion Attacks that Exploit Confidence Information and Basic Countermeasures. In CCS, 2015.\n\n- It is reasonable to compare ETML to DANN since both methods have conceptually similar characteristics: matching the data distribution and learning the posterior probabilities of the label given a sample. However, as the authors referred to in the main text, several methods have similar characteristics and much better performance than DANN. It is good to know if ETML is complementary to those methods through further experiments.\n- Experiments on a single real dataset are difficult to convince about the generality of UDA performance. Additional experiments on other datasets such as visual ones can strengthen the generality.\n\n**Overall rating**\n\nThe reviewer is leaning toward rejection, although the motivation is clear. The rating can be upgraded if the authors can solve the cons above.\n\n**Additional comment after rebuttal**\n\nHappy to hear that the authors plan to upgrade their draft. Since the submitted paper is not updated, the reviewer keeps the first rating but also looks forward to read a revised version in another conference or journal.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This work proposes an EM type of approach for domain adaptation under covariate shift. The approach well motivated and developed and experimentally evaluated on synthetic data.

Pro:
- The EM type of framework is simple and natural and  promising direction for DA, which should be explored and analyzed further.

Con:
- The presentation is highly overselling the results. Both in terms of the generality of the findings and in terms references to privacy preserving properties. Both would need a solid formal analysis which this submission does not provide.
- Several reviewers have stated that, while the authors promised updates to their manuscript during the author response phase, no such updated submission has been made.
- The work bases their approach by referring to a well known theoretical DA bound by Ben-David et al (2010). The theorem is not stated correctly. The most important component in that work is to restrict the models to a class of bounded capacity.
- The claim of the authors of "solving the problem" under covariate shift are overstated. It is reasonable to expect that the authors provide a more thorough analysis of the limitations or their approach, that is, clearly state the conditions under which it would succeed and fail. Below are some references on lower bounds of DA under covariate shift.
- Given that the theoretical analysis is limited, a more thorough experimental exploration would be expected.

Refs on difficulty of DA learning under covariate shift and bounded d_H distance:

Shai Ben-David, Ruth Urner:
On the Hardness of Domain Adaptation and the Utility of Unlabeled Target Samples. ALT 2012: 139-153

Shai Ben-David, Tyler Lu, Teresa Luu, Dávid Pál:
Impossibility Theorems for Domain Adaptation. AISTATS 2010: 129-136 
Decision: Reject