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Beyond Observed Auxiliary Relations: Environment-Conditioned Modeling for Multi-Behavior Recommendation
Seunghan Lee, Hyunsik Yoo, Jian Kang, Susik Yoon, SeongKu Kang
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Abstract:Multi-behavior recommendation (MBR) leverages auxiliary behavioral signals, such as clicks and add-to-cart, to enhance target behavior prediction like purchases. While recent graph neural network-based approaches have achieved strong performance by systematically propagating auxiliary behavior signals, they still suffer from two fundamental challenges inherent to auxiliary behaviors: (1) missing auxiliary signals, which hinder generalization to items without auxiliary observations, and (2) unreliable auxiliary signals, which amplify noise misaligned with the target behavior. To address these challenges in a unified manner, we propose BOAR, an environment-conditioned MBR framework that addresses missing and unreliable auxiliary signals through two complementary modules conditioned on auxiliary observability. Extensive experiments demonstrate that BOAR consistently outperforms state-of-the-art baselines, achieving up to 7.82% gains in HR@10 overall and up to 44.2% gains for target items without auxiliary observations, highlighting its ability to capture hidden preferences beyond observed auxiliary relations. Our code is available at: this https URL.
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- Source: https://arxiv.org/abs/2608.22920v1
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Beyond Observed Auxiliary Relations: Environment-Conditioned Modeling for Multi-Behavior RecommendationConference: Conference; June 03–05, 2018; Woodstock, NYISBN: 978-1-4503-X-X/2018/06Conference: Proceedings of the 35th ACM International Conference on Information and Knowledge Management; November 07–11, 2026; Rome, ItalyProceedings of the 35th ACM International Conference on Information and Knowledge Management (CIKM ’26), November 07–11, 2026, Rome, ItalyDOI: 10.1145/3799682.3841033ISBN: 979-8-4007-2539-5/2026/11CCS: Information systems Recommender systems Seunghan Lee Affiliation: Korea University , Seoul , Republic of Korea email: seunghanlee@korea.ac.kr , Hyunsik Yoo Affiliation: University of Illinois Urbana-Champaign , Champaign , IL , USA email: hy40@illinois.edu , Jian Kang Affiliation: MBZUAI , Abu Dhabi , United Arab Emirates email: jian.kang@mbzuai.ac.ae , Susik Yoon Affiliation: Korea University , Seoul , Republic of Korea email: susik@korea.ac.kr and SeongKu Kang Note: Corresponding author Affiliation: Korea University , Seoul , Republic of Korea email: seongkukang@korea.ac.kr 2026; © c Abstract. Multi-behavior recommendation (MBR) leverages auxiliary behavioral signals, such as clicks and add-to-cart, to enhance target behavior prediction like purchases. While recent graph neural network–based approaches have achieved strong performance by systematically propagating auxiliary behavior signals, they still suffer from two fundamental challenges inherent to auxiliary behaviors: (1) missing auxiliary signals, which hinder generalization to items without auxiliary observations, and (2) unreliable auxiliary signals, which amplify noise misaligned with the target behavior. To address these challenges in a unified manner, we propose , an environment-conditioned MBR framework that addresses missing and unreliable auxiliary signals through two complementary modules conditioned on auxiliary observability. Extensive experiments demonstrate that outperforms state-of-the-art baselines, achieving up to 7.82% gains in HR@10 overall and up to 44.2% gains for target items without auxiliary observations, highlighting its ability to capture hidden preferences beyond observed auxiliary relations. Our code is available at: https://github.com/LSH0411/BOAR. Keywords: Multi-behavior recommendation, Graph Rewiring, Observation bias †c-license: by 1. Introduction Users in modern recommender systems rarely express their preferences through a single type of behavior. Before making a purchase, users typically engage in multiple auxiliary behaviors, such as clicking items, adding them to carts, or saving them to wishlists. These auxiliary behaviors provide indirect yet complementary signals that reflect users’ latent preferences with respect to the final target behavior, such as purchasing. Accordingly, multi-behavior recommendation (MBR) aims to jointly model auxiliary behaviors together with the target behavior to better capture user preferences and improve recommendation accuracy (Jin et al. 2020; Chen et al. 2021). In multi-behavior recommendation, a wide range of approaches has been explored, from traditional matrix factorization (Zhao et al. 2015; Loni et al. 2016) to deep neural network-based models (Gao et al. 2019). Among these approaches, graph neural networks (GNNs) have recently emerged as a core paradigm by explicitly modeling the relational structure between users and items, rather than treating interactions independently (Wang et al. 2019). By representing different behavior types as distinct relations on graphs, GNNs systematically propagate auxiliary behavior signals to enhance target behavior prediction (Jin et al. 2020). This relational modeling is particularly effective in real-world settings, where target behaviors are extremely sparse. Furthermore, combined with advanced learning paradigms such as self-supervised and multi-task learning, recent methods learn richer and more generalizable representations and achieve state-of-the-art performance (Gu et al. 2022; Xu et al. 2023; Zhai et al. 2025). Figure 1. Overview of MBR setup. The left panel illustrates training and inference spaces. The right panel presents an empirical evaluation with two settings, reporting HR@10: (a) Hidden preference test, removing auxiliary interactions from observed target-positive items; and (b) Noise injection test, adding auxiliary interactions to target-negative items. Target-positive and target-negative items are defined with respect to the target behavior. Detailed setups are provided in Section 5.2.2. Although effective, existing GNN-based methods still face two fundamental challenges arising from the nature of auxiliary behaviors. Specifically, auxiliary signals are often missing or unreliable: their absence does not necessarily indicate the absence of user interest, while their presence does not necessarily indicate target intent. • C1: Missingness of Auxiliary Signals. In practice, users do not necessarily exhibit auxiliary behaviors for all items they may purchase. Importantly, such missingness is not at random (Li et al. 2023; Schnabel et al. 2016): auxiliary behaviors are observed for a skewed subset of items, driven by factors such as platform exposure policies and item popularity, rather than solely reflecting the absence of user interest.11 1 The observability of interactions is influenced by multiple factors. Decomposing these causes is not our focus, and we collectively refer to this skew as observation bias (Chen et al. 2023). GNNs propagating over such biased relations may overrepresent frequently exposed items with many interactions (Zhou et al. 2023). However, MBR must generalize beyond observed auxiliary relations and accurately recommend target behavior items without any auxiliary behaviors. This is illustrated by Figure 1 (left). The training space defined by observed interactions covers only a limited subset of the inference space, while hidden target items, i.e., target-positive items without auxiliary interactions, may appear outside the training space. Failure to capture such hidden preferences confines recommendations to items that users have already encountered, limiting their practical effectiveness. • C2: Unreliability of Auxiliary Signals. Observed auxiliary behaviors do not necessarily imply purchase intent. While auxiliary behaviors may indicate potential interest, they can also be triggered by accidental clicks or casual browsing. As a result, auxiliary behaviors contain noisy signals that are misaligned with the target behavior. GNNs are structurally vulnerable to such noise because message passing can propagate and amplify incorrect information (Dai et al. 2022; Wang et al. 2024). In MBR, auxiliary relation-driven propagation may amplify signals loosely related to the target behavior, thereby hindering target behavior prediction. Figure 1 (left) illustrates this issue: not all items with auxiliary behaviors are target-positive items that eventually lead to purchase, as auxiliary interactions can also involve target-negative items. While prior methods partially address these challenges, they remain insufficient to fully resolve them, and to our knowledge, no existing method addresses both simultaneously. Regarding C1, MEMBER (Kim et al. 2025a) introduces self-supervised learning to reduce over-reliance on observed auxiliary interactions; however, it still relies solely on the observed auxiliary signals, without explicitly uncovering hidden preferences. Regarding C2, several studies (Lee et al. 2024; Zhang et al. 2025) regulate message passing to suppress noise propagation; however, the regulation is only weakly guided by the target behavior, which limits its effectiveness in filtering unreliable auxiliary signals. Indeed, in Figure 1 (right), we evaluate recent GNN-based methods relevant to each challenge (MEMBER (Kim et al. 2025a) for C1, and MuLe (Lee et al. 2024), HGIB (Zhang et al. 2025) for C2) under settings that intensify each challenge by increasing the ratio of hidden target items and auxiliary noise, respectively. Existing methods show substantial performance degradation, suggesting limited capability to handle challenges arising from auxiliary behaviors. As a solution, we propose (Beyond Observed Auxiliary Relations), a framework that rewires given auxiliary relations into target-informative signals instead of using them as-is. This rewiring requires different strategies depending on auxiliary observability: when auxiliary observability is lacking, C1 calls for debiased densification to surface hidden preferences, whereas with strong auxiliary observability, C2 calls for selective pruning to filter unreliable signals. Although both challenges could in principle be addressed within a single model, doing so may yield suboptimal signals under training data imbalance: target interactions without auxiliary interactions form only a small fraction (e.g., 16.1%16.1\% in Tmall), causing the model to focus overly on auxiliary-observed cases and limiting hidden preference discovery (see Section 3.2 for detailed analysis). This naturally motivates an environment-conditioned modular design: two lightweight modules that focus on complementary rewiring strategies, i.e., recovering hidden preferences and pruning unreliable signals. To coordinate these specialized modules, auxiliary observability as an environment condition and probabilistically adjusts their contributions to each target behavior prediction, addressing the two challenges in a complementary manner. Our main contributions are: • We highlight two fundamental challenges of auxiliary behaviors that have not yet been fully addressed and, to our knowledge, present the first attempt to jointly address both. • We propose , an environment-conditioned modular MBR framework that overcomes the limitations of observed auxiliary relations through conditional modular modeling. • Extensive experiments show that consistent gains, especially for target behavior without auxiliary observations, with efficiency comparable to state-of-the-art baselines. 2. Related Works Multi-Behavior Recommendation (MBR). MBR leverages diverse auxiliary behaviors, such as clicks and add-to-cart, to alleviate the data sparsity of target behaviors like purchases. Early studies explored various techniques, including matrix factorization (Loni et al. 2016; Zhao et al. 2015), deep neural networks (Gao et al. 2019), and attention-based approaches (Guo et al. 2019; Xia et al. 2020; Xia et al. 2021a). Recently, GNN-based methods have emerged as the dominant approach (Jin et al. 2020; Xia et al. 2021b; Xia et al. 2021a; Wei et al. 2022; Xu et al. 2023). Existing methods adopt diverse graph encoding strategies, including integrating multiple behavior types into a unified graph (Jin et al. 2020; Zhang et al. 2020; Xia et al. 2021b; Chen et al. 2021; Xia et al. 2021a), explicitly modeling natural behavioral sequences (e.g., click→cart→purchase) (Yan et al. 2023a; Cheng et al. 2023; Meng et al. 2023a; Gong et al. 2025), and encoding each behavior graph in parallel (Yan et al. 2023b; Lee et al. 2024; Meng et al. 2023b; Zhang et al. 2025; Kim et al. 2025a). Furthermore, advanced training techniques such as self-supervised learning (Gu et al. 2022; Xu et al. 2023; Zhai et al. 2025; Zhang et al. 2025; Kim et al. 2025a) and multi-task learning (Yan et al. 2023a; Xu et al. 2023; Meng et al. 2023a; Meng et al. 2023b; Gong et al. 2025; Zhai et al. 2025) have been incorporated to enhance target behavior prediction. However, reliance on auxiliary behaviors introduces the two aforementioned challenges, and recent GNN-based methods have partially addressed them. For C1, MEMBER (Kim et al. 2025a) adopts a mixture-of-experts framework to distinguish purchases with auxiliary behaviors from those without. It employs self-supervised learning to reduce the dominance of purchases preceded by auxiliary behaviors, encouraging the model to better capture purchase patterns that occur independently of auxiliary signals. For C2, MuLe (Lee et al. 2024) introduces attention-based message passing to adjust the contributions of auxiliary behaviors and mitigate uncertainty during propagation, while HGIB (Zhang et al. 2025) applies the information bottleneck principle to filter out information weakly related to the final prediction. Despite these advancements, substantial room for improvement remains in addressing each challenge, and to our knowledge, no prior method has addressed both challenges simultaneously. Specifically, no dedicated effort has been made to explicitly handle bias or uncover hidden target items, limiting the ability to address C1. Meanwhile, existing methods control propagation primarily based on similarity induced by auxiliary behaviors, with limited guidance from target behaviors, which remains insufficient to resolve C2. Graph Rewiring for Imperfect Graphs. In MBR, the two aforementioned challenges are fundamentally manifested as structural imperfections in auxiliary behavior graphs. Graph rewiring (GR) is a core methodology in graph structure learning that reconstructs edge connections to address such imperfections (Linkerhägner et al. 2025; Gao et al. 2024; Ding et al. 2022; Guo et al. 2023a). GR methods typically jointly optimize graph structure and node representations to recover missing edges and mitigate unreliable connections. A common approach infers potential connections based on learned node embedding similarities, followed by pruning unreliable edges or adding new ones (Yang et al. 2021; Tian et al. 2022; Jiang et al. 2024; Bi et al. 2024). These techniques have proven effective across domains, including single-behavior recommendation (Yang et al. 2021; Tian et al. 2022), graph classification (Bi et al. 2024), and social recommendation (Jiang et al. 2024). Such improvements are attributed to pruning unreliable connections and strengthening homophilic relationships, enhancing graph quality. However, applying existing GR strategies to MBR may be suboptimal, as auxiliary behavior graphs contain imperfections tied to auxiliary observability. In particular, conventional similarity-based rewiring may amplify these imperfections: densification may favor items with abundant auxiliary interactions, failing to surface hidden preferences without auxiliary evidence (C1), while pruning may mistakenly remove informative auxiliary connections as noise (C2). This calls for an auxiliary observability-conditioned graph rewiring strategy tailored to the unique challenges of MBR. 3. Preliminaries 3.1. Problem Formulation Let U and ℐI denote the sets of users and items. We consider a set of behaviors ℬ=b1,…,b|ℬ|B=\b_1,…,b_|B|\ (e.g., click, cart, … , buy), where btar=b|ℬ|b_tar=b_|B| is the target behavior and ℬ∖btarB \b_tar\ are auxiliary behaviors. For each behavior b∈ℬb , we define a bipartite graph b=(∪ℐ,ℰb)G_b=(U ,E_b), where ℰb⊆×ℐE_b ×I denotes the observed interactions. Let ℰaux=⋃b∈ℬ∖btarℰbE_aux= _b \b_tar\E_b denote the set of all observed auxiliary interactions. For each pair (u,i)(u,i), we define a binary random variable Rui∈0,1R_ui∈\0,1\ indicating whether user u engages in the target behavior on item i. Our goal is to learn a function f(u,i)f(u,i) that estimates the probability that a user u exhibits the target behavior with item i: (1) f(u,i)≈[Rui∣u,i]=P(Rui=1∣u,i).f(u,i)\ ≈\ E\! [R_ui u,i ]\ =\ P(R_ui=1 u,i). We focus on addressing the aforementioned challenges, C1: missing auxiliary signals and C2: unreliable auxiliary signals, arising from the nature of auxiliary behaviors. Figure 2. Training-set imbalance between target interactions with (Obs) and without (Unobs) auxiliary observations (left), and the resulting performance gap between the two groups (right). Results are reported on the Tmall dataset. 3.2. Environment-Conditioned Modeling As discussed above, the two challenges require distinct graph rewiring strategies: densifying the target behavior graph to recover hidden target items when auxiliary evidence is lacking, and pruning the auxiliary behavior graphs to suppress noise misaligned with target behavior when auxiliary evidence is abundant. Empirical Evidence. Figure. 2 (left) shows that among target interactions, those with auxiliary interactions substantially outnumber those without. This imbalance can make GNN-based models overly focus on items with auxiliary observations. Indeed, Figure. 2 (right) shows that even state-of-the-art methods consistently suffer a significant performance drop on hidden target items. This suggests that a single model may struggle to derive rewiring signals suited to both cases, particularly for hidden preference discovery. This motivates a design that incorporates auxiliary observability, so that rewiring signals can be effectively derived for both auxiliary-observed and -unobserved cases. Environment-Conditioned Modeling. The above observations motivate a modular formulation in which target preference estimation is conditioned on auxiliary observability. By the law of total expectation, the target preference can be decomposed as: (2) [Rui∣u,i]=∑e∈0,1P(Eui=e∣u,i)[Rui∣u,i,Eui=e],E[R_ui u,i]= _e∈\0,1\P(E_ui=e u,i)\ E[R_ui u,i,E_ui=e], where EuiE_ui denotes the auxiliary-observability environment of the pair (u,i)(u,i). In the interaction log, the presence of auxiliary interactions provides an observable basis for this condition: Eui=0E_ui=0 for (u,i)∉ℰaux(u,i) _aux and Eui=1E_ui=1 for (u,i)∈ℰaux(u,i) _aux. Note that Eui=0E_ui=0 only means that no auxiliary behavior is observed in the current log; such pairs may become observable as user activity continues. Thus, the final prediction uses P(Eui=e∣u,i)P(E_ui=e u,i) as a soft weight, rather than relying solely on the observed environment information. Accordingly, we model the conditional expectations with two environment-conditioned sub-modules: (3) f0(u,i) f_0(u,i) ≈[Rui∣u,i,Eui=0], [R_ui u,i,E_ui=0], f1(u,i) f_1(u,i) ≈[Rui∣u,i,Eui=1]. [R_ui u,i,E_ui=1]. Here, f0f_0 focuses on rewiring under the auxiliary-unobserved condition, where hidden preferences should be recovered. In contrast, f1f_1 focuses on rewiring under the auxiliary-observed condition, where unreliable auxiliary relations should be filtered. The final prediction is obtained by softly aggregating the conditional predictions as f(u,i)=∑e∈0,1P(Eui=e∣u,i)fe(u,i)f(u,i)= _e∈\0,1\P(E_ui=e u,i)\,f_e(u,i). Here, P(Eui=e∣u,i)P(E_ui=e u,i) controls the contribution of each sub-module. This modular formulation enables coordination of complementary rewiring strategies across auxiliary-observability conditions, while behavior-specific importance is further taken into account within the modules as appropriate. The concrete instantiation is introduced in the subsequent section. 4. Proposed Method Figure 3. (Left) Overview of the . (Right) Construction of densification and refinement signals in each module, which are subsequently incorporated via self-supervised contrastive learning. Best viewed in color. We present BOAR, an environment-conditioned MBR framework (Figure 3). constructs disentangled input representations (Sec. 4.1), and then applies two modules, with distinct graph rewiring and learning strategies conditioned on the environment: • (Sec.4.2) Debiased densification module f0f_0 identifies hidden target items by mining hidden preferences while debiasing auxiliary relations, providing densification signals. • (Sec.4.3) Target-guided refinement module f1f_1 prunes auxiliary relations misaligned with the target behavior, refining auxiliary graphs toward target-consistent relations. Section 4.4 describes modular integration for learning and inference. 4.1. Input Representation Construction We construct input user/item representations using both the global graph and the target-behavior graph via a LightGCN (He et al. 2020) encoder. Let ∈ℝ(||+|ℐ|)×dE ^(|U|+|I|)× d denote the initial embedding matrix that stacks user and item embeddings. We obtain: (4) glo=LGCN(glo,),tar=LGCN(tar,),E^glo=LGCN\! (A_glo,E ), ^tar=LGCN\! (A_tar,E ), where gloA_glo denotes the adjacency matrix of the global graph gloG_glo constructed from all behaviors, i.e., ℰglo=⋃b∈ℬℰbE_glo= _b E_b, and tarA_tar denotes the adjacency matrix of the target behavior graph. The global embeddings capture comprehensive multi-behavior signals and provide warm-start features for subsequent behavior-specific encoding (Lee et al. 2024; Zhang et al. 2025; Kim et al. 2025b). In each module, the global embedding gloE^glo serves as a source of densification (f0f_0) or refinement (f1f_1) signals, while the target embedding tarE^tar is used for final prediction. Note that, to enable environment-conditioned modeling without increasing model capacity, we split the initial embedding into two halves and encode each half with both the global and target graph encoders. 4.2. Debiased Densification Module The densification module f0f_0 mines hidden preferences to provide densification signals for the target behavior graph, and consists of three components: (1) auxiliary popularity adversarial learning to adversarially suppress popularity-driven observation bias from item representations, (2) target hidden preference miner to efficiently discover hidden preferences, and (3) debiased densification learning to safely inject the mined knowledge into the final prediction. 4.2.1. Auxiliary Popularity Adversarial Learning. To reduce popularity-driven observation bias before hidden preference mining, we adapt the adversarial debiasing (Zhu et al. 2025; Zhang et al. 2024; Wu et al. 2021). The key idea is to prevent item representations from overly encoding auxiliary behavior popularity, so that they can better reflect user preference. This is achieved through adversarial learning with a gradient reversal layer (GRL) (Ganin et al. 2016). Following (Zhu et al. 2025), we construct a proxy signal for popularity-driven bias from the total number of auxiliary interactions each item receives: yi=1y_i=1 if the count exceeds the median interaction count across all items, and yi=0y_i=0 otherwise. This signal construction strategy provides practical and balanced adversarial supervision.22 2 While more complex signals could be considered, prior work has shown that this median-based strategy can effectively suppress popularity-related information (Zhu et al. 2025). We then define the adversarial supervision set as adv=(i,yi)S_adv=\(i,y_i)\. A discriminator fϕf_φ, implemented as a linear layer with a sigmoid function, is trained to predict the proxy signal yiy_i of each item i. Since popularity is an item-level property, the discriminator takes the item embedding as input and outputs y^i=σ(fϕ(GRL(iglo))) y_i=σ(f_φ(GRL(e^glo_i))), where σ is the sigmoid function. The discriminator is trained adversarially via the GRL, such that the item representations are updated to hinder accurate popularity signal prediction: (5) ℒadv=(i,yi)∼adv[−yilogy^i−(1−yi)log(1−y^i)] gatheredL_adv=E_(i,y_i) _adv [-y_i y_i-(1-y_i) (1- y_i) ] gathered This adversarial objective reduces the influence of popularity information in item representations, providing a basis for hidden preference mining. 4.2.2. Target Hidden Preference Miner. Based on the popularity-suppressed representations, we uncover potential target relations. Existing graph rewiring methods (Jiang et al. 2024; Guo et al. 2023a) typically rely on exhaustive pairwise similarity computation to identify highly similar node pairs, incurring prohibitive costs. To address this, we propose an efficient three-stage mining strategy: (i) locality-sensitive hashing (LSH)-based candidate retrieval, (i) learnable edge selection, and (i) weighted graph augmentation. Angular LSH-based Candidate Retrieval. To efficiently retrieve candidate items, we adopt angular LSH (Andoni et al. 2015; Kitaev et al. 2020), which enables fast approximate nearest-neighbor search based on angular similarity. Unlike traditional LSH, angular LSH constructs hash buckets using a single random projection followed by an argmax operation, which can be efficiently computed on GPUs via matrix multiplication. Moreover, its collision probability is monotonically related to angular similarity rather than L2 distance, making it better aligned with the inner product–based prediction. We first draw a random projection matrix ∈ℝd×d′R ^d× d with i.i.d. entries Rpq∼(0,1)R_pq (0,1), where d′d is the number of hash buckets and each column defines a random angular direction in the embedding space. For each node x, we normalize its representation and project it onto these directions, obtaining projection scores xs_x that reflect its angular alignment.33 3 Concretely, we compute x=[(xglo/|xglo|)⊤∥(xglo/|xglo|)⊤(−)]∈ℝ2d′s_x=[(e^glo_x/|e^glo_x|) R\,\|\,(e^glo_x/|e^glo_x|) (-R)] ^2d . The concatenation with −-R allows us to capture both positive and negative angular directions using a single argmax operation, resulting in 2d′2d effective buckets (Kitaev et al. 2020). Each node is then assigned to a hash bucket corresponding to the dominant projection direction as: h(x)=argmaxk(x)kh(x)= _k(s_x)_k. This procedure ensures that nodes with similar angular directions are likely to be grouped into the same bucket, without explicitly computing pairwise similarities. For each user u, we find unobserved items from the same bucket, i.e., i∣h(i)=h(u)\i h(i)=h(u)\, and retrieve top-n items by cosine similarity to form the candidate set ℰcand(u)E_cand(u). The total candidates across all users are aggregated as: ℰcand=⋃u∈(u,i)∣i∈ℰcand(u)E_cand= _u \(u,i) i _cand(u)\. Learnable Edge Selection. Not all candidates are equally reliable for target behavior prediction. To selectively retain edges truly aligned with this objective, we adopt a learnable edge selection strategy jointly optimized with the prediction task. For each candidate pair (u,i)∈ℰcand(u,i) _cand, we predict a binary add-or-skip decision using faddf_add, implemented as a linear layer. Specifically, we compute logits uiadd=fadd([uglo∥iglo])∈ℝ2z^add_ui=f_add([e^glo_u ^glo_i]) ^2, where each output dimension corresponds to add and skip decisions, respectively. We apply the Gumbel-Softmax (Jang et al. 2017) to obtain differentiable selection mask:44 4 GumbelSoftmax(,τ)k=exp((zk+gk)/τ)∑jexp((zj+gj)/τ)GumbelSoftmax(z;τ)_k= ((z_k+g_k)/τ) _j ((z_j+g_j)/τ), where gk=−log(−log(uk))g_k=- (- (u_k)) with uk∼Uniform(0,1)u_k (0,1). As τ→0τ→ 0, the output approaches a one-hot discrete sample. (6) muiadd=GumbelSoftmax(uiadd,τ)[0]∈0,1, splitm^add_ui&=GumbelSoftmax(z^add_ui;τ)[0]∈\0,1\,\\ split The selection mask is learned via backpropagation using the Gumbel-Softmax relaxation, favoring edges beneficial for target behavior prediction while suppressing misaligned ones. We guide readers unfamiliar with the Gumbel-Softmax to (Jang et al. 2017). Weighted Graph Augmentation. We construct the densified target graph using the selected edges. To reflect their varying importance, each edge is assigned a weight wuiw_ui that combines angular similarity and magnitude consistency (Chen et al. 2025): (7) wui=12(1+cos(uglo,iglo))⋅exp(−12σ2‖uglo−iglo‖22).w_ui= 12 (1+ (e^glo_u,e^glo_i) )· \! (- 12σ^2\|e^glo_u-e^glo_i\|_2^2 ). Original target edges are retained with unit weight, while selected candidates are added with weight wuiw_ui, where σ controls the sensitivity to the Euclidean distance between embeddings, and is set to 20 following (Chen et al. 2025). We define the weighted adjacency matrix ~tar A_tar as: (8) ~tar(u,i)=1,(u,i)∈ℰtar,wui,(u,i)∈ℰcandandmuiadd=1,0,otherwise. A_tar(u,i)= cases1,&(u,i) _tar,\\ w_ui,&(u,i) _cand\ \,\,and\,\,\ m^add_ui=1,\\ 0,&otherwise. cases The matrix ~tar A_tar is symmetric, as wui=wiuw_ui=w_iu by construction. 4.2.3. Debiased Densification Learning While the augmented graph reveals hidden preferences, it inevitably contains noise from the mining process, making direct use for final prediction suboptimal. We propose a strategy to robustly exploit densification signals. Preference Densification Learning. We adopt self-supervised contrastive learning with an InfoNCE loss (Oord et al. 2018) to encourage consistency between embeddings from the original and augmented graphs. This alignment emphasizes signals consistently supported by both views, enabling a more stable learning than directly treating augmented edges as ground-truth relations. Specifically, we align the original target embeddings tarE^tar (input of this module) with the augmented embeddings ~tar=LGCN(~tar,glo) E^tar=LGCN( A_tar,E^glo). The user-side loss for densification is defined as: (9) ℒuserdense=−∑u∈logexp(sim(utar,~utar)/τ)∑u′∈exp(sim(utar,~u′tar)/τ).L^user_dense=- _u (sim(e^tar_u, e^tar_u)/τ ) _u (sim(e^tar_u, e^tar_u )/τ ). where sim(⋅,⋅)sim(·,·) denotes a similarity function and τ is the temperature hyperparameter. The item-side loss ℒdenseitemL^item_dense is defined analogously, and the overall objective is ℒdense=ℒdenseuser+ℒdenseitemL_dense=L^user_dense+L^item_dense. Debiased BPR Loss. While auxiliary popularity adversarial learning mitigates popularity-driven bias at the representation level, it does not fully eliminate observation bias in preference learning. For a more explicit debiasing, we adopt the self-normalized inverse propensity score (SNIPS) (Schnabel et al. 2016) to reweight positive interactions in the ranking loss. SNIPS reduces the variance of the IPS estimator by normalizing importance weights, thereby stabilizing optimization even under imperfect propensity estimates. For each (u,i)(u,i), we first estimate the auxiliary-observation propensity p^ui=P^(Eui=1∣u,i) p_ui= P(E_ui=1 u,i), which measures how likely the pair is to be observed in auxiliary behaviors.55 5 Following (Schnabel et al. 2016; Li et al. 2023; Zhu et al. 2025), we adopt a simple propensity estimator trained via binary cross-entropy on auxiliary-observed (positive) and -unobserved (negative) pairs: ℒBCE=1||∑(u,i)∈BCE(p^ui,Eui)L_BCE= 1|D| _(u,i) BCE( p_ui,\,E_ui). Details are provided in Appendix A.1.2. The debiasing weight is defined by the normalized inverse propensity, ωui∝1/p^ui _ui 1/ p_ui, which reduces estimator variance. Intuitively, ωui _ui assigns smaller weights to pairs with high auxiliary observation propensity and larger weights to those with low propensity, correcting the skew induced by observation bias. Let sui=taru⊤taris_ui=e^tar_u e^tar_i denote the prediction score for target behavior. The debiased BPR loss is: (10) ℒrank=−∑(u,i)∈ℰtar∑j:(u,j)∉ℰtarωui⋅logσ(sui−suj)L_rank=- _(u,i) _tar _j:\,(u,j) _tar _ui· σ(s_ui-s_uj) Overall Learning Objective. The total loss of f0f_0 is as follows: (11) ℒf0=ℒrank+λadvℒadv+λdenseℒdense,L_f_0=L_rank+ _advL_adv+ _denseL_dense, where λadv _adv and λdense _dense are loss-balancing hyperparameters. Theoretical grounding for f0f_0 is provided in Appendix A.1.1: (i) Theorem A.1 establishes unbiasedness of the IPS-weighted loss; (i) Lemma A.2 shows that the GRL objective suppresses popularity-driven bias from item representations; and (i) Corollary A.3 shows that both are jointly necessary for hidden preference recovery. 4.3. Target-Guided Refinement Module The refinement module f1f_1 leverages target behavior signals to prune the auxiliary behavior graphs, and consists of two components: (1) a target-guided auxiliary graph refiner, which selectively prunes and reweights auxiliary relations, and (2) target-guided preference learning that provides target-aligned signals for auxiliary refinement and optimizes target behavior prediction. 4.3.1. Target-guided Auxiliary Graph Refiner We perform target-guided refinement on each auxiliary behavior graph bG_b using a behavior-specific LGCN encoder, with input auxiliary embeddings b=gloe^b=e^glo. The target embeddings tarE^tar serve as fixed anchors, and for notational simplicity, we denote x=detach(xtar)t_x=detach(e^tar_x). Target-Guided Feature Refinement. First, to handle feature-level misalignment, we introduce a feature-wise gate that adaptively balances auxiliary representations with target anchors. For each node x, the gate b∈(0,1)d α^b∈(0,1)^d determines, at each feature dimension, how much to rely on the auxiliary signals versus the target anchor. The gate is computed by conditioning on both representations: (12) xb α^b_x =σ(b1[bx∥x]+b1)∈(0,1)d =σ (W^b_1 [e^b_x\,\|\,t_x ]+b^b_1 )∈(0,1)^d xb ^b_x =xb⊙xb+(−xb)⊙x, = α^b_x ^b_x+ (1- α^b_x ) _x, where x∈∪ℐx . This adaptive interpolation selectively filters auxiliary features inconsistent with the target preference space. Selective Edge Pruning. Beyond feature-level refinement, we further refine the auxiliary graph structure by selectively pruning uninformative edges. The goal is to prevent auxiliary edges inconsistent with the target preference from propagating noise. For each edge (u,i)∈ℰb(u,i) _b, we predict a binary keep-or-drop decision using fdropf_drop, implemented as a linear layer. Specifically, we compute logits uib=fdrop([ub∥ib])∈ℝ2z^b_ui=f_drop([g^b_u\ \|\ g^b_i]) ^2, where each output dimension corresponds to keep and drop decisions, respectively. We apply Gumbel-Softmax to obtain differentiable pruning: (13) muib=GumbelSoftmax(uib,τ)[0]∈0,1,m^b_ui=GumbelSoftmax(z^b_ui;τ)[0]∈\0,1\, Optimized by the target-guided learning introduced in the subsequent subsection, edges misaligned with the target behavior are suppressed. An edge is retained if muib=1m^b_ui=1; otherwise it is pruned. Weighted Graph Refinement. We construct a refined adjacency matrix ~b A_b by selecting a subset of edges with the weighting scheme in Eq. (7), substituting refined representations ubg^b_u, ibg^b_i for ugloe^glo_u, igloe^glo_i: (14) ~b(u,i)=wuib,(u,i)∈ℰb′,0,otherwise, A_b(u,i)= casesw^b_ui,&(u,i) _b ,\\ 0,&otherwise, cases where ℰb′=(u,i)∈ℰb∣muib=1E_b =\(u,i) _b m^b_ui=1\ denotes the set of retained edges after pruning. The resulting matrix ~b A_b is symmetric, as wuib=wiubw^b_ui=w^b_iu by construction. Given the refined adjacency matrix ~b A_b, we obtain propagated representations as: ~b=LGCN(~b,glo) E^b=LGCN( A_b,E^glo). The same refinement procedure is applied to each auxiliary behavior b. 4.3.2. Target-Guided Preference Learning We introduce a target-guided preference learning that leverages contrastive alignment to provide supervisory signals for refining auxiliary graphs and strengthening target behavior prediction. Auxiliary Refinement Learning. Specifically, we treat the target embedding xt_x as a fixed anchor to prevent auxiliary noise from contaminating the target space, and align each auxiliary representation with it via contrastive learning. The user-side refinement loss for auxiliary behavior b is: (15) ℒuser,brefine=−∑u∈logexp(sim(u,ub)/τ)∑u′∈exp(sim(u,u′b)/τ).L^user,b_refine=- _u (sim(t_u,e^b_u)/τ ) _u (sim(t_u,e^b_u )/τ ). The item-side loss is defined analogously, and we sum both losses over all auxiliary behaviors: ℒrefine=∑b∈ℬ∖btarℒrefineuser,b+ℒrefineitem,bL_refine= _b \b_tar\L^user,b_refine+L^item,b_refine. Target Preference Learning. We aggregate behavior-specific embeddings according to their relevance to the target behavior. Let a(⋅):ℝ2d→ℝa(·):R^2d denote an attention function, implemented as a linear layer, over the concatenation of two embeddings. (16) αub=exp(a(utar,ub))∑b′∈ℬexp(a(utar,ub′)),aggu=∑b∈ℬαubbu, _u^b= \! (a(e^tar_u,e^b_u) ) _b \! (a(e^tar_u,e^b _u) ), ^agg_u= _b _u^b\,e^b_u, Based on the aggregated representations, we compute the preference score as sui=uagg⊤iaggs_ui=e_u^agg e_i^agg, and optimize the BPR loss on the target behavior: ℒrank=−∑(u,i)∈ℰtar∑j:(u,j)∉ℰtarlogσ(sui−suj)L_rank=- _(u,i) _tar _j:(u,j) _tar σ(s_ui-s_uj). Overall Learning Objective. The total loss of f1f_1 is as follows: (17) ℒf1=ℒrank+λrefineℒrefine,L_f_1=L_rank+ _refineL_refine, where λrefine _refine is a loss-balancing hyperparameter. 4.4. Unified Modular Learning and Inference According to our environment-conditioned design (Sec. 3.2), the final objective is expressed as a conditional expectation over module-wise objectives. Under this formulation, the auxiliary-observation propensity p^ui=P^(Eui=1∣u,i) p_ui= P(E_ui=1 u,i) and its complement 1−p^ui1- p_ui naturally serve as the assignment probabilities for the two modules:66 6 With a slight abuse of notation, ℒfe(u,i)L_f_e(u,i) denotes the loss induced by (u,i)(u,i) under fef_e. (18) ℒBOAR=(u,i)∈ℰtar[(1−p^ui)ℒf0(u,i)+p^uiℒf1(u,i)]L_BOAR=E_(u,i) _tar\! [(1- p_ui)\,L_f_0(u,i)+ p_ui\,L_f_1(u,i) ] Thus, each target interaction is softly assigned to the two modules according to its auxiliary-observation likelihood. An alternative design is to use a gating network, jointly optimized with the model, to assign module weights from user–item embeddings. However, we found that propensity-based assignment is more effective, as it is grounded in the environment signal of auxiliary observation rather than learned as an unconstrained gate. We highlight that propensity estimation is well established, and its stability and robustness to noise have been extensively studied (Rosenbaum and Rubin 1983; Schnabel et al. 2016; Li et al. 2023; Zhu et al. 2025). Inference. Consistent with the training objective, the final prediction is obtained as: (19) f(u,i)=(1−p^ui)f0(u,i)+p^uif1(u,i).f(u,i)=(1- p_ui)\,f_0(u,i)+ p_ui\,f_1(u,i). This soft assignment allows each instance to leverage complementary signals from both modules. Alternatively, replacing p^ui p_ui with the hard indicator EuiE_ui yields a deterministic 0-1 assignment, where each instance is routed exclusively to one module and cannot benefit from the other. We validate this design choice in Sec. 5.2.3. 5. Experiments 5.1. Experimental Settings Table 1. Dataset statistics. Hidden (%) denotes the proportion of hidden target items without auxiliary behaviors. Dataset #Users #Items #Clicks #Collects #Carts #Buys Hidden (%)* Tmall 41,738 11,953 1,813,498 221,514 1,996 255,586 16.38 Taobao 48,749 39,493 1,548,162 – 193,747 211,022 35.78 JData 93,334 24,624 1,681,430 45,613 49,891 321,883 18.55 * Computed over test interactions, distinct from the training-set ratio in Fig 2. Datasets. We use three widely used benchmark MBR datasets: (1) Tmall, from Alibaba platform and involving four behaviors (click, collect, cart, and buy); (2) Taobao, from Taobao and consisting of three behaviors (click, cart, and buy); and (3) JData, released by JD.com and containing four behaviors (click, collect, cart, and buy). For all datasets, we treat buy as the target behavior. Dataset statistics are summarized in Table 1. Table 2. Performance comparison. ∗ indicates statistical significance for p<0.05p<0.05 under t-test compared to the best baseline. Dataset Types Metric Single-Behavior Multi-Behavior MF-BPR LGCN LGCN-G MB-GMN CIGF CRCGN BCIPM MB-HGCN COPF MuLe HGIB MEMBER SHaRe BOAR Tmall General HR@10 0.0427 0.0391 0.1339 0.0452 0.0621 0.0813 0.1407 0.1462 0.1640 0.2088 0.2415 0.3507 0.2158 0.3739∗ NDCG@10 0.0210 0.0201 0.0681 0.0228 0.0320 0.0428 0.0765 0.0778 0.0883 0.1158 0.1274 0.1730 0.1133 0.1821∗ Unobserved HR@10 0.0662 0.0692 0.0835 0.0298 0.0474 0.0327 0.0548 0.0422 0.0426 0.0549 0.0420 0.0948 0.0727 0.1367∗ NDCG@10 0.0335 0.0382 0.0425 0.0154 0.0250 0.0174 0.0282 0.0223 0.0224 0.0289 0.0213 0.0594 0.0398 0.0812∗ Observed HR@10 0.3996 0.3773 0.4256 0.3740 0.3617 0.3841 0.4075 0.3762 0.4048 0.4557 0.4527 0.4554 0.4438 0.4587∗ NDCG@10 0.1891 0.1765 0.2044 0.1834 0.1705 0.1832 0.1953 0.1796 0.1935 0.2224 0.2202 0.2226 0.2149 0.2232∗ Taobao General HR@10 0.0239 0.0178 0.1049 0.0498 0.0649 0.1174 0.1270 0.1639 0.1543 0.2121 0.2480 0.3183 0.1842 0.3432∗ NDCG@10 0.0130 0.0095 0.0591 0.0198 0.0327 0.0655 0.0711 0.0950 0.0895 0.1346 0.1572 0.1640 0.1113 0.1886∗ Unobserved HR@10 0.0167 0.0124 0.0212 0.0182 0.0206 0.0177 0.0148 0.0153 0.0182 0.0248 0.0252 0.0221 0.0246 0.0352∗ NDCG@10 0.0093 0.0068 0.0105 0.0094 0.0102 0.0092 0.0077 0.0088 0.0075 0.0137 0.0147 0.0118 0.0140 0.0196∗ Observed HR@10 0.4587 0.4619 0.4938 0.4736 0.4849 0.4941 0.5024 0.4840 0.5343 0.5661 0.5797 0.5279 0.5435 0.5697 NDCG@10 0.2252 0.2290 0.2458 0.2318 0.2327 0.2530 0.2503 0.2446 0.2695 0.3050 0.3238 0.2738 0.2883 0.3061 Jdata General HR@10 0.3674 0.2779 0.4163 0.2815 0.3650 0.4872 0.5252 0.5234 0.4497 0.5837 0.6502 0.6589 0.6378 0.6971∗ NDCG@10 0.2234 0.1704 0.2411 0.1652 0.2272 0.2894 0.3158 0.3402 0.2723 0.4209 0.4667 0.4332 0.4584 0.4937∗ Unobserved HR@10 0.3610 0.2679 0.3671 0.2295 0.2971 0.3671 0.3740 0.4164 0.2610 0.4749 0.4566 0.3947 0.4641 0.5438∗ NDCG@10 0.2002 0.1432 0.2103 0.1415 0.1854 0.2029 0.2272 0.2588 0.1743 0.2975 0.2769 0.2403 0.3018 0.3665∗ Observed HR@10 0.7324 0.7189 0.7100 0.6937 0.7150 0.7452 0.7363 0.7234 0.7279 0.7693 0.7826 0.7383 0.7663 0.7757 NDCG@10 0.3751 0.4461 0.4372 0.4396 0.4427 0.4586 0.4458 0.4419 0.4648 0.4749 0.5340 0.4920 0.5235 0.5273 Baselines. We compare 13 baselines, covering both single- and multi-behavior recommendation methods. Specifically, we include two single-behavior models—MF-BPR (Rendle et al. 2009) and LightGCN (LGCN) (He et al. 2020)—and eleven multi-behavior methods: LGCN-G (He et al. 2020), MB-GMN (Xia et al. 2021b), CIGF (Guo et al. 2023b), CRGCN (Yan et al. 2023a), BCIPM (Yan et al. 2024), MB-HGCN (Yan et al. 2023b), COPF (Zhai et al. 2025), MuLe (Lee et al. 2024), HGIB (Zhang et al. 2025), MEMBER (Kim et al. 2025a), and SHaRe (Jiang et al. 2024). Among these, HGIB and SHaRe are graph rewiring-based methods; HGIB focuses on pruning-based rewiring to filter noisy auxiliary interactions, while SHaRe applies similarity-based rewiring to MBR, performing pruning on auxiliary behavior graphs and densification on the target behavior graph, serving as a direct rewiring baseline. Evaluation Protocol. We closely follow the evaluation protocols of prior work (Xia et al. 2021b; Guo et al. 2023b; Yan et al. 2024; Zhai et al. 2025; Lee et al. 2024; Zhang et al. 2025). We follow the widely adopted leave-one-out protocol, which holds out the most recent interaction of each user, together with all uninteracted items, as the test set. The second most recent interaction is reserved as the validation set. We evaluate performance using two ranking metrics: Hit Ratio (HR@k) and Normalized Discounted Cumulative Gain (NDCG@k). Following prior studies (Yan et al. 2024; Lee et al. 2024; Zhang et al. 2025; Zhai et al. 2025), we set k=10k=10 in all experiments. We report the average performance of five independent runs, each of which uses different random seeds. Performance Breakdown by Auxiliary Observability. For a more comprehensive evaluation, we report performance under three settings based on the observability of training auxiliary behaviors: general, observed, and unobserved. The general setting averages performance over all test instances. The observed setting reports the average over test instances with at least one auxiliary interaction (i.e., observed target items), whereas the unobserved setting does so over test instances without auxiliary interactions (i.e., hidden target items). This breakdown is intended to clearly reveal performance on hidden target items, which is essential for assessing generalization in real-world scenarios; strong performance in the unobserved setting indicates the model’s ability to discover novel target items beyond those that users have already interacted with. Hyperparameter Settings. Following (Yan et al. 2024; Lee et al. 2024; Zhang et al. 2025; Zhai et al. 2025; Yan et al. 2023a; Yan et al. 2023b), we set the embedding dimension d to 6464 and the batch size to 10241024 for all compared methods to ensure a fair comparison; in , each module uses 3232-dimensional embeddings, keeping the total dimension at 6464. We use the Adam optimizer (Kingma and Ba 2015), where the learning rate was tuned in 5⋅10−4,10−4\5· 10^-4,10^-4\ and the weight decay was tuned in 0,10−6\0,10^-6\. All hyperparameters are tuned via grid search on the validation set. For BOAR, the adversarial loss weight λadv _adv was fixed to 0.010.01, while the remaining loss weights λdense _dense and λrefine _refine were tuned in the range of [0.1,1.0][0.1,1.0]. The contrastive temperatures were tuned in the range of [0.1,5.0][0.1,5.0]. The number of GNN layers was fixed to 22 for all graph encoders, and the LSH projection dimension d′d was set to 3232, and the candidate retrieval size n was tuned in 10,20,30,40,50\10,20,30,40,50\. The Gumbel-Softmax temperature was fixed to 0.20.2. For baseline methods, we closely follow the hyperparameter search ranges reported in the original papers. 5.2. Results and Analysis 5.2.1. Overall Performance Comparison. Table 2 reports the performance of all methods under three evaluation settings: General, Unobserved, and Observed. the best performance in the general and unobserved settings across all datasets and metrics, with particularly large gains in the auxiliary-unobserved setting. In the general setting, outperforms all baselines, achieving improvements of up to 7.82% in HR@10 and 15.00% in NDCG@10 over the best baseline. Notably, in the unobserved setting, substantially larger margins, with maximum improvements of 44.2% in HR@10 and 36.7% in NDCG@10. In the observed setting, BOAR achieves the best or second-best performance across datasets; on datasets where HGIB leads, HGIB’s exclusive focus on pruning-based rewiring allows it to concentrate full capacity on the observed environment. This reflects a design trade-off: BOAR distributes model capacity across both environments, gaining substantially on unobserved items at a modest cost in the observed setting, and ultimately achieving the best overall performance in the general setting. Furthermore, SHaRe consistently underperforms BOAR, confirming that conventional similarity-based rewiring may amplify MBR imperfections rather than resolving them. Figure 4. Performance drop in the hidden-preference test. Figure 5. Performance drop in the noise-injection test. 5.2.2. Stress-Testing of Graph Rewiring Modules. To verify that each rewiring module operates as intended, we design two controlled settings that directly examine the accuracy of densification and pruning: (i) hidden preference test, in which hidden target items are increased by removing a fraction k of auxiliary interactions overlapping with target-positive pairs, placing greater demand on the densification of f0f_0; and (i) noise injection test, in which auxiliary noise is increased by injecting false auxiliary interactions into a fraction k of target-negative pairs, placing greater demand on the pruning of f1f_1. A method with accurate rewiring should exhibit minimal performance degradation as k increases, since correct densification recovers the removed signal and correct pruning suppresses the injected noise.77 7 Because interaction logs do not contain ground-truth purchase labels for every edge, direct edge-level evaluation is infeasible. Therefore, we use hidden preference discovery and noise suppression as proxy tests for graph rewiring accuracy, reflecting the respective goals of densification and pruning. In the hidden-preference test (Figure 4), as k increases, all methods degrade, indicating poor generalization when target-positive items lack auxiliary evidence. the smallest performance drop across k, reflecting its ability to recover hidden preferences beyond observed auxiliary relations via debiased densification. In the noise-injection test (Figure 5), injecting noisy auxiliary interactions degrades all methods, highlighting the vulnerability of auxiliary-driven message passing. the least degradation as k increases, owing to its target-guided refinement that suppresses noise propagation in auxiliary signals. These observations collectively show the effectiveness of robust target prediction beyond auxiliary behavior imperfections, consistent with Taobao results in Figure 1 (right). Table 3. Ablation study of BOAR on HR@10. Setting Unobserved Observed Datasets Tmall Taobao JData Tmall Taobao JData BOAR 0.1367 0.0352 0.5438 0.4587 0.5697 0.7757 Ablation on densification w/o ℒdenseL_dense 0.1081 0.0307 0.4768 0.4566 0.5607 0.7740 w/o ℒadvL_adv 0.1328 0.0335 0.5320 0.4549 0.5592 0.7753 w/o SNIPS 0.1214 0.0338 0.5352 0.4565 0.5645 0.7730 Ablation on refinement w/o TAGR 0.1321 0.0342 0.5391 0.4497 0.5623 0.7691 w/o ℒrefineL_refine 0.1329 0.0344 0.5256 0.4445 0.5445 0.7636 w/o Stop-gradient 0.1290 0.0334 0.5345 0.4395 0.5426 0.7587 Ablation on assignment Hard assignment 0.1315 0.0295 0.5408 0.4559 0.5607 0.7738 Learnable assignment 0.0609 0.0202 0.4698 0.4360 0.5411 0.7689 5.2.3. Ablation Study. We present a detailed ablation study on the densification (f0f_0) and refinement module (f1f_1). Table 3 reports the ablation results under both settings, with f0f_0 focusing on the unobserved setting and f1f_1 on the observed setting. For f0f_0, removing ℒdenseL_dense leads to a large performance drop, confirming that hidden-preference mining is the primary source of gains in the unobserved environment. In addition, incorporating both ℒadvL_adv and SNIPS yields the best performance, indicating a strong synergy between debiasing at the representation and prediction levels. For f1f_1, replacing the target-guided auxiliary graph refiner (TAGR) with the standard LGCN encoder without refinement consistently degrades performance. Moreover, ℒrefineL_refine, together with the stop-gradient operation that treats target embeddings as a fixed anchor, proves effective, supporting the validity of our design. Lastly, we validate the modular integration design: replacing soft assignment with hard assignment degrades performance, as exclusive assignment prevents each instance from leveraging complementary signals across both modules; substituting a learnable assignment88 8 We replace [p^ui, 1−p^ui][ p_ui,\,1- p_ui] with [gθ(u,i), 1−gθ(u,i)]=softmax(g[u∥i]+g)[g_θ(u,i),\,1-g_θ(u,i)]=softmax(W_g[e_u\|e_i]+b_g), where u,i∈e_u,e_i are the raw initial embeddings and g∈ℝ2×2dW_g ^2× 2d is jointly optimized with the prediction objective. causes a far more severe drop, as it lacks a principled basis for module assignment, and thus jointly optimizing the assignment with the prediction objective leads to unstable training. 5.2.4. Hyperparameter Study. We provide an analysis to guide hyperparameter selection for . Here, we investigate three key hyperparameters, while the remaining ones (e.g., λadv _adv) are fixed. Loss-balancing weights. Figure 6 presents the performance of f0f_0 and f1f_1 under varying λdense _dense and λrefine _refine, respectively. Overall, stable performance with respect to λdense _dense, while small values of λrefine _refine are more effective across datasets. A sufficiently large λdense _dense is necessary to effectively exploit hidden preference supervision, whereas excessively increasing λrefine _refine can degrade observed-item ranking due to over-regularization. In practice, moderate values consistently yield strong and stable performance. Candidate Retrieval Size. In Figure 7, we analyze the sensitivity of the candidate size n in angular LSH-based retrieval. Overall, limited sensitivity to n, with only minor performance variations across datasets, indicating that learnable edge selection effectively filters unreliable candidates even as n increases. When n is too small, the effect is limited, as potential hidden target-positive candidates may be missed. (a) Effect of λdense _dense on f0f_0 in the unobserved setting. (b) Effect of λrefine _refine on f1f_1 in the observed setting. Figure 6. Effects of loss-balancing hyperparameters. Figure 7. HR@10 under different candidate retrieval sizes n. Table 4. Efficiency analysis: comparison of per-epoch training time (sec), inference time (sec), and model size (in millions). Training time denotes the average time per epoch, and inference time denotes the time to generate recommendations for all users. Dataset Metric MuLe HGIB MEMBER BOAR Train time / epoch 81.47s 86.46s 164.85s 57.93s Tmall Inference time 12.52s 11.24s 12.82s 8.05s #Parameters 3.44M 3.44M 3.44M 3.44M Train time / epoch 70.22s 64.43s 116.97s 49.86s Taobao Inference time 21.69s 19.66s 22.84s 16.99s #Parameters 5.65M 5.66M 5.65M 5.65M Train time / epoch 100.12s 90.83s 178.49s 95.86s JData Inference time 5.92s 5.69s 6.22s 4.68s #Parameters 7.55M 7.56M 7.55M 7.56M 5.3. Complexity Analysis Table 4 compares the per-epoch training time and inference time, as well as the model size, of state-of-the-art MBR methods (Lee et al. 2024; Zhang et al. 2025; Kim et al. 2025a). All experiments are conducted using PyTorch with CUDA on an RTX 5000 Ada GPU and an Intel Xeon Gold 6338 CPU. introduces only a small number of additional parameters, mainly a few lightweight linear layers, and since the densification process is applied only during training, it achieves the fastest inference latency among all compared methods. achieves the fastest per-epoch training time on Tmall and Taobao, and remains comparable to HGIB on JData. Overall, strong performance with efficiency comparable to state-of-the-art methods, supporting its practical applicability. 6. Conclusion In this work, we first highlight two fundamental challenges of auxiliary behaviors: missing auxiliary signals and unreliable auxiliary signals. As a solution, we propose , an environment-conditioned framework that addresses missing and unreliable auxiliary signals through two complementary modules conditioned on auxiliary observability. Extensive experiments show that particularly strong gains on hidden target items without auxiliary behaviors, validating its effectiveness in capturing hidden preferences beyond observed auxiliary relations. Future work may explore streaming settings with continuously evolving user behaviors and interactions. Appendix A Appendix A.1. Supplementary Proofs and Method Details A.1.1. On the Validity of Debiasing in We provide a theoretical interpretation of two debiasing techniques underlying : Inverse Propensity Scoring (IPS) (Schnabel et al. 2016) and adversarial learning (Ben-David et al. 2010). IPS weighted BPR Loss (Eq. (10)). Let Oui∈0,1O_ui∈\0,1\ denote whether the target-behavior pair (u,i)(u,i) is observable in the training data (e.g., due to platform exposure), with propensity pobs(u,i)=Pr(Oui=1∣u,i)p_obs(u,i)= (O_ui=1 u,i). We assume target-positive signals are observed only when Oui=1O_ui=1. For an observed positive (u,i)∈ℰtar(u,i) _tar, we sample j∼q(j∣u)j q(j u) (independent of OuiO_ui) and use ℓθ(u,i,j)=−logσ(sui−suj) _θ(u,i,j)=- σ(s_ui-s_uj). Although Oui=1O_ui=1 for all logged pairs in practice, OuiO_ui is treated as a Bernoulli random variable in the data-generating process, enabling importance weighting. Define the ideal risk ℒideal=[ℓθ(u,i,j)]L_ideal=E[ _θ(u,i,j)] and the IPS-weighted risk ℒIPS=[Ouip^obs(u,i)ℓθ(u,i,j)]L_IPS=E\! [ O_ui p_obs(u,i) _θ(u,i,j) ]. Theorem A.1 (Unbiasedness under correct propensities (Schnabel et al. 2016)). Assume pobs(u,i)>0p_obs(u,i)>0. If p^obs(u,i)=pobs(u,i) p_obs(u,i)=p_obs(u,i) almost surely, then Bias[ℒIPS]=|ℒIPS−ℒideal|=0Bias[L_IPS]=|L_IPS-L_ideal|=0. Proof. Since ℓθ(u,i,j) _θ(u,i,j) is deterministic given (u,i,j)(u,i,j), ℒIPS=u,i,j[ℓθ(u,i,j)⋅[Ouip^obs(u,i)|u,i]].L_IPS=E_u,i,j\! [ _θ(u,i,j)·E\! [ . O_ui p_obs(u,i) |u,i ] ]. Since [Oui∣u,i]=pobs(u,i)=p^obs(u,i)E[O_ui u,i]=p_obs(u,i)= p_obs(u,i), the inner expectation is 11, yielding ℒIPS=ℒidealL_IPS=L_ideal. ∎ Connection to Eq. (10). Eq. (10) approximates ℒIPSL_IPS via Monte-Carlo over logged positives. Following (Li et al. 2023; Schnabel et al. 2016; Zhu et al. 2025), we estimate the pair-level propensity pobs(u,i)p_obs(u,i) via a lightweight binary classifier p^ui p_ui (Appendix A.1.2), trained to predict whether a user–item pair is auxiliary-observed. We set ωui∝1/p^ui _ui 1/ p_ui (SNIPS), up-weighting pairs with low auxiliary-observation propensity and down-weighting those with high propensity to correct observation bias. Auxiliary Popularity Adversarial Learning (Eq. (5)). Let aboveD_above and belowD_below denote the item distributions over items whose auxiliary interaction count exceeds (yi=1y_i=1) or falls below (yi=0y_i=0) the median (Sec. 4.2.1). Let z=gψ(i)∈z=g_ψ(i) be the learned item representation, with induced distributions Pψ:=gψ#aboveP_ψ:=g_ψ\#D_above and Qψ:=gψ#belowQ_ψ:=g_ψ\#D_below. Let ℋdH_d be binary classifiers h:→0,1h:Z→\0,1\ where h(z)=1h(z)=1 predicts above-median and h(z)=0h(z)=0 predicts below-median. Lemma A.2 (ℋH-divergence (Ben-David et al. 2010)). dℋd(Pψ,Qψ):=2suph∈ℋd|Prz∼Pψ[h(z)=1]−Prz∼Qψ[h(z)=1]|.d_H_d(P_ψ,Q_ψ):=2 _h _d | _z P_ψ[h(z)=1]- _z Q_ψ[h(z)=1] |. With εd∗:=minhεd(h) _d^*:= _h _d(h) where εd(h):=12PrPψ[h=0]+12PrQψ[h=1] _d(h):= 12 _P_ψ[h=0]+ 12 _Q_ψ[h=1], and ℋdH_d complement-closed, dℋd(Pψ,Qψ)=2(1−2εd∗)d_H_d(P_ψ,Q_ψ)=2(1-2 _d^*). Hence εd∗→12 _d^*→ 12 implies dℋd(Pψ,Qψ)→0d_H_d(P_ψ,Q_ψ)→ 0. Proof. For any h∈ℋdh _d, PrPψ[h=1]−PrQψ[h=1]=1−2εd(h) _P_ψ[h=1]- _Q_ψ[h=1]=1-2 _d(h). Since ℋdH_d is complement-closed, suph|PrPψ[h=1]−PrQψ[h=1]|=1−2εd∗ _h| _P_ψ[h=1]- _Q_ψ[h=1]|=1-2 _d^*. Multiplying by 22 yields the claim. ∎ Corollary A.3 (Why both GRL and IPS are required). If the GRL-based adversarial objective drives εd∗≈12 _d^*≈ 12, then by Lemma A.2, dℋd(Pψ,Qψ)≈0d_H_d(P_ψ,Q_ψ)≈ 0, meaning gψg_ψ suppresses popularity-driven observation bias from item representations in gloE^glo. This encourages LSH-based candidate retrieval (Sec. 4.2.2) to surface preference-aligned hidden items rather than popularity-similar ones; these candidates are incorporated into ~tar A_tar and propagated to tarE^tar via LGCN, supporting reliable recovery of hidden preferences at the representation level. However, even with suppressed popularity-driven observation bias in representations, optimizing the ranking objective on logged positives can still be dominated by frequently auxiliary-observed items, since they appear more often in ℰtarE_tar regardless of their representation. The propensity-weighted objective in Theorem A.1 corrects this prediction-level bias, while SNIPS provides variance-stabilized approximations. Therefore, removing either component leaves (i) popularity-driven observation bias in gloE^glo, degrading candidate quality for hidden items (without GRL), or (i) observation bias in the ranking objective toward frequently auxiliary-observed items (without IPS), both of which hinder reliable recovery of hidden preferences. A.1.2. Propensity Computation Following (Schnabel et al. 2016; Li et al. 2023; Zhu et al. 2025), we estimate p^ui=σ(uaux⊤iaux) p_ui=σ(e^aux_u e^aux_i) via a lightweight binary classifier, where aux=LGCN(aux,)E^aux=LGCN(A_aux,E) and auxA_aux is constructed from ℰauxE_aux. The classifier is trained via binary cross-entropy with EuiE_ui (Sec. 3.2) as supervision labels, where D is constructed by sampling positives from ℰauxE_aux and, for each user u, randomly sampling auxiliary-unobserved items as negatives at a 1:1 ratio: (20) ℒBCE=−∑(u,i)∈[Euilogp^ui+(1−Eui)log(1−p^ui)].L_BCE=- _(u,i) [E_ui p_ui+(1-E_ui) (1- p_ui) ]. We then apply self-normalized inverse propensity scoring: (21) ωui=p^ui−1∑(u′,i′)∈ℰtarp^u′i′−1, _ui= p_ui^-1 _(u ,i ) _tar p_u i ^-1, where p^ui p_ui is clipped to max(p^ui,10−5) ( p_ui,10^-5) for numerical stability. Acknowledgements. This work was supported by a Korea University Grant, IITP grants funded by the Korea government (MSIT): the ICT Creative Consilience Program (IITP-2026-RS-2020-I201819) and the Artificial Intelligence Star Fellowship (IITP-2026-RS-2025-02304828). 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