# On Training Knowledge Graph Embedding Models

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## Abstract

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## 1. Introduction

- We provide a comprehensive analysis of training loss functions as used in several state-of-the-art KGE models in Section 3. We also preform an empirical evaluation of different KGE models with different loss functions and we show the effect of these losses on the KGE models predictive accuracy.
- We study negative sampling strategies and we examine their effects on the accuracy and scalability of KGE models.
- We study the effects of changes in the different hyperparameters and their effects on the accuracy and scalability of KGE models during the training process.

## 2. Background

#### 2.1. Loss Functions in Learning to Rank

- Pointwise approach. The loss function is defined in terms of the difference between the element’s predicted score and its actual label value. The formula is as follows:$$\underset{\mathrm{pointwise}}{\mathcal{L}}(f;X,l)=\sum _{i=1}^{n}\varphi (f\left({x}_{i}\right)-l\left({x}_{i}\right)),$$
- Pairwise approach. The loss is defined as the summation of the differences between the predicted score of an element and the scores of other elements with a smaller labels’ value. The formula is as follows:$$\underset{\mathrm{pairwise}}{\mathcal{L}}(f;X,l)=\sum _{i=1}^{n-1}\sum _{j=1,l\left({x}_{j}\right)<l\left({x}_{i}\right)}^{n}\varphi (f\left({x}_{i}\right)-f\left({x}_{j}\right)),$$
- Listwise approach. The loss is defined as a comparison between the rank permutation probabilities of model scores and values of actual labels [21]. Let $\varphi \left(x\right)$ be an increasing and strictly positive function. We define the probability of an object being ranked on the top (a.k.a. top one probability), given the scores of all the objects as:$$P(f,{x}_{i})=\frac{\varphi \left(f\left({x}_{i}\right)\right)}{{\sum}_{j=i}^{n}\varphi \left(f\left({x}_{j}\right)\right)},$$$$\underset{\mathrm{listwise}}{\mathcal{L}}(f;X,l)=\sum _{i=1}^{n}{\mathcal{L}}_{m}(P(f,{x}_{i}),P(l,{x}_{i})),$$

#### 2.2. Knowledge Graph Embedding Process

#### 2.2.1. Negative Sampling

#### 2.2.2. Embedding Interactions

- Distance-based embeddings’ interactions

- Factorisation-based embedding interactions

#### 2.3. Ranking Evaluation Metrics

- Mean Average Precision (MAP). MAP is a ranking measure that evaluates the quality of a rank depending on the whole rank of its true (relevant) elements. First, we need to define Precision at position k (denoted as $P@k$):$$P@k(q,l)=\frac{{\sum}_{i\le k}I(l,{x}_{i})}{k},$$The Average Precision (AP) is defined by:$$AP(q,l)=\frac{{\sum}_{i=1}^{n}P@k(q,l)\xb7I(l,{x}_{i})}{{n}_{1}},$$$$\mathrm{MAP}(Q,l)=\frac{{\sum}_{i=1}^{n}AP(q,l)}{n}.$$
- Mean Reciprocal Rank (MRR). The Reciprocal Rank (RR) is a statistical measure used to evaluate the response of ranking models depending on the rank of the first correct answer. The MRR is the average of the reciprocal ranks of results for different queries in Q. Formally, MRR is defined as:$$\mathrm{MRR}=\frac{1}{n}\sum _{i=1}^{n}\frac{1}{\mathcal{R}({x}_{i},f)},$$
- Hits@k. This metric represents the number of correct elements predicted among the top-k elements in a rank, where we use Hits@1, Hits@3 and Hits@10. This metric indicates that the model’s probability of ranking a relevant (true) fact in the top-k element scores in the rank.

#### 2.4. Experimental Evaluation

- ●
- Benchmarking Datasets. In our experiments, we use five knowledge graph benchmarking datasets:
- PSE: a poly-pharmacy side-effects dataset [33] containing facts about drug combinations and their related side-effects. The dataset was introduced by Zitnik et al. [33] to study modelling poly-pharmacy side-effects using knowledge graph embedding models. Since the dataset is significantly larger than the available standard benchmark we use it to study the effects of hyperparameters and accuracy of the knowledge graph embedding models.

- ●
- Evaluation Protocol. We evaluate the KGE models using a unified protocol that assesses their performance in the task of link prediction. Let X be the set of facts, i.e., triples; ${\Theta}_{E}$ be the embeddings of entities E, and ${\Theta}_{R}$ be the embeddings of relations R. The KGE evaluation protocol works in three steps:
- (1)
- Corruption: For each $x=(s,p,o)\in X$, x is corrupted $2\left|E\right|-1$ times by replacing its subject and object entities with all the other entities in E. The corrupted triples can be defined as:$${x}_{\mathrm{corr}}=\bigcup _{{s}^{\prime}\in E}({s}^{\prime},p,o)\cup \bigcup _{{o}^{\prime}\in E}(s,p,{o}^{\prime})$$
- (2)
- Scoring: Both original triples and corrupted instances are evaluated using a model-dependent scoring function. This process involves looking up embeddings of entities and relations, and computing scores depending on these embeddings.
- (3)
- Evaluation: Each triple and its corresponding corruption triples are evaluated using the RR ranking metric as a separate query, where the original triples represent true objects and their corruptions false ones. It is possible that corruptions of triples may contain positive instances that exist among training or validation triples. In our experiments, we alleviate this problem by filtering out positive instances in the triple corruptions. Therefore, MRR and Hits@k are computed using the knowledge graph original triples and non-positive corruptions only [1].

## 3. Loss Functions in KGE Models

#### 3.1. KGE Pointwise Losses

- Pointwise square error loss (SE). It is a pointwise ranking loss function used in RESCAL [4]. It models training losses with the objective of minimising the squared difference between model predicted scores for triples and their true labels:$$\underset{{\mathrm{SE}}_{Pt}}{\mathcal{L}}=\frac{1}{2}\sum _{i=1}^{n}{(f\left({x}_{i}\right)-l\left({x}_{i}\right))}^{2}.$$The optimal score for true and false facts is 1 and 0, respectively. A nice to have characteristic of the SE loss is that it does not require configurable training parameters, shrinking the search space of hyperparameters compared to other losses (e.g., the margin parameter of the hinge loss).
- Pointwise hinge loss. Hinge loss can be interpreted as a pointwise loss, where the objective is to generally minimise the scores of negative facts and maximise the scores of positive facts to a specific configurable value. This approach is used in HolE [9], and it is defined as:$$\underset{{\mathrm{hinge}}_{Pt}}{\mathcal{L}}=\sum _{x\in X}{[\lambda -l\left(x\right)\xb7f\left(x\right)]}_{+},$$
- Pointwise logistic loss. The ComplEx [8] model uses a logistic loss, which is a smoother version of pointwise hinge loss without the configurable margin parameter. Logistic loss uses a logistic function to minimise the negative triples score and maximise the positive triples score. This is similar to hinge loss, but uses a smoother linear loss slope defined as:$$\underset{{\mathrm{logistic}}_{Pt}}{\mathcal{L}}=\sum _{x\in X}log(1+exp(-l\left(x\right)\xb7f\left(x\right))),$$

#### 3.2. KGE Pairwise Losses

- Pairwise hinge loss. Hinge loss is a linear learning to rank loss that can be implemented in both a pointwise or pairwise loss settings. In both, the TransE [1] and DistMult [7] models the hinge loss is used in its pairwise form and defined as follows:$$\underset{{\mathrm{hinge}}_{Pr}}{\mathcal{L}}=\sum _{x\in {X}^{+}}\sum _{{x}^{\prime}\in {X}^{-}}{[\lambda +f\left({x}^{\prime}\right)-f\left(x\right)]}_{+},\phantom{\rule{-3.0pt}{0ex}}$$

- Pairwise logistic loss. Logistic loss can also be interpreted as pairwise margin based loss following the same approach as in hinge loss. The loss is then defined as:$$\underset{{\mathrm{logistic}}_{Pr}}{\mathcal{L}}=\sum _{x\in {X}^{+}}\sum _{{x}^{\prime}\in {X}^{-}}log(1+exp(f\left({x}^{\prime}\right)-f\left(x\right))),\phantom{\rule{-3.0pt}{0ex}}$$

#### 3.3. KGE Multi-Class Losses

- Binary cross entropy loss (BCE). The authors of the ConvE model [14] proposed a new binary cross entropy multi-class loss to model the training error of KGE models in link prediction. In this setting, the whole vocabulary of entities is used to train each positive fact such that for a triple $(s,p,o)$, all facts $(s,p,{o}^{\prime})$ with ${o}^{\prime}\in E$ and ${o}^{\prime}\ne o$ are considered false. The BCE loss can be defined as follows:$${\mathcal{L}}_{spo}^{\mathrm{BCE}}=-\frac{1}{N}\sum _{i}({l}_{spo}\xb7log\left({\varphi}_{spo}\right)+(1-{l}_{spo})\xb7log(1-{\varphi}_{sp{o}^{\prime}})),\phantom{\rule{-3.0pt}{0ex}}$$
- Negative-log softmax loss (NLS). In a recent work, Lacroix et al. [15] introduced a soft-max regression loss to model training error of the ComplEx model as a multi-class problem. In this approach, the objective for each $(s,p,o)$ triple is to minimise the following loss:$$\begin{array}{cc}\hfill {\mathcal{L}}_{spo}^{\mathrm{NLS}}& ={\mathcal{L}}_{spo}^{{o}^{\prime}}+{\mathcal{L}}_{spo}^{{s}^{\prime}},\mathrm{where}\hfill \\ \hfill {\mathcal{L}}_{spo}^{{o}^{\prime}}& =-{\varphi}_{spo}+log({\sum}_{{o}^{\prime}}exp\left({\varphi}_{sp{o}^{\prime}}\right)\hfill \\ \hfill {\mathcal{L}}_{spo}^{{s}^{\prime}}& =-{\varphi}_{spo}+log({\sum}_{{s}^{\prime}}exp\left({\varphi}_{{s}^{\prime}po}\right)\phantom{\rule{-3.0pt}{0ex}}\hfill \end{array}$$

#### 3.4. Effects of Training Objectives on Accuracy

#### 3.5. Effects of Training Objectives on Scalability

## 4. KGE Training Hyperparameters

#### 4.1. Training Hyperparameters Effects on KGE Scalability

#### Analysis of the Predictive Scalability Experiments

- The results confirm that KGE models have linear time complexity as shown in Figure 5 where the models’ runtime grows linearly corresponding to increase of both data size and embedding vector sizes.
- The results also confirm that models such as the TriModel and Complex model which have more than one embedding vector for each entity, and relations require more training time compared to models with only one embedding vector.
- The results also show that the training batch size has a significant effect on the training runtime, therefore, using larger batch sizes are suggested to significantly decrease the training runtime of the training of KGE models.

#### 4.2. Training Hyperparameters’ Effects on KGE Accuracy

- Changes on the embedding vectors’ size have the biggest effect on the predictive accuracy of KGE models. Thus, we suggest to carefully select this parameter by searching through a larger search space, which can help with ensuring that the models can reach their best representations of the knowledge graph.
- The increased number of training iterations can sometimes have a negative effect on the outcome predictive accuracy. Thus, we suggest using early stopping techniques to decide when to stop model training before accuracy starts to decrease.
- Both the number of negative samples and batch sizes showed a small effect on the predictive accuracy of KGE models. Thus, this parameter can either be assigned fixed values or be found using a small search spaces to help decrease the hyperparameters search space, and thus the hyperparameters’ tuning runtime.

## 5. Discussion

#### 5.1. The Compromise between Scalability and Accuracy

#### 5.2. The Relationship between Datasets and Embedding Interaction Functions

#### 5.3. Compatibility between Scoring and Loss Functions

#### 5.4. Limitations of Knowledge Graph Embedding Models

**Lack of interpretability.**In knowledge graph embedding models, the learning objective is to model nodes and edges of the graph using low-rank vector embeddings that preserve the graph’s coherent structure. The embedding learning procedure operates mainly by transforming noise vectors to useful embeddings using gradient decent optimisation on a specific objective loss. These procedures, however, work as a black box that is hard to interpret compared to other association rule mining and graph traversal approaches that can be interpreted based on the features they use.**Sensitivity to data quality.**KGE models generate vector representations of entities according to their prior knowledge. Therefore, the quality of this knowledge affects the quality of the generated embeddings.**Hyperparameter sensitivity.**The outcome predictive accuracy of KGE embeddings is sensitive to their hyperparameters [16]. Therefore, minor changes in these parameters can have significant effects on the outcome predictive accuracy of KGE models. The process of finding the optimal parameters of KGE models is traditionally achieved through an exhausting brute-force parameter search. As a result, their training may require rather time-consuming grid search procedures to find the right parameters for each new dataset.

#### 5.5. Future Directions

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**An illustration of the process of training a knowledge graph embedding model over an example triple $x=(s,p,o)$—the original triple—and ${x}^{\prime}$ refers to a corrupted (negative) version of it.

**Figure 2.**Plot of the loss growth of different types of pointwise knowledge graph embedding loss functions.

**Figure 3.**Plot of the loss growth of different types of pairwise knowledge graph embedding loss functions. The term Pr. is an abbreviation for pairwise.

**Figure 4.**A set of plots that describe the relation between the training runtime (in seconds) and the dataset size for the multi-class and ranking losses for different models on the YAGO10 dataset. The results reported in this figure are acquired by training KGE models with a small embedding of dimension 10 for 20 iterations. The TransE model’s plot reports only results for ranking loss functions.

**Figure 5.**A set of line plots which describe the changes of training data sizes and training hyperparameters and their effects on the training runtime of the TransE, DistMult, TriModel and ComplEx models on the PSE dataset. The runtime is reported in second for all the plots.

**Figure 6.**A set of plots which describe the effects of training hyperparameters of KGE models and their effects on the models’ accuracy in terms of MRR on different benchmarking datasets. The base hyperparameters for our experiments are: embedding size ($k=150$), negative samples per positive (n = 2), batch size ($b=2048$), number of epochs ($e=500$), optimizer (AMSgrad), learning rate ($\mathrm{lr}=0.01$).

**Table 1.**Statistics of entities, relations, and triples count per split of the benchmarking datasets which we use in this study.

Dataset | Entity Count | Relation Count | Train | Valid | Test |
---|---|---|---|---|---|

NELL239 | 48K | 239 | 74K | 3K | 3K |

WN18RR | 41K | 11 | 87K | 3K | 3K |

FB15k-237 | 15K | 237 | 272K | 18K | 20K |

YAGO10 | 123K | 37 | 1M | 5K | 5K |

PSE | 32K | 967 | 3.7M | 459K | 459K |

**Table 2.**Link prediction results for KGE models with different loss functions on standard benchmarking datasets. The abbreviations $MC$, $Pr$, $Pt$ stand for multi-class, pairwise and pointwise, respectively. The * mark is used to denote the model’s original loss function as first proposed by the authors. In the ranking losses, the best results are computed per model and highlighted using bold font, and underlined values represent the best result in each respective loss approach.

Model | Loss | NELL239 | WN18RR | Fb15k-237 | |||||
---|---|---|---|---|---|---|---|---|---|

MRR | H10 | MRR | H10 | MRR | H10 | ||||

Ranking Loss | TransE | Pr | * Hinge | 0.28 | 0.43 | 0.20 | 0.47 | 0.27 | 0.43 |

Logistic | 0.27 | 0.43 | 0.21 | 0.48 | 0.26 | 0.43 | |||

Pt | Hinge | 0.19 | 0.32 | 0.12 | 0.34 | 0.12 | 0.25 | ||

Logistic | 0.17 | 0.31 | 0.11 | 0.31 | 0.01 | 0.23 | |||

SE | 0.01 | 0.02 | 0.00 | 0.00 | 0.01 | 0.01 | |||

DistMult | Pr | Hinge | 0.20 | 0.32 | 0.40 | 0.45 | 0.10 | 0.16 | |

Logistic | 0.26 | 0.40 | 0.39 | 0.45 | 0.19 | 0.36 | |||

Pt | * Hinge | 0.25 | 0.41 | 0.43 | 0.49 | 0.21 | 0.39 | ||

Logistic | 0.28 | 0.43 | 0.43 | 0.50 | 0.20 | 0.39 | |||

SE | 0.31 | 0.48 | 0.43 | 0.50 | 0.22 | 0.40 | |||

ComplEx | Pr | Hinge | 0.24 | 0.38 | 0.39 | 0.45 | 0.20 | 0.35 | |

Logistic | 0.27 | 0.43 | 0.41 | 0.47 | 0.19 | 0.35 | |||

Pt | Hinge | 0.21 | 0.36 | 0.41 | 0.47 | 0.20 | 0.39 | ||

* Logistic | 0.14 | 0.24 | 0.36 | 0.39 | 0.13 | 0.28 | |||

SE | 0.35 | 0.52 | 0.47 | 0.53 | 0.22 | 0.41 | |||

Multi-class losses | CP | MC | BCE | - | - | - | - | - | - |

NLS | - | - | 0.08 | 0.12 | 0.22 | 0.42 | |||

DistMult | MC | BCE | - | - | 0.43 | 0.49 | 0.24 | 0.42 | |

NLS | 0.39 | 0.55 | 0.43 | 0.50 | 0.34 | 0.53 | |||

ComplEx | MC | BCE | - | - | 0.44 | 0.51 | 0.25 | 0.43 | |

NLS | 0.40 | 0.58 | 0.44 | 0.52 | 0.35 | 0.53 |

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Mohamed, S.K.; Muñoz, E.; Novacek, V.
On Training Knowledge Graph Embedding Models. *Information* **2021**, *12*, 147.
https://doi.org/10.3390/info12040147

**AMA Style**

Mohamed SK, Muñoz E, Novacek V.
On Training Knowledge Graph Embedding Models. *Information*. 2021; 12(4):147.
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**Chicago/Turabian Style**

Mohamed, Sameh K., Emir Muñoz, and Vit Novacek.
2021. "On Training Knowledge Graph Embedding Models" *Information* 12, no. 4: 147.
https://doi.org/10.3390/info12040147