Local-Global History-aware Contrastive Learning for Temporal Knowledge Graph Reasoning

Reasoning on KGs has been a hot research topic that infers unknown facts based on known facts. Existing reasoning methods over KGs can be divided into two categories: static KG (SKG) reasoning and temporal KG reasoning.

Contrastive Learning as a self-supervised learning paradigm, due to its impressive performance in distinguishing instances of different categories, has gained significant attention in recent years. The basic idea of contrastive learning [37, 13] is to learn the similarity between different views of the same instances through training, while maximizing the dissimilarity between different views of instances. In self-supervised contrastive learning, the augmented instances are derived by randomly sampling a mini-batch of $N$ instances. The origin instances and the augmented instances are used to optimize the following loss function given a pair of positive instances $(i,j)$. The contrastive loss is expressed as:

where $2N$ is the sum of the number of the original and augmented instances, $\mathbf{x}_{i}$ is the projection embedding of the instance $i$, $\tau$ is a temperature parameter helping the model learn from hard negatives, and $\cdot$ is dot product that is used to compute the similarity of instances between different views. Recently, contrastive learning has been applied to static KG reasoning to alleviate the long-tail data problem but is rarely explored in TKG reasoning. The recent CENET [29] is a single-view historical contrastive learning method designed to enhance the representation of entities with less historical interactions. Different from CENET, we propose a local-global query contrast module to mitigate noise interference and enhance model robustness.

A TKG $\mathcal{G}$ is formalized as a sequence of KG snapshots, i.e., $\mathcal{G}=\{\mathcal{G}_{1},\mathcal{G}_{2},...,\mathcal{G}_{|\mathcal{T}|}\}$. Each KG snapshot $\mathcal{G}_{t}=(\mathcal{E},\mathcal{R},\mathcal{F}_{t})$ actually is a directed multi-relational graph that is a set of valid facts at $t$. A fact (or an event) is denoted as a quadruple ($e_{s},r,e_{o},t$) where the subject entity $e_{s}\in\mathcal{E}$ and the object entity $e_{o}\in\mathcal{E}$ is connected by a relation $r\in\mathcal{R}$ at time $t\in\mathcal{T}$. The primary notations and their meanings used in this paper are described in Table I.

TKG extrapolation task involves forecasting a missing object entity (or subject entity ) given a query $(e_{q},r_{q},?,t_{q})$ (or $(?,r_{q},e_{q},t_{q})$) according to previous historical KG snapshots $\{\mathcal{G}_{0},\mathcal{G}_{1},...,\mathcal{G}_{t_{q}-1}\}$. Without loss of generality, the inverse relation quadruples $(e_{o},r^{-1},e_{s},t)$ are added to the TKG dataset. So the TKG extrapolation task can be reduced to object entities predictions.

As shown in Fig. 3, the overall framework of LogCL is composed of three components: global entity-aware attention encoder, local entity-aware recurrent encoder and local-global query contrast module. Local entity-aware attention recurrent encoder adopts an entity-aware attention mechanism to capture the necessary information related to queries for prediction. If facts in KG snapshots at the latest timestamps contain semantic information relevant to the queries, these facts will help to improve the probability of entity prediction. Global entity-aware attention encoder takes into account the global historical facts related to queries and avoids overlooking important historical facts that do not appear in the recent local KG snapshots. Local-global query contrast module is introduced to guide a better integration of global and local historical information and to enhance the robustness of LogCL. Finally, LogCL combines local and global historical embedding representations for entity prediction.

As discussed in the introduction, each KG snapshot at the latest $m$ timestamp (i.e. $\{\mathcal{G}_{t_{q}-m+1},...,\mathcal{G}_{t_{q}-1}\}$) doesn’t provide equally contributions for predicting the query $q=(e_{t_{q}},r_{t_{q}},?,t_{q})$. The importance of KG snapshot historical information relevant to the query needs to be modeled to further optimize the evolution of entity and relation in adjacent time. To this end, we propose a local entity-aware attention recurrent encoder that efficiently captures historical information in KG snapshots relevant to the query. The local entity-aware attention recurrent encoder involves a KG snapshot aggregation pipeline and a KG snapshot sequences evolution pipeline. Specifically, the aggregation of the KG snapshot is to learn the spatial semantic structure of concurrent facts by the Graph Convolution Network (GCN) from the spatial view, while the evolution of KG snapshot sequences is to capture sequential dependencies of entities and relations through the recurrent mechanism from the temporal view. Note that, the inputs to entities and relations on the first timestamp are randomly initialized.

The global entity-aware attention encoder aims to capture the longer historical facts that are not considered in the local KG snapshot sequence. For each query $(e_{q},r_{q},?,t_{q})$, unlike CyGNet [8] and TiRGN[11] that only use one-hop target object entities associated with queries to learn the repetition of facts in further history, we consider candidate multi-hop historical facts that provide more semantic information of historical facts to help entities prediction. For example, meetings that are held periodically are preceded by different hosting processes, and these different hosting processes can be of great help to future meetings. Therefore, we construct the historical query subgraph by sampling the historical facts relevant to queries to obtain rich semantic information.

For a TKG $\mathcal{G}_{<t_{q}}=\{\mathcal{G}_{1},\mathcal{G}_{2},...,\mathcal{G}_{t_{q}-1}\}$, we first sample the one-hop historical facts $\mathcal{G}^{\prime}_{g_{1}}$ containing the query subject entity $s$ in the given query. Next, we extract the one-hop target object entity associated with the query entity-relation pair, and then proceed to sample the one-hop facts $\mathcal{G}^{\prime}_{g_{2}}$ containing the one-hop target object entity. Finally, we integrate the two collected historical fact sets to obtain the most relevant historical subgraph to the query, $\mathcal{G}^{\prime}_{g}=\mathcal{G}^{\prime}_{g_{1}}\cup\mathcal{G}^{\prime}_{g_{2}}$. It is worth noting that, the historical query subgraph is a static KG that does not involve temporal information and can change along the query time.

After obtaining the historical query subgraph, we update the global entity representation by modeling the structural semantic representation of the historical query subgraph. To encode the historical query subgraph that is formed by sampling two-hop neighbors, we adopt another R-GCN to perform message aggregation. Since the historical query subgraph does not consider temporal information, we directly use the randomly initialized embedding of entities and relations as input to R-GCN. The specific formula is expressed as follows:

where $\mathbf{h}_{g}^{l+1}$ denotes the output embedding of entities in the $l$-th layers of the R-GCN in the historical query subgraph. For simplicity, the final layer of the output of R-GCN is denoted as $\mathbf{H}_{g}^{Agg}$. For global historical information, we also use the entity-aware attention mechanism to learn the historical facts related to queries. The specific formula is expressed as follows:

where the $\beta$ is the attention score, $W_{6}$ is the learnable weight matrice, and $\vec{\mathbf{h}}_{g}^{e_{q}}$ denotes the final global representation of entities through global entity-aware attention encoder.

Clearly, the local and global encoders defined above well capture the local and global dependency for queries. However, during the actual training process, the input data is disturbed by extraneous noise, leading to a sharp decline in reasoning performance. Existing methods cannot address this challenge well. Inspired by the unsupervised contrastive learning [13, 29], we propose a local-global query contrast module that identifies highly correlated local and global features of entities and relations at the query level, enabling noise filtering. Specifically, for each query at timestamp $t_{q}$, The purpose of the local-global query contrast module is to learn the local and global contrastive representations of queries by minimizing the supervised contrastive loss, which makes the local and global representations of the same query to be as close as possible in the semantic space, while different queries are separated. Formally, Formally, the query embedding representations of local and global are obtained by using the concatenation operation:

where $\mathbf{z}_{t}$ is the representation of the local query at timestamp $t$, $\mathbf{z}_{g}$ is the representation of the global query, and $MLP$ is used to normalize and project the embedding of queries onto the unit sphere for further contrastive training. It is worth noting that since the historical query subgraph does not consider temporal information, the origin relation embeddings are used to encode the global embedding representation of queries.

In our work, the global and local query representations can be considered as the two-view representations derived from encoding historical facts in TKGs. Thus, if taking the query representation generated by the local encoder as the anchor, the global encoder can be viewed as an augmented encoding process. The local and global representations of the same query are used as positive pairs at timestamp $t$, i.e.,($\mathbf{z}_{t,i}$, $\mathbf{z}_{g,i}$), the local and global of representation of different queries are used as negative pairs i.e.,($\mathbf{z}_{t,i}$, $\mathbf{z}_{g,k}$). Formally, the computation of the supervised contrastive loss $\mathcal{L}_{lg}$ at timestamp $t_{q}$ is as follows:

where $Q_{t_{q}}$ and $N_{t_{q}}$ denote the minibatch that is the set of queries and the number of queries at timestamp $t_{q}$, separately; $\tau$ is the temperature parameter. The objective of $\mathcal{L}_{lg}$ is to make the representations of the same category closer, enhancing the common essential features of the local and global encoders required for predicting the query, thus alleviating the influence of noise and improving the robustness of the model. Similarly, the supervised loss $\mathcal{L}_{gl}$ can be obtained by using the representation generated by the global encoder as an anchor. To make the LogCL further distinguish different semantic representations of queries in the semantic space, we adopt Eq. 17 to constrain the local and global query representations and obtain two supervised losses $\mathcal{L}_{ll}$ and $\mathcal{L}_{gg}$. In this way, we can obtain the four supervised contrastive losses: $\mathcal{L}_{lg}$, $\mathcal{L}_{gl}$, $\mathcal{L}_{ll}$ and $\mathcal{L}_{gg}$. Thus, the final supervised contrastive loss is computed as $\mathcal{L}_{cl}=(\mathcal{L}_{lg}+\mathcal{L}_{gl}+\mathcal{L}_{ll}+\mathcal{L}_{gg})/4$.

ConvTransE[22] is a strong score function that is widely adopted for the recent TKG reasoning task. In this work, we also adopt ConvTransE to perform the entity prediction task at timestamp $t_{q}$. Therefore, the entity prediction score of ConvTransE is as follows:

where $\lambda$ is a variable factor that is set at [0,1], which is used to trade off global and local representations of entities. In this paper, the entity prediction task can be considered as a multi-label learning problem. Thus, the loss of entity prediction $\mathcal{L}_{tkg}$ is formalized as:

where $\phi\left(e_{s},r,e,t\right)$ is the entity prediction probabilistic scores. $\mathbf{y}_{t}^{e}\in\mathbb{R}^{|\mathcal{E}|}$ is the label of which the element is 1 if the fact occurs, otherwise 0. The final loss function is computed as:

In the work, the contrastive supervised loss $\mathcal{L}_{cl}$ and the entity prediction of loss $\mathcal{L}_{tkg}$ are trained simultaneously.

Note that, since reverse quadruples are added in the training or testing process, the inverse query sets are generated by the original query sets at timestamp $t_{q}$. During each epoch, the KG snapshots in history are built using the combination of the original quadruples and the inverse quadruples. If the combination sets are directly used for training, the entity-aware attention will perceive the target subject entity and target object entity, resulting in data leakage. To avoid such potential data leakage, we present a two-phase forward propagation strategy. In detail, the original query set is first considered in the forward propagation, and then the second forward propagation is performed on the query inverse sets during each epoch. In this way, the prediction scores and loss can be obtained for training or testing from the two-phase forward propagation. The detailed training procedure of LogCL is summarized in Algorithm 1.

We analyze the computational complexity of our proposed LogCL model. For the local entity-aware attention recurrent encoder, the time complexities of entity and relation evolution are $O(m(|\mathcal{E}|+P))$ and $O(m|\mathcal{R}|)$, respectively, where $P$ is the complexity of entity attention mechanism at $t$ timestamp. For the global entity-aware encoder, the time complexity of generating the historical query subgraph is $O(2|\mathcal{T}||\mathcal{F}_{t}|)$, the time complexity of historical query subgraph aggregation is $O(|\mathcal{E}|+P)$. The time complexity of the local-global query contrast module is $O(4mC)$. Thus, the time complexity of LogCL is $O(m(|\mathcal{E}+\mathcal{R}+P|)+|\mathcal{E}|+P+2|\mathcal{T}||\mathcal{F}_{t}|+4mC)$.

The overall experimental results of LogCL and baselines on four benchmark datasets are reported in Table III. The best results are marked in bold, and the second-best results are reported using underlining. From the experimental results, we have the following observation:

• •

  The performance of LogCL consistently is better than the state-of-the-art methods on four benchmark datasets. Specifically, when comparing the second-best results, LogCL achieves the most significant improvements of 5.2%, 4.9%, 7.9%, and 7.9% in MRR on ICEWS14, ICEWS18, ICEWS05-15 and GDELT datasets.

• •

  For the state-of-the-art models Hismatch and TIRGN, although HisMatch also considers the local historical information related to the query, it fails to effectively consider the importance of the local historical fact information related to the query and does not consider the global historical information, resulting in weaker performance than LogCL. TiRGN takes into account both global and local historical information, but ignoring query-related historical information leads to lower performance than Hismath. Although CENET considers historical contrastive learning with our model LogCL, its performance is lower than LogCL due to the lack of evolutionary modeling of facts. In addition, CENET uses historical contrastive learning to enhance entities with less historical interactions by contrast, while we use contrastive learning from a local and global view to improve the robustness of the model.

• •

  An interesting phenomenon on all models is that the results on ICEWS14 and ICEWS18 datasets perform better than GDELT and ICEWS05-15 datasets. The reason for such phenomenon may be that more complex dynamic interactions among entities and relations exist in ICEWS18 and GDELT datasets. Such complex dynamic facts association are difficult to be modeled and lead to the worse results on ICEWS18 and GDELT datasets. The excellent performance exhibited by LogCL on ICEWS18 and GDELT datasets demonstrates the ability of LogCL to model complex temporal facts interactions.

• •

  Extrapolation models on all datasets perform better than interpolation and static models, mainly because static models do not consider temporal information and are difficult to capture the dynamic changes of entities and relations. For interpolation models, they only focus on the completion of historical missing facts and lack the ability to model the evolution of entities and relationships over time and predict future unseen facts.

To further better analyze each part of LogCL that contributes to the prediction results, we conduct ablation studies on all datasets in terms of MRR and Hits@1/5/10. The results of the LogCL variants are presented in Table IV.

In this section, we conduct experiments on ICEWS14 and 18 datasets to further analyze the impact of parameters in LogCL, including the parameter $\lambda$ in the prediction part, the temperature coefficient $\tau$ in the local-global query contrast module, and GNN aggregation encoder.

To further study the impact of different kinds of GNNs in the local entity-aware attention recurrent encoder and the global entity-aware attention encoder, R-GCN is replaced in these two encoders with CompGCN[24] and KBGAT[25]. The MRR and Hits@1 results of ICEWS14, ICEWS18, and ICEWS05-15 are reported in Table V. It can be seen that LogCL (RGCN) gets the best results on ICEWS05-15 dataset and shows strong performance in ICEWS14 and ICEWS18 datasets.

To investigate the role of the two-phase propagation in our LogCL, we manage experiments to evaluate the results of two forward propagations separately. Log-$\mathcal{FP}$ denotes a variant that predicts the object entities on the origin query set and only performs the first phase propagation. Log-$\mathcal{SP}$ that predicts the subjected entities on the reverse query set and only considers the second phase propagation. The results are shown in Table VII. It can be seen that the performance of Log-$\mathcal{FP}$ is superior to the Log-$\mathcal{SP}$. The reason behind this phenomenon is that the dataset composed of inverse relations may introduce a certain bias, which subsequently impacts the performance of the model. In contrast, the original dataset provides a more comprehensive reflection of the real relations between entities, resulting in better performance in the experimental results.

Since the evolutionary pattern changes with emerging facts, exploring how to adjust the model to emerging facts is crucial. Following the CEN [33] and RETIA [45], we perform a batch of experiments to learn emerging facts on ICEWS14, ICEWS18 and ICEWS05-15 datasets under the online setting. The results are shown in the Fig.10. The results of CEN, RETIA and LogCL under the online setting outperform the results in Table III under the offline setting. It is because that historical facts in the test set are updated under the online setting. In addition, LogCL achieves greater improvements than the CEN and RETIA under the online setting. This proves that LogCL can effectively solve domain obstacles such as time-varying problems.

We provide the case study for LogCL, LogCL-w/o-eatt, and LogCL-w/o-cl on ICEWS14 dataset. Two queries with the top-5 prediction entities and scores are reported in Table VI.

For the query (China, Sign_formal_agreement, ?, 2014-12-4), we can observe that LogCL and logCL-w/o-cl can predict the correct answer, while the top-5 results given by LogCL-w/o-eatt do not contain the correct answer. For the query (Iran, Engage_in_diplomatic_cooperation, ?, 2014-11-30), although LogCL, LogCL-w/o-eatt, and LogCL-w/o-eatt can give answers prepared and the answers are top-1, the prediction scores of LogCl and LogCL-w/o-eatt for correct answers are much higher than that of LogCL-w/o-eatt. The reason for the above phenomenon is that the reasoning of LogCL-w/o-eatt primarily relies on historical facts that are in close proximity in time and weakly associated with the query. As a result, LogCL-w/o-eatt fails to capture the historical facts that have a strong relevance to the query, leading to lower accurate prediction results. These two cases can further prove the effectiveness of the entity-aware attention mechanism and the strong reasoning ability of LogCL.