Leveraging Distribution Alignment via Stein Path for Cross-Domain Cold-Start Recommendation

1 Introduction . Data sparsity and cold-start are long-standing problems in recommender systems [ 15 , 19 ] . With the development of internet techniques , most users always participant in many platforms or domains for different purposes . Therefore , Cross-Domain Recommendation ( CDR ) has emerged to utilise the relatively richer information from a source domain to improve the recommendation accuracy in a target domain [ 52 , 53 ] . Most existing CDR models can tackle the data sparsity problem in the target domain by assuming the existence of overlapped users or items with similar tastes or attributions across domains [ 5 ] . Instead of focusing on solving the data sparsity problem , we consider cold-start item recommendation under the CDR setting . Specifically , we concentrate on the Cross-Domain Cold-Start Recommendation ( CDCSR ) problem , that is , two domains share the same user set but different items , and both domains have auxiliary representations such as item profiles or descriptions . The prime challenge is how to leverage the information from the source domain , where the items are ‘ warm ’ , to improve the recommendation performance of the target domain , where the items are ‘ cold ’ . The CDCSR problem popularly exists in practice , for instance , a movie marketing platform newly launches a book renting service where there is no user-book interaction yet , as shown in Figure 1 . Existing researches on cold-start recommendation and CDR can not solve the above problem well . On the one hand , existing cold-start recommendation models assume that the distributions of cold items should be consistent with the warm ones as they are homogeneous [ 54 , 18 , 12 ] . On the other hand , ∗Corresponding Author 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . existing CDR models assume that both source and target domains have user-item interactions for learning the mapping functions [ 25 ] . Since the cold and warm items are heterogeneous with different latent embedding distributions in practice , and there is no user-item interaction in the target domain , conventional cold-start and CDR models can not properly suitable to the CDCSR problem . Similar to the transfer learning task , the key to the CDCSR problem is to reduce the discrepancy between the latent embedding distributions across domains . However , both the warm and cold item representations are scattered and complicated due to the fact that the latent embeddings may represent diverse information . Thus , existing transfer learning based domain adaptation approaches [ 23 , 45 , 37 ] can not achieve good alignment results , which limits their performances . To address the aforementioned issue , in this paper , we propose DisAlign , a cross-domain recommendation framework for the CDCSR problem . In order to better align the complicated latent embedding distributions and make high quality of rating predictions , we utilize two modules in DisAlign , i.e. , rating prediction module and embedding distribution alignment module , as will be shown in Figure 2 . The rating prediction module aims to capture user and item collaborative preferences in the source domain , and we propose metric-based contrastive learning for modelling . The goal of distribution alignment module is to properly match the latent embedding distributions across domains , and we propose two techniques for it , i.e. , Stein path alignment and its improved version called proxy Stein path alignment . Specifically , inspired by the particle-based inference algorithm Stein Variational Gradient Descent ( SVGD ) [ 21 , 13 , 20 ] , we first propose Stein path alignment to minimize the domain discrepancy through the particle-moving process , which can take both the source probability and target intra-domain structure into account . Although Stein path can obtain satisfying performance , it has to involve all the target samples during the training process , which is time consuming when data size is large . Thus , we further propose proxy Stein path alignment which only needs to exploit typical samples to represent the target data distribution , and thus can accelerate the operation speed . The comparison and visualization results in experiments will show the reliability and efficiency of DisAlign . We summarize the main contributions of this paper as follows : ( 1 ) We propose a novel framework , i.e. , DisAlign , for the CDCSR problem , which can utilize both rating and auxiliary representations from the source domain to improve the recommendation performance of the target domain . ( 2 ) To our best knowledge , this is the first attempt in literature to propose Stein path alignment for aligning the latent embedding distributions across domains , and we also propose its improved version , i.e. , proxy Stein path , for higher efficiency . ( 3 ) Empirical studies on Douban and Amazon datasets demonstrate that DisAlign significantly improves the state-of-the-art models under the CDCSR setting . 2 The proposed model . 2.1 Framework of DisAlign First , we describe notations . We assume there are two domains , i.e. , a source domain S and a target domain T . We assume both domains S and T have NU users , S has NS warm items , and T has NT cold items . Let RS ∈ RNU×NS be the warm rating matrix in S and RT ∈ RNU×NT be the cold rating matrix in T . In CDCSR setting , RT is absence during training and will be only used for test , since items are cold in T . We also assume that the warm items and the cold items have auxiliary representations XW ∈ RNS×Z and XV ∈ RNT×Z , respectively , with Z denoting the dimension of auxiliary representations . The auxiliary representations usually include useful sideinformation , e.g. , themes , reviews , profiles in a movie domain . Our purpose is to predict the absent RT in T by leveraging RS in S and the auxiliary representations in both S and T . Then , we introduce the overview of our proposed DisAlign framework , as is illustrated in Figure 2 . DisAlign model mainly has two modules , i.e. , rating prediction module and embedding distribution alignment module . To avoid the error superimposition problem [ 54 ] , the rating prediction module in the source domain mainly provides end-to-end joint training of modelling user and item collaborative embeddings and matching the item collaborative embeddings with item auxiliary embeddings . The embedding distribution alignment module aligns the distributions between the warm and cold items across domains , minimizing the discrepancy between auxiliary latent feature embeddings in the source and target domains sufficiently . We will introduce these two modules in details later . 2.2 Rating prediction module . We first introduce the rating prediction module of DisAlign . For the i-th user and the j-th item in the source ( warm ) domain , we define their corresponding one-hot ID vectors as XUi and X V j , respectively . For the j-th warm item , we also define its auxiliary representation as XWj . The rating prediction module mainly has three purposes , including ( 1 ) exploiting user and item collaborative embeddings based on ratings , which is the prime purpose ; ( 2 ) matching the item collaborative embeddings with item auxiliary embeddings ; and ( 3 ) obtaining more discriminative item collaborative embeddings using unsupervised clustering method . First , user and item collaborative embeddings should accurately represent the corresponding rating interactions . We obtain the user and item collaborative embeddings by FU ( XU ) = U ∈ RN×D and FV ( XV ) = V ∈ RN×D , respectively . Here , FU and FV denote the user and item encoding networks respectively , N is batch size , and D is the dimension of collaborative embeddings . After that , we use pairwise ranking loss LBE based on metric-based contrastive learning [ 10 , 14 ] : minLBE = − ∑ ( Ui , Vj , Vk ) ∈D log exp〈Ui , Vj〉 exp〈Ui , Vj〉+ exp〈Ui , Vk〉 , ( 1 ) where D : = { ( Ui , Vj , Vk ) |RSij > RSik } denotes the original preference pairs [ 32 ] , and 〈· , ·〉 denotes the inner product . The loss function LBE can pull the positive items close and push the negative items away for a certain user according to his/her preference . Second , a item ’ s collaborative embedding should be similar to its auxiliary embeddings to avoid the error superimposition problem [ 54 ] . We utilize a networkGV to translate the auxiliary representations into auxiliary embeddings as GV ( XW ) = W ∈ RN×D . After it , the item embedding matching loss is given by LBM = ||W − V ||22 . Third , similar item collaborative embeddings should be clustered in order to obtain more discriminative latent features . We adopt deep unsupervised K-Means clustering approach [ 24 , 48 ] for this , and the corresponding loss is minFFT=I LBK = [ Tr ( V V T ) − Tr ( FV V TF T ) ] , where F ∈ RK×N is the cluster indicator matrix and K denotes the cluster number . In summary , the loss of the rating prediction module is a combination of the three losses , that is : min FFT=I LB = LBE + ηLBM + ζLBK , ( 2 ) where ζ and η represent the balance hyper-parameters . The optimization procedure is given as below : ( 1 ) Fixing the other variables except F , we update F through minFFT=I Tr ( FV V TF T ) with singular value decomposition algorithm ; ( 2 ) Fixing F , we update other variables through gradient descent methods for several iterations then go back to step ( 1 ) until it convergences . For the sake of stability , in practice , we update F every 15 iterations . 2.3 Embedding distribution alignment module . 2.3.1 Overview . We then introduce the embedding distribution alignment module of DisAlign . We use GV ( XCj ) = C ∈ RN×D to denote the auxiliary embeddings of the cold items in the target domain . Specifically , GV ( · ) is a two-stream siamese network with shared weights for encoding both warm item auxiliary representation XW and cold item auxiliary representation XC . We denote pW and pC as the warm and cold item auxiliary embedding probability distributions , respectively , and denote pV as the warm item collaborative embedding probability distribution . In CDCSR setting , pW 6= pC and pV 6= pC , because the embeddings generated from the source ( warm ) domain and the target ( cold ) domain are heterogeneous , which leads to the domain discrepancy problem . Let us consider a case where the source domain has user-book interactions while the target domain has user-movie interactions . Although books and movies share some similar characteristics , the auxiliary representations of the Book domain usually include authors and writing styles , while the auxiliary representations of the Movie domain include directors and actors , which brings discrepancy . Without embedding distribution alignment , a recommender system may recommend horror books instead of history books to a user who likes history movies rather than horror movies due to domain discrepancy , as illustrated in the left of Figure 3 . After alignment , the history movies/books and horror movies/books are aligned , as is shown in the right of Figure 3 , and thus the recommender system can provide more reliable results . In order to reduce the distribution discrepancy between the source and target domains , we introduce two approaches , i.e. , Stein path alignment and Proxy stein path alignment . 2.3.2 Stein path alignment As mentioned in Section 1 , since the latent embeddings from two domains in CDCSR are scattered and complicated due to the fact that they may represent diverse information , previous domain adaptation methods can not be effectively utilized to solve the distribution discrepancy problem . Therefore , we propose a new distribution alignment approach , named Stein path alignment . Stein path alignment can prompt the target samples move to the source domain through proper paths , according to the target intra-domain structure and the probability distribution of the source domain . Stein path alignment relies on Stein Variational Gradient Descent ( SVGD ) [ 21 , 42 ] , a variational inference method that starts from a set of initial particles and iteratively updates them with an approximate steepest direction , whose main iteration process is : zi , l = zi , l−1 + φp ( zi , l−1 ) , φp ( z ) = 1 N N∑ j=1 [ k ( z , zj ) ∇z log p ( z ) +∇zk ( z , zj ) ] , ( 3 ) where zi , l denotes the i-th original target sample at the l-th iteration , p ( z ) denotes the source probability distribution , denotes the step size , and k ( x , y ) = exp ( − ( ||x− y||22 ) /σ2 ) is the Gaussian kernel function with σ denoting the bandwidth . Existing researches [ 21 , 7 , 2 ] have proved that mean field theory can guarantee the rigorous theoretical convergence of SVGD , that is , the gradient dynamics at particle level will approach to zero : limt→+∞ ( 1/N × ∑N i=0 φp ( zi , t ) ) → 0 . Stein path distance . We denote zi , t as the Stein mirror point of zi,0 when SVGD convergences at the t-th iteration . We propose Stein path distance as below : PT→S ( Z ) : = 1 N N∑ i=0 ||zi , t − zi,0||22 = 1 N N∑ i=0 ||zi , t−1 + φS ( zi , t−1 ) − zi,0||22 . ( 4 ) Stein path distance quantifies the discrepancy between the source domain S and the target domain T by taking the average length of all paths from T to S through the t-th iteration . Stein path considers the source probability distribution and intra-domain structures , and thus can avoid negative transfer arisen from coarsely pairwise matching by traditional methods . Meanwhile Stein path is also explainable . Let zi,0 denote the auxiliary embedding of i-th book in the Book domain , zi , t could be taken as a similar movie in the Movie domain , e.g. , the movie is based on the story of the book . The calculation of Stein path distance mainly has three steps . First , adopting kernel density estimation [ 30 , 29 , 39 ] with radial basis function kernel to estimate the probabilities of W and V . Second , finding the Stein mirror point of the cold item auxiliary embeddings through SVGD by Equation ( 3 ) . Third , calculating the Stein path distance using Equation ( 4 ) . The calculation details will be given in Appendix A.1 . Stein path loss . In summary , the better the source and target domains are aligned , the smaller the Stein path distance . Therefore , we innovatively propose Stein path loss to align the cold item auxiliary embedding C with warm item collaborative preference V and auxiliary embedding W as below : minLSP = PC→W ( C ) + PC→V ( C ) + ||PC→W ( C ) − PC→V ( C ) ||22 , ( 5 ) where the first two terms denote the Stein path distances from C to W and C to V , respectively , and the third term reinforces that these two distances should be similar . 2.3.3 Proxy Stein path alignment Although Stein path alignment achieves satisfying performance , it has scalability problem when facing large dataset . Because all the cold items in each batch need to be used for calculating Stein path distance . Therefore , it is urgent to reduce the computation cost to accelerate the optimization process . To do this , we propose proxy Stein path approach which only needs to choose the most typical cold item proxies to represent the global properties in order to speed up the alignment process through SVGD . We now describe the main steps for finding the proxies and the optimization procedure . Multiple-proxies algorithm . We first introduce a highly efficient multiple-proxies algorithm , which aims to find typical proxy samples in the target domain . Suppose there exists M ∈ RH×D typical proxies for the cold item auxiliary embeddings C , where H denotes the number of proxies . Let Ψ ∈ RN×H be the similarity matrix between the cold items in the target domain and the proxies . Inspired by [ 1 , 28 ] , we formulate the multiple-proxies optimization problem as min M , ψi1=1 , ψij≥0 N∑ i=1 H∑ j=1 ψij ||ci −mj ||22 + α N∑ i=1 H∑ j=1 ψij logψij , ( 6 ) where ci denotes the i-th cold item auxiliary embedding and mj denotes the j-th corresponding proxy . The entropy norm regularization term ∑N i=1 ∑H j=1 ψij logψij is set to avoid trivial solution with α denoting the regularization strength . Compared with the square norm || · ||2 , the entropy norm can not only obtain a nonnegative and nonlinearly representational similarity matrix but also reduce the computational cost [ 1 ] . In summary , the multiple-proxies optimization algorithm has two main steps , i.e. , ( 1 ) updating Ψ , which has closed-form solution , and ( 2 ) updating M as mj = ∑N i=1 ψijci/ ∑N i=1 ψij . The optimization could be done by repeating ( 1 ) and ( 2 ) until Ψ and M convergence . We will present the optimization details in Appendix A.2 . Proxy Stein path distance . After finding the typical proxies M and setting Ψ as a constant , we propose proxy Stein path distance according to the original Stein path as below : P∗T→S ( M ) = 1 H H∑ i=0 ||mi , t −mi,0||22 = 1 H H∑ i=0 ||mi , t−1 + φS ( mi , t−1 ) −mi,0||22 , ( 7 ) where mi , t denotes the i-th proxy mi at the t-th iteration . Notably , in each batch , proxy Stein path only needs to move the number of proxy samples ( H ) in the target domain rather than the total number of samples ( N ) . Since H < N , proxy Stein path can reduce the time consumption on calculating the Stein path distance . Proxy Stein path loss . Similarly to the Stein path loss , the proxy Stein path loss is given by : minLPSP = P∗C→W ( M ) + P∗C→V ( M ) + ||P∗C→W ( M ) − P∗C→V ( M ) ||22 . ( 8 ) The optimization of proxy Stein path alignment mainly has four steps . The first step is adopting the multiple-proxies algorithm to figure out the typical proxies M in the target domain . The following three steps are similar as Stein path alignment mentioned in the Section 2.3.2 , except that we are moving proxies M rather than C. We will present the optimization details in Appendix A.3 . Time complexity analysis . The time complexity of Stein path alignment is O ( N3t1 ) , where t1 is the iteration number . The time complexities of the multiple-proxies algorithm and proxy Stein path alignment are O ( NHt2 ) and O ( H2Nt1 ) , respectively , where t2 is the inner-loop iteration number . Therefore , the total time complexity of proxy Stein path alignment is O ( NHt2 ) + O ( H2Nt1 ) = O ( NHt2 +H 2Nt1 ) . Since H < N , proxy Stein path alignment is much cheaper than Stein path alignment . Empirically , we set H∗ = [ N2 ] .

Summary: This paper proposes a framework for cross-domain item cold-start recommendation. The main idea follows the algorithm of Stein Variational Gradient Descent, by iteratively transporting the auxiliary embeddings of cold items in the target domain to match the embedding distribution of warm items in the source domain. The authors further propose to reduce computational time costs by selecting typical proxies of cold items.