Sequential Recommendation with Controllable Diversification: Representation Degeneration and Diversity

In the SR problem, there are a set of users $\mathcal{U}$, a set of items $\mathcal{V}$, and interactions feedback between users and items. Each interaction is associated with its timestamp. For each user $u\in\mathcal{U}$, we sort the interacted items chronologically and format the user as the sequence of interacted items, denoted as $\mathcal{S}^{u}={[v^{u}_{1},v^{u}_{2},\dots,v^{u}_{|\mathcal{S}^{u}|}]}$. Specifically, $v^{u}_{i}\in\mathcal{V}$ represents the $i$-th interacted item in the sequence $\mathcal{S}^{u}$ of the user $u$. SR tackles the problem of predicting the next item based on the sequence of user’s historical interactions, which is formalized as predicting the probability of the next item $p\left(v_{|\mathcal{S}^{u}|+1}^{(u)}=v\left|\mathcal{S}^{u}\right.\right)$. In the recommendation generation stage, the next item prediction scores over all items are ranked to generate the top-N recommendation list for the user $u$.

Given a user interaction sequence $\mathcal{S}^{u}$, we pre-process it by either truncating earliest interactions or padding with zeros such that the sequence length becomes the maximum sequence length $n$ to get a fixed length sequence $s=(s_{1},s_{2},\dots,s_{n})$. An item embedding matrix $\mathbf{M}\in\mathbb{R}^{|\mathcal{V}|\times d}$ is randomly initialized and will be optimized, where $d$ is the latent dimension size. A trainable positional embedding $\mathbf{P}\in\mathbb{R}^{n\times d}$ is added to the sequentially ordered item embedding matrix as $\mathbf{E}_{\mathcal{S}^{u}}$. Formally, we formulate the sequence encoder as follows:

where $\mathbf{H}_{u}\in\mathbb{R}^{n\times d}$ denotes sequential output embeddings of $\mathcal{S}^{u}$, and for each timestep $t$, and $\textbf{H}_{(u,t)}$ encodes the predicted next-item ($v_{t+1}$) representation.

The next item preference score prediction is given as:

where $\mathbf{E}_{v}$ denotes the embedding of the item $v$. The typical optimization process adopts the classical sampled cross-entropy (sampled CE) loss or bayesian personalized ranking loss (BPR) (Rendle et al., 2012), which is maximizing the probability of users ranking positive items $v^{+}_{t+1}$ with greater preference scores over sampled negative items $v^{-}_{t+1}$. The optimization can be formulated as a sampled cross-entropy loss minimization as:

where $\mathbf{M}_{v^{-}_{t+1}}^{j}$ denotes the $j$-th negatively sampled item, $N^{-}_{u}$ denotes the number of negatively sampled items from the item set without interaction with the user $u$. The value of $N^{-}_{u}$ affects the recommendation performance significantly (Ding et al., 2019; Chen et al., 2017; Mao et al., 2021). The typical choice in BPR and binary cross-entropy losses uses one negatively sampled item $N^{-}_{u}=1$.

We propose a simple, intuitive, and straightforward metric to measure the degree of singular spectrum smoothness, which is calculated as the Area Under Singular value Curve (AUSC). The area under singular curve (AUSC) is calculated as the sum of normalized singular values as follows:

When the singular value curve decays fast, as observed in the degenerated examples in Fig. (2), the area under the singular curve (AUSC) is typically small because singular values other than the largest one are all small and all singular values are normalized. We can also see that when all singular values approximate the largest singular value, i.e., $|\sigma_{i}-\sigma_{\text{max}}|\approx 0$ for all $\sigma_{i}$, the AUSC is maximized, in which the singular curve becomes flatter. For example, our toy example shown in Fig. (3) compares two singular curves, in which the left one is the fast decaying curve and the right one is an improved and more flat singular curve. The left fast decaying curve has an AUSC value of 13.64, while the right smoothed curve has an AUSC value of 26.73.

We build upon the AUSC to propose a singular spectrum smoothing regularization as a surrogate of optimizing AUSC. With the discussion in Section 3.1, item and sequence degeneration issues are intertwined. Thus, we deploy the proposed singular spectrum smoothing regularization to both sequence output and item embeddings, which is referred to as dual smoothing. Specifically, the singular spectrum smoothing losses for both sequence $\mathcal{L}_{\text{seq}}$ and item $\mathcal{L}_{\text{item}}$ are conducted as follows:

where $\mathbf{H}$ is the sequence output embedding matrix of all users, and The singular spectrum smoothing loss is the lower bound of our proposed AUSC smoothness metric as follows:

The loss minimization of either $\mathcal{L}_{\text{seq}}$ or $\mathcal{L}_{\text{item}}$ achieves the joint effects of

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  maximizing the nuclear norm (encouraging the sum of singular values),

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  minimizing the Frobenius norm (suppressing the largest singular value),

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  and enlarges more tail singular values.

When $\mathcal{L}_{\text{seq}}$ and $\mathcal{L}_{\text{item}}$ are minimized, the nuclear norm is maximized, improving the determinant and diversity of $\mathbf{H}$ and $\mathbf{M}$. We adopt the Frobenius norm instead of the largest singular value for more stable training. The minimization of $\mathcal{L}_{\text{seq}}$ and $\mathcal{L}_{\text{item}}$ maximizes the AUSC of sequence embeddings and item embeddings, i.e., more smooth singular curves are introduced. Specifically, when the singular spectrum smoothing loss is minimized, the Frobenius norm is minimized so that the infinite norm of item embedding $\mathbf{M}$ is alleviated. The largest singular value is suppressed.

Computational Efficiency: the computational complexity of singular values calculation can be reduced when $\mathcal{L}_{\text{seq}}$ and $\mathcal{L}_{\text{item}}$ are conducted in batch, which is $O(\text{min}(B^{2}d,Bd^{2}))$ (Cui et al., 2020), and $B$ denotes the batch size and $d$ is the dimension size. Different from (Cui et al., 2021, 2020; Chen et al., 2022), we present the nuclear norm divided by the Frobenius norm on embeddings of sequence and item sides while only nuclear norm of the prediction outputs is maximized in (Cui et al., 2021, 2020; Chen et al., 2022).

With singular spectrum smoothing on both sequence output embeddings side $\mathcal{L}_{\text{seq}}$ and item side $\mathcal{L}_{\text{item}}$, the final optimization loss $\mathcal{L}$ with the recommendation loss in Eq. (3) to be minimized is defined as follows:

where $\lambda$ is the hyper-parameter for controlling the sequence output embeddings side smoothing weight, and $\beta$ is the hyper-parameter for controlling the item side smoothing weight contribution to the final loss minimization. The final recommendation list is generated by calculating the scores defined in Eq. (2) on all items for each user. The recommendation scores on all items are sorted in descending order to produce the top-$N$ list.

We present the statistics of the dataset in Table 1. We evaluate all models in four benchmarks public Amazon review datasets¹¹1http://deepyeti.ucsd.edu/jianmo/amazon/index.html. We choose Beauty, Toys and Games (Toys), Tools and Home (Tools), and Office Products (Office) categories in our experiments as these four categories are widely adopted benchmark categories (Tang and Wang, 2018; Kang and McAuley, 2018; Sun et al., 2019; Fan et al., 2021; Li et al., 2020; Fan et al., 2022). We treat the existence of user-item reviews as positive user-item interactions. For each user, we sort the interacted items chronologically to form the user interaction sequence. The last interaction of the user sequence is used for testing and the second to last one for validation. We adopt the standard 5-core pre-processing setting on users (Tang and Wang, 2018; Kang and McAuley, 2018; Sun et al., 2019; Fan et al., 2021; Li et al., 2020; Fan et al., 2022) to filter out users with less than five interactions. The detailed dataset statistics are presented in Table 1.

We compare the proposed SPMRec with state-of-the-art sequential recommendation methods. We only include the static method BPRMF (Rendle et al., 2012) due to the page limitation. For sequential methods, we compare Caser (Tang and Wang, 2018), SASRec (Kang and McAuley, 2018), DT4SR (Fan et al., 2021), BERT4Rec (Sun et al., 2019), FMLP-Rec (Zhou et al., 2022), and STOSA (Fan et al., 2022). DuoRec (Qiu et al., 2022) addresses the degeneration issue in contrastive learning for SR. Note that we use only one training negative sample (i.e., $N^{-}_{u}=1$) for models with the Cross-Entropy loss for fair comparisons (Ding et al., 2019; Chen et al., 2017; Mao et al., 2021).

We generate the top-N recommendation list for each user based on the dot-product between sequence output embedding and the entire item set in descending order. We rank all items for all models so that no item sampling bias is introduced in evaluation for fair comparisons (Krichene and Rendle, 2020). The evaluation includes classical top-N ranking metrics, Recall@N and NDCG@N. We present the averaged results over all test users. The test results are reported based on the best validation results. We report metrics in multiple Ns, including $N=\{5,10,40\}$, which are widely adopted in most existing SR methods (Kang and McAuley, 2018; Sun et al., 2019; Fan et al., 2022).

We evaluate the diversity of the recommendation list for all test users in two widely adopted metrics, including the Intra-list diversity (Zhang and Hurley, 2008; Ziegler et al., 2005) and the Coverage@100 of item categories for each user’s top-N recommendation list. We focus on the top-10 ranking list for intra-list diversity calculation and top-100 for the coverage metric. The intra-list diversity is calculated as follows:

where $\mathcal{I}_{u}^{N}$ denotes the top-$N$ recommendation list for the user $u$, $\text{cos}(\cdot,\cdot)$ represents the cosine similarity, $\mathbf{M}$ is the item embedding table defined in Section 2.2, and the reported diversity is the average diversity over all users. The larger value of diversity indicates higher intra-list diversity for the recommendation. Note that, we also found similar improvements in other larger $N$s, such as $N=\{20,40\}$.

We present the overall performance comparison with relevant state-of-the-art baselines in Table 2. From Table 2, we can easily conclude the superiority of SPMRec in providing sequential recommendations in various datasets and achieving not only better performance but also more diversity. We have several observations to conclude as follows:

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  SPMRec achieves the best performance and diversity, and outperforms SASRec with large improvement margins. In most performance metrics and all datasets, SPMRec significantly outperforms the second-best baseline with 0.52% to 12.33% in ranking performance and 4.71% to 14.41% in diversity. The improvements demonstrate the effectiveness of SPMRec in the sequential recommendation, while the recommendation diversity is also noticeably improved. Furthermore, as SPMRec is built on top of the original implementation of SASRec (Kang and McAuley, 2018), we observe 6.92% to 22.53% relative improvements over SASRec. We argue that the improvements originate from the singular spectrum smoothings on both sequence and item sides.

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  SPMRec can achieve better balance between recommendation quality and diversity. Comparing the diversity of SPMRec and baselines, we can observe that SPMRec consistently achieves better diversity results while having better performances, indicating the better capability of SPMRec in balancing the recommendation quality and diversity.

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  SPMRec achieves performance larger improvements in sparser datasets. Comparing SPMRec and SASRec, we can observe at least 17% improvements in both Beauty and Tools categories datasets, which have lower densities of 0.05% and 0.07%, respectively. The recommendation diversity is also significantly improved, especially for tools with a 14.43% improvement. However, in Toys and Office datasets with higher densities of 0.08% and 0.44%, the improvements are at most 14%. This observation demonstrates that the proposed singular spectrum smoothing SPMRec benefits sparser datasets, especially for datasets with a large portion of short sequences.

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  The consideration of Frobenius norm benefits the recommendation and diversity. STOSA adopts the sum of two L2 distances on mean and covariance embeddings and DuoRec utilizes the uniformity loss based on the exponential L2 distance. We can observe that STOSA and DuoRec both outperform the dot-product-based SASRec framework in recommendation quality and diversity. This observation further demonstrates the need to consider the Frobenius norm in embedding learning. Moreover, SPMRec performs better than STOSA and DuoRec, demonstrating the necessity of considering the regularization from the singular spectrum perspective.

Item Spectrum Smoothing controlled by $\beta$ is closely related to the recommendation diversity. We demonstrate a strong connection between our item spectrum smoothing and the recommendation diversity by showing the relationship of $\beta$ and the diversity, as shown in Fig. (4). In Fig. (4), the x-axis denotes the value of $\beta$. As the $\beta$ becomes larger, we can observe that the diversity consistently increases, in all four datasets, demonstrating the strong connection between the recommendation diversity and our proposed item spectrum smoothing component.

Moreover, the recommendation performance drops when we demand too high recommendation diversity. Another observation is that the NDCG@10 drops significantly when we set $\beta$ to a too-large value, which also indicates a higher diversity. This observation demonstrates that a proper selection of $\beta$ can achieve satisfactory recommendation performance and diversity simultaneously. However, demanding higher diversity can hurt the recommendation performance.

We compare the proposed SPMRec with spectrum smoothing regularization with existing methods for alleviating the degeneration issue in word embeddings learning, including the cosine regularization (Gao et al., 2019) as follows:

and the Euclidean method (Zhang et al., 2020) as follows:

We apply the cosine and Euclidean regularization methods to both sides. We report the best grid searched results of all regularization methods in Table 3. Note that the uniformity property adopted by DuoRec and DirectAU (Qiu et al., 2022; Wang et al., 2022b) falls into the Euclidean methods category.

As shown in Table 3, neither cosine regularization or Euclidean regularization can consistently outperform the original SASRec without any regularization. Arguably, the potential reason is that neither of these two methods constrains the Frobenius norm of embeddings, which is the upper bound of the largest singular value, as discussed in Section 3.1. When cosine regularization and the Euclidean method are minimized, the naive solution is $||\mathbf{E}_{i}||_{2}\rightarrow\infty$ for all $i$ if we do not consider the supervised recommendation loss. Moreover, different from the word embedding task, sequences in the recommendation task are shorter on average, posing a greater challenge in addressing the degeneration in SR. However, our proposed singular spectrum smoothing method SPMRec minimizes the Frobenius norm while maximizing the nuclear norm, achieving the goals of smoothing the singular value curve and suppressing the largest singular value simultaneously.

We conduct the ablation study to show the effectiveness of two important components, including singular spectrum smoothing for sequence output and item embeddings. We investigate the smoothing weights $\lambda$ (for sequence output embeddings smoothing) and $\beta$ (for item embeddings smoothing) in Fig. (6) and Fig. (7), respectively. We include the SASRec (without any regularization) and DuoRec (with uniformity regularization). Note that the special case ($\lambda=0$ and $\beta=0$) is included, which is SASRec without any regularization. Additionally, the sequence outputs and item embeddings spectrum smoothing losses are in different scales ($10^{-1}$ for sequence embeddings and $10^{-4}$ for item embeddings). Hence, the scales of $\lambda$ and $\beta$ are different with the purpose of balancing the contributions of two regularizations.

We visualize the singular values curve of sequence output embedding for our proposed SPMRec and SASRec, as shown in Fig. (10). We can observe that the proposed SPMRec has a smoother decaying curve than SASRec. Moreover, for SPMRec, the degrees of spectrum smoothness for all four datasets are 26.34 for Beauty, 20.76 for Tools, 20.25 for Toys, and 19.24 for Office. However, SASRec has lower degrees for spectrum smoothness as 19.54 for Beauty, 17.10 for Tools, 17.00 for Toys, and 13.67 for Office. SPMRec consistently has larger degrees of spectrum smoothness than SASRec, which demonstrates the effectiveness of SPMRec in preventing sequences from the degeneration for the sequential recommendation.

Sequential Recommendation (SR) is a technique that encodes the temporal interactions of each user as a sequence and uses the encoded behaviors to infer their preferences. The main idea behind SR is to model the transitions between items within sequences. The earliest SR methods used Markovian approaches, such as FPMC (Rendle et al., 2010) and Fossil (He and McAuley, 2016), with the assumption that the item prediction depends only on the previous interacted items Another line of work utilizes the Recurrent Neural Network (RNN) as the base model for modeling sequential interactions. Several works adopt RNNs to address the SR problem, including GRU4Rec (Hidasi et al., 2015), HGN (Ma et al., 2019; Peng et al., 2021), HRNN (Quadrana et al., 2017), and RRN (Wu et al., 2017). Different variants of RNNs are proposed, such as the original GRU in GRU4Rec and hierarchical RNN in HGN and HRNN. The third line of methods to solve the SR problem is using the Convolution Neural Network (CNN), with the interpretation of a sequence of item embeddings as an image, including Caser (Tang and Wang, 2018), CosRec (Yan et al., 2019), and NextItNet (Yuan et al., 2019). The recent success of the self-attention module (Vaswani et al., 2017) inspires the adaptation of Transformer to the sequential recommendation task. The self-attention module captures the sequential high-order item transitions by modeling attention from all items in the sequence. SASRec (Kang and McAuley, 2018) is the earliest work of applying Transformer to SR. BERT4Rec (Sun et al., 2019) builds upon BERT (Devlin et al., 2019) with bidirectional Transformers for SR. FMLP-Rec (Zhou et al., 2022) proposes an all-MLP architecture in the Transformer for SR. Several variants of Transformers (Fan et al., 2021; Li et al., 2020; Wu et al., 2020; Liu et al., 2021) are further proposed.

The representation degeneration problem describes the issue of learned embeddings being narrowly distributed in a small cone (Gao et al., 2019). The degeneration issue is firstly proposed in the word representations learning problem. As shown in (Gao et al., 2019), words from text data follow the long-tail distribution, which degenerates the representation learning process. Theoretically, it causes the narrow distribution of words in the representation latent space (Gao et al., 2019). Moreover, an empirical analysis (Yu et al., 2022) shows that the learning of tail words degenerates the representation optimization of frequent words. An important analysis in (Wang et al., 2020) shows that degeneration is closely related to the fast decaying singular values distribution. In other words, the largest singular value is significantly larger than all other singular values.

Distinct from the natural language processing (NLP) area, SR still has several unique characteristics. For example, sequences are short (data sparsity in recommendation systems), while sentences/paragraphs in NLP are typically long. This distinct characteristic introduces additional challenges for SR, where the degeneration issue can also happen in the sequence output embeddings. DuoRec (Qiu et al., 2022) observes that the representation degeneration issue also exists in the self-supervised SR, with data augmentations. DuoRec and DirectAU (Wang et al., 2022b) optimizes the uniformity property in the in-batch representations to enforce the representations to be uniformly distributed, where the uniformity is based on the Euclidean distance. This work connects the item embeddings degeneration with the degeneration of sequence representations. The intertwining of both sequence and item rank degeneration is rarely discussed before in SR. Moreover, some solutions from NLP to the degeneration problem only apply to the item side, including the CosReg (Gao et al., 2019) based on cosine similarities, Euclidean regularization based on learned word similarities (Zhang et al., 2020), spectrum control (Wang et al., 2020; Yan et al., 2022), and gradient gating (Yu et al., 2022).

As discussed in (de Souza Pereira Moreira et al., 2021), natural language processing (NLP) and sequential recommendation (SR) share multiple similarities, including both learning discrete tokens learning (words in NLP and items in SR) and long-tail distribution of frequency for both words and items. With this inspiration, DuoRec (Qiu et al., 2022) observes that the representation degeneration issue also exists in the self-supervised SR, with data augmentations. DuoRec optimizes the uniformity property in the in-batch augmented representations to alleviate the degeneration issue in contrastive learning, where the uniformity is based on the Euclidean distance.