MB-HGCN: A Hierarchical Graph Convolutional Network for Multi-behavior Recommendation

Following previous works [14, 12, 15, 34], we initialize the ID of user $u\in\mathcal{U}$ and item $i\in\mathcal{I}$ as $d$-dimensional embedding vector $\bm{e_{u}}^{0}$ and $\bm{e_{i}}^{0}$, respectively. Let $\bm{P}\in\mathbb{R}^{M\times d}$ and $\bm{Q}\in\mathbb{R}^{N\times d}$ be the embedding matrices for the user and item embedding initialization, where $M$ and $N$ represent the number of users and items, respectively. Each user and item ID is represented as a unique embedding. Given the one-hot embedding matrix $\bm{ID}^{\mathcal{U}}$ and $\bm{ID}^{\mathcal{I}}$ for all users and items, the embeddings of user $u$ and item $i$ are initialized as:

where $\bm{ID}_{u}^{\mathcal{U}}$ and $\bm{ID}_{i}^{\mathcal{I}}$ represent the user $u$’s and the item $i$’s one-hot vector, respectively.

As mentioned, our model adopts a hierarchical GCN structure to exploit the multi-behavior for embedding learning. The interaction information of all behaviors are integrated into the unified graph $\mathcal{G}$ to learn the general user preferences and item features, denoted as the global embedding $\bm{e}_{u}^{g}$ and $\bm{e}_{i}^{g}$ for user $u$ and item $i$, respectively. The learned global embedding is then fed into each behavior-specific graph $\mathcal{G}_{k}$ to learn the behavior-specific embedding $\bm{e}_{u}^{k}$ and $\bm{e}_{i}^{k}$ for user $u$ and item $i$ in $\mathcal{G}_{k}$, respectively. For the embedding learning in each graph, we employ the LightGCN [12] model, which is a lightweight CF-based single-behavior recommendation model. It simplifies the standard GCN and only retains the core neighborhood aggregation component, which has proven to be effective and superior in performance. It is worth mentioning that other GCN models can also be adopted, such as Ultra-GCN [34] and SVD-GCN [35]. The core of LightGCN is to recursively aggregate information from neighboring nodes for embedding update of the target node. The graph convolution operation in LightGCN is:

where $\frac{1}{\sqrt{\left\lvert N_{u}\right\rvert}\sqrt{\left\lvert N_{i}\right\rvert}}$ denotes the normalization coefficient, $N_{u}$ represents the set of items that are interacted with the user $u$, and $N_{i}$ is the same. After l-layer propagation, LightGCN combines the embeddings obtained at each layer as the final user and item representation. Given the total number of layers as $L$, the representation of user $u$ and item $i$ after the LightGCN process are as follows:

where $\alpha_{l}$ is a hyperparameter represents the importance of the $l$-th layer embedding, and $\bm{e}_{u}^{(0)}(\bm{e}_{i}^{(0)})$ is the initial embedding of user $u$ (item $i$).

As shown in Fig. 1, LightGCN with the same settings is used in both the unified graph $\mathcal{G}$ and each behavior-specific graph $\mathcal{G}_{k}$ to learn the embeddings of users and items.

Global embedding. Following the Eq. 2, we respectively obtain L embeddings to describe a user $\{\bm{e}_{u}^{(1)},\bm{e}_{u}^{(2)},\cdots,\bm{e}_{u}^{(L)}\}$ and an item $\{\bm{e}_{i}^{(1)},\bm{e}_{i}^{(2)},\cdots,\bm{e}_{i}^{(L)}\}$ after L-layer propagation. Before combining these embeddings, normalization is adopted to alleviate the impact of the embedding scale [12]. In our model, we apply $L_{2}$ normalization for simplicity:

where $\bm{e}^{g}_{u}$ and $\bm{e}^{g}_{i}$ are the learned global embedding from $\mathcal{G}$, they are also the sharing input of the following behavior-specific graph $\mathcal{G}_{k}$. $\bm{e}_{u}^{(0)}$ and $\bm{e}_{i}^{(0)}$ are the input of graph $\mathcal{G}$ (i.e., the initialized embedding $\bm{e}_{u}^{0}$ and $\bm{e}_{i}^{0}$). Intuitively, the farther neighbors have less important, thus, we set the $\alpha_{l}$ as $1/(l+1)$.

Behavior-specific embedding. Similarly, taking $\bm{e}^{g}_{u}$ and $\bm{e}^{g}_{i}$ as the initial embeddings for the embedding learning in each behavior-specific graph $\mathcal{G}_{k}$, we can obtain $K$ behavior-specific embeddings for each user $u$ and item $i$ ( i.e., $\{\bm{e}_{u}^{1},\bm{e}_{u}^{2},\cdots,\bm{e}_{u}^{K}\}$ and $\{\bm{e}_{i}^{1},\bm{e}_{i}^{2},\cdots,\bm{e}_{i}^{K}\}$).

Through the embedding learning process, for each user $u$ and item $i$, we obtain their global embeddings $\bm{e}^{g}_{u}$ and $\bm{e}^{g}_{i}$, as well as behavior-specific embedding set $\{\bm{e}_{u}^{1},\bm{e}_{u}^{2},\cdots,\bm{e}_{u}^{K}\}$ and $\{\bm{e}_{i}^{1},\bm{e}_{i}^{2},\cdots,\bm{e}_{i}^{K}\}$. In the next, we would like to aggregate the above user and item embeddings respectively by adopting different strategies to obtain the final embedding for recommendation.

User embedding aggregation. Considering different behaviors may convey some distinct information of user preference, to distill valuable information from different behavior-specific embeddings for the target behavior prediction, we design a novel weighting scheme for user embedding aggregation. Taking the user $u$ as example, the aggregation is formulated as:

where $||$ denotes the operation of stacking vectors to form a matrix. $\bm{U}\in\mathbb{R}^{d\times K}$ is the matrix by stacking user embeddings learned from each behavior, in which $d$ represents the embedding size and $K$ is the number of behaviors. $\bm{\delta}$ is the weight vector calculated based on the similarity between the embedding of each behavior and that of the target behavior. Formally, it is computed as

In this equation, ${\bm{e}_{u}^{k}}^{\top}\bm{U}$ calculates the embedding similarity between the $k$-th behavior and other behaviors. The denominator $\sqrt{d}$ is used to prevent the vanishing gradient problem, and $softmax(\cdot)$ is adopted for normalization.

The underlying rationality of the aggregation strategy is that the behavior with more similar embeddings contain more relevant preference information with the target behavior, and thus contribute more to the target behavior in the aggregation. With this aggregation strategy, our model can adaptively extract valuable information from other behaviors for the target behavior prediction. Moreover, together with the multi-task learning, it also avoids the behavior-specific embeddings to be optimized towards the target behavior in the learning process. Through the above operation, we can obtain the embedding set $\{\tilde{\bm{e}}_{u}^{1},\tilde{\bm{e}}_{u}^{2},\cdots,\tilde{\bm{e}}_{u}^{K}\}$.

Item embedding aggregation. Since item features are consistent across different behaviors, we simply apply a linear combination to aggregate the embeddings learned from different behaviors. In different types of behaviors, the users who interacted with the items are different and the total number of interactions (i.e., the number of users who interacted with the items) are also different. Intuitively, with more interactions, the learned features should be more comprehensive. Accordingly, the weight assigned to the $k$-th behavior-specific embedding for an item $i$ is defined as:

where $w_{k}$ is a learnable parameter for the $k$-th behavior; $n_{ik}$ denotes the number of users that interacted with item $i$ in the $k$-th behavior. The behavior-specific embeddings of item $i$ are aggregated as:

Notice that both user embedding $\tilde{\bm{e}}_{u}^{k}$ and item embedding $\tilde{\bm{e}}_{i}$ are obtained from the behavior-specific information. In order to obtain more comprehensive representation, we combine them with the global embeddings to obtain the final embeddings:

where $\oplus$ denotes the element-wise sum.

MTL [36] is a learning strategy for jointly optimizing different-yet-related tasks. To better exploit the multiple-behavior information in user and item embedding learning, we treat each behavior as an independent training task. The inner product of user and item embeddings is adopted to estimate the prediction score. Take the $k$-th behavior as an example:

The pairwise Bayesian Personalized Ranking (BPR) [7] loss is adopted in optimization for each task:

where $\mathcal{O}=\{(u,i,j)|(u,i)\in\mathcal{R}^{+},(u,j)\in\mathcal{R}^{-}\}$ is defined as positive and negative sample pairs, $\mathcal{R}^{+}$ ($\mathcal{R}^{-}$) denotes the observed (unobserved) samples in the $k$-th behavior, and $\sigma(\cdot)$ is the sigmoid function. Following the Eq. 12, we obtain the loss function for all the $K$ tasks, i.e., $\{\mathcal{L}_{1},\mathcal{L}_{2},\cdots,\mathcal{L}_{K}\}$, then the $K$ loss functions are summed for joint optimization. Intuitively, the contribution of different tasks should be different. Assigning different weights to different losses may enhance the final performance, however, this is not our main focus in this study. Here we simply treat them equally to focus on studying the effectiveness of our embedding learning strategy and leave the study of different weights in the loss function as a future work. The final loss function is formulated as:

where $\bm{\Theta}$ represents all trainable parameters in our model and $\beta$ is the coefficient that controls the strength of the $L_{2}$ normalization to prevent over-fitting. To improve the generalization ability, two widely used dropout strategies [15, 37, 38] are also adopted in training: node dropout and message dropout, which are used to randomly drop out nodes in the graph and information in the embedding, respectively.

Three real-world datasets are adopted for experiments:

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  Tmall. This dataset is collected from Tmall²²2https://www.tmall.com/, which is one of the largest e-commerce platforms in China. It contains 41,738 users and 11,953 items with 4 types of behaviors, i.e., view, collect, cart, and buy.

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  Beibei. This dataset is collected from Beibei³³3https://www.beibei.com/, which is the largest infant product retail e-commerce platform in China. This dataset contains 21,716 users and 7,977 items with three types of behaviors, i.e., view, cart, and buy.

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  Jdata. This dataset is collected from JD⁴⁴4https://www.jd.com/, which is one of the most popular and influential e-commerce websites in the Chinese e-commerce field. This dataset contains 93,334 users and 24,624 items with 4 types of behaviors, i.e., view, collect, cart, and buy.

For the above datasets, we follow the previous work to remove the duplicated records by keeping the earliest one [14, 15]. The statistical information of the three datasets is summarized in Table I.

We adopt the widely used leave-one-out strategy for model evaluation [39, 15, 14]. In training stage, the last postive item for each user is selected to construct the validation set for hyper-parameter tuning. In the evaluation stage, all the items in the test set are ranked according to the predicted scores by recommendation models. Meanwhile, two representative evaluation metrics in recommendation: Hit Ratio (HR@K) [40] and Normalized Discounted Cumulative Gain (NDCG@K) [41] are adopted to evaluate the performance:

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  HR@K: a performance metric used to evaluate the accuracy of a recommender system by measuring the proportion of test items for which the correct recommendation appears within the top K positions of the ranked list.

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  NDCG@K: a metric that measures the quality of the recommended items by considering both their relevance and their position in the ranked list.

To demonstrate the performance of our model, we compare our MB-HGCN with several representative recommendation models, including three single-behavior models and six multi-behavior models.

Single-behavior model:

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  MF-BPR [7]. BPR is a widely used optimization strategy, which assumes that the predicted scores of positive samples are higher than that of negative ones. MF-BPR has been widely used as a baseline to evaluate the performance of newly proposed models.

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  NCF [14]. It is a representative model combining neural network and CF, which combines shallow generalized matrix factorization model and deep multi-layer perceptron model to learn the interaction between users and items.

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  LightGCN [12]. It removes the feature transformation and nonlinear activation components in the standard GCN model, and only keeps the core neighborhood aggregation component, which simplifies the model structure and achieves a significant performance improvement over its counterpart.

Multi-behavior model:

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  R-GCN [17]. R-GCN differentiates the relations between nodes via edge types in the graph and designs different propagation layers for different types of edges to model the relation information. This model can adapt to the multi-behavior recommendation.

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  NMTR [14]. It is a deep learning model for multi-behavior recommendation, which designs a neural network for each behavior. It sequentially passes the interaction score among behaviors and also adopts multi-task learning for joint optimization.

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  MBGCN [15]. This model constructs a heterogeneous graph to learn user preferences through user-item propagation and adopts a linear aggregation for feature fusion. In addition, item-item propagation is exploited to enhance item embedding learning.

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  GNMR [18]. This model designs a relation aggregation network to model interaction heterogeneity and attempts to explore the dependencies among different types of behaviors via recursive embedding propagation over the heterogeneous graph.

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  S-MBRec [28]. This model consists of a supervised and a self-supervised learning task, which separately learns the user and item embeddings from each behavior and adopts a star-style contrastive learning strategy to construct a contrastive view pair for the target and each auxiliary behavior.

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  CRGCN [32]. This model designs a cascaded residual network to explore the connection between different behaviors from the perspective of embedding propagation. The multi-task learning is also adopted for joint optimization.

Our model is implemented by Pytorch⁵⁵5https://pytorch.org/. In the implementation of all methods, the mini-batch size and embedding size are set to 1024 and 64, respectively [12]. Adam [42] optimizer is adopted for the optimization. In addition, we employ grid search to tune the learning rate and regularization weights (i.e., $\beta$) in the $[1e^{-2},3e^{-3},1e^{-3},1e^{-4}]$ and $[1e^{-2},1e^{-3},3e^{-4},1e^{-4}]$ ranges, respectively. Meanwhile, we carefully tune the hyperparameters in the baselines according to their original papers, and an early stop strategy is adopted in the training stage.

We design a hierarchical graph convolutional network for embedding learning, where a unified graph $\mathcal{G}$ is utilized to learn a coarse-grained global embedding, which is then used as a shared initialization for refining embeddings in behavior-specific graphs. To validate the effectiveness of learning coarse-grained global embeddings, we conduct an experiment where we remove the unified graph component and compare the results to the original model that retain the unified graph. Specifically, we train the model without the unified graph $\mathcal{G}$ and leverage the initialized embedding (i.e., $\bm{e_{u}}^{0}$ and $\bm{e_{i}}^{0}$) as the initialization of the behavior-specific graph $\mathcal{G}_{k}$ ($k\in[1,K]$). Experimental results are reported in Table III.

The experimental results suggest that removing the unified graph component leads to a significant decrease in performance. This result is attributed to the coarse-grained global embeddings learned in the unified graph component can provide better initialization for refining embeddings in behavior-specific graphs, which allows for more accurate learning in those graphs. Moreover, it is observed that there is a tremendous performance difference between the two models with and without the unified graph. In fact, without the unified graph, the model degenerates to a variant of R-GCN, where the difference is the embedding aggregation strategy. Compared with the results in Table II, the model w/o. $\mathcal{G}$ significantly outperformed R-GCN. The results provide evidence for the effectiveness of the proposed embedding aggregation strategies for users and items, and further verifies the validity of the unified graph design.

Our intuition for designing user embedding aggregation strategies is that user interests vary across behaviors, and thus, not all user preferences contribute to the prediction of the target behavior. Therefore, we design a simple adaptive embedding aggregation strategy for user embeddings. To verify the effectiveness of the design for adaptive user embedding aggregation strategy, we conduct three experiments: 1) sum agg., we remove our adaptive user embedding aggregation module and directly sum different behavior-specific embeddings for information aggregation. 2) linear agg., we replace our adaptive user embedding aggregation module with linear aggregation, which assigns different weights based on the number of interactions for each behavior (the same to the item aggregation strategy). 3) adaptive agg., our proposed adaptive embedding aggregation strategy. Experimental results are reported in Table IV.

The experimental results demonstrate that adopting the adaptive aggregation strategy achieves the best performance. The sum agg. method yields poor performance due to the varying interests that users exhibit in different behaviors, and the aggregation strategy lacks consideration of the importance of each behavior. Although the linear agg. method considers the importance of different behaviors, behaviors with more interactions may not necessarily reflect more accurate user preferences. In contrast, our adaptive aggregation strategy aggregates relevant information at the feature level based on the similarity between different behaviors, resulting in better aggregation of relevant information. It is worth mentioning that our aggregation scheme does not introduce any additional parameters into the model. This avoids the potential risks of negative impact on the embedding learning process from additional parameters introduced by the aggregation scheme.

To verify this point, we perform an additional experiments. We pre-train our model to keep the optimal embeddings that learned from each behavior and remove the aggregation of global embeddings (i.e., the operation of Eq. 10) to eliminate the effects of global embeddings. The goal is to only retain the training of target behaviors to avoid the effects of multi-task learning. On this basis, we compare the performance of the following three variants: 1) $M_{unfix}$, which employs linear aggregation strategy for user embedding aggregation. 2) $M_{fix}$, which fixes the parameters in the embedding learning process based on the first experiment. 3) $M_{adap}$, which adopts our adaptive aggregation strategy for user embedding aggregation. The experimental results are reported in Fig. 2.

According to Fig. 2, we can observe that for the two linear aggregation methods, the one with fixed parameters significantly outperforms the one with unfixed parameters. This is because the supervision signal cannot be transferred to the embedding learning process in the method of fixed parameters, in which only the parameters of linear aggregation process are optimized. This suggests that optimization of aggregation parameters may lead to locally optimal solutions for embedding learning. Furthermore, the method that adopts the adaptive aggregation strategy is better than the two methods which employ linear aggregation. The reason is that our adaptive aggregation strategy does not introduce any parameters, facilitating the embedded learning to be optimized on the right direction.

In this experiment, we evaluate the effectiveness of a linear aggregation strategy for item embeddings aggregation. Considering that behaviors with more interactions may reflect more comprehensive features of items, we assign weights to each behavior based on its number of interactions (as shown in Eq. 8). To verify the effectiveness of this design, we conduct the following experiments: 1) fix $\gamma_{ik}$, we assign the same weight (i.e., set $\gamma_{ik}=1$) to each behavior for embedding aggregation. 2) w/o. $w_{k}$, we remove the learnable parameter $w_{k}$ and strictly assign weights based on the number of interactions for each behavior. 3) w. $w_{k}$, which keeps the learnable parameter $w_{k}$ allows for fine-tuning the importance of different behaviors (our approach). The experimental results are reported in Table V.

The results in Table V indicate a significant improvement for the weight allocation method (w/o. $w_{k}$) over the non-weight allocation method (fix $\gamma_{ik}$), which supports our viewpoint that behaviors with more interactions reflect more comprehensive item features. In the two weight allocation methods, w/o. $w_{k}$ and w. $w_{k}$, the method that fine-tuned the weights using the learnable parameter achieved better performance, indicating that the contribution of different behaviors varies for different items. Therefore, fine-tuning the weights via the learnable parameter can better aggregate the representation of items, further validating the effectiveness of our proposed strategy.

Aggregate global embeddings into the final embedding, as shown in Eq. 10, is to obtain more comprehensive representation. We conduct an ablation study to verify this point by comparing it with the variant without considering the global embeddings. The experimental results are reported in Table VI.

It is shown that the method with global embedding aggregation, our model can gain a relative improvement of 16.23% and 15.79% on Tmall, 6.91% and 9.59% on Beibei, 15.89% and 18.22% on Jdata for HR@10 and NDCG@10, respectively. This demonstrates that aggregate global embedding can indeed improve performance. Global embeddings reflect coarse-grained user preferences, while behavior-specific embeddings reflect fine-grained user preferences. Combining these two types of embeddings can provide a comprehensive and hierarchical representation of user preferences, which can further improve recommendation performance. It again justifies the effectiveness of our hierarchical design by learning embedding from both global and behavior-specific levels.

We adopt a multi-task learning (MTL) framework for joint optimization. To verify its effectiveness, we compare the method with and without MTL, in which the method without multi-task learning train the target behavior in a single-task. The experimental results are reported in Table VII.

The experimental results, reported in Table VII, demonstrate that the MTL method outperforms the single-task method across all three datasets, indicating the effectiveness of MTL. The underlying reason for its effectiveness lies in the fact that our model treats each behavior as an independent task during training, and the better the behavior-specific embedding fits the user preferences exhibited in current behavior during the training process, the more accurately relevant information can be aggregated in the adaptive aggregation stage, leading to more precise predictions. It is worth noting that the scale of interaction data for different behaviors may affect the performance of multi-task learning. Therefore, considering the importance of different behaviors is necessary, but it is not the focus of our current research. We plan to research it in future work.