DsMtGCN: A Direction-sensitive Multi-task framework for Knowledge Graph Completion

Link prediction or knowledge graph completion aims to predict missing triples by scoring candidate triples with embeddings of entities and relations leaned by well-designed models. Some non-neural approaches were first proposed in this field, they usually take relations as transfer vectors to convert head entities to tail entities, different models differ in score functions and representation spaces. For example, TransE[6], TransH[7], and TransR[8] are known as typical additive models, since they project entities and relations to vector space, then optimize the distance-based score function with the goal of making the head entity added by relation as close as possible to the tail entity. On the other hand, multiplicative models use semantic similarity-based score function to measure the plausibility of given triplets, enabling them to maintain good expression ability when the scale of the knowledge graph grows. Specifically, RESCAL[9] and DistMult[10] introduce bilinear operation in point-wise space, while Complex[11] and QuatE[12] represent embeddings in complex-valued space and four dimensional space respectively to capture complicated patterns like composition and antisymmetry of relations.

With the wide application of neural networks in various fields, several neural network based models have been proposed to improve expression ability by capturing more complex interactions among triples. Considering that the simplicity and expressiveness of Multi-Layer Perception (MLP) are suitable for enhancing score functions, NTN[23] and SME[13] explore nonlinear modeling with a fully-connected layer and activation function, while NAM[14] adopts deeper hidden layers and utilizes information from the hidden encoding. Convolutional Neural Networks (CNN) have also been employed to improve expression ability by learning deeper features with fewer shared parameters. For instance, ConvE[15] applies 2D convolution over a two-dimensional matrix reshaped from stacked head entities and relations to model the interactions among them, Conv-KB[16] adopts 1D filters on the same dimension of entities and relations directly to keep better transitional characteristic, ConvR[17] replaces global filters used by ConvE with relation-specific ones, InteractE[18] proposes several skills including feature permutation, checkered reshaping as well as circular convolution to maximize the number of heterogeneous and homogeneous interactions between entity and relation features. The information used by most methods is limited to the triple level as mentioned above, in order to make better decisions, RSN[24] tries to capture long-term path dependence with Recurrent Neural Networks (RNN), while transformer-based models like CoKE[25] and KG-BERT[26] choose to mine contextual information from knowledge graphs.

Although knowledge graphs are fundamentally multi-relational graphs, the aforementioned methods, which are more suitable for Euclidean data, tend to overlook the structural information inherent in them. To make up for it, recent works introduce Graph Neural Networks (GNN) for graph context modeling under the encoder-decoder framework. R-GCN[19] proposes relation specific filters to take different relations into account while modeling the message passing on knowledge graphs. SACN[20] learns adaptive weights of adjacent nodes with same relation instead of taking the neighborhood of each entity equally, so that node structure, node attributes and relation types can contribute to the performance gains. VR-GCN[21] explicitly learns the representation of entities and relations to make full use of relation information. CompGCN[22] introduces entity-relation composition operations while aggregating neighbors, which is proven to generalize previous GCN-based models including KipfGCN[27], R-GCN[19], D-GCN[28] and SACN[20]. KBGAT[29] claims to surpass others by learning graph attention based embeddings, but its results have been questioned by another study[30]. Nevertheless, none of above studies suppose that neighbors in different directions should be specifically focused under different sub-tasks, which inspires our work.

Multi-task Learning (MTL) is a specially designed network structure to leverage information across tasks instead of learning each task in isolation [31]. Benefiting from its simplicity and efficiency, MTL has contributed to computer vision [32, 33], natural language processing [34], recommendation system [35] and so on.

In the field of link prediction, existing MTL models mainly focus on introducing additional tasks or datasets. Specifically, SENN [36] utilizes relation prediction to promote link prediction, TransMTL [37] learns embeddings of multiple knowledge graphs simultaneously to mine shared structure patterns among them. However, the aforementioned methods bring much extra computational overhead, and it’s not easy to find suitable sub-tasks or external datasets. In this paper, we define sub-tasks by decomposing existing tasks, which reduces the number of parameters and avoids introducing extra noise.

We denote KGs as $\mathcal{G}=\{(h,r,t)\mid(h,r,t)\in\mathcal{T},h,t\in\mathcal{E},r\in\mathcal{R}\}$, where $h,t$ are head and tail entity, $r$ represents the relation connecting $h$ and $t$, $\mathcal{T,E,R}$ denote the set of triples, entities, and relations. The train, valid and test set of triples are denoted as $\mathcal{T}^{train}$, $\mathcal{T}^{valid}$ and $\mathcal{T}^{test}$ respectively.

Given $(h,r)$, the link prediction task aims to predict missing $t$ on $\mathcal{G}$. Let $e_{h},e_{r},e_{t}\in R^{d}$ be corresponding embeddings with dimension $d$, a score function defined as $\psi:{R}^{d}\times{R}^{d}\times{R}^{d}\rightarrow{R}$ takes $e_{h},e_{r},e_{t}$ as input and outputs a score measuring the plausibility of triplet $(h,r,t)$, then $t$ with the highest score is selected as the prediction result.

Following [38], the information in KGs is required to flow in backward and self-loop directions besides forward direction. For triple $(h,r,t)$, we generate its backward version $(t,r^{-1},h)$ and self-loop version $(h,r^{0},h)$ to augment original data. After that, the new relation set is $\mathcal{R}^{\prime}=\mathcal{R}\cup\mathcal{R}^{-1}\cup\{r^{0}\}$, where $\mathcal{R}^{-1}=\{r^{-1}\mid r\in\mathcal{R}\}$; the new triple set is $\mathcal{T}^{\prime}=\mathcal{T}\cup\mathcal{T}^{-1}\cup\mathcal{T}^{0}$, where $\mathcal{T}^{-1}=\{(t,r^{-1},h)\mid(h,r,t)\in\mathcal{T}\}$, $\mathcal{T}^{0}=\{(h,r^{0},h)\mid h\in\mathcal{E}\}$, the prediction tasks can be divided into forward sub-tasks like $(h,r,?)$ and backward sub-tasks like $(h,r^{-1},?)$.

For one entity $h$, its neighbors $\mathcal{N}_{h}$ can be grouped into forward neighbors $\mathcal{N}^{F}_{h}$ and backward neighbors $\mathcal{N}^{B}_{h}$, where ${N}^{F}_{h}=\{(r,t)\mid(h,r,t)\in\mathcal{T}\}$, ${N}^{B}_{h}=\{(r^{-1},t)\mid(h,r^{-1},t)\in\mathcal{T}^{-1}\}$, and they are considered to contribute differently to the two sub-tasks. Furthermore, the embedding representation learned from $\mathcal{N}^{F}_{h}$ and $\mathcal{N}^{B}_{h}$ are denoted as $e^{f}_{h}$ and $e^{b}_{h}$ respectively, the set of $e^{f}_{h}$ and $e^{b}_{h}$ can be rewritten in the matrix form as $E^{f},E^{b}\in R^{|\mathcal{E}|\times d}$.

Inspired by CompGCN, we use entity-relation composition operation $\phi$ and message passing function $M$ to model the information flow from neighbors in different directions.

Specifically, the features of entity $h$ and relation $r$ are initialized as $v_{h},v_{r}\in R^{d_{i}}$. Given one known triple $(h,r,t)$, the operation $\phi$ is applied to get the distributed representation of $(r,t)$, which is implemented in two ways including Multiplication (Mult) [10] and Circular-correlation (Corr) [39] defined as follows, where $\circ$ denotes Hadamard product and $\star$ means compressing the tensor product.

The representations are then aggregated by $M$, where $W^{F},W^{B},$ $W^{L}\in R^{d_{i}\times d}$ are direction-specific weights.

In essence, $e^{f}_{h}$ and $e^{b}_{h}$ can be regarded as two views of $h$, which are expected to satisfy the principle of consensus and difference. Specifically, the consensus principle means $e^{f}_{h}$ and $e^{b}_{h}$ should be close to each other appropriately considering that they share similar semantics, the difference principle claims that embeddings of different entities from the same view should be distinguished as a whole.

For keeping the two principles, we impose distance constrains $dis(E^{f},E^{b})$ to bring them closer, the conicity geometric constrains $con(E^{f})$ and $con(E^{b})$ are used to increase the distinguishability of embeddings between different entities. The conicity term is originally defined as a geometric attribute [40], while it is regarded as an explicit constraint here. The formula of $dis$ and $con$ are shown as follows, where $E^{f}_{i}$ stands for the $i$-th row of $E^{f}$, $\|x-y\|_{2}$ and $cos(x,y)$ denote the Euclidean distance and cosine similarity.

We illustrate the effect of constraints in Fig. 2(b), the conicity constraints make it easier to distinguish points on the same plane, while the distance constraint can help to align points of the same color on different planes, the combination of them can achieve the superb result.

Considering the specific requirements for neighbor information in different directions under forward and backward sub-tasks, the multi-head self-attention mechanism [41] at entity level is applied.

For simplicity, we first expound how to calculate $e^{F}_{h}$ for the forward sub-tasks with single head. Suppose the dimension of attention vectors is $d_{a}$, given stacked embedding $E_{h}=[e_{h}^{f};e_{h}^{l};e_{h}^{b}]\in R^{3\times d}$, we obtain the query vector $Q_{h}^{F}\in R^{3\times d_{a}}$ and the key vector $K_{h}^{F}\in R^{3\times d_{a}}$ as follows, where $W^{F}_{Q},W^{F}_{K}\in R^{d\times d_{a}}$ are task-specific weights.

Then the scaled dot-product result $S^{F}\in R^{3\times 3}$ can be formulated as:

The final attention $A^{F}\in R^{3}$ can be obtained after softmax and mean-pooling ($mp$), the $mp$ operator aims to save mean value of each column of the matrix to be processed, note that the sum of $A^{F}$ equals 1, the $m$-th element is denoted as $A^{F}_{m}$:

To get richer latent information and enhance the stability of results, we can use more heads to compute $A^{F}$. If the number of heads is $n_{h}$, the updated formula is written down below, where ${Q_{h}^{F_{i}}}=E_{h}{W_{Q}^{F_{i}}}$, ${K_{h}^{F_{i}}}=E_{h}{W^{F_{i}}_{K}}$:

In the end, $e_{h}^{F}\in R^{d}$ is the weighted sum of $e_{h}^{f},e_{h}^{l},e_{h}^{b}$ as follows, and $e_{h}^{B}$ can be calculated with similar process as shown in Fig. 2(c).

In this work, we use ConvE to model the interaction between entities and relations. Specifically, the score function $\psi$ is defined as follows, the convolution operation is applied on the 2D reshaping of $e_{h},e_{r}$ with filter $\omega$, and $g$ denotes the nonlinear activation function.

Taking into account the different sub-tasks, one triple is scored as follows, where $\sigma$ is the sigmoid function mapping score to the interval of $(0,1)$, $\alpha$ is the indicator variable, whose value is 1 if current prediction task belongs to the forward prediction sub-tasks otherwise 0.

For most previous works, the target of training is to minimize such standard binary cross-entropy loss for $i$-th triple, where $s_{i}$ is the predicted score, the label $t_{i}$ indicates the existence or non-existence of the triple, and $l$ controls the degree of label smoothing.

To model the triple uncertainty, we propose a new label smoothing mechanism as follows to prevent the model from giving too low scores to potential targets, where $u_{i}=|\{t|(h_{i},r_{i},t)\in\mathcal{T}^{train}\}|$ ranging from 1 to $|\mathcal{E}|$ is the measurement of uncertainty of $i$-th triple, and $k$ controls the scale of adjustment:

The final loss function $\mathcal{L}$ is defined as follows, where $\lambda_{1}$ and $\lambda_{2}$ are trade-off parameters:

We compare our DsMtGCN model with various baselines, the results are summarized in Table 2, it can be observed that with the model structure becomes more complex, the overall effect improves, CNN based models surpass non-neural models, and GNN based models achieve better performance with the leverage of graph structures. Compared with all of the above models, DsMtGCN outperforms them under all metrics on FB15k-237 and 2 out of 4 metrics on WN18RR, indicating the strong competitiveness of our model.

Compared with other GCN models like R-GCN, SACN and CompGCN, the improvement of our model can be attributed to the well designed multi-task framework, on the other hand, the superior performance over previous MTL models can be explained with the introduce of divided sub-tasks and structure information aggregated from neighbors, which demonstrates the effectiveness of our model.

We can also find several differences between different metrics and datasets from the above results. In particular, MR is not a proper metric as it always fails to keep pace with other metrics, a smaller MR doesn’t means larger MRR and Hits@k, as the simple averaging it adopts will inevitably be affected by extreme values. Besides, the performance of overly complex models on WN18RR is limited[30], as it only contains 11 relations, the dataset is too simple to avoid over-fitting. In view of the complexity of knowledge graph in reality as well as the fairness of comparison, our follow-up experiments will pay more attention to the performance of metrics other than MR on FB15k-237.

To further investigate the performance of DsMtGCN under different sub-tasks and relation types, we present the results of our model compared with ConvE [15], InteractE [18], SACN [20] and CompGCN [22] on FB15k-237 in Table 3. Following previous work, the relations are divided into four categories including one-to-one (1-1), one-to-many (1-N), many-to-one (N-1) and many-to-many (N-N) by counting the average number of tails per head and heads per tail [7]. It can be observed that both CompGCN and DsMtGCN outperform other models with significant improvements on all four relation types, which can be attributed to the full exploitation of graph structure information. When it comes to DsMtGCN, it gets the highest MRR for all relation types especially complex ones like 1-N, N-1 and N-N, which indicates that aggregating the information from neighbors in different directions specifically can help to handle complex relations under both forward and backward sub-tasks.

Since our model has outperformed other competitors in the aforementioned comparative experiments, to further probe into the contribution of each module to the final performance, we conduct ablation experiments of DsMtGCN on FB15k-237. Considering that the main innovations in this work can be divided into three modules including geometric constraints, multi-head fine-grained self-attention mechanism and label smoothing with triple uncertainty, we abbreviate them as GC, MHSA and TU respectively for the convenience of explanation. In order to analyze the role of each module, we remove them from the original model in turn to get following sub-models: DsMtGCN w/o GC, DsMtGCN w/o MHSA and DsMtGCN w/o TU, where w/o means without specific module or mechanism. Specifically, we set $\lambda_{1}$ and $\lambda_{2}$ to 0 to obtain DsMtGCN w/o GC, we replace $A^{F}$ with the vector filled with 1 to obtain DsMtGCN w/o MHSA, and $k$ is initialized to 0 to get DsMtGCN w/o TU, these sub-models are trained under the same settings, the results are shown in Table 4.

It’s obvious that the removal of MHSA brings the greatest decrease in effect, as it enables our model to make full use of structural information from neighbors in different directions at entity level, which is the core of our direction-sensitive multi-task framework. The effect of GC is evident as well, because it can help to improve the geometric distribution of embeddings. Lastly, the introduction of TU proves to be indispensable for further improvements by considering the number of possible tail entities.

In order to show the effect of GC module in a more explicit way, we visualize the cosine similarity and geometric distribution of $E^{f}$ and $E^{b}$ in Fig.3, the mean value of cosine similarity (i.e., conicity) is marked by bold dotted line, and the scatter graph shows the distribution of embeddings reduced to two dimensions by principal component analysis (PCA). From Fig.3(a)(b)(d)(e), we can find that curves with GC is more gentle and the center is more left, which means that better diversity is preserved among embeddings. Besides, The points in Fig.3(c) are more evenly distributed than those in Fig.3(f), and the dots with different colors can cover each other more completely in Fig.3(c), indicating the validity of GC module.

To further verify the effectiveness of MHSA module, we replace it with multi-head adaptive attention mechanism (MHAA) and multi-head parameter-generated attention mechanism (MH-PA). In particular, MHAA implements attention weights with randomly initialized learnable vectors, MHPA applies linear transformation from the embedding matrix to attention weights, the attention distribution generated by above models and corresponding prediction effects are shown in Fig. 4, r/w means to replace MHSA with above substitutes. It’s clear that while dealing with different sub-tasks, the preference for neighbor information in different directions changes evidently, the backward neighbors are paid more attention under the forward sub-tasks, while the opposite happens for backward sub-tasks. Besides, there exists intersections between curves, which means that even for the same sub-tasks, different entities have different preferences, demonstrating the importance of adopting attention at entity level. Ultimately, the prediction effect in the graph increases gradually from left to right, and the overlapping parts between different curves become smaller, indicating that the MHSA module can better allocate attention weights according to different sub-tasks and entities.

We employ case study to further analyze why DsMtGCN works, several examples are shown in Table 5, where tasks and corresponding targets come from $\mathcal{T}^{test}$, forward and backward neighbors are taken from $\mathcal{T}^{train}$. Generally speaking, backward neighbors can provide more information to forward sub-tasks, and forward neighbors are more beneficial to backward sub-tasks, because they can provide more complementary information in most cases. For example, while dealing with (Who Framed Roger Rabbit, genre, ?), the neighbors in same direction like (genre, Parody), (genre, Buddy film) are unable to provide additional information about Fantasy as they are at the same level, while (distributor^-1,Disney) gives a strong signal; when the query is (Charles S.Dutton, student^-1, ?), knowing Charles S.Dutton is a student of Yale University is helpless, but his deep connection with Baltimore can remind us of the Towson University in Baltimore. On the other hand, this rule is not true for queries about abstract concepts like (Soul music, artists, ?) or (Earth, area^-1, ?), as the opposite direction usually means abstract hierarchy, while the answer is concrete.