Multi-Sample based Contrastive Loss for Top-k Recommendation

The contrastive loss has become an excellent tool in unsupervised representation learning. It aims to maximize the similarities of positive pairs and minimize that of negative pairs [19, 20, 25, 26, 27]. The contrastive loss function is widely used for many kinds of data, such as images, text, audio, graphs, etc. It has been applied in the field of recommendation[28, 29].

Broadly speaking, functions that use pairwise contrastive learning processes are contrastive loss functions which have many forms. BPR and triplet loss are the basic contrast-based and widely used loss functions. BPR [18] loss aims to maximize the distance between positive pair and negative pair, which is proposed for the ranking task and widely used in the top-k recommendation. Triplet loss [30, 31, 32] can be used to train samples with small differences, especially for human faces. The samples are triplets (anchor, positive, negative). The triplet loss is calculated by optimizing the distance between the anchor and positive samples to be smaller than the distance between the anchor and negative samples.

However, they employ limited pairs of samples. Contrastive loss functions based on multiple pairs of samples are more efficient which contain Multi-class N-pair loss, InfoNCE loss, Non-Parametric Softmax Classifier, NT-Xent loss. Multi-class N-pair loss [33] is proposed from a deep metric learning perspective, which greatly improves the triplet loss by jointly pushing out multiple negative samples at each update. InfoNCE loss [34, 35, 36] is proposed by maximizing a lower bound on mutual information based on Noise-Contrastive Estimation. Non-Parametric Softmax Classifier [37] is presented by maximizing distinction between instances via a novel non-parametric softmax formulation in an unsupervised feature learning approach. They come from different fields and formula derivation but share a similar form. NT-Xent loss (the normalized temperature-scaled cross entropy loss) [19, 24] is proposed on these bases, but with the minor difference that the denominator does not contain positive samples. All of them are widely used in contrastive learning framework and always obtain the state-of-the-art results.

The contrastive loss is used in the recommendation system in the contrastive learning framework for recommendation recently. SGL[28] is proposed for top-k recommendation, CL4SRec [29] is used for sequential recommendation. But none of them has improved CL to fit the recommendation field.

The user-item interaction records in recommendation systems are naturally viewed as bipartite graphs. Graph-based data augmentation is widely used and studied in graph contrastive learning, which contains both traditional subgraph sampling methods and recently proposed methods [38, 39, 28, 24, 40, 41]. User and item are inherently linked and dependent on each other in the user-item bipartite graph. Data augmentation for GCN is also challenging due to the complex, non-Euclidean structure of the graph, and few works study the data augmentation of graphs. Therefore, graph data augmentation must be tightly integrated with the graph rather than replicating the methods used in Computer Vision and Natural Language Processing domains.

Graph data augmentation conforms to the basic assumptions of graph data processing. Node dropping assumes that edge vertex missing does not alter semantics. Edge perturbation is considered to improve the robustness of the semantics against connectivity changes. Masking node features enhance semantic robustness by losing some attributes for each node. Subgraphs assume that local structure can hint the complete semantics [24].

The graph augmentation can be divided into two types, feature-space augmentations and structure-space augmentations: (1) feature-space augmentations are realized by modifying initial node features, such as masking features or adding Gaussian noise, and (2) structure-space augmentations operate on graph structure by adding or removing nodes or edges (edge perturbation), sub-sampling or subgraphs by random walk, or generating different views using shortest distances or diffusion matrices [40].

Recently, Zhu et al. [41] proposes adaptive graph augmentation to design augmentation schemes that tend to keep important structures and attributes unchanged while the unimportant links and features are perturbed. Zhao et al. [42] utilizes a neural edge predictor to predict likely edges for graph augmentation to improve node classification performance. For top-k recommendations, the latest work, SGL [28], uses three operators on the graph structure, namely node dropout, edge dropout and random walk. And experimental results show that the edge dropout performs the best.

Differences with existing CL functions: Many works are done just using the CL function in the contrastive Learning framework. SGL in the recommendation system employs a multi-tasking mechanism with joint use of CL and BPR. Despite some improvements are proposed on CL, such as soft contrastive loss [43], debiased contrastive loss [44], they all treat the weights of the positive sample and negative samples as the same. And the problem of imbalanced positive and negative samples is still not be concerned. More importantly, how to adapt CL to recommendation systems is a new topic worth investigating, especially to emphasize the importance of positive samples on sparser datasets. Thus, the proposed importance-aware CL is different from the previous works.

Differences with existing graph data augmentation: Existing graph data augmentations in the recommendation system are common methods in the graph field. How to make full use of the limited positive samples to obtain better results, especially for the recommendation system is an important task and challenge for data augmentation. We randomly sampled a fixed number within one-hop neighbors which is not the same as random dropout or subgraph by random walk on multiple hops. More importantly, our method is a structure-space based augmentation, and traditional structure-space based methods are generally work in the aggregation process of GCNs. The traditional augmented data are used one by one under the same loss which is a serial approach. We use multiple positive samples at the same time and combine them explicitly with the loss function which is a parallel way for better constraints.

Recently, graph related methods have shown excellent performances, which treat user-item interactions as graph structures and adopt graph convolution network. Combined with collaborative filtering, NGCF [14], LR-GCCF [15], LightGCN [16], etc. are excellent models for top-k recommendation.

LightGCN is the state-of-the-art method and is introduced here as our main baseline. This model includes only the most essential component in GCN - neighborhood aggregation - for collaborative filtering which is much easier to implement and train and gains substantial improvements. Then neighborhood aggregation is defined as follows:

where $u,i$ denote the user and the item in the user-item graph, ${e}_{u}^{\left(k\right)},{e}_{i}^{\left(k\right)}$ respectively denote embeddings of $u,i$ of the $k$-th layer. Specially, $k=0$ represents the initialized latent vector; $\mathcal{N}_{u}$ and $\mathcal{N}_{i}$ represent the set of the neighbors of target $u$ and $i$, relatively. The final embedings of users and items are:

where $K$ is the numbers of layers; $\alpha_{k}$ denotes the importance of the $k$-th layer embedding, and they can be treated as a hyper-parameter to be tuned manually, or as a model parameter to be optimized automatically. Following the original paper of LightGCN, the mean of embeddings from all layers are adopted as the final embeddings, that is $\alpha_{k}$=1/($K$+1).

LightGCN-single, a variant of LightGCN are also proposed in the paper, where only the $k$-th embeddings, ${e}_{u}^{(k)},{e}_{i}^{(k)}$, are used as final embeddings. This variant, instead of the original LightGCN, is used here for its better performance and is named sLightGCN for short.

The BPR loss is used for training in LightGCN. We present it here for comparison with MSCL.

where $\sigma(\cdot)$ is the logistic sigmoid function.

The model prediction is defined as the inner product of the user and item final embeddings:

Based on this prediction, the top-k most similar items are recommended to the user.

The proposed MSCL is used for model training, and the method is named as sLightGCN_MSCL. MSCL replaces the BPR loss which is used in the original LightGCN. Except for the proposed loss function, our method remains the same as the LightGCN. The L2 regularization for all parameters is also used following LightGCN, and it is omitted here for clarity.

In this subsection, we analyze the complexity of sLightGCN_MSCL following SGL [28]. Since sLightGCN_MSCL does not introduce trainable parameters and there is no change of model prediction, the spatial complexity and the time complexity of the model inference are the same as LightGCN. The complexity of sLightGCN_MSCL can be divided into two parts, that of sLightGCN and MSCL, and they are $O(2|E|)+O(2|E|Lds|E|/N)$, $O(mN|E|ds)$, where $E$ is the edge in the user-item interaction graph, $L,s,m$ denote the number of GCN layers, the number of epochs, the number of multiple positive samples, and $d,N$ denote the embedding size, the batch size, respectively. For comparison, that of the BPR loss is $O(2|E|ds)$.

In fact, the overall amount of calculation is significantly reduced because the number of training epoch is substantially reduced due to better convergence performance as shown in training efficiency in section V. MSCL is $O(mN/2)$ times larger than the computational cost of BPR, but this is a simple inner product which is directly accelerated by matrix operations through the GPU. Therefore, there is no significant increase in training time in each epoch as can be seen in section V.

To evaluate the effectiveness of MSCL, we conduct experiments on three benchmark datasets: Yelp2018 [16, 28], Amazon-Book [16, 28], and Alibaba-iFashion [28, 2].

Yelp2018: Yelp2018 is adopted from the 2018 edition of the Yelp challenge. The local businesses like restaurants and bars are viewed as the items.

Amazon-book: Amazon-review is a widely used dataset for product recommendation and Amazon-book from the collection is selected.

Alibaba-iFashion: Alibaba-iFashion is a large and rich dataset for Fashion Outfit recommendation. 300k users and all their interactions over the fashion outfits are randomly sampled by SGL [28]. It is quite sparse, which is a significant difference from the first two datasets.

Three datasets vary significantly in domains, size, and sparsity. The statics of the processed datasets are summarized in Table I. For comparison purposes, we directly use the split data provided in SGL[28].

Following NGCF, LightGCN, and SGL [14, 16, 28], two widely used evaluation metrics, Recall@K and NDCG@K where K=20, are used to evaluate the performance of top-k recommendation. Recall measures the number of items that the user likes in the test data that has been successfully predicted in the top-k ranking list. NDCG considers the positions of the items and higher scores are given if the items are ranked higher. It is a metric about ranking and thus is important for the top-k recommendation. The larger the values, the better the performance for both metrics.

We implement our proposed method on top of the official code of LightGCN¹¹1https://github. com/gusye1234/LightGCN-PyTorch based with Pytorch. We replace the loss function and follow LightGCN’s settings as much as possible. The embedding size is fixed to 64 and the default batch size is 2048 for all models. The learning rate and L2 regularization coefficients are chosen by grid search in the range of $\{0.0001,0.001,0.01\}$ and $\{1e-5,1e-4,\cdots,1e-2\}$. These are hyper-parameters of the original LightGCN. We adjust hyper-parameters of MSCL, $M$ and $\tau$, in the ranges $\{1,3,5,\cdots,15\}$, $\{0.1,0.2,0.5,1.0\}$, respectively. And $\tau$ is 0.1 or 0.2 usually. The weight $\alpha$ is adjust in [0.4,0.7].

To demonstrate the performance of our method, we select many strong baselines for comparison. NGCF [14], LR-GCCF [15], LightGCN [16] are the competing baselines with GCN of top-k recommendation recently which having shown to outperform several methods including GC-MC [45], PinSage [46], NeuMF [9] since the previous works [14, 16, 15]. The latest methods, SGL [28], is also selected which is a self-supervised based method. In addition, the basic method and the variable autoencoder-based methods, MF and Mult-VAE, are compared.

MF: This is a traditional method based on matrix factorization which is based only on the embeddings of users and items, namely ${e}_{u}$ and ${e}_{i}$.

NGCF [14]: NGCF integrates the bipartite graph structure into the embedding process based on the graph convolutional network. It explicitly exploits the collaborative signal in the form of high-order connectivities by propagating embeddings on the graph structure.

LR-GCCF [15]: This method enhances the recommendation performance with less complexity by removing the non-linearity. The final embeddings are the same as NGCF.

LightGCN and sLightGCN [16]: LightGCN is the state-of-the-art GCN based collaborative filtering model, and sLightGCN is a variant. They are described in detail in Section III.

Mult-VAE [47]: Mult-VAE extends variational autoencoders (VAEs) to collaborative filtering and uses a multinomial likelihood for the data distribution. Besides, it introduces an additional regularization parameter for optimization. It can be seen as a special case of self-supervised learning(SSL) for recommendation.

SGL [28]: SGL is the latest baseline for top-k recommendations. It introduces self-supervised learning into the recommendation system based on the contrastive learning framework. It is implemented on LightGCN and uses a multi-task approach that unites the contrastive loss and the BPR loss function. SGL mainly benefits from graph contrastive learning to reinforce user and item representations. Following the paper, the edge drop based SGL achieving the best performance are adopted here.

The performance comparison on the three datasets is shown in Table II. The best results are shown in bold while underlined scores are the second best. We follow the experimental results of SGL [28], except for MF, LR-GCCF, and our methods. After statistical analysis, the standard deviations on Recall and NDCG are not big than $\pm 0.0002$ under different initialization seeds. We have the following observations:

MF is the most basic and simplest method and performs the worst. NGCF, LR-GCCF, and LightGCN are GCN based methods. NGCF achieved improvements relative to MF by introducing the GCN method into top-k recommendations, especially on the Yelp2008 dataset. LR-GCCF, LightGCN and sLightGCN can be seen as improvements of NGCF. Their performances are better than NGCF, and these results are consistent with the performance in the original paper. These three methods show the significant role of graph convolution methods in recommendation systems. LightGCN is the strongest baseline and becomes the basis for subsequent methods, such as SGL and our method. LightGCN removes the nonlinear activation layer and learning parameters making the model more applicable to recommendation systems rather than simply employing GCN which illustrates that the GCN method should be modified to fit the recommendation system.

Mult-VAE and SGL are methods that belong to self-supervised learning (SSL). The results of Mult-VAE are generally better than NGCF, indicating that the variational auto-encoder based method and self-supervised learning is competitive for recommendation. The results of SGL show that it has a clear boost compared with LightGCN and suboptimal results are obtained on two datasets which demonstrates the advancement of contrastive learning methods.

The proposed sLightGCN_MSCL is the best among all methods. LightGCN_MSCL is also listed as a variant of our method, which is also superior to other methods. Compared to the latest and best method SGL, the improvements of sLightGCN_MSCL on Yelp2018, Amazon-Book, and Alibaba-iFashion are 2.37%, 21.34%, 6.67% on Recall, 2.34%, 22.96%, 7.43% on NDCG, respectively. SGL uses CL and BPR jointly in the multi-task learning approach without exploiting the potential of CL. Our approach is simpler and consumes less time which can be seen in Training Efficiency in Section V. This shows the correctness of improving CL.

MSCL combines two components, the different importance of positive and negative samples and the use of multiple positive samples. ICL, MCL denote importance-aware CL and multiple positive samples based CL respectively. They are shown in Equations (8)-(10).

To show the applicability of MSCL, we apply it to many methods and compare it with the BPR loss, and methods with these two loss are named as “*-MSCL”, “*-BPR”.

The results are shown in Table IV, and the percentage of improvements relative to BPR are also presented. MSCL-based methods outperform the BPR-based methods on all results on the three datasets, and have significant improvements on MF, NGCF and LR-GCCF. sLightGCN_MSCL consistently obtains the best results on all datasets and has desirable improvements. In particular, the improvement on the Amazon-Book dataset is still about 25%.

Furthermore, MF is the most fundamental method just based on embeddings in the recommendation field, and many methods can be seen as developments of MF. Theoretically, MSCL is suitable for all embedding based methods. Thus, the effectiveness of MF shows that MSCL can be widely used in recommendation systems. The above experimental results and analysis show that our proposed MSCL is model-agnostic and widely adaptable.

We think that MSCL is more suitable for the Top-k recommendation task. This can be illustrated by theoretical analysis and experimental results.

Theoretically, MSCL is compared with the loss function BPR. The common goal of both BPR and MSCL is to learn better feature representation by comparing between positive and negative samples. BPR uses a limited number of comparisons, usually one or several while MSCL employ thousands. Moreover, MSCL improves the quality of comparison by distinguishing the importance of positive and negative samples and makes better use of the few positive samples. MSCL makes the similarities between positive and negative samples more accurate through more and better comparisons. The top-k recommendation is a ranking task, and BPR is proposed specifically for ranking tasks. MSCL outperforms BPR in terms of theoretical and experimental results. Therefore, MSCL could get better ranking results and makes more sense for top-k recommendations.

Experimentally, the improvement of the evaluation metric NDCG is more obvious. NDCG is a ranking related metric which is more meaningful for ranking and top-k recommendation task compared to Recall. The following two observations support us well: (1) In Table IV, we found that the NDCG boost generally more than Recall. On average, the improvements are 19.98%, 32.95%, 8.64% on Recall while that are 21.34%, 35.61%, 11.86% on NDCG for the three datasets respectively. This shows the superiority of MSCL for top-k recommendation on different methods. (2) Fig. 6 shows the more significant improvement of MSCL over BPR on NDCG compared to Recall with different $K$. This shows that the ranking performance of NDCG is consistently higher than Recall even as $K$ changes.

The performance of MF_MSCL is particularly noteworthy in Table IV.

(1) MF_MSCL gains the most significant improvement, among all MSCL-based versus BPR-based methods. The results are even better than those of all BPR-based methods, sLightGCN_BPR included. It indicates MSCL with the most basic and simple method is significantly better than excellent methods recently proposed, even the state-of-the-art GCN methods. Thus, to some extent, a good loss function works better than new models.

(2) In addition, we find that the results MF_MSCL are also competitive. They are close to or better than that of NGCF_MSCL and LR-GCCF_MSCL on all datasets, and are close to sLightGCN_MSCL on the Alibaba-iFashion dataset. This indicates that MSCL is also effective in directly optimizing embeddings without a complex model, such as GCN.

These observations also indicate that MSCL can achieve competitive results in the simplest baseline, which is also consistent with the latest research Graph-MLP [48]. Graph-MLP indicates that it is sufficient for learning discriminative node representations only by MLP and graph based CL, without the complex GCN. Compared with Graph-MLP, MF_MSCL is more concise and simple. It is based only on embeddings and improved CL functions, which is still effective even without MLP. Graph-MLP does not optimize the CL loss function which is what we do, this shows the great potential of MSCL.

Three other points need to be highlighted. (1) As the most basic and simple method, MF_MSCL can be widely used in various tasks of recommendation systems, not only the top-k tasks. The applicability of MSCL is best illustrated by MF_MSCL. (2) MF_MSCL also has other advantages of MSCL presented in this section, such as being more suitable for top-k recommendation and fast convergence. (3) Importantly, it is valuable for applications with high space and time requirements or industrial applications at a large scale.

The training efficiency of MSCL is also significantly improved as shown in Fig. 7. Because of the large difference in loss values, following LightGCN, SGL, the test performance on three datasets are used to show the convergence speed. In terms of the number of training epochs required to achieve optimal performance, more than 900 epochs are required for BPR, while MSCL achieves the best performance at 46, 3, and 90 epochs on the three datasets respectively. BPR requires too many epochs for convergence while MSCL converges earlier, so we adopt the same number of epoch as MACL for comparison.

For the first two datasets, MSCL converges directly to the high values approximating the final performance with slight fluctuations while BPR converges slowly at lower values. On the third dataset, it is slightly more difficult to converge due to the sparsity of the dataset. MSCL converges sharply by about 5 epochs to the value that approximates the final performance. This all shows that MSCL has a fast convergence capability. The training efficiency is improved at least tens of times on different datasets in terms of train epochs as mentioned before. The main reason for the high training efficiency is that multiple samples are learned at the same time as demonstrated in [38].

Moreover, in terms of actual training time, MSCL does not significantly increase the training time per epoch. Table V shows the average time consumption in each epoch in seconds. MSCL takes less time than BPR when one positive sample is used as shown in the first two columns of the table. Because BPR requires negative sampling while MSCL needs not and the computation with multiple negative samples is accelerated by the GPU. When the number of positive samples increases by 1, the average time increased on the three data sets is 0.8s. Such time consumption is completely negligible. When $M$=5, MSCL and BPR consume the same time, but the performance is much better than BPR. Relative to the latest SGL [28] based on the contrastive learning framework, it takes about 3.7x larger than LightGCN while ours is about 1.5 times of LightGCN.

The above analyses demonstrate that MSCL has remarkable improvement in convergence speed and training efficiency than BPR. And there is no significant increase in time consumption per epoch, which is an advantage over SGL in terms of performance and time consumption.

The proposed MSCL combines ICL and MCL aiming to solve the problem of different importance of positive and negative samples and insufficient use of positive samples. The problems and solutions are also fit for BPR. Therefore, we modify the loss function of the multi-sample based BPR in the same way and present the MSBPR function. In this case, the same sampling method of MSCL is used by MSBPR. The formula of MSBPR is as follows:

where $\sigma$ is the logistic sigmoid. We use $f(u,i)$ instead of $\hat{y}_{ui}$ by drawing on comparative learning because the $\hat{y}_{ui}$ based approach doesn’t work.

The results are shown in Table VI, and the baseline is sLightGCN. With the only exception in all results that the recall of MSBPR is worse than BPR on Amazon-Book, the overall MSBPR-based methods are better than BPR. It shows that our proposed idea can be extended to BPR and other pair-wise based loss functions. In addition, we find that MSCL works better than MSBPR, especially on the Amazon-Book dataset, which shows the superiority of the CL function again and the correctness of improving CL in this paper. Therefore, MSCL is better than BPR and MSBPR in the field of recommendation systems.