Unsupervised domain adaptation with imputation

1 INTRODUCTION . When dealing with machine learning applications in the real world , data usually come with several imperfections that make classical algorithms hardly deployable . One of these issues is that data are often incomplete . Typically , while capturing data coming from different locations with several sensors per location , a sensor may randomly fail or even may be just missing at a given location . Such a situation can also occur in disease diagnosis in multi-modal medical imaging where one of the modalities fails or is not available ; for example the positron emission tomography ( PET ) modality which reveals metabolic information for clinical tests requires ingesting a radioactive tracer which poses health risks and is often missing ( Cai et al. , 2018 ) . Similarly , in computational advertising applications , information is missing for users who do not have a prior history on a merchant ’ s website while their global clicking behavior across websites may be known . Another common issue is that data used for training and deployment may differ in their generation process : data may be collected on different devices , background noise or compression schemes may affect differently training or deployment data , leading to a shift in data distribution . This has given rise to the important literature of Domain Adaptation ( Pan & Yang , 2010 ; Kouw & Loog , 2019 ) . These two issues are usually independently addressed by developing models handling only the missing data or the domain adaptation problem . In this paper , motivated by practical advertising applications , we consider unsupervised domain adaptation ( i.e . labels are not available in the target domain ) for classification when ( 1 ) part of input data is missing in the target domain thus requiring some form of imputation , ( 2 ) there is no possible supervision in the target domain for imputation thus requiring indirect supervision from the source domain , and ( 3 ) there exists a domain shift between the source and target distributions requiring domain adaptation . More precisely we consider this adaptation-imputation setting for nonstochastic missing data , i.e . when the same features are missing for all target samples . This contrasts with many imputation problems which take benefit of stochasticity in missing features . We propose a model that handles unsupervised domain shift and missing data assuming nonstochastic missing data in the target domain . The model learns to perform imputation for the target domain while aligning the distributions of the source and target domains in a latent space , thus going beyond the simple juxtaposition of a data imputation module followed by a domain-invariant feature representation learning module . Imputation makes use of an indirect supervision from the complete source domain . This key property allows us to handle non-stochastic missing data , while satisfying the constraints related to adaptation and to the classification objective . The imputation process plays several roles in our global architecture as it provides us with information about the missing data for the target domain while contributing to the domain-invariant loss and the reconstruction loss . Extensive empirical evidence on handwritten digits and Click-Through-Rate prediction ( CTR ) domain adaptation problems illustrate the benefit of the proposed model . The original contributions are the following : • We introduce a new problem : joint unsupervised domain adaptation and imputation for classification motivated by practical applications ; • We propose a new model for handling the problem end-to-end . It learns to generate relevant missing information while aligning source and target distributions in a latent space and to classify source instances ; • We evaluate the model not only on academic benchmarks but also on challenging real world advertising data . 2 RELATED WORK . We review below typical related work for domain adaptation and data imputation . 2.1 UNSUPERVISED DOMAIN ADAPTATION ( UDA ) . A number of shallow learning methods approach Domain Adaptation by weighting individual observations during training . These methods focus either on data importance-weighting ( Cortes & Mohri , 2014 ; Zadrozny , 2014 ) or on class importance-weighting ( Z. Lipton & Smola , 2018 ) . Recent deep learning methods try to align the distributions of the two domains , for example by embedding them in a joint latent space . There are two main directions for learning joint embeddings . One is based on adversarial training , making use of GAN extensions . The other one directly exploits explicit distance measures between distributions such as Wasserstein or Maximum Mean Discrepancy ( MMD ) . For the former , the seminal work of Ganin & Lempitsky ( 2015 ) learns to map source and target domains onto a common embedding space , by optimizing a double objective : on the one hand they minimize an approximation of the H-divergence between the source and target embeddings via adversarial training , on the other hand they learn to classify the source data embeddings . This influential work has been followed by several extensions and variants . ADDA ( Tzeng et al. , 2017 ) advocates the use of two different mappings for the source and the target domains based on the argument that this is more suitable when the marginals are different in the two domains . Liu & Tuzel ( 2016 ) trains coupled generative adversarial network ( CoGAN ) for learning a joint distribution of multi-domain images , that can be used for UDA . Bousmalis et al . ( 2017 ) use a generator to map the source to the target domain while training the classifier on the learned representations using source labels . CDAN ( Long et al. , 2017 ) improves the domain discriminator by conditioning it on classifier predictions . A second family of approaches proposes metric based divergences such as MMD ( Long et al. , 2015 ) for measuring the loss between source and target representations . DeepJDOT ( Damodaran & Kellenberger , 2018 ) makes use of an optimal transport formulation to align the joint distributions in a latent space . In addition to feature alignment they perform label distribution alignment following Courty et al . ( 2017 ) . All these works rely on the assumption of covariate shift and consider that full input data is available for both source and target domains . Our two models ( ADV and OT ) can be seen respectively as extensions of Ganin & Lempitsky ( 2015 ) and Damodaran & Kellenberger ( 2018 ) for the missing data problem . 2.2 IMPUTATION . Data imputation is a classical problem addressed by several methods ( Little & Rubin , 2002 ; Van Buuren , 2018 ; Murray , 2018 ) . The usual setting is different from ours since it considers reconstructing the whole missing data in the input space , while we consider 1 ) reconstruction in a latent space and 2 ) partial reconstruction since we are interested in the information relevant to the classification task only . Recent generative models like GANs ( Goodfellow et al. , 2014 ) or VAEs ( Kingma & Welling , 2013 ; Rezende et al. , 2014 ) have been adapted for data imputation in Yoon et al . ( 2018 ) and Mattei & Frellsen ( 2019 ) respectively . GAIN ( Yoon et al. , 2018 ) is an extension of conditional GANs where the generator takes as input an incomplete data and performs imputation while the discriminator is trained to guess for each sample if each variable is original or imputed . Mattei & Frellsen ( 2019 ) suggests a method based on deep latent variable models and importance sampling that offers tighter likelihood bound compared to the standard VAE bound . Most approaches consider a supervised setting where 1 ) paired complete and incomplete data are available and 2 ) missingness corresponds to a stochastic process ( e.g . a mask distribution for tabular data ) , 3 ) imputation is performed in the original feature space . Note that this is different from our setting where there is no direct supervision ( supervision is only provided indirectly through the source domain ) and missingness is non-stochastic which makes the problem harder since one can not compute statistics on different incomplete samples . The general approach with generative models is to learn a distribution over imputed data which is similar to the one of plain data . This comes in many different instances and usually , generative training alone is not sufficient ; additional loss terms are often used . In paired problems , i.e . when each missing datum is associated to a plain version of the datum , these additional terms consist of a reconstruction term imposed by a MSE contraint ( Isola et al. , 2016b ) . In unpaired problems a cycle-consistency loss is imposed as in Zhu et al . ( 2017 ) . Li et al . ( 2019 ) ; Pajot et al . ( 2019 ) are among the very few approaches addressing unsupervised imputation in which full instances are never directly used . Both extend the AmbientGAN ( Bora et al. , 2018 ) framework and consider stochastic missingness . Our imputation problem is closer to the ones addressed in some forms of inpainting or for multimodality missing data . The former problem is addressed e.g . in Pathak et al . ( 2016 ) who proposes an encoder-decoder model trained according to a joint reconstruction and adversarial loss . The latter is addressed in Cai et al . ( 2018 ) who considers the case of multi-modality when one or more modalities are systematically absent , but they do not consider adaptation . They propose to learn to reconstruct the missing modality distribution conditionally to the observed one . Both approaches are fully supervised . Ding et al . ( 2014 ) is the only paper we are aware of that considers imputation as we do . Their approach is based on low rank constraints and dictionary learning to guide the transfer between domains . We do not use this method as a baseline due to the complexity and running time of this method which relies on singular value decompositions and dictionnary learning . 3 PROBLEM DEFINITION . Let us denote respectively ( xS , yS ) ∈ Rn × R and ( xT , yT ) ∈ Rn × R , data from the source and target domains where x− is an input , y− the associated label and n is the dimension of the input space . “ − ” holds for either source S or target T . The joint distribution on each domain is denoted respectively pS ( X , Y ) and pT ( X , Y ) . We consider that x− has two components , x− = ( x−1 , x−2 ) . The problem we address is Unsupervised Domain Adaptation ( UDA ) with missing features in the target domain . More precisely , we make the following hypotheses . Missingness : We assume that features are the same across domains and that source features xS = ( xS1 , xS2 ) are always available while in the target domain only xT1 is available and xT2 is systematically missing . For advertising applications for example , x would characterize the user browsing behaviour on merchant sites ; x−1 characterizes global user features aggregated over his navigation history , which are known for all users ; x−2 characterizes user history on a target merchant site . Source domain would consist of all users who already visited this merchant site and target domain of users who never visited this site . UDA : we assume that source labels yS are available whereas target labels yT are unknown . Covariate shift : we assume covariate shift as in most UDA papers e.g . Ganin & Lempitsky ( 2015 ) .

Summary: .* The paper presents and addresses the problem of performing domain adaptation when the target domain is systematically (i.e., not the result of a stochastic process) missing subsets of the data. The issue is motivated by applications where one modality of data becomes unavailable in the target domain (e.g., when deciding which ads to serve to new users, the predictor may have access to behavior across other websites but not on a specific merchant's website). The proposed method learns to map source and target data to a latent space where the representations for the source and target are aligned, the missing components of the target can be inferred, and classification can be performed successfully. These are achieved by adversarial/optimal transport loss on source and target features, a mean-squared error and adversarial loss on latent generation/imputation, and a cross entropy loss on source label prediction, respectively. Experiments are performed on digits and click-through rate (CTR) prediction and include a thorough set of baselines/oracles for comparison.