Title: Adapting Neural Link Predictors for Data-Efficient Complex Query Answering
Venue: NeurIPS 2023
Accepted: yes
Keywords: complex query answering

neural link prediction

knowledge graph embeddings

knowledge graphs

relational learning

adapters
Abstract: Answering complex queries on incomplete knowledge graphs is a challenging task where a model needs to answer complex logical queries in the presence of missing knowledge. Prior work in the literature has proposed to address this problem by designing architectures trained end-to-end for the complex query answering task with a reasoning process that is hard to interpret while requiring data and resource-intensive training.  Other lines of research have proposed re-using simple neural link predictors to answer complex queries, reducing the amount of training data by orders of magnitude while providing interpretable answers. The neural link predictor used in such approaches is not explicitly optimised for the complex query answering task, implying that its scores are not calibrated to interact together. We propose to address these problems via CQD$^{\mathcal{A}}$, a parameter-efficient score \emph{adaptation} model optimised to re-calibrate neural link prediction scores for the complex query answering task. While the neural link predictor is frozen, the adaptation component -- which only increases the number of model parameters by $0.03\%$ -- is trained on the downstream complex query answering task. Furthermore, the calibration component enables us to support reasoning over queries that include atomic negations, which was previously impossible with link predictors. In our experiments, CQD$^{\mathcal{A}}$ produces significantly more accurate results than current state-of-the-art methods, improving from $34.4$ to $35.1$ Mean Reciprocal Rank values averaged across all datasets and query types while using $\leq 30\%$ of the available training query types. We further show that CQD$^{\mathcal{A}}$ is data-efficient, achieving competitive results with only $1\%$ of the complex training queries and robust in out-of-domain evaluations. Source code and datasets are available at https://github.com/EdinburghNLP/adaptive-cqd.

Adapting Neural Link Predictors for Data-Efficient Complex Query Answering

Abstract

Answering complex queries on incomplete knowledge graphs is a challenging task where a model needs to answer complex logical queries in the presence of missing knowledge. Prior work in the literature has proposed to address this problem by designing architectures trained end-to-end for the complex query answering task with a reasoning process that is hard to interpret while requiring data and resource-intensive training. Other lines of research have proposed re-using simple neural link predictors to answer complex queries, reducing the amount of training data by orders of magnitude while providing interpretable answers. The neural link predictor used in such approaches is not explicitly optimised for the complex query answering task, implying that its scores are not calibrated to interact together. We propose to address these problems via CQD A , a parameter-efficient score adaptation model optimised to re-calibrate neural link prediction scores for the complex query answering task. While the neural link predictor is frozen, the adaptation component - which only increases the number of model parameters by 0 . 03% - is trained on the downstream complex query answering task. Furthermore, the calibration component enables us to support reasoning over queries that include atomic negations, which was previously impossible with link predictors. In our experiments, CQD A produces significantly more accurate results than current state-of-the-art methods, improving from 34 . 4 to 35 . 1 Mean Reciprocal Rank values averaged across all datasets and query types while using ≤ 30% of the available training query types. We further show that CQD A is data-efficient, achieving competitive results with only 1% of the complex training queries, and robust in out-of-domain evaluations. Source code and datasets are available at https://github.com/EdinburghNLP/adaptive-cqd .

1 Introduction

A Knowledge Graph (KG) is a knowledge base representing the relationships between entities in a relational graph structure. The flexibility of this knowledge representation formalism allows KGs to be widely used in various domains. Examples of KGs include general-purpose knowledge bases such as Wikidata [Vrandeˇ ci´ c and Kr¨ otzsch, 2014], DBpedia [Auer et al., 2007], Freebase [Bollacker et al., 2008], and YAGO [Suchanek et al., 2007]; application-driven graphs such as the Google Knowledge Graph, Microsoft's Bing Knowledge Graph, and Facebook's Social Graph [Noy et al., 2019]; and domain-specific ones such as SNOMED CT [Bodenreider et al., 2018], MeSH [Lipscomb, 2000], and Hetionet [Himmelstein et al., 2017] for life sciences; and WordNet [Miller, 1992] for linguistics.

Answering complex queries over Knowledge Graphs involves a logical reasoning process where a conclusion should be inferred from the available knowledge.

Neural link predictors [Nickel et al., 2016] tackle the problem of identifying missing edges in large KGs. However, in many domains, it is a challenge to develop techniques for answering complex queries involving multiple and potentially unobserved edges, entities, and variables rather than just single edges.

Prior work proposed to address this problem using specialised neural networks trained endto-end for the query answering task [Hamilton et al., 2018, Daza and Cochez, 2020, Ren et al., 2020, Ren and Leskovec, 2020, Zhu et al., 2022], which offer little interpretability and require training with large and diverse datasets of query-answer pairs. These methods stand in contrast with Complex Query Decomposition [CQD, Arakelyan et al., 2021, Minervini et al., 2022], which showed that it is sufficient to re-use a simple link prediction model to answer complex queries, thus reducing the amount of training data required by orders of magnitude while allowing the possibility to explain intermediate answers. While effective, CQD does not support negations, and fundamentally, it relies

on a link predictor whose scores are not necessarily calibrated for the complex query answering task. Adapting a neural link predictor for the query answering task while maintaining the data and parameter efficiency of CQD, as well as its interpretable nature, is the open challenge we take on in this paper.

We propose CQD A , a lightweight adaptation model trained to calibrate link prediction scores, using complex query answering as the optimisation objective. We define the adaptation function as an affine transformation of the original score with a few learnable parameters. The low parameter count and the fact that the adaptation function is independent of the query structure allow us to maintain the efficiency properties of CQD. Besides, the calibration enables a natural extension of CQD to queries with atomic negations.

An evaluation of CQD A on three benchmark datasets for complex query answering shows an increase from 34 . 4 to 35 . 1 MRR over the current state-of-the-art averaged across all datasets while using ≤ 30% of the available training query types. In ablation experiments, we show that the method is data-efficient; it achieves results comparable to the state-of-the-art while using only 1% of the complex queries. Our experiments reveal that CQD A can generalise across unseen query types while using only 1% of the instances from a single complex query type during training.

2 Related Work

Link Predictors in Knowledge Graphs Reasoning over KGs with missing nodes has been widely explored throughout the last few years. One can approach the task using latent feature models, such as neural link predictors [Bordes et al., 2013, Trouillon et al., 2016, Yang et al., 2014, Dettmers et al., 2018, Sun et al., 2019, Balaˇ zevi´ c et al., 2019, Amin et al., 2020] which learn continuous representations for the entities and relation types in the graph and can answer atomic queries over incomplete KGs. Other research lines tackle the link prediction problem through graph feature models [Xiong et al., 2017, Das et al., 2017, Hildebrandt et al., 2020, Yang et al., 2017, Sadeghian et al., 2019], and Graph Neural Networks [GNNs, Schlichtkrull et al., 2018, Vashishth et al., 2019, Teru et al., 2020].

Complex Query Answering Complex queries over knowledge graphs can be formalised by extending one-hop atomic queries with First Order Logic (FOL) operators, such as the existential quantifier ( ∃ ), conjunctions ( ∧ ), disjunctions ( ∨ ) and negations ( ¬ ). These FOL constructs can be represented as directed acyclic graphs, which are used by embedding-based methods that represent the queries

using geometric objects [Ren et al., 2020, Hamilton et al., 2018] or probabilistic distributions [Ren and Leskovec, 2020, Zhang et al., 2021, Choudhary et al., 2021] and search the embedding space for the answer set. It is also possible to enhance the properties of the embedding space using GNNs and Fuzzy Logic [Zhu et al., 2022, Chen et al., 2022]. A recent survey [Ren et al., 2023] provides a broad overview of different approaches. Recent work [Daza and Cochez, 2020, Hamilton et al., 2018, Ren and Leskovec, 2020] suggests that such methods require a large dataset with millions of diverse queries during the training, and it can be hard to explain their predictions.

Our work is closely related to CQD [Arakelyan et al., 2021, Minervini et al., 2022], which uses a pre-trained neural link predictor along with fuzzy logical t-norms and t-conorms for complex query answering. A core limitation of CQD is that the pre-trained neural link predictor produces scores not calibrated to interact during the complex query-answering process. This implies that the final scores of the model are highly dependent on the choice of the particular t-(co)norm aggregation functions, which, in turn, leads to discrepancies within the intermediate reasoning process and final predictions. As a side effect, the lack of calibration also means that the equivalent of logical negation in fuzzy logic does not work as expected.

With CQD A , we propose a solution to these limitations by introducing a scalable adaptation function that calibrates link prediction scores for query answering. Furthermore, we extend the formulation of CQD to support a broader class of FOL queries, such as queries with atomic negation.

3 Background

A Knowledge Graph G ⊆ E × R × E can be defined as a set of subject-predicate-object ⟨ s, p, o ⟩ triples, where each triple encodes a relationship of type p ∈ R between the subject s ∈ E and the object o ∈ E of the triple, where E and R denote the set of all entities and relation types, respectively. A Knowledge Graph can be represented as a First-Order Logic Knowledge Base, where each triple ⟨ s, p, o ⟩ denotes an atomic formula p ( s, o ) , with p ∈ R a binary predicate and s, o ∈ E its arguments.

First-Order Logical Queries We are concerned with answering logical queries over incomplete knowledge graphs. We consider queries that use existential quantification ( ∃ ) and conjunction ( ∧ ) operations. Furthermore, we include disjunctions ( ∨ ) and atomic negations ( ¬ ). We follow Ren et al. [2020] by transforming a logical query into Disjunctive Normal Form [DNF, Davey and Priestley, 2002], i.e. a disjunction of conjunctive queries, along with the subsequent extension with atomic negations in [Ren and Leskovec, 2020]. We denote such queries as follows:

̸

In Equation (1), the variable A is the target of the query, V 1 , . . . , V m denote the bound variable nodes , while c ∈ E represent the input anchor nodes , which correspond to known entities in the query. Each e i denotes a logical atom, with either one ( p ( c, V ) ) or two variables ( p ( V, V ′ ) ).

The goal of answering the logical query Q consists in finding the answer set J Q K ⊆ E such that a ∈ J Q K iff Q [ a ] holds true. As illustrated in Figure 1, the dependency graph of a conjunctive query Q is a graph where nodes correspond to variable or non-variable atom arguments in Q and edges correspond to atom predicates. We follow Hamilton et al. [2018] and focus on queries whose dependency graph is a directed acyclic graph, where anchor entities correspond to source nodes, and the query target A is the unique sink node.

Example 3.1 (Complex Query) . Consider the question ' Which people are German and produced the music for the film Constantine? '. It can be formalised as a complex query Q ≡ ? T : country ( Germany , T ) ∧ producerOf ( Constantine , T ) , where Germany and Constantine are anchor nodes, and T is the target of the query, as presented in Figure 1. The answer J Q K corresponds to all the entities in the knowledge graph that are German composers for the film Constantine.

Continuous Query Decomposition CQD is a framework for answering EPFO logical queries in the presence of missing edges [Arakelyan et al., 2021, Minervini et al., 2022]. Given a query Q , CQD

defines the score of a target node a ∈ E as a candidate answer for a query as a function of the score of all atomic queries in Q , given a variable-to-entity substitution for all variables in Q .

Each variable is mapped to an embedding vector that can either correspond to an entity c ∈ E or to a virtual entity . The score of each of the query atoms is determined individually using a neural link predictor [Nickel et al., 2016]. Then, the score of the query with respect to a given candidate answer Q [ a ] is computed by aggregating all of the atom scores using t-norms and t-conorms - continuous relaxations of the logical conjunction and disjunction operators.

Neural Link Predictors A neural link predictor is a differentiable model where atom arguments are first mapped into a d -dimensional embedding space and then used to produce a score for the atom. More formally, given a query atom p ( s, o ) , where p ∈ R and s, o ∈ E , the score for p ( s, o ) is computed as ϕ p ( e s , e o ) , where e s , e o ∈ R d are the embedding vectors of s and o , and ϕ p : R d × R d ↦→ [0 , 1] is a scoring function computing the likelihood that entities s and o are related by the relationship p . Following Arakelyan et al. [2021], Minervini et al. [2022], in our experiments, we use a regularised variant of ComplEx [Trouillon et al., 2016, Lacroix et al., 2018] as the neural link predictor of choice, due to its simplicity, efficiency, and generalisation properties [Ruffinelli et al., 2020]. To ensure that the output of the neural link predictor is always in [0 , 1] , following Arakelyan et al. [2021], Minervini et al. [2022], we use either a sigmoid function or min-max re-scaling.

T-norms and Negations Fuzzy logic generalises over Boolean logic by relaxing the logic conjunction ( ∧ ), disjunction ( ∨ ) and negation ( ¬ ) operators through the use of t-norms, t-conorms, and fuzzy negations. A t-norm ⊤ : [0 , 1] × [0 , 1] ↦→ [0 , 1] is a generalisation of conjunction in fuzzy logic [Klement et al., 2000, 2004]. Some examples include the G¨ odel t-norm ⊤ min ( x, y ) = min { x, y } , the product t-norm ⊤ prod ( x, y ) = x × y , and the Łukasiewicz t-norm ⊤ Luk ( x, y ) = max { 0 , x + y -1 } .

Analogously, t-conorms are dual to t-norms for disjunctions - given a t-norm ⊤ , the complementary t-conorm is defined by ⊥ ( x, y ) = 1 -⊤ (1 -x, 1 -y ) . In our experiments, we use the G¨ odel t-norm and product t-norm with their corresponding t-conorms.

Fuzzy logic also encompasses negations n : [0 , 1] ↦→ [0 , 1] . The standard n stand ( x ) = 1 -x and strict cosine n cos = 1 2 (1 + cos( πx )) are common examples of fuzzy negations[Kruse and Moewes, 1993]. To support a broader class of queries, we introduce the standard and strict cosine functions to model negations in CQD A , which was not considered in the original formulation of CQD.

Continuous Query Decomposition Given a DNF query Q as defined in Equation (1), CQD aims to find the variable assignments that render Q true. To achieve this, CQD casts the problem of query answering as an optimisation problem. The aim is to find a mapping from variables to entities S = { A ← a, V 1 ← v 1 , . . . , V m ← v m } , where a, v 1 , . . . , v m ∈ E are entities and A,V 1 , . . . , V m are variables, that maximises the score of Q :

̸

where ⊤ and ⊥ denote a t-norm and a t-conorm - a continuous generalisation of the logical conjunction and disjunction, respectively - and ϕ p ( e s , e o ) ∈ [0 , 1] denotes the neural link prediction score for the atom p ( s, o ) .

Complex Query Answering via Combinatorial Optimisation Following Arakelyan et al. [2021], Minervini et al. [2022], we solve the optimisation problem in Equation (2) by greedily searching for a set of variable substitutions S = { A ← a, V 1 ← v 1 , . . . , V m ← v m } , with a, v 1 , . . . , v m ∈ E , that maximises the complex query score, in a procedure akin to beam search . We do so by traversing the dependency graph of a query Q and, whenever we find an atom in the form p ( c, V ) , where p ∈ R , c is either an entity or a variable for which we already have a substitution, and V is a variable for which we do not have a substitution yet, we replace V with all entities in E and retain the topk entities t ∈ E that maximise ϕ p ( e c , e t ) - i.e. the most likely entities to appear as a substitution of V according to the neural link predictor. As we traverse the dependency graph of a query, we keep a beam with the most promising variable-to-entity substitutions identified so far.

Example 3.2 (Combinatorial Optimisation) . Consider the query ' Which musicians M received awards associated with a genre g ? , which can be rewritten as ? M : ∃ A. assoc ( g, A ) ∧ received ( A,M ) . To answer this query using combinatorial optimisation, we must find the topk awards a that are candidates to substitute the variable A in assoc ( g, A ) . This will allow us to understand the awards associated with the genre g . Afterwards, for each candidate substitution for A , we search for the topk musicians m that are most likely to substitute M in received ( A,M ) , ending up with k 2 musicians. Finally, we rank the k 2 candidates using the final query score produced by a t-norm. ■

4 Calibrating Link Prediction Scores on Complex Queries

The main limitation in the CQD method outlined in Section 3 is that neural link predictors ϕ are trained to answer simple, atomic queries, and the resulting answer scores are not trained to interact with one another.

Example 4.1. Consider the running example query ' Which people are German and produced the music for the film Constantine? ' which can be rewritten as a complex query Q ≡ ? T : country ( Germany , T ) ∧ producerOf ( Constantine , T ) . To answer this complex query, CQD answers the atomic sub-queries Q 1 = country ( Germany , T ) and Q 2 = producerOf ( Constantine , T ) using a neural link predictor, and aggregates the resulting scores using a t-norm. However, the neural link predictor was only trained on answering atomic queries, and the resulting scores are not calibrated to interact with each other. For example, the scores for the atomic queries about the relations country and producerOf may be on different scales, which causes problems when aggregating such scores via t-norms. Let us assume the top candidates for the variable T coming from the atomic queries Q 1 , Q 2 are A 1 ← Sam Shepard and A 2 ← Klaus Badelt , with their corresponding neural link prediction scores 1 . 2 and 8 . 9 , produced using ϕ country and ϕ producerOf . We must also factor in the neural link prediction score of the candidate A 1 for query Q 2 at 7 . 4 and vice versa at 0 . 5 . When using the G¨ odel t-norm ⊤ min ( x, y ) = min { x, y } , the scores associated with the variable assignments A 1 , A 2 are computed as, min(8 . 0 , 0 . 5) = 0 . 5 min(7 . 4 , 1 . 2) = 1 . 2 . For both answers A 1 and A 2 , the scores produced by ϕ country for Q 1 are always lower than the scores produced with ϕ producerOf for Q 2 , meaning that the scores of the latter are not considered when producing the final answer. This phenomenon can be broadly observed in CQD, illustrated in Figure 2. ■

To address this problem, we propose a method for adaptively learning to calibrate neural link prediction scores by back-propagating through the complex query-answering process. More formally, let ϕ p denote a neural link predictor. We learn an additional adaptation function ρ θ , parameterised by θ = { α, β } , with α, β ∈ R . Then, we use the composition of ρ θ and ϕ p , ρ θ ◦ ϕ p , such that:

Here, the function ρ defines an affine transformation of the score and when the parameters α = β = 0 , the transformed score ρ θ ( ϕ p ( e V , e V ′ )) recovers the original scoring function. The parameters θ can be conditioned on the representation of the predicate p and the entities V and V ′ , i.e. θ = ψ ( e V , e p , e V ′ ) ; here, ψ is an end-to-end differentiable neural module with parameters W . e V , e p , e V ′ respectively denote the representations of the subject, predicate, and object of the atomic query. In our experiments, we consider using one or two linear transformation layers with a ReLU non-linearity as options for ψ .

The motivation for our proposed adaptation function is twofold. Initially, it is monotonic, which is desirable for maintaining the capability to interpret intermediate scores, as in the original formulation of CQD. Moreover, we draw inspiration from the use of affine transformations in methodologies such as Platt scaling [Platt et al., 1999], which also use a linear function for calibrating probabilities and have been applied in the problem of calibration of link prediction models [Tabacof and Costabello, 2020]. Parameter-efficient adaptation functions have also been applied effectively in other domains, such as adapter layers Houlsby et al. [2019] used for fine-tuning language models in NLP tasks.

Training For training the score calibration component in Equation (3), we first compute how likely each entity a ′ ∈ E is to be an answer to the query Q . To this end, for each candidate answer a ′ ∈ E , we compute the answer score as the complex query score assuming that a ′ ∈ E is the final answer as:

4

2

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0.0

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Density

Distribution of neural link prediction scores

1 = gender(male, T)

2 = producerOf(Constantine, T)

Tmin

(

1,

2)

Score

Training Queries

2in

3in

3i

2i

n

n

1p

2p

3p

pi

u

u

2u

n

inp

ip

u

u

up

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pni

Variable Node

Target Node

u

Disjunction

n

Negation

Equation (4) identifies the variable-to-entity substitution S that 1) maximises the query score score ( Q , S ) , defined in Equation (2), and 2) associates the answer variable A with a ′ ∈ E , i.e. A ← a ′ ∈ S . For computing S with the additional constraint that A ← a ′ ∈ S , we use the complex query answering procedure outlined in Section 3. We optimise the additional parameters W introduced in Section 4, by gradient descent on the likelihood of the true answers on a dataset D = { ( Q i , a i ) } |D| i =1 of query-answer pairs by using a 1-vs-all cross-entropy loss, introduced by Lacroix et al. [2018], which was also used to train the neural link prediction model:

In addition to the 1-vs-all [Ruffinelli et al., 2020] loss in Equation (5), we also experiment with the binary cross-entropy loss, using the negative sampling procedure from Ren and Leskovec [2020].

5 Experiments

Datasets To evaluate the complex query answering capabilities of our method, we use a benchmark comprising of 3 KGs: FB15K [Bordes et al., 2013], FB15K-237 [Toutanova and Chen, 2015] and NELL995 [Xiong et al., 2017]. For a fair comparison with previous work, we use the datasets of FOL queries proposed by Ren and Leskovec [2020], which includes nine structures of EPFO queries and 5 query types with atomic negations, seen in Figure 3. The datasets provided by Ren and Leskovec [2020] introduce queries with hard answers, which are the answers that cannot be obtained by direct graph traversal; in addition, this dataset does not include queries with more than 100 answers, increasing the difficulty of the complex query answering task. The statistics for each dataset can be seen in Table 1. Note that during training, we only use 2i , 3i , 2in , and 3in queries, corresponding to ≤ 30% of the training dataset, for the adaptation of the neural link predictor. To assess the model's ability to generalise, we evaluate it on all query types.

Evaluation Protocol For a fair comparison with prior work, we follow the evaluation scheme in Ren and Leskovec [2020] by separating the answer of each query into easy and hard sets. For test and validation splits, we define hard queries as those that cannot be answered via direct traversal along the edges of the KG and can only be answered by predicting at least one missing link, meaning non-trivial reasoning should be completed. We evaluate the method on non-trivial queries by calculating the rank r for each hard answer against non-answers and computing the Mean Reciprocal Rank (MRR).

Baselines We compare CQD A with state-of-the-art methods from various solution families in Section 2. In particular, we choose GQE [Hamilton et al., 2018], Query2Box [Ren et al., 2020], BetaE [Ren and Leskovec, 2020] and ConE [Zhang et al., 2021] as strong baselines for query embedding methods. We also compare with methods based on GNNs and fuzzy logic, such as FuzzQE [Chen et al., 2022], GNN-QE [Zhu et al., 2022], and the original CQD [Arakelyan et al., 2021, Minervini et al., 2022], which uses neural link predictors for answering EPFO queries without any fine-tuning on complex queries.

Model Details Our method can be used with any neural link prediction model. Following Arakelyan et al. [2021], Minervini et al. [2022], we use ComplEx-N3 [Lacroix et al., 2018]. We identify the optimal hyper-parameters using the validation MRR. We train for 50 , 000 steps using Adagrad as an optimiser and 0.1 as the learning rate. The beam-size hyper-parameter k was selected in k ∈ { 512 , 1024 , . . . , 8192 } , and the loss was selected across 1-vs-all [Lacroix et al., 2018] and binary cross-entropy with one negative sample.

Parameter Efficiency We use the query types 2i , 3i , 2in , 3in for training the calibration module proposed in Section 4. We selected these query types as they do not require variable assignments other than for the answer variable A , making the training process efficient. As the neural link prediction model is frozen, we only train the adapter layers that have a maximum of W ∈ R 2 × 2 d learnable weights. Compared to previous works, we have ≈ 10 3 times fewer trainable parameters, as shown in Table 3, while maintaining competitive results.

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Average MRR with 1% and 100% training data

average MRR with 1%

average MRR with 100%

5.1 Results

Complex Query Answering Table 2 shows the predictive accuracy of CQD A for answering complex queries compared to the current state-of-the-art methods. Some methods do not support queries that include negations; we leave the corresponding entries blank. We can see that CQD A increases the MRR from 34 . 4 to 35 . 1 averaged across all query types and datasets. In particular, CQD A shows the most substantial increase in predictive accuracy on NELL995 by producing more accurate results than all other methods for all query types. CQD A can achieve these results using ≤ 30% of the complex query types during training while maintaining competitive results across each dataset and query type. For queries including negations, CQD A achieves a relative improvement of 6 . 8% to 37 . 1% , which can be attributed to the fact that the adaptation is completed with query types 2in and 3in that include negation, which allows for learning an adaptation layer that is robust for these types of queries. In our experiments, we found that calculating the neural adaptation parameters θ of the adaptation function ρ θ in Equation (3) as a function of the predicate representation yields the most accurate results followed by computing θ as a function of the source entity and predicate representation, which is strictly more expressive. In Appendix A, we show the impact of the adaptation layers on the neural link prediction scores.

The adaptation process does not require data-intensive training and allows the model to generalise to query types not observed during training. This prompts us to investigate the minimal amount of data samples and query types required for adaptation.

Data Efficiency To analyse the data efficiency of CQD A , we compare the behaviour of the pre-trained link predictors tuned with 1% and 100% of the training complex query examples in FB15K-237, presented in Table 4. For adapting on 1% of the training complex queries, we used the same hyper-parameters we identified when training on the full dataset. Even when using 1% of the complex training queries ( 3290 samples) for tuning, the model still achieves competitive results, with an average MRR difference of 2 . 2 compared to the model trained using the entire training set. CQD A also produces higher test MRR results than GNN-QE with an average MRR increase of 4 . 05 .

We can also confirm that the adaptation process converges after ≤ 10% of the training epochs as seen in Figure 4. The convergence rate is not hindered when using only 1% of the training queries. This shows that CQD A is a scalable method with a fast convergence rate that can be trained in a data-efficient manner.

Out-of-Distribution Generalisation To study the generalisation properties of CQD A , we trained the adaptation layer on all atomic queries and only 1% of samples for one training query type 2i , one of the simplest complex query types. We see in Table 4 that CQD A can generalise to other types of complex queries not observed during training with an average MRR difference of 2 . 9 compared to training on all training query types. CQD A also produces significantly higher test MRR results

Table 3: Number of parameters used by different complex query answering methods - values for GNN-QE are approximated using the backbone NBFNet [Zhu et al., 2021], while the remaining use their original studies.

than GNN-QE, with an average increase of 5 . 1 MRR. The greatest degradation in predictive accuracy occurs for the queries containing negations, with an average decrease of 2 . 7 . This prompts us to conjecture that being able to answer general EPFO queries is not enough to generalise to the larger set of queries, which include atomic negation. However, our method can generalise on all query types, using only 1% of the 2i queries, with 1496 overall samples for adaptation.

Fine-Tuning All Model Parameters One of the reasons for the efficiency of CQD A is that the neural link predictor is not fine-tuned for query answering, and only the parameters in the adaptation function are learned. We study the effect of fine-tuning the link predictor using the full training data for CQD and CQD A on FB15K-237. We consider several variants: 1) CQDF, where we F ine-tune all neural link predictor parameters in CQD; 2) CQD A F , where we fine-tune all link predictor parameters in CQD A , 3) CQDR, where we learn a transformation for the entity and relation embeddings and we use it to R eplace the initial entity and relation representations, and 4) CQDC, where we learn a transformation for the entity and relation embeddings, and we C oncatenate it to the initial entity and relation representations.

It can be seen from Table 5 that CQD A yields the highest test MRR results across all query types while fine-tuning all the model parameters produces significant degradation along all query types, which we believe is due to catastrophic forgetting [Goodfellow et al., 2013] of the pre-trained link predictor.

6 Conclusions

In this work, we propose the novel method CQD A for answering complex FOL queries over KGs, which increases the averaged MRR over the previous state-of-the-art from 34 . 4 to 35 . 1 while using ≤ 30% of query types. Our method uses a single adaptation layer over neural link predictors, which allows for training in a data-efficient manner. We show that the method can maintain competitive predictive accuracy even when using 1% of the training data. Furthermore, our experiments on training on a subset ( 1% ) of the training queries from a single query type ( 2i ) show that it can generalise to

new queries that were not used during training while being data-efficient. Our results provide further evidence for how neural link predictors exhibit a form of compositionality that generalises to the complex structures encountered in the more general problem of query answering. CQD A is a method for improving this compositionality while preserving computational efficiency. As a consequence, rather than designing specialised models trained end-to-end for the query answering task, we can focus our efforts on improving the representations learned by neural link predictors, which would then transfer to query answering via efficient adaptation, as well as other downstream tasks where they have already proved beneficial, such as clustering, entity classification, and information retrieval.

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Distribution of neural link prediction scores

Before Score Adaptation

After Score Adaptation

A Impact of adaptation

We investigate the effect of the adaptation process in CQD A by comparing the score of the neural link predictor before and after applying the adaptation layer. As we see from Figure 5, the scores before adaptation have a variation of 5 . 04 with the boundaries at [ -8 , 12] . This makes them problematic for complex query answering as discussed in Section 4. The Adapted scores have a smaller variation at 0 . 03 while the maximum and minimum lie in the [0 , 1] range.

B On data efficiency and generalisation of CQD A

We conduct a series of experiments comparing the performance of CQD A trained while only using 1% of the training data to the complete training set in FB15K-237. We see from Figure 6 (top) that the model maintains a strong performance on the complex reasoning task with a 2 . 4 averaged MRR degradation compared to using the complete data. This phenomenon is even more pronounced when we use only 1% of the queries while using 2i as our training query types Figure 6 (bottom). In this data and query type constrained mode CQD A maintains competitive performance with a degradation of 2 . 7 averaged MRR for complex queries compared to training with complete data. This highlights the data-efficient nature of CQD A and shows that training a score adaptation layer does not require significant data for training. This is also accompanied by the observation that CQD A is able to generalise to unseen query types while training only on 2i. This behaviour is not inherent, as seen from Table 4.

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MRR for training on 1% of 2i and All of training queries. Average difference: 2.7

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the challenging task of answering complex queries on incomplete knowledge graphs, where missing knowledge introduces additional complexity. Previous approaches either employed end-to-end architectures with opaque reasoning processes or relied on simple neural link predictors, sacrificing information gain for computational efficiency. To overcome these limitations, the authors propose a parameter-efficient score adaptation model optimized for recalibrating neural link prediction scores in complex query answering. While the neural link predictor remains frozen, the adaptation component, with minimal additional parameters, is trained on the downstream task. The proposed method significantly improves accuracy compared to state-of-the-art approaches, achieving higher Mean Reciprocal Rank values while using only a fraction of the available training query types.\n\nSTRENGTHS:\n### Originality\nThe paper introduces a novel approach to addressing the task of answering complex queries on incomplete knowledge graphs. The use of a parameter-efficient score adaptation model, optimized for recalibrating neural link prediction scores, sets it apart from previous methods. This original approach demonstrates innovation in the field of complex query answering and contributes to advancing the understanding of addressing missing knowledge in knowledge graphs.\n\n### Quality\nThe paper provides thorough evaluations and comparisons with state-of-the-art methods, demonstrating that the proposed approach produces significantly more accurate results. The proposed method significantly improves accuracy compared to state-of-the-art approaches, achieving higher Mean Reciprocal Rank values while using only a fraction of the available training query types.\n\n### Clarity\nThe paper effectively communicates the proposed approach and its components, such as the parameter-efficient score adaptation model and the frozen neural link predictor. The writing is clear, concise, and well-structured.\n\n### Significance\nThe paper addresses an important and challenging task in the field of complex query answering on incomplete knowledge graphs. By introducing the approach and demonstrating its superior performance compared to existing methods, the paper makes a significant contribution to the advancement of techniques for handling missing knowledge in knowledge graphs.\n\nWEAKNESSES:\nThe authors haven't discussed the limitation of this work in the paper.\n\nQUESTIONS:\nN/A\n\nLIMITATIONS:\nThis is a solid work.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a score adaptation model called CQDA for efficient complex query answering on incomplete knowledge graphs. The authors address the problems in existing methods that are either hard to interpret or require intensive training. CQDA is a parameter-efficient model that recalibrates neural link prediction scores for the complex query answering task. The authors demonstrate that CQDA outperforms state-of-the-art methods in terms of accuracy, data efficiency, and robustness.\n\nSTRENGTHS:\n- The paper addresses an important problem in the field of complex query answering on incomplete knowledge graphs. The proposed CQDA model provides a solution that improves accuracy while maintaining computational efficiency.\n- The paper introduces a parameter-efficient score adaptation model that recalibrates neural link prediction scores. This approach reduces the need for extensive training data and resources.\n- Experimental results show that CQDA achieves significantly better results than current state-of-the-art methods. The model is also shown to be data-efficient and robust in out-of-domain evaluations.\n\nWEAKNESSES:\nI think there is a discrepancy between Figure 1 and the statement, and thus the paper lacks an important baseline: **normalizing scores then merging**.\n\nIn Figure 1, the paper shows that the score scale may be different for different subqueries, which makes me think that the original COD may be without normalization which maps the score from any scale to [0,1]. However, the paper also mentions that they follow previous work to add normalization as below:\n> To ensure that the output of the neural link predictor is always in [0, 1], following Arakelyan et al. [2021], Minervini et al. [2022], we use either a sigmoid function or min-max re-scaling.\n\n So, what would happen if we do a merge after normalization on scores for all subqueries? Please correct me if I miss anything.\n\nQUESTIONS:\n- Figure 1 shows the scores over all entities but does not clearly present if the score is from a score function, or from a linear layer. If I understand correctly, the score before adaptation should come from the score function $\\phi$, right?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a novel approach called CQD^A for answering complex queries on incomplete knowledge graphs. The authors address the challenge of answering complex logical queries in the presence of missing knowledge by re-calibrating neural link prediction scores. They introduce an adaptation component that is trained on the downstream complex query answering task, while the neural link predictor is frozen. The proposed model, CQD^A, significantly improves the Mean Reciprocal Rank values compared to existing state-of-the-art methods. It achieves this by increasing the accuracy of answers while using a smaller amount of training data. Additionally, CQD^A is shown to be data-efficient and robust in out-of-domain evaluations.\n\nTo sum up, the proposed method is simple yet efficient. I have no reason to reject.\n\nSTRENGTHS:\n1. Novel Approach: The paper introduces a unique and novel approach for answering complex queries on incomplete knowledge graphs. The use of neural link predictors and the adaptation component allows for efficient complex query answering while providing interpretable answers. Well motivation.\n2. Improved Accuracy: The proposed model, CQD^A, outperforms current state-of-the-art methods. This improvement demonstrates the effectiveness of the re-calibrated neural link prediction scores in producing more accurate results.\n3. Data Efficiency: CQD^A achieves competitive results even with only 1% of the training data. This data efficiency is advantageous as it reduces the computational and resource requirements for training the model.\n4. Robustness: The model's robustness is demonstrated through out-of-domain evaluations. The ability to perform well in different domains further strengthens the applicability and effectiveness of CQD^A.\n\nWEAKNESSES:\n1. Interpretability of Results: While the paper mentions that the proposed approach provides interpretable answers, it would be beneficial to provide some concrete examples or explanations to illustrate this interpretability.\n2. Further Analysis of Training Data Reduction: The paper mentions that CQD^A achieves competitive results with only 1% of the training data. It would be interesting to see a more detailed analysis of the impact of reducing the training data on the model's performance across different query types and datasets.\n3. Future Directions: It would be valuable to discuss potential future directions for the research. This could include exploring the applicability of CQD^A in different domains or investigating the scalability of the approach to larger knowledge graphs.\n\nQUESTIONS:\nsee weaknesses.\n\nLIMITATIONS:\nLimitations are ignored in this paper.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-06"}

comments: SUMMARY:
This paper proposes an adaptation model on top of neural link predictors to learn score re-calibration to suit complex query answering task.Shows empirical evaluation on standard benchmark datasets to show its value. One of the benefits is a simple calibration model with lesser training data on complex questions gives better results than ones present in the literature.

STRENGTHS:
Paper is well written and easy to follow. Contributions are clearly spelled out. 
Experimental results cover complex query answering benchmarks and show the value.

WEAKNESSES:
Contribution seems incremental on top of the existing works. Seems like a small extension to CQD. 
Overall results show benefits only on certain query cases compared to state of the art.

QUESTIONS:
Model is fine tuned with 2i and 3i queries, and test results on those two query patterns don't show any win in two datasets. Any comment on why its happening?

Method seems to get clear benefit on NELL but not on other datasets. any comments on why its happening.

LIMITATIONS:
Yes
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-07

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes CQD$^\\mathcal{A}$ an adaptive variant of the complex query decomposition (CQD) approach. The authors identify that the neural link predictors used in CQD produce uncalibrated scores that can be in very different ranges of each other. The authors show that CQD$^\\mathcal{A}$ can adaptively learn an adaptive function that alleviates this problem.\nIn an extensive evaluation the authors show that this adaptive calibration imbues CQD with favorable properties such as being more data efficient, robust to OOD and being able to support negation.\n\nSTRENGTHS:\nThe paper addresses an important limitation of CQD, wherein the Neural link predictors can produce vastly different ranges of scores leading reduced performance. The authors provide a very simple fix that (inspired from Platt's scaling) allows these scores to be calibrated. \nThe authors show that this allows the model to boast better results on three benchmarks. The authors also show that this adaptive setting allows the modelling for negations that was previously not feasible.  (**Significance** and **originality**)\n\nThe paper is generally well written and easy to follow. (**clarity**)\n\nThe authors also provide extensive analysis with regards to their proposed approach which can give useful insights regarding what helps improve performance. (**quality**)\n\nWEAKNESSES:\nI think one weakness is that the approach lacks a great deal of novelty. The proposed method is a minor augmentation to an existing approach. However, I still believe that the authors have identified an important problem with CQD and have provided a technically sound solution.\n\nQUESTIONS:\nPerhaps the part about negation could be better explained in the paper.  The authors say that negations are modelled by either a standard or strictly cosine functions. In theory this could be added to vanilla CQD as well. Was this experimented with? Maybe improved results ny CQD$^\\mathcal{A}$  over vanilla CQD would bolster the importance of calibration for supporting negations.\n\nLIMITATIONS:\nThe authors have not addressed the limitations or potential societal impacts of their work.\n Perhaps the authors could talk about how CQD$^\\mathcal{A}$ can produce explainable solutions to complex questions which can create systems that can be verified.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-08"}