Graph-Relational Domain Adaptation

1 INTRODUCTION . Generalization of machine learning methods hinges on the assumption that training and test data follows the same distribution . Such an assumption no longer holds when one trains a model in some domains ( source domains ) , and tests it in other domains ( target domains ) where data follows different distributions . Domain adaptation ( DA ) aims at improving performance in this setting by aligning data from the source and target domains so that a model trained in source domains can generalize better in target domains ( Ben-David et al. , 2010 ; Ganin et al. , 2016 ; Tzeng et al. , 2017 ; Zhang et al. , 2019 ) . Existing DA methods tend to enforce uniform alignment , i.e. , to treat every domain equally and align them all perfectly . However , in practice the domains are often heterogeneous ; one can expect DA to work well when the source domains are close to the target domains , but not when they are too far from each other ( Zhao et al. , 2019 ; Wang et al. , 2020 ) . Such heterogeneity can often be captured by a graph , where the domains realize the nodes , and the adjacency between two domains can be captured by an edge ( see Fig . 1 ) . For example , to capture the similarity of weather in the US , we can construct a graph where each state is treated as a node and the physical proximity between two states results in an edge . There are also many other scenarios where the relation among domains can be naturally captured by a graph , such as the taxonomies of products in retail or connections among research fields of academic papers . Given a domain graph , we can tailor the adaptation of domains to the graph , rather than dictating the data from all the domains to align perfectly regardless of the graph structure . One naïve DA method for such graph-relational domains is to perform DA for each pair of neighboring domains separately . Unfortunately , due to the strict alignment between each domain pair , this method will still lead to uniform alignment so long as the graph is connected . To generalize DA to the graphrelational domains , we argue that an ideal method should ( 1 ) only enforce uniform alignment when the domain graph is a clique ( i.e. , every two domains are adjacent ) , and ( 2 ) more importantly , relax uniform alignment to adapt more flexibly across domains according to any non-clique domain graph , thereby naturally incorporating information on the domain adjacency . In this paper , we generalize adversarial DA methods and replace the traditional binary ( or multi-class ) discriminator with a novel graph discriminator : instead of distinguishing among different domains , our graph discriminator takes as input the encodings of data to reconstruct the domain graph . We show that our method enjoys the following theoretical guarantees : it recovers classic DA when the the domain graph is a clique , and realizes intuitive alignments for other types of graphs such as chains and stars ( see Fig . 4 ) . We summarize our contributions as follows : • We propose to use a graph to characterize domain relations and develop graph-relational domain adaptation ( GRDA ) as the first general adversarial DA method to adapt across domains living on a graph . • We provide theoretical analysis showing that at equilibrium , our method can retain the capability of uniform alignment when the domain graph is a clique , and achieve non-trivial alignment for other types of graphs . • Empirical results on both synthetic and real-world datasets demonstrate the superiority of our method over the state-of-the-art DA methods . 2 RELATED WORK . Adversarial Domain Adaptation . There have been extensive prior works on domain adaptation ( Pan & Yang , 2009 ; Pan et al. , 2010 ; Long et al. , 2018 ; Saito et al. , 2018 ; Sankaranarayanan et al. , 2018 ; Zhang et al. , 2019 ; Peng et al. , 2019 ; Chen et al. , 2019 ; Dai et al. , 2019 ; Nguyen-Meidine et al. , 2021 ) . Typically they aim to align the distributions of the source and target domains with the hope that the predictor trained on labeled source data can generalize well on target data . Such alignment can be achieved by either directly matching their distributions ’ statistics ( Pan et al. , 2010 ; Tzeng et al. , 2014 ; Sun & Saenko , 2016 ; Peng et al. , 2019 ; Nguyen-Meidine et al. , 2021 ) or training deep learning models with an additional adversarial loss ( Ganin et al. , 2016 ; Zhao et al. , 2017 ; Tzeng et al. , 2017 ; Zhang et al. , 2019 ; Kuroki et al. , 2019 ; Chen et al. , 2019 ; Dai et al. , 2019 ) . The latter , i.e. , adversarial domain adaptation , has received increasing attention and popularity because of its theoretical guarantees ( Goodfellow et al. , 2014 ; Zhao et al. , 2018 ; Zhang et al. , 2019 ; Zhao et al. , 2019 ) , its ability to train end-to-end with neural networks , and consequently its promising empirical performance . These methods typically treat every domain equally and enforce uniform alignment between source-domain data and target-domain data ; this is done by generating domain-invariant encodings , where domain invariance is achieved by training the encoder to fool the discriminator that classifies the domain index . In contrast , we naturally relax such uniform alignment using a graph discriminator to reconstruct a domain graph that describes domain adjacency . Domain Adaptation Related to Graphs . There are also works related to both DA and graphs . Usually they focus on adaptation between two domains where data points themselves are graphs . For example , ( Pilancı & Vural , 2019 ; Pilanci & Vural , 2020 ) use frequency analysis to align the data graphs between the source domain and the target domains , and ( Alam et al. , 2018 ; Ding et al. , 2018 ) perform label propagation on the data graph . In contrast , GRDA considers a setting completely different from the above references . Instead of focusing on adapting between two domains with data points in the form of graphs ( e.g. , each data point itself is a node ) , GRDA adapts across multiple domains ( e.g. , with each state in the US as a domain ) according to a domain graph ( i.e. , each domain is a node ) . Therefore the methods above are not applicable to our setting . Note that ( Mancini et al. , 2019 ) uses metadata-weighted batch normalization to propagate information among domains with similar metadata , but it is not an adversarial domain adaptation method . It is orthogonal to GRDA and can be used as a backbone network to further improve GRDA ’ s performance ( see the Appendix for empirical results ) . It is also worth noting that , in this paper , we assume the domain graph is given . It would be interesting future work to combine GRDA with domain relation inference methods ( e.g. , domain embeddings ( Peng et al. , 2020 ) ) when there is a natural but unobserved graph relation among domains . 3 METHOD . In this section , we will first briefly introduce the problem setting and then elaborate our domain adaptation method . 3.1 PROBLEM SETTING AND NOTATION . We focus on the unsupervised domain adaptation setting with N domains in total . Each domain has a discrete domain index u ∈ U = [ N ] , { 1 , . . . , N } , belonging to either the source domain index set Us or the target domain index set Ut . The relationship between domains is described by a domain graph with the adjacency matrix A = [ Aij ] , where i and j index nodes ( domains ) in the graph . Given labeled data { ( xsl , ysl , usl ) } nl=1 from source domains ( usl ∈ Us ) , unlabeled data { xtl , utl } ml=1 from target domains ( utl ∈ Ut ) , and the domain graph described by A , we want to predict the label { ytl } ml=1 for data from target domains . Note that the domain graph is defined on domains with each domain ( node ) containing multiple data points . 3.2 GRAPH-RELATIONAL DOMAIN ADAPTATION ( GRDA ) . Overview . We use an adversarial learning framework to perform adaptation across graph-relational domains . The adversarial game consists of three players : ( 1 ) an encoder E , which takes as input a datum xl , the associated domain index ul , and the adjacency matrix A to generate an encoding el = E ( xl , ul , A ) , ( 2 ) a predictor F , which makes predictions based on the encoding el , and ( 3 ) a graph discriminator D , which guides the encoding to adapt across graph-relational domains . Specifically , the discriminator D takes in a mini-batch of B encodings el ( l ∈ [ B ] ) , and tries to reconstruct the domain graph A . By letting the encoder E play adversarially against the discriminator D , the graph-relational information of domains will be removed from the encoding el in order to make the discriminator D incapable of reconstructing the graph . Note that the graph discriminator in our adversarial game is different from classic discriminators which classify the domain index , as shown in Fig . 2 . Formally , GRDA performs a minimax optimization with the following loss function : min E , F max D Lf ( E , F ) − λdLd ( D , E ) , ( 1 ) where Lf ( E , F ) is the predictor loss and Ld ( D , E ) is the discriminator loss , and λd is a hyperparameter balancing them two . Below we discussed these two terms in detail . Predictor . In Eqn . 1 , the predictor loss Lf ( E , F ) is defined as Lf ( E , F ) , Es [ hp ( F ( E ( xl , ul , A ) ) , y ) ] , where the expectation Es is taken over the sourcedomain data distribution ps ( x , y , u ) . hp ( · , · ) is a predictor loss function for the task ( e.g. , L2 loss for regression ) . Encoder and Node Embeddings . Given an input tuple ( xl , ul , A ) , the encoder E first computes a graph-informed domain embedding zul based on the domain index ul and the domain graph A . Then we feed zul along with xl into a neural network to obtain the final encoding el . Formally we have el = E ( xl , ul , A ) = f ( xl , zul ) , ( 2 ) where f ( · , · ) is a trainable neural network . In theory , any embeddings for node ( domain ) indices should work equally well so long as they are distinct from one another ( thus forming a bijection to the set of domains [ N ] ) . Here we pre-train the embeddings by a reconstruction loss for simplicity , and our intuition is that good embeddings of nodes should inform us of ( thus reconstruct ) the graph structure . Suppose the nodes indices i and j are sampled independently and identically from the marginal domain index distribution p ( u ) ; the reconstruction loss is written as Lg = Ei , j∼p ( u ) [ −Aij log σ ( z > i zj ) − ( 1−Aij ) log ( 1− σ ( z > i zj ) ) ] , where σ ( x ) = 11+e−x is the sigmoid function . Note that in general we could use any node embedding methods ( Grover & Leskovec , 2016 ; Tang et al. , 2015 ; Kipf & Welling , 2016b ) , but this is not the focus of this paper . For fair comparison , we use exactly the same encoder , i.e. , E ( x , u , A ) , for all the methods in the experiments of Sec . 5 . Graph Discriminator . The discriminator loss Ld ( D , E ) in Eqn . 1 is defined as Ld ( D , E ) , E ( x1 , u1 ) , ( x2 , u2 ) [ h ( x1 , u1 , x2 , u2 ) ] , ( 3 ) h ( x1 , u1 , x2 , u2 ) =−Au1 , u2 log σ ( ẑ > 1 ẑ2 ) − ( 1−Au1 , u2 ) log ( 1− σ ( ẑ > 1 ẑ2 ) , where ẑ1 = D ( E ( x1 , u1 , A ) ) , ẑ2 = D ( E ( x2 , u2 , A ) ) are the discriminator ’ s reconstructions of node embeddings . The expectation E is taken over a pair of i.i.d . samples ( x1 , u1 ) , ( x2 , u2 ) from the joint data distribution p ( x , u ) . The discriminator loss Ld ( D , E ) essentially quantifies how well the reconstructed node embedding ẑ1 , ẑ2 preserve the information of the original connections , or , equivalently , A . We refer readers to the Appendix for detailed model architectures . Due to the adversarial nature of how the discriminator D and the encoder E engage with the loss , the discriminator D would aim to recover the domain graph via the adjacency structure ( A ) , while the encoder E would prevent the discriminator D from doing so . Intuitively , if the discriminator D is powerful enough to uncover any information regarding the domain graph in the encoding el , the optimal encoder E will have to remove all the information regarding graph-relational domains in the encoding el , thus successfully adapting across graph-relational domains . We will formally elaborate the above arguments further in the next section .

Summary: This work optimizes the adversarial domain adaption model by generalizing the uniform domain alignment with a domain encoding graph. The proposed method is simple and easy to follow, and the overall idea is clearly described. Some experiments show the effectiveness of the proposed method on both synthetic and real-world datasets.