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Probabilistic Residual Learning for Online Recommendations
Wenyuan Wang, Yusong Zhao, Zihao Xu, Hengyi Wang, Qi Xu, Zhigang Hua, Yan Xie, Yi Wang, Zihao Zhao, Bo Long, Chengzhi Mao, Shuang Yang, Hengguan Huang, Hao Wang
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
Last extracted: 7/24/2026, 2:22:32 AM
Summary
The paper introduces Probabilistic Residual Learning (PRL), a causal Bayesian framework designed to enhance existing deep learning recommender systems. PRL operates as a plug-and-play module that models the residual between ground-truth ratings and base model predictions. It employs probabilistic clustering to group users, models domain-level confounders to address cross-domain shifts, and uses do-calculus to aggregate cluster-specific residual predictions, thereby improving cold-start performance and interpretability.
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Relation Signals (5)
Probabilistic Residual Learning → enhances → Deep Learning Recommender Systems
confidence 92% · compatible with various base deep learning recommender systems, improving their performance
Probabilistic Residual Learning → addresses → Cold-Start
confidence 90% · Experiments across datasets and base recommenders demonstrate improved cold-start cross-domain recommendation.
Probabilistic Residual Learning → models → Domain Confounders
confidence 88% · models domain-level confounders that influence user and item representations
Probabilistic Residual Learning → discovers → User Clusters
confidence 87% · automatically discovering meaningful user clusters
Probabilistic Residual Learning → uses → Do-Calculus
confidence 85% · aggregates cluster-specific residual predictions over the confounders using do-calculus.
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Abstract
Abstract:Modern recommender systems are typically based on deep learning (DL) models, where a dense encoder learns representations of users and items. As a result, these systems often suffer from the black-box nature and computational complexity of the underlying models, making it difficult to systematically enhance their recommendation capabilities. To address this problem, we propose Probabilistic Residual Learning (PRL), a causal Bayesian recommendation model that models the residual between ground-truth and base predictions, enabling targeted refinement of existing systems. Specifically, PRL (1) probabilistically groups users for localized residual modeling, (2) models domain-level confounders that influence user and item representations, and (3) aggregates cluster-specific residual predictions over the confounders using do-calculus. Experiments demonstrate that our plug-and-play PRL is compatible with various base deep learning recommender systems, improving their performance while automatically discovering meaningful user clusters.
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- Source: https://arxiv.org/abs/2607.20863v1
- Canonical: https://arxiv.org/abs/2607.20863v1
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by Probabilistic Residual Learning for Online Recommendations Wenyuan Wang Rutgers UniversityPiscatawayNew JerseyUnited States w462@scarletmail.rutgers.edu , Yusong Zhao Rutgers UniversityPiscatawayNew JerseyUnited States yz1635@scarletmail.rutgers.edu , Zihao Xu Rutgers UniversityPiscatawayNew JerseyUnited States zihao.xu@rutgers.edu , Hengyi Wang Rutgers UniversityPiscatawayNew JerseyUnited States hw514@scarletmail.rutgers.edu , Qi Xu MetaSunnyvaleCaliforniaUnited States xuqi0511@gmail.com , Zhigang Hua MetaSunnyvaleCaliforniaUnited States zhua@meta.com , Yan Xie MetaSunnyvaleCaliforniaUnited States yanxie@meta.com , Yi Wang Rutgers UniversityPiscatawayNew JerseyUnited States yw1013@scarletmail.rutgers.edu , Zihao Zhao Rutgers UniversityPiscatawayNew JerseyUnited States z1009@rutgers.edu , Bo Long MetaSunnyvaleCaliforniaUnited States bolong@meta.com , Chengzhi Mao Rutgers UniversityPiscatawayNew JerseyUnited States chengzhi.mao@rutgers.edu , Shuang Yang MetaSunnyvaleCaliforniaUnited States shuangyang@meta.com , Hengguan Huang University of CopenhagenCopenhagenDenmark hengguan.huang@sund.ku.dk and Hao Wang Rutgers University & UIUCPiscatawayNew JerseyUnited States hwangml@illinois.edu (2026) Abstract. Modern recommender systems are typically based on deep learning (DL) models, where a dense encoder learns representations of users and items. As a result, these systems often suffer from the black-box nature and computational complexity of the underlying models, making it difficult to systematically enhance their recommendation capabilities. To address this problem, we propose Probabilistic Residual Learning (PRL), a causal Bayesian recommendation model that models the residual between ground-truth and base predictions, enabling targeted refinement of existing systems. Specifically, PRL (1) probabilistically groups users for localized residual modeling, (2) models domain-level confounders that influence user and item representations, and (3) aggregates cluster-specific residual predictions over the confounders using do-calculus. Experiments demonstrate that our plug-and-play PRL is compatible with various base DL recommender systems, improving their performance while automatically discovering meaningful user clusters. Bayesian Deep Learning, Probabilistic Graphical Model †journalyear: 2026†copyright: c†conference: 20th ACM Conference on Recommender Systems; September 27–October 02, 2026; Minneapolis, MN, USA†booktitle: 20th ACM Conference on Recommender Systems (RecSys ’26), September 27–October 02, 2026, Minneapolis, MN, USA†doi: 10.1145/3773078.3831754†isbn: 979-8-4007-2284-4/2026/09†ccs: Information systems Recommender systems Figure 1. Overview of PRL. During training, PRL models the residuals of a fixed base recommender, probabilistically clusters users, and learns cluster-specific correction models. During inference, it predicts a confounder-adjusted residual using the inferred cluster and adds it to the base prediction. Overview of PRL. A fixed base recommender first produces base rating predictions. PRL computes residual ratings, probabilistically groups users into clusters, and learns cluster-specific residual models conditioned on domain factors. At inference time, the predicted residual is added to the base prediction to obtain the final rating. 1. Introduction Over the past decade, personalized recommendations have significantly improved user experiences in domains such as e-commerce and social media. The recommender systems driving these advancements often rely on sophisticated deep learning (DL) models (ChungGCB14; vaswani2017attention; srgnn) capable of handling vast amounts of data, enabling highly accurate predictions and personalized interactions. Despite their effectiveness, these models are often costly or difficult to systematically adapt to new domains, particularly under cold-start domain shifts such as changes in user markets. Cold-start scenarios, a critical problem in recommendation systems, exacerbate these issues due to the presence of heterogeneous features and the influence of diverse and spurious patterns. As a result, existing models exhibit notably low performance in such settings. Existing work (DBLP:conf/sigir/Yuan0KZ20; DBLP:conf/sigir/WuYCLH020; DBLP:conf/sigir/BiSYWWX20a; li2019zero; DBLP:conf/sigir/Hansen0SAL20; DBLP:conf/sigir/LiangXYY20; DBLP:conf/sigir/ZhuSSC20; liu2020heterogeneous; kweon2024perk) often addresses domain shift through shared users or items. However, such overlap is often unavailable in practice; for example, users and items may be disjoint across countries. This setting also requires modeling shared confounders, as exposure and popularity effects may induce domain-specific correlations that transfer poorly. Additionally, existing methods often overlook latent user clusters, missing opportunities for user-cluster-targeted enhancement. To address these problems, we draw inspiration from the hierarchical Bayesian DL framework (BDL; BDLSurvey) to propose causal hierarchical Bayesian DL model, dubbed Probabilistic Residual Learning (PRL), as a plug-and-play framework to improve and potentially interpret any base recommender systems in cross-domain settings. Fig. 1 shows the simplified overview of our framework. During the training stage, given any base recommender (e.g., DLRM (DLRM), CDL (cdl), or NCF (NCF)), PRL is learned by jointly (1) computing its predicted ratings and the residual ratings (i.e., the difference between the ground-truth ratings and the base model’s predicted ratings), (2) dividing users into latent clusters based on the residual ratings using our probabilistic clustering method, (3) training a sub-model for each latent user cluster. During the inference stage, once PRL is learned, given a new user-item pair, PRL can then (1) estimate the user’s cluster ID to select the proper sub-model, (2) use this sub-model to perform causal inference to debias potential confounders and predict the residual rating, and (3) add this predicted residual rating to the base predicted rating to obtain the final predicted rating, thereby producing the final recommendation. Notably, PRL is plug-and-play, i.e., it is compatible with any base DL recommendation model and can enhance the original model’s performance. Our contributions are as follows: • We formulate cross-domain adaptation as a plug-and-play probabilistic residual correction problem for fixed base recommenders. • We develop a hierarchical Bayesian model that combines probabilistic user clustering with causal adjustment of domain confounders. • Experiments across datasets and base recommenders demonstrate improved cold-start cross-domain recommendation. 2. Probabilistic Residual Learning 2.1. Problem Setting and Notations Consider a recommendation dataset containing I users and J items. A DL encoder fv(⋅):ℝd→ℝhf_v(·):R^d→R^h encodes each item j’s raw features jv∈ℝd x_j^v∈R^d into fv(jv)f_v( x_j^v). For a given user i and an item j, there is a ground-truth rating Rij∈ℝR_ij∈R, a base predicted rating R^ij∈ℝ R_ij∈R provided by a base recommender, and a residual rating R~ij=Rij−R^ij R_ij=R_ij- R_ij. There is a latent cluster ID k (k∈1,…,Kk∈\1,...,K\) that indicates which user group user i belongs to. We assume that there exists a user latent vector i∈ℝh u_i∈R^h for each user i and an item latent vector j∈ℝh v_j∈R^h for each item j; they are both impacted by a causal confounder ∈ℝg s∈R^g, where g≪hg h. Our goal is to predict the final rating R using the residual R~ R, i.e., R=R^+R~R= R+ R, where R R represents the rating from the original (base) DL recommender. When the original recommender is provided, R R is fixed; therefore we only need to learn R~ R in order to predict the final rating R. For generality, we assume M domains, where mim_i and mjm_j denote the domain ID of user i and item j, respectively. Figure 2. Probabilistic graphical model of PRL. The cluster variable πi _i selects the user prior, while the domain factor m s_m influences i u_i, j v_j, and the residual rating R~ij R_ij. The probabilistic graphical model of PRL. The user cluster variable pi selects a cluster-specific prior for the user latent variable. The domain factor influences the user latent variable, item latent variable, and residual rating. Item features are encoded by a deep learning model to inform the item latent variable. 2.2. Method Overview PRL jointly models cluster-specific users, content-informed items, and confounder-adjusted residuals. Fig. 2 shows their dependencies and variational formulation. Generative Process. Below we describe the generative process of PRL shown in Fig. 2. For each domain m, PRL draws a confounder m∼(,) s_m N( 0, I). For each item j, it draws (1) j∼p(j∣jv,m)=PoG(fv(jv),vm,Λv−1,λv−1). v_j p( v_j x_j^v, s_m)=PoG\! (f_v( x_j^v), W^v s_m, _v^-1 I, _v^-1 I ). For each user i, PRL draws a cluster assignment πi∼p(πi∣θ) _i p( _i θ) and i∼(πi+um,πi). u_i N( $μ$_ _i+ W^u s_m, $ $_ _i). Finally, it draws the residual rating R~ij∼(i⊤j+R⊤m,λR~ij−1). R_ij N\! ( u_i v_j+ w^R s_m, _ R_ij^-1 ). Here, u W^u, v W^v, and R w^R are globally shared parameters. The product-of-Gaussians prior in Eqn. 1 combines the content-based estimate fv(jv)f_v( x_j^v) and the domain-conditioned estimate vm W^v s_m. The resulting distribution is also Gaussian (pog), with (2) pog $μ$_pog =Λvfv(jv)+λvvmΛv+λv, = _vf_v( x_j^v)+ _v W^v s_m _v+ _v, λpog _pog =Λv+λv. = _v+ _v. Model Factorization. As shown in Fig. 2, we factorize the generative model into four conditional distributions: p(i,j,π,R~ij|k,kk=1K,jv,m) p( u_i, v_j,π, R_ij|\ $μ$_k, $ $_k\_k=1^K, x_j^v, s_m) (3) = =~ p(R~ij|i,j,m)p(i|k,kk=1K,m,π)p(j|jv,m)p(π). p( R_ij| u_i, v_j, s_m)p( u_i|\ $μ$_k, $ $_k\_k=1^K, s_m,π)p( v_j| x_j^v, s_m)p(π). p(π|θ)p(π|θ) is the prior distribution for π. Each of the remaining distributions is assumed as a Gaussian distribution and is shown as follows: (4) p(R~ij|i,j,m) p( R_ij| u_i, v_j, s_m) =(i⊤j+R⊤m,λR~ij−1), =N( u_i v_j+ [rgb]0,0,0 w^R s_m, _ R_ij^-1), (5) p(i|k,kk=1K,m,π) p( u_i|\ $μ$_k, $ $_k\_k=1^K, s_m,π) =(πi+um,πi), =N( $μ$_ _i+ [rgb]0,0,0 W^u s_m, $ $_ _i), (6) p(j|jv,) p( v_j| x_j^v, s) =PoG(fv(jv),vm,Λv−1,λv−1), =PoG(f_v( x_j^v), W^v s_m, _v^-1 I, _v^-1 I), where i and j refer to the user index and the item index, respectively. We employ variational distributions q(i,j|jv)q( u_i, v_j| x_j^v) to approximate the posterior distributions of i u_i and j v_j. (7) q(ii=1I,jj=1J)=∏i=1Iq(i)∏j=1Jq(j). q(\ u_i\_i=1^I,\ v_j\_j=1^J)=Π _i=1^Iq( u_i)Π _j=1^Jq( v_j). More specifically, we assume q(j)q( v_j) follows a gaussian distribution: (8) q(j) q( v_j) =(j,Λv−1). =N( $μ$_ v_j, _v^-1 I). Here, j is the item index, Λv∈ℝ _v∈R refers to the precision. Similarly: (9) q(i) q( u_i) =(i,Λu−1), =N( $μ$_ u_i, _u^-1 I), where i is the user index, and Λu∈ℝ _u∈R is the precision. Different users and items have different approximate posteriors. We also use a categorical variational distribution q(π)q(π) to approximate the posterior distribution of π (more details below). Learning Objective. We maximize an evidence lower bound (ELBO) as our learning objective for both the generative and inference models. ℒELBO(jv,R~ij) L_ELBO( x^v_j, R_ij) = =~ q(i)q(j)[logp(i,j,R~ij|k,kk=1K,jv,m,π)] E_q( u_i)q( v_j) [ p( u_i, v_j, R_ij|\ $μ$_k, $ $_k\_k=1^K, x^v_j, s_m,π) ] +q(π)[p(π|θ)]−q(π)[q(π)] +E_q(π)[p(π|θ)]-E_q(π)[q(π)] −q(i)q(j)[logq(j)] -E_q( u_i)q( v_j) [ q( v_j)] (10) −q(i)q(j)[logq(i)]. -E_q( u_i)q( v_j) [ q( u_i)]. Combining Eqn. 3 and Eqn. 7, we obtain the following decomposition: ℒELBO(jv,R~ij) L_ELBO( x^v_j, R_ij) (11) = =~ −DKL(q(i)∥p(i|k,kk=1K,m,π)) [rgb]0,0,0-D_KL (q( u_i)\|p( u_i|\ $μ$_k, $ $_k\_k=1^K, s_m,π) ) (12) +q(i)q(j)q(π)[logp(R~ij|i,j,m,π)] +E_q( u_i)q( v_j)q(π) [ p( R_ij| u_i, v_j, s_m,π) ] (13) −DKL(q(j)∥p(j|jv,m)), -D_KL (q( v_j)\|p( v_j| x_j^v, s_m) ), (14) −DKL(q(π)∥p(π|θ)), -D_KL (q(π)\|p(π|θ) ), where DKL(⋅∥⋅)D_KL(·\|·) is the Kullback-Leibler (KL) divergence. For Eqn. 11, we compute the log likelihood for each cluster k as logp(ii∈Ik∣k,k,m,π) p(\ u_i\_i∈ I_k \ $μ$_k, $ $_k\, s_m,π) = =~ −12∑i∈Ik[log|k|+(i−k−um)⊤k−1(i−k−um)] - 12 _i∈ I_k[ | $ $_k|+( u_i- $μ$_k- [rgb]0,0,0 W^u s_m) $ $_k^-1( u_i- $μ$_k- W^u s_m)] (15) +C, +C, where i is the user index, IkI_k is the set of user index that belongs to cluster k, and C is a constant. Similarly, all the other terms can be expanded as: (16) logp(R~ij|i,j,)=−λR~ij2(R~ij−i⊤j−R⊤m)2+C, p( R_ij| u_i, v_j, s)=- _ R_ij2 ( R_ij- u_i v_j- [rgb]0,0,0 w^R s_m )^2+C, DKL(q(j)∥p(j|jv,m))=−λv2∥j−vm∥2 D_KL (q( v_j)\|p( v_j| x_j^v, s_m) )=- _v2\| [rgb]0,0,0 $μ$_ v_j- [rgb]0,0,0 W^v s_m\|^2 (17) −Λv2‖j−fv(jv)‖2+C. - _v2\| [rgb]0,0,0 $μ$_ v_j-f_v(x_j^v)\|^2+C. Here C can be omitted because we treat scalars like λv∈ℝ _v , λu∈ℝ _u , Λv∈ℝ _v , and Λu∈ℝ _u as constants. Intuition for Each Term in Eqn. 2.2. Below, we describe the intuition of each term in Eqn. 2.2: (1) Regularize Latent Variable i u_i (Eqn. 11). The KL term DKL(q(i)∥p(i|k,kk=1K,m,π))D_KL (q( u_i)\|p( u_i|\ $μ$_k, $ $_k\_k=1^K, s_m,π) ) aims to regularize user i’s latent variable i u_i, ensuring i u_i is close to the center of its corresponding user cluster πi _i, and therefore close to other users’ latent embeddings in the same cluster. (2) Predict Residual Rating R~ij R_ij from i u_i and j v_j (Eqn. 12). p(R~ij|i,j,m)p( R_ij| u_i, v_j, s_m) uses the inferred i u_i, j v_j, and the causal confounder m s_m to predict the residual rating, thereby encouraging i u_i and j v_j to retain more information to maximize prediction performance. (3) Regularize Latent Variable j v_j (Eqn. 13). Similar to Eqn. 11, DKL(q(j)∥p(j|jv,m))D_KL(q( v_j)\|p( v_j| x^v_j, s_m)) is the KL divergence term between the inference model q(⋅|jv)q(·| x^v_j) and the generative model p(⋅|m)p(·| s_m); this encourages the inferred posterior q(j|jv)q( v_j| x^v_j) to be close to the prior distribution p(j|m)p( v_j| s_m). (4) Regularize Latent Variable π (Eqn. 14). DKL(q(π)∥p(π|θ))D_KL (q(π)\|p(π|θ) ) is the KL divergence term between the categorical variational distribution q(π)q(π) and the prior p(π|θ)p(π|θ); this encourages the inferred posterior q(π)q(π) to be close to the prior p(π|θ)p(π|θ). 2.3. Inference and Learning In our framework, we need to learn several parameters, including the Gaussian parameters k,kk=1K\ $μ$_k, $ $_k\_k=1^K, user latent u, item latent v, and the parameters of the functions fx(⋅)f_x(·) and fv(⋅)f_v(·), as well as uW^u, vW^v, and Rw^R. The following sections detail the learning process for all these parameters. The complete algorithm is outlined in Algorithm 1. 1) μk,kk=1K\ $μ$_k, $ $_k\_k=1^K. To optimize k,kk=1K\ $μ$_k, $ $_k\_k=1^K, we take derivatives of Eqn. 2.2 w.r.t. k $μ$_k and k $ $_k as follows: (18) ∂ℒ∂k=k−1(i−k−um), ∂L∂ $μ$_k= $ $_k^-1 ( u_i- $μ$_k- [rgb]0,0,0 W^u s_m ), (19) ∂ℒ∂k=12k−1[(i−k−um)(i−k−um)⊤−k]k−1. ∂L∂ $ $_k= 12 $ $_k^-1 [ ( u_i- $μ$_k- [rgb]0,0,0 W^u s_m ) ( u_i- $μ$_k- [rgb]0,0,0 W^u s_m ) - $ $_k ] $ $_k^-1. Setting Eqn. 18 and Eqn. 19 to zero leads to the following update rules, respectively: (20) k=1|Ik|∑i∈Ik(i−um), $μ$_k= 1|I_k| _i∈ I_k ( u_i- [rgb]0,0,0 W^u s_m ), (21) k=1|Ik|∑i∈Ik(i−k−um)(i−k−um)⊤, $ $_k= 1|I_k| _i∈ I_k ( u_i- $μ$_k- [rgb]0,0,0 W^u s_m ) ( u_i- $μ$_k- [rgb]0,0,0 W^u s_m ) , where k I_k is the set of user index i that belongs to cluster k. 2) i,j u_i, v_j, and πi _i. After computing the gradients of Eqn. 2.2 w.r.t. the means of i∼(i,Λu−1) u_i N( $μ$_ u_i, _u^-1 I) (i.e., i $μ$_ u_i) and j∼(j,Λv−1) v_j N( $μ$_ v_j, _v^-1 I) (i.e., j $μ$_ v_j), we obtain the following update rules: i= $μ$_ u_i= (πiλR~(i,:)⊤+)−1[πi+um ( $ $_ _i V _ R_(i,:) V + I)^-1 [ $μ$_ _i+ [rgb]0,0,0 W^u s_m (22) +πiλR~(i,:)(~(i,:)−R⊤m)], + $ $_ _i V _ R_(i,:)( R_(i,:)- [rgb]0,0,0 w^R s_m I) ], j= $μ$_ v_j= [λR~(:,j)⊤+(λv−Λv)]−1 [ U _ R_(:,j) U +( _v- _v) I ]^-1 ⋅[λvvm−Λvfv(jv) · [ _v [rgb]0,0,0 W^v s_m- _vf_v( x_j^v) (23) +λR~(:,j)(~(:,j)−R⊤m)]. + U _ R_(:,j)( R_(:,j)- [rgb]0,0,0 w^R s_m I) ]. PRL is efficient; Eqn. 22 uses only the user’s rated items, so its cost scales with rated items rather than the full catalog. Note that here U and V refer to user latent matrix (i)i=1I( u_i)_i=1^I and item latent matrix (j)j=1J( v_j)_j=1^J. ~(i,:):=(R~i1,⋯,R~iJ)⊤ R_(i,:):=( R_i1,·s, R_iJ) , ~(:,j):=(R~1j,⋯,R~Ij)⊤ R_(:,j):=( R_1j,·s, R_Ij) . λR~(i,:):=diag(λR~i1,⋯,λR~iJ) _ R_(i,:):=diag( _ R_i1,·s, _ R_iJ), λR~(:,j):=diag(λR~1j,⋯,λR~Ij) _ R_(:,j):=diag( _ R_1j,·s, _ R_Ij). With a uniform prior distribution on πi _i, i.e., p(πi=k|θ)=1Kp( _i=k|θ)= 1K, the estimation of πi _i is similar to that of a Gaussian mixture model (GMM). Specifically, we can approximate p(πi=k|ui,vj,xjv,μk,Σkk=1K)p( _i=k|u_i,v_j,x_j^v,\ _k, _k\^K_k=1) using (24) q(πi=k)=(i;μk+Wusmi,Σk)∑l=1K(i;μl+Wusmi,Σl), q( _i=k)= N( u_i; _k+W_us_m_i, _k) _l=1^KN( u_i; _l+W_us_m_i, _l), which is obtained by maximizing the ELBO Eqn. 11∼ 14. We then choose the optimal (most probable) cluster for each user i, i.e., the cluster assignment for user i is set to πi=argmaxkq(πi=k) _i= argmax_kq( _i=k), which is used to compute Eqn. 22∼ 23. Algorithm 1 Learning Algorithm of PRL Input: Raw item features v x^v, initialized fx(⋅)f_x(·) and fv(⋅)f_v(·) parameters, u,v,R [rgb]0,0,0 W^u, [rgb]0,0,0 W^v, [rgb]0,0,0 w^R, initialized Gaussian parameters k,kk=1K\ $μ$_k, $ $_k\_k=1^K, and the number of epochs T. for t=1:Tt=1:T do for m=1:Mm=1:M do Update i u_i and j v_j using Eqn. 22 and Eqn. 23 update the variational distribution of πi _i using Eqn. 24. Update u,v,R [rgb]0,0,0 W^u, [rgb]0,0,0 W^v, [rgb]0,0,0 w^R using Eqn. 25, Eqn. 26 and Eqn. 27. Update the parameters of fv(⋅)f_v(·) using gradient ascent of ℒL in Eqn. 2.2. Update k,kk=1K\ $μ$_k, $ $_k\_k=1^K using Eqn. 20 and Eqn. 21, respectively. Output: fv(⋅)f_v(·) parameters, u,v,R [rgb]0,0,0 W^u, [rgb]0,0,0 W^v, [rgb]0,0,0 w^R, and Gaussian parameters k,kk=1K\ $μ$_k, $ $_k\_k=1^K. 3) u [rgb]0,0,0 W^u, v [rgb]0,0,0 W^v, R [rgb]0,0,0 w^R. The update rules for u [rgb]0,0,0 W^u, v [rgb]0,0,0 W^v, and R [rgb]0,0,0 w^R are as follows: (25) u=1I(∑i=1Ii−∑k=1K|Ik|k)m⊤(mm⊤)−1, [rgb]0,0,0 W^u= 1I( _i=1^I u_i- _k=1^K|I_k| $μ$_k) s_m ( s_m s_m )^-1, (26) v=1J∑j=1Jjm⊤(mm⊤)−1, [rgb]0,0,0 W^v= 1J _j=1^J v_j s_m ( s_m s_m )^-1, (27) R=∑i,jλR~ij(R~ij−i⊤j)∑i,jλR~ij(mm⊤)−1m. [rgb]0,0,0 w^R= _i,j _ R_ij( R_ij- u_i v_j) _i,j _ R_ij( s_m s_m )^-1 s_m. 4) Parameters of fx(⋅)f_x(·) and fv(⋅)f_v(·). We use gradient ascent of ℒL in Eqn. 2.2 to update these parameters. Inference. Inference includes the E-Step in Algorithm 1, where PRL infers latent variables i u_i and j v_j, and updates the parameters of encoder model fv(⋅)f_v(·) using gradient ascent of ℒL in Eqn. 2.2. Learning. Learning includes the iteration between the E-Step and M-Step in Algorithm 1 until convergence. In each M-Step, we update the Gaussian parameters k,kk=1K\ $μ$_k, $ $_k\_k=1^K following the update rules from Eqn. 20 and Eqn. 21, respectively; we update learnable parameters u,v [rgb]0,0,0 W^u, [rgb]0,0,0 W^v, R [rgb]0,0,0 w^R following the update rules from Eqn. 25, Eqn. 26, and Eqn. 27, respectively. 2.4. Plug-and-Play PRL Below we discuss key components of our plug-and-play PRL after learning all parameters with Algorithm 1. Inferring User Clusters. Given the learned mixture parameters k,kk=1K\ $μ$_k, $ $_k\_k=1^K, PRL assigns each user to the cluster πi _i with the largest posterior probability in Eqn. 24, similar to a hard EM algorithm. Isolating Causal Confounders m s_m. With the structured causal model (SCM), we estimate a domain-level confounder representation m s_m for each domain m by approximating its posterior distribution p(m|R~,jv,k,kk=1K)p( s_m| R, x_j^v,\ $μ$_k, $ $_k\_k=1^K) via variational domain indexing (VDI) (VDI). In this way, we can minimize the bias introduced by the causal confounder m s_m when inferring i u_i and j v_j using Eqn. 22 and Eqn. 23, respectively. In practice, m s_m may be instantiated by an observed proxy or by a learned domain index, depending on the dataset. When VDI is used, the domain index sms_m can be thought of as the embedding for each domain m. For example, in the dataset XMRec where each of the 18 domains contains items and users from one market/country (e.g., France or US), sms_m can be thought of as a “country” embedding. Interestingly, our preliminary results show that similar countries tend to have similar domain embedding sms_m (i.e., domain index) (VDI) . In other words, sms_m captures the similarities among different domains and therefore provides valuable information for our recommender systems. Debiasing the Causal Confounders. Under our PRL framework, for each inferred user cluster k, we perform causal inference for each user i in this cluster to predict the residual R~ij R_ij (for each item j) while debiasing the causal confounders s. Specifically, with the inferred i u_i and j v_j (using Eqn. 22 and Eqn. 23) and m s_m, we can predict R~ij R_ij by do-calculus as (28) p(k)(R~ij|do(i),do(j))=∑m=1Mp(k)(R~ij|i,j,m)p(m), p^(k)( R_ij|do( u_i),do( v_j))=Σ _m=1^Mp^(k)( R_ij| u_i, v_j, s_m)p( s_m), where p(k)(R~ij|i,j,)p^(k)( R_ij| u_i, v_j, s) represents the k’th sub-model trained from the k’th cluster’s user data. In practice, we use k=πik= _i (πi _i is user i’s cluster) when predicting user i’s rating R~ij R_ij. Figure 3. Causal inference in PRL is equivalent to removing the influence of confounder s on i u_i and j v_j. The causal graphical model after intervention on the user and item latent variables. The incoming edges from the domain confounder $ $ to $ _i$ and $ _j$ are removed, while $ _i$ and $ _j$ remain connected to the residual rating $ R_ij$. Note that performing causal inference by intervening (i,j)( u_i, v_j) effectively cuts the relations between the causal confounders s and (i,j)( u_i, v_j). Fig. 3 demonstrates the do-calculus that PRL performs for debiasing the causal confounder s. Intuition behind Do-Calculus: Why Marginalize over the Domain Factor m s_m in Eqn. 28. Estimating interventional distributions typically requires actively intervening in a recommender system to collect data, which is often costly and impractical. Instead, PRL leverages do-calculus to estimate the interventional effect p(k)(R~ij∣do(i),do(j))p^(k)( R_ij do( u_i),do( v_j)) from observational data by marginalizing over the confounder s. This operation reduces reliance on spurious domain-specific correlations. In datasets such as XMRec, where each domain corresponds to a country, m s_m can act as a confounder, inducing biases such as exposure or popularity effects. A predictor conditioned on a specific domain may overfit these shortcuts, attributing preferences to country-specific patterns rather than true user-item relationships. For example, a model may associate Bollywood films with cricket gear due to their co-occurrence in a market, not due to intrinsic preference. By marginalizing over m s_m as in Eqn. 28, PRL removes such confounding effects and retains residual signals that are more stable and transferable across domains. Summary. To summarize, for each user i, PRL causally infers the residual rating R~i R_i as follows: (1) Infer the user cluster πi _i by approximating its posterior p(πi|i,j,jv,k,kk=1K)p( _i| u_i, v_j, x^v_j,\ $μ$_k, $ $_k\_k=1^K). (2) Infer the residual rating R~ij R_ij by causal Bayesian model averaging defined in Eqn. 28. (3) Predict the final rating as R=R~+R^R= R+ R, where R R is the base recommender’s prediction. Table 1. Performance of PRL with different base models on XMRec. The best results are marked with bold face. Data Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 France, Italy, India → Japan, Mexico CDL (Base Model) 0.0143 0.0016 0.0028 0.0009 0.0009 PRL (Full) 0.1091 0.0128 0.0463 0.0108 0.0068 DLRM (Base Model) 0.0044 0.0004 0.0004 0.0002 0.0002 PRL (Full) 0.0295 0.0035 0.0048 0.0018 0.0018 PerK (Base Model) 0.1098 0.0128 0.0512 0.0112 0.0068 PRL (Full) 0.1635 0.0192 0.0637 0.0151 0.0102 NCF (Base Model) 0.0131 0.0148 0.0026 0.0008 0.0008 PRL (Full) 0.1137 0.0137 0.0309 0.0090 0.0073 LightGCN (Base Model) 0.0182 0.0021 0.0050 0.0014 0.0011 PRL (Full) 0.1003 0.0121 0.0316 0.0084 0.0064 Mexico, Spain, India → Japan, Germany CDL (Base Model) 0.1127 0.0135 0.0301 0.0086 0.0072 PRL (Full) 0.1761 0.0230 0.0593 0.0163 0.0123 DLRM (Base Model) 0.0756 0.0093 0.0085 0.0041 0.0049 PRL (Full) 0.2017 0.0246 0.0545 0.0156 0.0131 PerK (Base Model) 0.1443 0.0177 0.0601 0.0143 0.0094 PRL (Full) 0.2750 0.0335 0.1086 0.0263 0.0179 NCF (Base Model) 0.0096 0.0012 0.0022 0.0007 0.0007 PRL (Full) 0.1558 0.0202 0.0280 0.0107 0.0108 LightGCN (Base Model) 0.0165 0.0022 0.0061 0.0016 0.0012 PRL (Full) 0.1064 0.0138 0.0278 0.0087 0.0077 Table 2. Performance of PRL with different base models on MovieLens. The best results are marked with bold face. Data Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 1, 18, 35, 45, 50, 56 → 25 CDL (Base Model) 0.0179 0.0274 0.0045 0.0581 0.0587 PRL (Full) 0.0252 0.0409 0.0072 0.1071 0.1076 DLRM (Base Model) 0.0714 0.1096 0.0285 0.2433 0.2366 PRL (Full) 0.0716 0.1101 0.0284 0.2431 0.2372 PerK (Base Model) 0.0682 0.1029 0.0290 0.2224 0.2107 PRL (Full) 0.0690 0.1037 0.0287 0.2190 0.2110 NCF (Base Model) 0.0050 0.0250 0.0011 0.0251 0.0251 PRL (Full) 0.0240 0.0387 0.0057 0.0947 0.1005 LightGCN (Base Model) 0.0081 0.0132 0.0019 0.0381 0.0358 PRL (Full) 0.0249 0.0402 0.0069 0.1076 0.1055 25 → 1, 18, 35, 45, 50, 56 CDL (Base Model) 0.0576 0.0848 0.0174 0.1602 0.1716 PRL (Full) 0.0645 0.0952 0.0202 0.1772 0.1897 DLRM (Base Model) 0.0848 0.1342 0.0382 0.3347 0.3225 PRL (Full) 0.0903 0.1405 0.0414 0.3455 0.3319 PerK (Base Model) 0.0746 0.1164 0.0324 0.2701 0.2661 PRL (Full) 0.0792 0.1225 0.0355 0.2821 0.2757 NCF (Base Model) 0.0140 0.0229 0.0030 0.0633 0.0652 PRL (Full) 0.0450 0.0694 0.0144 0.1639 0.1711 LightGCN (Base Model) 0.0093 0.0157 0.0022 0.0497 0.0482 PRL (Full) 0.0290 0.0480 0.0097 0.1493 0.1395 3. Experiments We evaluate whether PRL improves heterogeneous base recommenders and whether its causal component benefits cross-domain recommendation. Additional settings and results can be found in the supplementary material. 3.1. Datasets XMRec. XMRec (xmrec) contains 18 local markets (countries), 16 product categories, and 52.5M user-item interactions. For each item j, we use its description as the item feature jv x_j^v. We exclude users with fewer than three purchases. The three train-test splits are: France, Italy, India → Japan, Mexico; Mexico, Spain, India → Japan, Germany; and Germany, Italy, Japan → United States, India. We use product production countries as the causal confounders m s_m. MovieLens. MovieLens (harper2015movielens) contains movie ratings from users of different ages. We use movie titles and plots as item features jv x_j^v, and derive user features from the first three films each user rated. We exclude users with fewer than five ratings or with no rating above 3, leaving 6,034 users and 3,705 items. We use two age-based train-test splits: 1–18, 18–25, 35–45, 45–50, 50–56, 56+56^+ → 25–35; and 25–35 → all other groups. For brevity, we denote each age group by its starting age, e.g., “1” for “1–18”. We use normalized movie release years as causal confounders m s_m. In all experiments, we use a cold-start setting where each testing domain user has only one rating in the training set, making the recommendations extremely challenging. Table 3. Comparison between PRL w/o Causality and PRL (Full) on a specific domain with different base models on XMRec. The best results are marked with bold face. Data Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 France, Italy, India → Japan, Mexico CDL (Base Model) 0.0143 0.0016 0.0028 0.0009 0.0009 CDL PRL w/o Causality 0.1058 0.0126 0.0333 0.0088 0.0067 PRL (Full) 0.1091 0.0128 0.0463 0.0108 0.0068 DLRM (Base Model) 0.0044 0.0004 0.0004 0.0002 0.0002 DLRM PRL w/o Causality 0.0232 0.0026 0.0039 0.0014 0.0014 PRL (Full) 0.0295 0.0035 0.0048 0.0018 0.0018 PerK (Base Model) 0.1098 0.0128 0.0512 0.0112 0.0068 PerK PRL w/o Causality 0.1376 0.0160 0.0558 0.0129 0.0085 PRL (Full) 0.1635 0.0192 0.0637 0.0151 0.0102 NCF (Base Model) 0.0131 0.0148 0.0026 0.0008 0.0008 NCF PRL w/o Causality 0.1056 0.0126 0.0235 0.0074 0.0067 PRL (Full) 0.1137 0.0137 0.0309 0.0090 0.0073 LightGCN (Base Model) 0.0182 0.0021 0.0050 0.0014 0.0011 LightGCN PRL w/o Causality 0.0940 0.0112 0.0289 0.0076 0.0059 PRL (Full) 0.1003 0.0121 0.0316 0.0084 0.0064 Figure 4. Recall@N on all three XMRec splits, comparing each base model with PRL without causality and full PRL. Recall at different cutoff values on three XMRec train-test splits. Each plot compares the base recommender, PRL without causal adjustment, and full PRL for CDL, DLRM, or PerK. 3.2. Base Recommenders, Baselines, and Metrics PRL is a plug-and-play framework compatible with any base recommender. In this paper, we compare PRL with five representative recommendation models, including Collaborative Deep Learning (CDL) (cdl), Deep Learning Recommendation Model (DLRM) (DLRM), Top Personalized-K Recommendation. (PerK) (kweon2024perk), Neural Collaborative Filtering (NCF) (NCF), and Light Graph Convolutional Network (LightGCN) (lightgcn). These methods serve as both our baselines and our base recommenders to be enhanced by PRL. We evaluate all methods using Recall@20, Precision@20, F1@20, MAP@20, and NDCG@20, averaged over all users; formal metric definitions and training details are in the Appendix. 3.3. Results Results for Different Base Models. Table 1 and Table 2 show that PRL improves the base models overall. Gains are smaller on MovieLens, where release year provides a weaker confounding signal than XMRec’s market factors. Recall@N with Larger N. Fig. 4 shows that full PRL consistently outperforms its non-causal counterpart across base models, splits, and cutoff values. Visualizations of the Clusters. Fig. 5 visualizes the user latent u for all five base models on the XMRec dataset. Each visualization shows a distinct separation into 3 clusters, indicating successful user grouping of our model. Furthermore, Figure 6 illustrates the relationship between user clusters and items using the CDL-based PRL model on the same dataset. For each user, we selected the item with the highest rating they have given, recorded the item ID and its rating, and visualized the results. Different clusters are represented using distinct colors, effectively showcasing the distribution and preferences of users within each cluster. For instance, Cluster 1 (Red) shows pronounced preferences for 4-5 specific items, underscoring the impact of user clustering on improving PRL’s performance. Performance of Each Cluster Discovered by PRL. For a deeper understanding of the model performance, we include more fine-grained results for different clusters discovered by PRL in Appendix. Results show that our PRL can usually improve performance in most clusters. Figure 5. t-SNE projections of PRL user latents i u_i on XMRec. Colors denote cluster assignments πi _i inferred by Eqn. 24, rather than ground-truth labels. Five t-SNE projections of the learned user latent representations obtained with CDL, DLRM, PerK, NCF, and LightGCN. Points are colored according to the three clusters inferred by PRL. Figure 6. User clusters based on users’ highest rated items, using the CDL-based PRL model applied to the XMRec dataset. X-axis indicates the item ID, while Y-axis indicates the score of the item. Clusters are distinguished by different colors. A scatter plot of the highest-rated item selected for each user under the CDL-based PRL model. The horizontal axis gives item indices and the vertical axis gives rating scores. Points are colored according to the three user clusters inferred by PRL, revealing cluster-specific concentrations over items and scores. Case Study on Debiasing. To explore PRL’s debiasing capability and cross-domain generalization, we conducted a detailed analysis using the first domain pair of the XMRec dataset, which corresponds to the countries France, Italy, India, Japan, and Mexico. We first examined the top-20 recommendations for each user using the CDL baseline model. The resulting distributions showed notable country-specific biases: • Italian users: 34 Camera & Photo recommendations (3.70%) out of 920 total recommendations. • Indian users: 878 Camera & Photo recommendations (6.46%) out of 13,600 total recommendations. The bias ratio across countries was 1.75× (maximum: 6.46%, minimum: 3.70%), indicating that Indian users were recommended camera products 1.75 times more frequently than Italian users. After applying our PRL model, we examined the top-20 recommendations for each user. The resulting distributions were notably more balanced: • Italian users: 33 Camera & Photo recommendations (3.59%) out of 920 total recommendations. • Indian users: 427 Camera & Photo recommendations (3.14%) out of 13,600 total recommendations. The bias ratio across countries was reduced to 1.14× (maximum: 3.59%, minimum: 3.14%), representing a 38.5% reduction in country-specific bias. This demonstrates that PRL successfully mitigated the preference biases through probabilistic user clustering and causal debiasing, thereby enabling the model to learn more generalizable user-item interaction patterns. Ablation Study. The comparison in Table 3 and Fig. 4 highlights the performance difference between ‘PRL w/o Causality’ and ‘PRL (Full)’. The results consistently show that ‘PRL (Full)’ outperforms its counterpart, ‘PRL w/o Causality’, emphasizing the crucial role of causal inference in enhancing the effectiveness of the PRL model. Furthermore, a comparison between the base model and ‘PRL w/o Causality’ also reveals notable performance improvements, validating the efficacy of PRL’s user cluster discovery. Additional details and results can be found in Table S1∼ 15 of Appendix. 4. Related Work Domain-Dependent Recommendation. Previous work has explored in-domain recommendation scenarios. Early methods, including PMF (pmf) and BPR (bpr), applied collaborative filtering techniques to address challenges in recommendation. Sequential models such as GRU4Rec (gru4rec), SASRec (sasrec), UniSRec (unisrec), and CauseRec (causerec) use interaction histories; the latter two transfer item representations and augment counterfactual sequences, respectively. CDL (cdl; CRAE; CKE; ColVAE) instead uses item content for cold-start recommendation. Despite advances in in-domain recommendation, cross-domain recommendation remains understudied. Existing work uses domain adaptation techniques (VDI; TSDA; UDIL; GRDA; CIDA; DANN), often relying on shared users or items across source and target domains (DBLP:conf/sigir/Yuan0KZ20; DBLP:conf/sigir/WuYCLH020; DBLP:conf/sigir/BiSYWWX20a; li2019zero; DBLP:conf/sigir/Hansen0SAL20; DBLP:conf/sigir/LiangXYY20; DBLP:conf/sigir/ZhuSSC20; liu2020heterogeneous). On the other hand, some methods enhance recommendation performance in both source and target domains simultaneously (0008T20; HuZY18; ZhaoLF19). In contrast, PRL first infers user clusters and confounders before cluster-specific recommendation, improving generalization and robustness to domain shifts. Causal Inference for Recommendation. Causal inference (pearl2009causality; CounTS) and discovery (glymour2019review; ICL) have been widely applied to model cause-and-effect relationships between variables in the machine learning community. Recently, it has been employed to improve the performance of recommender systems (wang2020causal). PDA (zhang2021causal) uses causal intervention to address popularity bias in recommendations, while DICE (zheng2021disentangling) learns representations from user interactions based on the structured causal model (SCM). Additionally, some research focuses on debiasing recommendations without adopting a causal inference perspective (li2021debiasing; Wang0LZY022; chen2023bias). However, these approaches do not consider user groups within the SCM framework. In contrast, our method divides users into clusters based on a confounder variable and generates recommendations by aggregating user ratings through do-calculus, providing a more interpretable and sophisticated approach. 5. Conclusion In this paper, we address the problem of cross-domain recommendation by introducing a novel causal Bayesian framework, named Probabilistic Residual Learning (PRL). PRL generates recommendations by: (1) inferring the user cluster ID, (2) inferring the residual rating based on our causal debiasing framework, and (3) predicting the final rating as a correction to the base model’s prediction. PRL can enhance the performance of any base recommenders in a plug-and-play manner, and automatically discover meaningful user clusters. As a general probabilistic framework compatible with various recommendation systems, PRL can be extended to additional modalities beyond textual data in future research. 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Probabilistic Residual Learning for Online Recommendations (Appendix) Appendix A More Details on Experiments and Implementation A.1. Metrics mAP. mAP is defined as: (S1) APi=1|Ji|∑n=1Nreli,n×Precisioni@n, _i= 1|J_i| _n=1^Nrel_i,n×Precision_i@n, where N is the total number of recommended items, Precisioni@nPrecision_i@n is the precision at rank n, and |Ji||J_i| is the total number of relevant items for user i. The mean Average Precision (mAP) is then calculated by averaging APiAP_i over all users: (S2) mAP=1|I|∑i=1|I|APi, = 1|I| _i=1^|I|AP_i, where |I||I| is the total number of users. NDCG. NDCG@N is computed as follows. First, the Discounted Cumulative Gain (DCG@N) is calculated: (S3) DCGi@N=∑n=1N2reli,n−1log2(n+1), _i@N= _n=1^N 2^rel_i,n-1 _2(n+1), where reli,nrel_i,n denotes the relevance of the item at position n for user i. Next, the Ideal Discounted Cumulative Gain (IDCG@N), representing the maximum possible DCG (i.e., all relevant items ranked at the top), is calculated as: (S4) IDCGi@N=∑n=1min(N,|Ji|)21−1log2(n+1)=∑n=1min(N,|Ji|)1log2(n+1), _i@N= _n=1 (N,|J_i|) 2^1-1 _2(n+1)= _n=1 (N,|J_i|) 1 _2(n+1), where |Ji||J_i| denotes the total number of relevant items for user i. Finally, the Normalized Discounted Cumulative Gain is obtained by normalizing DCG@N by IDCG@N: (S5) NDCGi@N=DCGi@NIDCGi@N. _i@N= DCG_i@NIDCG_i@N. Here the logarithmic term log2(n+1) _2(n+1) discounts the relevance based on the item’s position in the ranked list, serving as the normalization factor. A.2. Training Configurations Following CDL (cdl), we set the hidden dimension h=50h=50 for all latent vectors, as well as for the encoder network. Similar to CDL, we also add an additional decoder to reconstruct v x^v, serving as a regularization term during training. During training, we use AdamW (Adam) as our optimizer, with a learning rate of 10−310^-3 and a batch size of 256256. The base models were trained for 100 epochs, while PRL was trained for 150 epochs. All experiments were conducted on an NVIDIA RTX A5000 GPU. A.3. Performance of Each Clusters Discovered by PRL Table S7, S8, S9, S10, S11 show PRL’s performance across different clusters on XMRec using CDL, DLRM, PerK, NCF, and LightGCN as base models. These results support the conclusion that PRL improves upon the base models even without incorporating the causality component. Furthermore, the full PRL consistently outperforms its non-causal counterpart across all configurations. For example, CDL, as the base model, achieves a recall@2020 of 0.02410.0241 for User Cluster 1 in the split of “France, Italy, India → Japan, Mexico”. When PRL without the causal inference component is applied, recall improves to 0.02780.0278. The full PRL further enhances performance for this metric, achieving a recall@2020 of 0.07080.0708. Table S1, S3, S4, S5, S6 show PRL’s performance across different clusters on MovieLens with the same five base models. Even with some fluctuations, the similar improvements are consistent with the results for XMRec. A.4. Ablation Study The performance comparison across Table S1-S11 shows that “PRL (Full)” generally outperforms “PRL w/o Causality”, highlighting the effectiveness of causal inference in PRL. Additionally, comparing the base model with “PRL w/o Causality” reveals performance enhancements, suggesting that PRL’s user cluster discovery significantly boosts performance. Simple Baseline. Moreover, we conducted experiments by clustering the user features and then performing per-cluster modeling to predict the residuals. Table S12 presents the results comparing this simple baseline with our full PRL, verifying that our latent variable modeling is highly effective, while per-cluster modeling alone yields limited improvements. Larger Base Models. We also conducted additional ablation experiments by scaling up the CDL baseline to approximately match the parameter size of the PRL-enriched network. Concretely, we expanded both the depth and width of the CDL architecture as suggested. The original CDL structure was 512 → 200 → 50, while the larger (deeper and wider) version used an architecture of 512 → 550 → 400 → 50, resulting in roughly 1.05M parameters, which is comparable to PRL’s 0.90M parameters. This ensures that the comparison isolates the effect of PRL from mere model capacity differences. The results are presented in Table S13. Although the deeper CDL exhibits a slight performance improvement over the base model, PRL still achieves substantially higher performance across most metrics. This demonstrates that PRL’s gains stem from its ability to capture cross-domain relational patterns, user-cluster-specific representations, rather than simply from an increased parameter count. More User Records in the Training Set. We conducted experiments using CDL as the base model to analyze the effect of incorporating a higher number of testing-domain user records into the training process. Specifically, we used the second domain pair of the XMRec dataset, involving users from Mexico, Spain, India, Japan, and Germany. we selected users with more than four interactions and varied the number of testing-domain user records included in the training set to construct 1-shot, 2-shot, and 3-shot scenarios, where n-shot means there are n records for each testing user in the training set. The performance of both the base CDL model and our proposed PRL model in these settings is summarized in Table S14 and Table S15 below. Table S1. Performance of PRL on different user clusters with CDL as the base model on MovieLens. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 1, 18, 35, 45, 50, 56 → 25 CDL (Base Model) 0.0 0.0 0.0 0.0 0.0 1 PRL w/o Causality 0.0 0.0 0.0 0.0 0.0 PRL (Full) 0.0 0.0 0.0 0.0 0.0 CDL (Base Model) - - - - - 2 PRL w/o Causality - - - - - PRL (Full) - - - - - CDL (Base Model) 0.0179 0.0274 0.0045 0.0581 0.0587 3 PRL w/o Causality 0.0186 0.0302 0.0056 0.0864 0.0802 PRL (Full) 0.0252 0.0409 0.0072 0.1071 0.1077 25 → 1, 18, 35, 45, 50, 56 CDL (Base Model) 0.0558 0.0861 0.0174 0.1758 0.1879 1 PRL w/o Causality 0.0317 0.0528 0.0095 0.1511 0.1572 PRL (Full) 0.0558 0.0861 0.0174 0.1759 0.1879 CDL (Base Model) 0.0651 0.0795 0.0173 0.0938 0.1020 2 PRL w/o Causality 0.0676 0.0880 0.0183 0.1159 0.1259 PRL (Full) 0.1016 0.1341 0.0319 0.1832 0.1972 CDL (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Table S2. Performance of PRL with different base models on MovieLens. The best results are marked with bold face. Data Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 1, 18, 35, 45, 50, 56 → 25 CDL (Base Model) 0.0179 0.0274 0.0045 0.0581 0.0587 PRL w/o Causality 0.0186 0.0302 0.0057 0.0863 0.0801 PRL (Full) 0.0252 0.0409 0.0072 0.1071 0.1076 DLRM (Base Model) 0.0714 0.1096 0.0285 0.2433 0.2366 PRL w/o Causality 0.0232 0.0026 0.0039 0.0014 0.0014 PRL (Full) 0.0716 0.1101 0.0284 0.2431 0.2372 PerK (Base Model) 0.0682 0.1029 0.0290 0.2224 0.2107 PRL w/o Causality 0.0582 0.0877 0.0212 0.1755 0.1787 PRL (Full) 0.0690 0.1037 0.0287 0.2190 0.2110 NCF (Base Model) 0.0050 0.0250 0.0011 0.0251 0.0251 PRL w/o Causality 0.0231 0.0374 0.0055 0.0927 0.0989 PRL (Full) 0.0240 0.0387 0.0057 0.0947 0.1005 LightGCN (Base Model) 0.0081 0.0132 0.0019 0.0381 0.0358 PRL w/o Causality 0.0248 0.0402 0.0070 0.1077 0.1053 PRL (Full) 0.0249 0.0402 0.0069 0.1076 0.1055 Table S3. Performance of PRL on different user clusters with DLRM as the base model on MovieLens. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 1, 18, 35, 45, 50, 56 → 25 DLRM (Base Model) - - - - - 1 PRL w/o Causality - - - - - PRL (Full) - - - - - DLRM (Base Model) 0.0714 0.1097 0.0285 0.2433 0.2367 2 PRL w/o Causality 0.0269 0.0434 0.0073 0.1078 0.1112 PRL (Full) 0.0716 0.1101 0.0284 0.2431 0.2372 DLRM (Base Model) 0.0 0.0 0.0 0.0 0.0 3 PRL w/o Causality 0.0 0.0 0.0 0.0 0.0 PRL (Full) 0.0 0.0 0.0 0.0 0.0 25 → 1, 18, 35, 45, 50, 56 DLRM (Base Model) 0.0790 0.1264 0.0343 0.3266 0.3146 1 PRL w/o Causality 0.0328 0.0548 0.0116 0.1716 0.1656 PRL (Full) 0.0848 0.1366 0.0396 0.3634 0.3505 DLRM (Base Model) 0.0882 0.1390 0.0405 0.3396 0.3271 2 PRL w/o Causality 0.0975 0.1382 0.0426 0.2572 0.2374 PRL (Full) 0.1119 0.1561 0.0486 0.2745 0.2583 DLRM (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Table S4. Performance of PRL on different user clusters with Perk as the base model on MovieLens. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 1, 18, 35, 45, 50, 56 → 25 Perk (Base Model) - - - - - 1 PRL w/o Causality - - - - - PRL (Full) - - - - - Perk (Base Model) 0.0686 0.1040 0.0295 0.2271 0.2150 2 PRL w/o Causality 0.0583 0.0884 0.0215 0.1788 0.1820 PRL (Full) 0.0683 0.1036 0.0288 0.2215 0.2136 Perk (Base Model) 0.0585 0.0745 0.0173 0.1053 0.1023 3 PRL w/o Causality 0.0550 0.0701 0.0139 0.0942 0.0967 PRL (Full) 0.0847 0.1074 0.0277 0.1559 0.1467 25 → 1, 18, 35, 45, 50, 56 Perk (Base Model) 0.0745 0.1179 0.0332 0.2868 0.2826 1 PRL w/o Causality 0.0319 0.0530 0.0140 0.1811 0.1563 PRL (Full) 0.0745 0.1179 0.0332 0.2870 0.2828 Perk (Base Model) 0.0750 0.1090 0.0292 0.2033 0.1995 2 PRL w/o Causality 0.0939 0.1338 0.0367 0.2399 0.2323 PRL (Full) 0.0984 0.1407 0.0446 0.2628 0.2469 Perk (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Table S5. Performance of PRL on different user clusters with NCF as the base model on MovieLens. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 1, 18, 35, 45, 50, 56 → 25 NCF (Base Model) 0.0 0.0 0.0 0.0 0.0 1 PRL w/o Causality 0.0 0.0 0.0 0.0 0.0 PRL (Full) 0.1964 0.1325 0.0230 0.0754 0.1000 NCF (Base Model) 0.0051 0.0087 0.0011 0.0282 0.0279 2 PRL w/o Causality 0.0271 0.0443 0.0067 0.1134 0.1210 PRL (Full) 0.0285 0.0463 0.0070 0.1159 0.1231 NCF (Base Model) 0.0047 0.0074 0.0009 0.0172 0.0177 3 PRL w/o Causality 0.0125 0.0192 0.0023 0.0386 0.0409 PRL (Full) 0.0120 0.0185 0.0022 0.0389 0.0410 25 → 1, 18, 35, 45, 50, 56 NCF (Base Model) 0.0149 0.0248 0.0032 0.0710 0.0729 1 PRL w/o Causality 0.0309 0.0515 0.0088 0.1494 0.1555 PRL (Full) 0.0306 0.0512 0.0087 0.1484 0.1551 NCF (Base Model) - - - - - 2 PRL w/o Causality - - - - - PRL (Full) - - - - - NCF (Base Model) 0.0098 0.0150 0.0021 0.0302 0.0319 3 PRL w/o Causality 0.0941 0.1316 0.0312 0.2094 0.2185 PRL (Full) 0.1071 0.1481 0.0392 0.2309 0.2402 Table S6. Performance of PRL on different user clusters with LightGCN as the base model on MovieLens. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 1, 18, 35, 45, 50, 56 → 25 LightGCN (Base Model) - - - - - 1 PRL w/o Causality - - - - - PRL (Full) - - - - - LightGCN (Base Model) 0.0081 0.0132 0.0019 0.0381 0.0358 2 PRL w/o Causality 0.0248 0.0402 0.0070 0.1075 0.1052 PRL (Full) 0.0248 0.0401 0.0069 0.1073 0.1053 LightGCN (Base Model) 0.0226 0.0224 0.0075 0.0227 0.0222 3 PRL w/o Causality 0.0214 0.0378 0.0115 0.1911 0.1611 PRL (Full) 0.0563 0.0884 0.0226 0.2219 0.2056 25 → 1, 18, 35, 45, 50, 56 LightGCN (Base Model) 0.0094 0.0157 0.0022 0.0498 0.0484 1 PRL w/o Causality 0.0300 0.0495 0.0101 0.1515 0.1416 PRL (Full) 0.0288 0.0477 0.0097 0.1492 0.1394 LightGCN (Base Model) 0.0068 0.0110 0.0011 0.0277 0.0294 2 PRL w/o Causality 0.0297 0.0428 0.0130 0.0953 0.0765 PRL (Full) 0.0531 0.0793 0.0204 0.1597 0.1559 LightGCN (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Table S7. Performance of PRL on different user clusters with CDL as the base model on XMRec. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 France, Italy, India → Japan, Mexico CDL (Base Model) 0.0241 0.0028 0.0062 0.0018 0.0015 1 PRL w/o Causality 0.1972 0.0238 0.0905 0.0197 0.0127 PRL (Full) 0.0708 0.0074 0.0652 0.0105 0.0039 CDL (Base Model) 0.0126 0.0014 0.0022 0.0007 0.0008 2 PRL w/o Causality 0.0902 0.0107 0.0236 0.0069 0.0057 PRL (Full) 0.1156 0.0138 0.0431 0.0109 0.0073 CDL (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Mexico, Spain, India → Japan, Germany CDL (Base Model) 0.1742 0.0225 0.0333 0.0123 0.0120 1 PRL w/o Causality 0.2114 0.0267 0.0707 0.0194 0.0142 PRL (Full) 0.1665 0.0222 0.0634 0.0170 0.0119 CDL (Base Model) 0.0903 0.0102 0.0289 0.0072 0.0054 2 PRL w/o Causality 0.1532 0.0187 0.0524 0.0136 0.0100 PRL (Full) 0.1796 0.0233 0.0579 0.0160 0.0124 CDL (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Germany, Italy, Japan → United States, India CDL (Base Model) 0.0262 0.0059 0.0079 0.0041 0.0033 1 PRL w/o Causality 0.0261 0.0063 0.0072 0.0044 0.0036 PRL (Full) 0.0266 0.0064 0.0062 0.0042 0.0037 CDL (Base Model) 0.0244 0.0054 0.0088 0.0042 0.0031 2 PRL w/o Causality 0.0166 0.0037 0.0041 0.00234 0.0021 PRL (Full) 0.0250 0.0055 0.0088 0.0042 0.0031 CDL (Base Model) 0.0277 0.0049 0.0066 0.0028 0.0027 3 PRL w/o Causality 0.0194 0.0045 0.0049 0.0030 0.0026 PRL (Full) 0.0278 0.0049 0.0067 0.0028 0.0027 Table S8. Performance of PRL on different user clusters with DLRM as the base model on XMRec. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 France, Italy, India → Japan, Mexico DLRM (Base Model) 0.0051 0.0005 0.0004 0.0002 0.0003 1 PRL w/o Causality 0.0246 0.0027 0.0039 0.0014 0.0014 PRL (Full) 0.0345 0.004 0.0056 0.0021 0.0021 DLRM (Base Model) 0.0000 0.0000 0.0000 0.0000 0.0000 2 PRL w/o Causality 0.0150 0.0017 0.0040 0.0010 0.0009 PRL (Full) 0.0000 0.0000 0.0000 0.0000 0.0000 DLRM (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Mexico, Spain, India → Japan, Germany DLRM (Base Model) 0.0000 0.0000 0.0000 0.0000 0.0000 1 PRL w/o Causality 0.3296 0.0416 0.0203 0.0153 0.0222 PRL (Full) 0.3074 0.0395 0.0213 0.0152 0.0211 DLRM (Base Model) 0.0780 0.0096 0.0087 0.0042 0.0051 2 PRL w/o Causality 0.1398 0.0174 0.0277 0.0096 0.0093 PRL (Full) 0.1984 0.0241 0.0555 0.0157 0.0128 DLRM (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Germany, Italy, Japan → United States, India DLRM (Base Model) 0.0023 0.0006 0.0003 0.0003 0.0003 1 PRL w/o Causality 0.0042 0.0011 0.0010 0.0007 0.0007 PRL (Full) 0.0046 0.0011 0.0009 0.0007 0.0006 DLRM (Base Model) 0.0018 0.0005 0.0003 0.0003 0.0003 2 PRL w/o Causality 0.0045 0.0012 0.0010 0.0007 0.0007 PRL (Full) 0.0045 0.0011 0.0012 0.0007 0.0007 DLRM (Base Model) 0.0036 0.0008 0.0005 0.0004 0.0004 3 PRL w/o Causality 0.0052 0.0015 0.0009 0.0009 0.0009 PRL (Full) 0.0141 0.0034 0.0075 0.0032 0.0019 Table S9. Performance of PRL on different user clusters with PerK as the base model on XMRec. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 France, Italy, India → Japan, Mexico PerK (Base Model) 0.1752 0.0204 0.1152 0.022 0.0108 1 PRL w/o Causality 0.2153 0.0260 0.1255 0.0252 0.0139 PRL (Full) 0.1782 0.0210 0.1162 0.0226 0.0114 PerK (Base Model) 0.0986 0.0115 0.0403 0.0094 0.0061 2 PRL w/o Causality 0.1243 0.0143 0.0440 0.0108 0.0076 PRL (Full) 0.1629 0.0189 0.0548 0.0138 0.0100 PerK (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Mexico, Spain, India → Japan, Germany PerK (Base Model) 0.1434 0.0176 0.0582 0.014 0.0094 1 PRL w/o Causality 0.2175 0.0262 0.0913 0.0217 0.0140 PRL (Full) 0.2905 0.0353 0.1157 0.0278 0.0188 PerK (Base Model) 0.1495 0.0184 0.0723 0.0166 0.0098 2 PRL w/o Causality 0.2783 0.0307 0.0964 0.0232 0.0163 PRL (Full) 0.1790 0.0224 0.0646 0.0167 0.0120 PerK (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Germany, Italy, Japan → United States, India PerK (Base Model) 0.0194 0.0043 0.0057 0.003 0.0024 1 PRL w/o Causality 0.0295 0.0066 0.0087 0.0046 0.0037 PRL (Full) 0.0308 0.0068 0.0086 0.0046 0.0038 PerK (Base Model) 0.0126 0.0028 0.0032 0.0018 0.0016 2 PRL w/o Causality 0.0155 0.0035 0.0040 0.0022 0.0020 PRL (Full) 0.0162 0.0037 0.0048 0.0025 0.0021 PerK (Base Model) 0.0261 0.0035 0.0091 0.0025 0.0019 3 PRL w/o Causality 0.0174 0.0027 0.0013 0.0012 0.0014 PRL (Full) 0.0266 0.0041 0.0102 0.0033 0.0022 Table S10. Performance of PRL on different user clusters with NCF as the base model on XMRec. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 France, Italy, India → Japan, Mexico NCF (Base Model) 0.0090 0.0010 0.0019 0.0005 0.0005 1 PRL w/o Causality 0.2013 0.0238 0.0537 0.0151 0.0127 PRL (Full) 0.1581 0.0176 0.0476 0.0122 0.0093 NCF (Base Model) 0.0165 0.0019 0.0032 0.0010 0.0010 2 PRL w/o Causality 0.0893 0.0107 0.0184 0.0061 0.0057 PRL (Full) 0.1062 0.0130 0.0280 0.0084 0.0069 NCF (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Mexico, Spain, India → Japan, Germany NCF (Base Model) 0.0097 0.0013 0.0022 0.0007 0.0007 1 PRL w/o Causality 0.1081 0.0142 0.0181 0.0073 0.0076 PRL (Full) 0.1560 0.0202 0.0280 0.0107 0.0108 NCF (Base Model) - - - - - 2 PRL w/o Causality - - - - - PRL (Full) - - - - - NCF (Base Model) 0.0 0.0 0.0 0.0 0.0 3 PRL w/o Causality 0.0 0.0 0.0 0.0 0.0 PRL (Full) 0.0 0.0 0.0 0.0 0.0 Germany, Italy, Japan → United States, India NCF (Base Model) 0.0020 0.0005 0.0006 0.0004 0.0003 1 PRL w/o Causality 0.0204 0.0051 0.0041 0.0030 0.0029 PRL (Full) 0.0214 0.0055 0.0039 0.0032 0.0031 NCF (Base Model) 0.0018 0.0005 0.0003 0.0003 0.0003 2 PRL w/o Causality 0.0064 0.0015 0.0008 0.0008 0.0009 PRL (Full) 0.0079 0.0021 0.0011 0.0011 0.0012 NCF (Base Model) 0.0 0.0 0.0 0.0 0.0 3 PRL w/o Causality 0.0 0.0 0.0 0.0 0.0 PRL (Full) 0.0 0.0 0.0 0.0 0.0 Table S11. Performance of PRL on different user clusters with LightGCN as the base model on XMRec. “-” means a cluster contains only training-set users, i.e., no test-set users to evaluate. The best results are marked with bold face. Data Cluster Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 France, Italy, India → Japan, Mexico LightGCN (Base Model) 0.0261 0.0034 0.0028 0.0015 0.0018 1 PRL w/o Causality 0.1742 0.0209 0.0749 0.0167 0.0111 PRL (Full) 0.1400 0.0154 0.0482 0.0115 0.0081 LightGCN (Base Model) 0.0168 0.0019 0.0054 0.0013 0.0010 2 PRL w/o Causality 0.0804 0.0095 0.0211 0.0060 0.0051 PRL (Full) 0.0936 0.0115 0.0288 0.0079 0.0061 LightGCN (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Mexico, Spain, India → Japan, Germany LightGCN (Base Model) 0.0093 0.0009 0.0046 0.0008 0.0005 1 PRL w/o Causality 0.0972 0.0097 0.0129 0.0045 0.0051 PRL (Full) 0.0 0.0 0.0 0.0 0.0 LightGCN (Base Model) 0.0170 0.0023 0.0062 0.0017 0.0012 2 PRL w/o Causality 0.1040 0.0135 0.0215 0.0077 0.0072 PRL (Full) 0.1790 0.0224 0.0646 0.0167 0.0120 LightGCN (Base Model) - - - - - 3 PRL w/o Causality - - - - - PRL (Full) - - - - - Germany, Italy, Japan → United States, India LightGCN (Base Model) 0.0016 0.0005 0.0002 0.0002 0.0003 1 PRL w/o Causality 0.0062 0.0017 0.0012 0.0010 0.0010 PRL (Full) 0.0066 0.0017 0.0014 0.0011 0.0010 LightGCN (Base Model) 0.0 0.0 0.0 0.0 0.0 2 PRL w/o Causality 0.0 0.0 0.0 0.0 0.0 PRL (Full) 0.0 0.0 0.0 0.0 0.0 LightGCN (Base Model) 0.0016 0.0004 .0002 0.0002 0.0002 3 PRL w/o Causality 0.0037 0.0009 0.0010 0.0006 0.0005 PRL (Full) 0.0039 0.0008 0.0012 0.0006 0.0005 Table S12. Results on the simple baseline and PRL. Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 CDL 0.0143 0.0016 0.0028 0.0009 0.0009 Clustering + Residuals 0.0156 0.0018 0.0023 0.0009 0.0010 PRL (Full) 0.1091 0.0128 0.0463 0.0108 0.0068 DLRM 0.0044 0.0004 0.0004 0.0002 0.0002 Clustering + Residuals 0.0163 0.0018 0.0029 0.0009 0.0010 PRL (Full) 0.0295 0.0035 0.0048 0.0018 0.0018 PerK 0.1098 0.0128 0.0512 0.0112 0.0068 Clustering + Residuals 0.1118 0.0129 0.0513 0.0113 0.0069 PRL (Full) 0.1635 0.0192 0.0637 0.0151 0.0102 NCF 0.0131 0.00148 0.0026 0.0008 0.0008 Clustering + Residuals 0.0164 0.0019 0.0029 0.0010 0.0010 PRL (Full) 0.1137 0.0137 0.0309 0.0090 0.0073 LightGCN 0.0182 0.0021 0.0050 0.0014 0.0011 Clustering + Residuals 0.0277 0.0031 0.0054 0.0017 0.0017 PRL (Full) 0.1003 0.0121 0.0316 0.0084 0.0064 Table S13. Performance comparison between larger CDL and PRL. Method Recall@20 F1@20 MAP@20 NDCG@20 Precision@20 CDL 0.0143 0.0016 0.0028 0.0009 0.0009 CDL (Larger) 0.0223 0.0026 0.0022 0.0011 0.0014 PRL (Full) 0.1091 0.0128 0.0463 0.0108 0.0068 Table S14. Results in n-shot settings for the CDL base model. Method Config Recall@20 Precision@20 F1@20 MAP@20 NDCG@20 CDL (Base Model) 1-shot 0.0647 0.0105 0.0181 0.0111 0.0098 CDL (Base Model) 2-shot 0.0700 0.0085 0.0152 0.0108 0.0078 CDL (Base Model) 3-shot 0.0817 0.0060 0.0112 0.0159 0.0106 Table S15. Results in n-shot settings for our proposed model PRL (Full) using CDL as the base model. Method Config Recall@20 Precision@20 F1@20 MAP@20 NDCG@20 PRL (Full) 1-shot 0.1080 0.0195 0.0359 0.0252 0.0208 PRL (Full) 2-shot 0.1507 0.0195 0.0356 0.0359 0.0216 PRL (Full) 3-shot 0.2178 0.0145 0.0272 0.0471 0.0251 References