Exploiting Positional Information for Session-based Recommendation

Markov chain is applied by many models (Shani et al., 2005; Zimdars et al., 2001) to learn the dependency of items in sequential data. Using probabilistic decision-tree models, Zimdars et al. (Zimdars et al., 2001) proposed to encode the state of the transition pattern of items. Shani et al. (Shani et al., 2005) made use of a Markov Decision Process (MDP) to compute item transition probabilities.

Deep learning models become popular since the widely use of recurrent neural networks (Hidasi et al., 2016; Li et al., 2017; Liu et al., 2018; Hidasi and Karatzoglou, 2018; Aliannejadi and Crestani, 2018; Qian et al., 2019; Li et al., 2019, 2021). There are three main branches of methods to perform the representation learning of the session, i.e., recurrent models (Hidasi et al., 2016; Hidasi and Karatzoglou, 2018; Li et al., 2017), attention models (Liu et al., 2018; Guo et al., 2019; Sun et al., 2019b) and graph models (Wu et al., 2019; Qiu et al., 2019; Xu et al., 2019; Qiu et al., 2020a, b; Wang et al., 2020; Xia et al., 2021). (1) For recurrent models, e.g., GRU4REC (Hidasi et al., 2016; Hidasi and Karatzoglou, 2018) and NARM (Li et al., 2017), Gated Recurrent Unit (Chung et al., 2014) and Long Short-Term Memory (Hochreiter and Schmidhuber, 1997) are applied respectively and the positional information is implicitly modeled by the recurrent computing procedure. The recurrent structure includes a strong inductive bias that the relationship between items is linear along with the position. (2) For attention models, e.g., NARM (a self-attention layer is applied after the recurrent layer) and STAMP (Liu et al., 2018) utilizes self-attention (Vaswani et al., 2017) over the last item to capture the relationship between the last item and the rest in the session. These attention-based methods only consider the importance of the last position while neglecting other positions. (3) In graph modeling, e.g., SR-GNN (Wu et al., 2019), GC-SAN (Xu et al., 2019), FGNN (Qiu et al., 2019) and MGNN-SPred (Wang et al., 2020), a session is converted into a graph and Graph Neural Networks (GNN) (Kipf and Welling, 2017; Li et al., 2016; Velickovic et al., 2018) captures the connectivity of items. Afterward, a readout function is applied to compute a session representation with the processed item representations. For SR-GNN and GC-SAN, the readout function is similar to attention-based models by performing a self-attention over the last item. While FGNN uses a Set2Set (Vinyals et al., 2016) module and computes a descriptive vector, which is considered as a latent description of items. MGNN-SPred makes use of the mean feature of the whole sequence to represent the user modeling. Consequently, these GNN-based methods only capture the relative position for the connected items, which does not satisfy the forward-awareness and backward-awareness. The proposed PosRec falls into the category of graph-based model. To enhance the exploitation of the positional information of the graph representation learning, the (L)DPE is included in the embedding of the items and the graph neural network is redesigned to have a position-aware module.

Sequential recommendation is a close research field to SBRS. In recent years, deep learning models are very popular (Tang and Wang, 2018; Kang and McAuley, 2018; Sun et al., 2019a; Fang et al., 2020; Wang et al., 2021a; Chen et al., 2020; Zhang et al., 2018). Caser (Tang and Wang, 2018) applies convolutional layers to process the embeddings of items in a sequence. SASRec (Kang and McAuley, 2018) and BERT4Rec (Sun et al., 2019a) use the Transformer (Vaswani et al., 2017) in a single direction style and a bidirection style respectively to model the sequential pattern in the interaction sequence.

Absolute positional encoding is firstly introduced with the attention structure to provide the access of sequential information for the permutation invariant computation (Vaswani et al., 2017). It assigns a fixed vector to each position in a sequence. The vector is computed either in a sinusoidal way or a learned style. For example, the language model BERT (Devlin et al., 2019) and the recommendation model BERT4Rec (Sun et al., 2019a), they both use the learned positional encoding. Relative positional encoding is later proposed to encode the relative position of two words, which is more meaningful for the natural language (Shaw et al., 2018; Yang et al., 2019; Dai et al., 2019; Wang et al., 2021b). For example, the language model XLNet (Yang et al., 2019) and Transformer-XL (Dai et al., 2019) propose different types of relative position encodings to represent the relative positional information between words in a sentence. Other positional encodings include different positional encoding schemes that are suitable for data structures other than one-dimensional sequence. For example, to apply the attention to images, there are 2D positional encoding schemes (Wang and Liu, 2019; Bello et al., 2019; Parmar et al., 2019; Lee et al., 2019; Carion et al., 2020) that provide either the absolute or the relative encoding. For example, the attention augmented network (Bello et al., 2019) designs a 2D relative positional encoding to encode the positional information in the activation map. For tree structures, Shiv and Quirk (Shiv and Quirk, 2019) proposed a specific scheme to encode the relationship between the root node and children nodes.

Recently, to enable neural networks to work on structured data (e.g., graph, point cloud, etc.), Graph Neural Networks (GNN) are widely investigated (Kipf and Welling, 2017; Velickovic et al., 2018; Li et al., 2016; You et al., 2019). Generally, the computation flow of GNN is called message passing, which is based on neighborhood aggregation. For example, GCN (Kipf and Welling, 2017), GAT (Velickovic et al., 2018) and GGNN (Li et al., 2016) are majorly different in the aggregation method. However, these GNN models could easily fall into a lack of representative ability since the message passing is performed on a narrow scope of nodes. Thus, PGNN (You et al., 2019) is proposed to include the information from randomly chosen anchor nodes to utilize extra structural information.

The positional encoding (PE) is introduced by (Vaswani et al., 2017) to enable the self-attention module to utilize the positional information of languages. Here, the sinusoidal positional encoding (SDE) $\bm{P}\in\mathbb{R}^{d\times 1}$ of a token at position $pos$ in the session of length $l$ is defined as:

where $i\in\{0,1,\dots,d/2-1\}$, $d$ is the dimension of the feature vector and $f(i)=10000^{2i/d}$. In the following, all $pos\in\{0,1,\dots,l-1\}$ if not specified.

To investigate the representation ability of a PE in a session, we define two features: forward-awareness and backward-awareness. If a PE is forward-aware, the PE of the first token is the same for all sequences. Furthermore, if a position $pos$ exists in any sequence, the PE for $pos$ is the same across these sequences. For example, if we assign the position itself as the PE, i.e., $P_{0}^{l}=0,P_{1}^{l}=1\dots$, then it is forward-aware. In contrast, if a PE is backward-aware, the PE of the last token is the same for all sequences. Furthermore, if an $h$-th last position $pos$ exists in any sequence, the PE for $pos$ is the same across these sequences. For example, if we assign the reverse position as the PE, i.e., $P_{l-1}^{l}=0,P_{l-2}^{l}=1\dots$, then it is backward-aware. A demonstration of forward-awareness and backward-awareness can be found in Fig. 1.

If there is a mapping between two PE in a session, the mapping also applies to other sessions that contain same positions. Similarly, the property of backward-aware PE is as the following:

These Definitions and Properties together give another important Properties of an absolute PE.

As discussed in the Introduction, the position in a session carries specific positional information in SBRS. The first item reflects the initial intention of the user while the last item is always considered more relevant to the latest preference of the user. And the items in-between usually represent the preference shift inside the session. For the forward-aware requirement, following Definition 3.1, two items at the same position of two different sessions always have the same slice of their PE. Following Property 3.1, the relationship between any position and the first position is the same across different sessions. As for the backward-aware requirement, the position in forward-aware requirement is changed into the reverse position following Definition 3.2 and Property 3.2.

This Theorem states that the sinusoidal positional encoding is not informative for positional information in session-based recommendation. As proved above, SPE is forward-aware because it is exactly calculated based on the position. Using SPE in any SBRS can only indicate how far an item is from the user’s initial intention. However, SPE is not backward-aware as it simply cannot tell if an item is at the last position of a session. In SBRS, it is crucial to know the preference shift within the session (Hidasi et al., 2016; Shani et al., 2005). Because SPE is not backward-aware, if a model uses SPE, there is no information about the closeness between an item and the user’s latest preference (i.e., the item at the last position).

Relative positional encoding (RPE) is designed to relax the assumption in language models that a word in an absolute position has the same meaning. RPE focuses more on the meaning of the relative position between two words. As proved above, RPE is neither forward-aware nor backward-aware, thus failing to meet both requirements of SBRS.

Empirically, the closer an item is to the last item, the more accurate it can reflect the user’s latest preference. As discussed in the Introduction, many methods consider the last item as the representation of the latest preference (usually referred to as short-term or local preference). Meanwhile, other items are treated with less importance (usually referred to as long-term or global preference). Following Theorem 3.1, SPE only contains the forward-awareness. Intuitively, we can modify the SPE to a reverse sinusoidal positional encoding (RSPE):

With RSPE rather than SPE, a model is theoretically able to utilize the positional information that can reflect how an item is different from the latest preference in the session. But obviously, RSPE neglects the positional information representing the relationship between an item and the initial intention.

In the content above, we focus on the positional encoding that only rolls out in a single direction. In the following, we will discuss the additional positional encoding and 2D positional encoding.

Direct addition of SPE and RSPE fails in this situation. Such an addition will create a symmetric positional encoding that for $pos_{a}$ and $pos_{b}$ in a length $l$ session, if $pos_{a}-0=l-pos_{b}-1$, $P_{pos_{a}}^{l}=P_{p-pos_{b}-1}^{l}$. This will break the Property 3.3.

If we exchange the $2i$ and $2i+1$ dimensions of RSPE in Eq. (4), and do the addition of SPE and RSPE, the resulted additional sinusoidal positional encoding (ASPE) follows Property 3.3 for uniqueness, but it is inconsistent with Definition 3.1 and Definition 3.2. Therefore, there is no guarantee on the positional information in SBRS according to Definition 3.3.

An interesting property about this ASPE is that the pattern of uniqueness is insufficient so that the attention model cannot easily infer the positions but only to memorize. In SPE and RSPE, there is a linear combination property between two positions (Vaswani et al., 2017; Shiv and Quirk, 2019). But the ASPE breaks this property, which leads to the attention model cannot learn the relationship between different positions. We prove this difference in Appendix A.

In the literature of computer vision that utilizes attention mechanism, there is a type of encoding for images called 2D sinusoidal positional encoding (2DSPE) (Wang and Liu, 2019; Bello et al., 2019; Parmar et al., 2019; Lee et al., 2019; Carion et al., 2020). If $pos$ and $l$ are considered as the height and width, then 2DSPE is similar to the ASPE that the encoding of each pair $(pos,l)$ is totally unique and thus, it does not follow the forward-aware and backward-aware requirements. Therefore, they are not eligible for SBRS. Detail of an example of 2DSPE is presented in Appendix B.

In SBRS, an item is denoted as $v$ and there is a unique item set $\mathcal{V}=\{v_{1},v_{2},v_{3},\ldots,v_{m}\}$, with $m$ being the number of items. A session sequence from an anonymous user is defined as an order list $\mathcal{S}=[v_{s,1},v_{s,2},v_{s,3},\ldots,v_{s,l}]$, $v_{s,*}\in\mathcal{V}$. $l$ is the length of the session $\mathcal{S}$. In this paper, a sequence has at least one item and $l\in\mathbb{Z}^{+}$. The goal of our model is to take an anonymous session $\mathcal{S}$ as input, and predict the next item $v_{s,l+1}$ that matches the current preference.

We propose a dual positional encoding (DPE) by concatenating of half of the SPE and half of the RSPE positional encoding. The DPE is defined as:

where $i\in\{0,1,\dots,d/4\}$ and for clarity, we assume $d/4\in\mathbb{Z}$ and all our results can be easily generalized to other cases.

This theorem states that the proposed DPE can represent the positional information of SBRS. For the $A=\{0,1,\dots,d/2-1\}$ dimensions of DPE, it is forward-aware. While for the rest $B=\{d/2,d/2+1,\dots d-1\}$ dimensions, it is backward-aware. For example, for the first item of any session with length $p$ and $q$, $P_{0,A}^{p}=P_{0,A}^{q}$ is always true while no guarantee that $P_{0,B}^{p}=P_{0,B}^{q}$. But $P_{p-1,B}^{p}=P_{q-1,B}^{q}$ is always true for DPE.

The proposed DPE has met the requirements of SBRS. This encoding scheme is parameter-free. In this situation, positions in a session can be considered linear because they are ordered with a consistent interval. However, in the real world, the position of each item actually comes from the timestamp of the interaction. The time intervals between interactions are neither consistent nor linear. Therefore, we propose the following learned dual positional encoding (LDPE) to improve the inductive bias injected into a session-based recommendation model:

where $\text{Embed}\in\mathbb{R}^{max(l)\times d}$ stands for a learned embedding matrix and $[\cdot]$ is the same as the slice operation for a list in Python. Similar to DPE, LDPE follows the Definition 3.1 and 3.2, and thus 3.3.

With DPE and LDPE ((L)DPE), we now build our Positional Recommender model (PosRec) based on the Position-aware Gated Graph Neural Network (PGGNN) and the bidirectional Transformer. Similar to GNN-based models (Wu et al., 2019; Qiu et al., 2019; Xu et al., 2019), a session is firstly converted into a weighted and directed session graph. Then PGGNN is applied to calculate the position-aware item embedding as the input of the bidirectional Transformer layer. (L)DPE is incorporated into the bidirectional Transformer layer to enhance the positional information. In the end, a single vector is computed as the representation of the session and used to predict the user’s next click.

We will discuss the relationship and difference between the proposed PosRec model and existing recommendation methods as well as the position encoding schemes.

In this section, we will describe the experimental setup in terms of datasets (Section 5.1.1), the preprocessing procedure (Section 5.1.2), baselines (Section 5.1.3), evaluation metrics (Section 5.1.4) and the implementation (Section 5.1.5).

The overall recommendation performance is demonstrated in Table LABEL:tab:baseline. We compare the PosRec with the following baselines: (1) shallow methods: POP, S-POP, Item-KNN (Sarwar et al., 2001), BPR-MF (Rendle et al., 2009) and FPMC (Rendle et al., 2010); (2) GRU-based methods: GRU4REC (Hidasi et al., 2016) and NARM (Li et al., 2017); (3) Attention-based method: STAMP (Liu et al., 2018); (4) GNN-based methods: SR-GNN (Wu et al., 2019), FGNN (Qiu et al., 2019) and GC-SAN (Xu et al., 2019) and (5) adapted sequential methods: SASRec (Kang and McAuley, 2018).

Our PosRec achieves the best performance compared with all baselines across all the datasets by four metrics. Compared with the previous state-of-the-art methods, the improvement is consistent. Our proposed PosRec method effectively exploits the positional information in the positional encoding module and the PGGNN module.

The traditional methods generally cannot compete with the current deep learning models. The popularity-based methods, POP and S-POP, simply recommend items based on the frequencies of appearance in the whole dataset and the current session respectively. These popularity-based approaches tend to recommend fixed items in a general situation, which is not able to learn to recommend the proper items. Although the S-POP model can make use of the session information to improve the performance compared with the POP, it still lacks the ability to learn the session pattern. PosRec performs much better compared with both of them. For shallow learning-based methods, BPR-MF and FPMC, they do not consider the session information in the recommendation, they do not have a competitive performance neither scale up well in the larger dataset Yoochoose $1/4$. Item-KNN outperforms all the traditional methods. Item-KNN only considers the closeness between items while avoiding sequential information. Generally, these traditional methods cannot compete with the neural network-based recommender systems.

GRU4REC is the first session-based model that uses a recurrent structure to capture sequential information. GRU4REC outperforms the shallow and popularity-based methods by a large margin, which can be viewed as a strong baseline. The sequential information is encoded explicitly by the recurrent calculation procedure. But the positional information of a session is only included implicitly along with the recurrent calculation. A big issue of the recurrent-based methods is that there is a catastrophic forgetting of the early information. For the attention-based baselines, NARM and STAMP, they apply the self-attention mechanism mainly over the last item, which considers the last item as the pivot item to represent the latest intention. This structure is a better inductive bias in positional information than linearly rolling out as RNN structure by outperforming the GRU4REC. This situation is also proved by the better performance of STAMP compared with NARM. STAMP completely excludes the recurrent structure and relies heavily on the last item. SASRec achieves a slightly higher result than STAMP since they both perform attention while SASRec includes a learned absolute positional encoding. Recent GNN-based methods, SR-GNN, FGNN and GC-SAN, have made improvements by utilizing the connectivity of items. These methods use a session graph to represent the session sequence by linking the interactions according to their chronological order. The relative positional information between interactions is included in the edge and in the graph readout calculation step, it resembles the attention mechanism to focus more on the last item. The positional information is only partially considered with the relative positional information and the backward-awareness.

To evaluate the effect of (L)DPE, we substitute (L)DPE with the following encoding schemes in the bidirectional Transformer module in PosRec: SPE (sinusoidal positional encoding), LPE (learned positional encoding), RSPE (relative sinusoidal positional encoding), LRPE (learned relative positional encoding), ASPE (additional sinusoidal positional encoding), ALPE (additional learned positional encoding), 2DSPE (2D sinusoidal positional encoding) and 2DLPE (2D learned positional encoding), where S indicates a sinusoidal scheme and L indicates a learned scheme. The result is presented in Fig. 3 and 4. To further verify the ubiquitous efficacy of LDPE, we integrate LDPE into the attention-based models, STAMP and SASRec. The result is shown in Table 2.

Among Fig. 3 and 4, the blue line in each chart indicates the base model that does not include any positional encoding in the PosRec model. Compared with other positional encoding schemes, LDPE can consistently improve the recommendation performance in all situations. For the learnable scheme, LPE and LRPE both achieve relatively good performance because they represent parts of the positional information. The LPE scheme is adopted by SASRec (Kang and McAuley, 2018) and BERT4Rec (Sun et al., 2019a) for the sequential recommendation as well. There is a small gap for LPE to outperform LRPE. It could be because LPE provides a further forward-aware information while the readout function has already included the last item as the backward-aware information. ALPE and 2DLPE are not theoretically considered as suitable for SBRS. But they still have a comparable result because their entries of the PE are unique so that the model can still learn the pattern of the encoding but in a harder way. As for the parameter-free scheme, LPE and RSPE achieve comparable results with DPE because they provide reasonable positional information to the model. ASPE has the worst result because the forward-aware and backward-aware information is entangled in the representation. In contrast, 2DSPE has a clearer connection between entries than ASPE.

In Table 2, it is shown that LDPE can consistently improve the performance of attention-based models. STAMP does not originally use any positional encoding. While SASRec has already utilized an LPE in its basic method. For both Yoochoose and Diginetica dataset, LDPE can improve the performance of LPE by a large margin with the same number of parameters, which verifies that LDPE can be integrated into different models.

To evaluate the effect of the anchor node aggregation in PGGNN, we add this module to GNN-based methods: SR-GNN, FGNN, GC-SAN and PosRec-ANC (indicating the PosRec model removing the anchor node aggregation). The result is presented in Fig. 5 an 6.

It is shown that with the anchor node aggregation, the performance of all GNN-based methods has an increase for Recall. While for the MRR metric, only GC-SAN and PosRec gain a steady improvement. Anchor nodes improve the model performance: (1) injecting the inductive bias of the distance between normal items and important items to the model; (2) increasing the connectivity of the session graph. On one hand, the position-aware node feature learning is proved by (You et al., 2019) that it can provide the positional information in addition to the structure information. On the other hand, session data is originally sparse in the aspect of connectivity. A large portion of session data is even a simple sequence without any repetitive items. After including the aggregation of anchor nodes, the connectivity is increased because there are additional edges between normal nodes and anchor nodes. GNN layers are more suitable to process such data.

In this experiment, we visualize how SPE, DPE, LPE, and LDPE look like and their characteristics. We firstly show the SPE and DPE themselves in Fig. 7. We examine the nearest neighbor relationship between encoding of different positions in Fig. 9. Example session lengths are set to 10 and 20 and the embedding size is set to 100. Fig. 7(a) and 7(b) show the SPE and DPE for length 10 respectively. We further present the heatmap of LPE and LDPE of length 10 and 20 in Fig. 11 and Fig. 12, which are learned based on the experiment on Yoo. 1/64.

The nearest neighbor heatmaps shown in Fig. 9 represent the dot product of the positional encoding of the same session length. SPE and DPE of lengths 10 and 20 are demonstrated respectively. By definition, the heatmap of RSPE is the same as SPE. We can see that within the same length, these positional encoding schemes do have a strong correlation to the same position of the same length.

To examine the correlation of the positional encoding from different session lengths, we demonstrate the results from the dot product between lengths of 10 and 20 in Fig. 10. For the case of SPE in Fig. 10(a),there is one diagonal strip that a strong correlation is only in between the same position but not the same reverse position. For example, there is no strong correlation between the last position of length 10 and length 20 (0.16 in the lower right corner). While for the case of DPE in Fig. 10(b), two diagonal strips indicate a strong correlations in both direction of positions.

From Fig. 7(a) and 8(a), it is clear to see the forward-awareness of SPE. For example, no matter what length is, the encoding for the first position will always be the same (the first row). While for the last position of lengths 10 and 20, as the max length and the embedding size vary, there will not be a shared same part, which is not backward-aware. For RSPE, the case is just the reverse, i.e., the heat map would be upside down of SPE. From Fig. 7(b) and 8(b), both forward-awareness and backward-awareness are demonstrated. The same positions and reverse positions will always share the same half of the embedding respectively.

The nearest neighbor heatmaps of LPE and LDPE are shown in Fig. 11 and Fig. 12 respectively. From the heatmaps between the same length for 10 and 20 in Fig. 11(a) and Fig. 11(b), the LPE focuses mostly on the initial intention with the most related encoding being the first position. From the heatmap between the length 10 and 20 in Fig. 11(c), it is obvious that LPE can only capture the forward positional information of the initial intention of sessions with different lengths. The position encodings close to the end of sessions are not carrying forward or backward positional information, which is shown at the bottom right corner of Fig. 11(c) with a similar closeness between different positions. Compared with LPE, the heatmaps of LDPE are shown in Fig. 12(a) and Fig. 12(b), from which the diagonal cells indicate that the LDPE can capture the positional information of distinct positions of the same length. While for Fig. 12(c), it can be seen that the LDPE can capture the backward positional information with the light cell at the bottom right corner. For the forward positional information, the heatmap at the upper left corner shows a diagonal pattern. While the diagonal pattern is more obvious on the right part in Fig. 12(c), which indicates that the backward positional information is more important and useful for SBRS.

In this experiment, we evaluate the effect of $\lambda_{0},\lambda_{1}$ and $\lambda_{2}$ in Eq. 10. We evaluate the relative scale between these three hyper-parameters. We fix $\lambda_{0}$ and $\lambda_{1}$ to 1 because they both represent the strength about the information from the last position. And we change $\lambda_{2}$ from $\{0.25,0.5,1,2\}$. In Fig. 13, we demonstrate the result of the sensitivity of $\lambda_{2}$ is shown. For the Yoochoose dataset in Fig. 13(a), the best performance is achieved when $\lambda_{2}=0.5$. Compared with $\lambda_{2}=0.25$, we can see that inadequate initial intent will do harm to the overall performance. When it comes to $\lambda_{2}=1\text{ or }2$, we can also imply that the initial intent is not as important as the latest preference. For the Diginetica dataset in Fig. 13(b), the best performance is achieved when $\lambda_{2}=1$. Compared with $\lambda_{2}=0.25\text{ or }0.5$, we can see that more initial intent will benefit the overall performance.