=Paper=
{{Paper
|id=Vol-3032/presentation2
|storemode=property
|title=Towards Explainable Relational Boosting via Propositionalization
|pdfUrl=https://ceur-ws.org/Vol-3032/SEDAMI2021-Paper3-BlazSkrlj.pdf
|volume=Vol-3032
|authors=Blaz Skrlj,Nada Lavrac
}}
==Towards Explainable Relational Boosting via Propositionalization==
https://ceur-ws.org/Vol-3032/SEDAMI2021-Paper3-BlazSkrlj.pdf
Towards Explainable Relational Boosting via
Propositionalization
Blaž Škrlj1,2[0000−0002−9916−8756] and Nada Lavrač1,2
1
Jožef Stefan Institute, Jamova 39, Ljubljana, SI
2
Jožef Stefan International Postgraduate School, Jamova 39, Ljubljana, SI
{blaz.skrlj,nada.lavrac}@ijs.si
Abstract. Many problems can be modeled by using relational struc-
tures, such as graphs or relational databases. Propositionalization refers
to the process of transforming a relational database into a propositional
(tabular) representation, suitable for a wide repertoire of machine learn-
ing algorithms, including contemporary black-box classifiers such as deep
neural networks or ensemble-based classifiers such as Gradient boosted
tree ensembles. Even though such black-box classifiers often outperform
symbolic learners, their results are hard to interpret by humans. This
paper explores the means to improve black-box classifier interpretabil-
ity in a relational setting. To this end, we conducted a series of experi-
ments to evaluate how relational features, constructed using Aleph, RSD
and Tertius propositionalization algorithms, impact the interpretability
and performance of black-box classifiers such as Gradient boosted tree
ensembles. We show how improved interpretability can be achieved by
combining XGBoost with SHAP, an algorithm that leverages the ideas
from coalitional game theory to assign importance scores to the obtained
relational features, offering both the state-of-the-art performance as well
as insights into the feature impacting individual predictions.
Keywords: Relational learning, boosting, propositionalization
1 Introduction
Relational data mining is widely used in many areas of science, including biology,
sociology, and more. The most common task solved by relational learners is entity
classification, which is the focus of this work. In relational data classification,
two conceptually different approaches are commonly used3 :
– In the first approach, a learner can learn directly from the given data. Here, a
relational learner, e.g., Aleph [20], is given a relational database, with class-
labelled training instances as input to the learner. The learner leverages the
relational structure of the database to construct a set of interpretable rules,
which cover the majority of positive examples and can be used as a classifier
(new instances can be classified using the learned rules).
3
Copyright © 2021 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
Škrlj and Lavrač.
– The second approach implements a two-step process named propositionaliza-
tion. In the first step, the relational data is transformed into a propositional
format, and in the second step, the classification of transformed tabular
data is solved by effective propositional learning algorithms, including state-
of-the-art deep neural networks [8] or Gradient boosting machines [3].
In this work we focus exclusively on the second group of algorithms, i.e.
propositionalization-based learners. Propositionalization algorithms take a rela-
tional database as input and transform it into a propositional table, consisting
of labeled instances described by features that have been automatically derived
from the underlying relational structure. These features correspond to simple
conjuncts of the form fi := p1 ∧ p2 · · · ∧ pk , where fi denotes the i-th rela-
tional feature (a conjunct of elementary relational features) and pj represents
the j-th elementary relational feature. A valid feature could be, for example,
fi := atomType(a) ∧ charge(b). Each elementary relational feature pj describes
some relational property of the dataset. For example, if instances correspond
to researchers who published papers in conferences, pj will return value true or
false for a given author. Property pj can be a rather complex relational query
involving multiple relations (as long as that query returns either true or false), or
the result of some other aggregation function. For example, the property could
be “does author X have a paper published at the ECML/PKDD conference?” or
“how many papers has author X published at the ECML/PKDD conference?”.
While such feature construction could be done manually by a data analyst, we
are only interested in automated propositionalization methods.4
The remainder of this work is structured as follows. First, we discuss some
of the related work focusing on the task of propositionalization. Next, we discuss
the notion of boosting and how it can be applied in combination with propo-
sitionalization. We next describe the experimental setting used to benchmark
a selection of existing propositionalization approaches alongside a number of
widely used propositional learners. Finally, we discuss the results and possible
implications of the proposed approach.
2 Background technologies
In the following section, we discuss the related work and techniques relevant to
this work.
2.1 Propositionalization
To be able to apply standard machine learning algorithms on multi-relational
data, the considered relational database should be transformed into a single
tabular data format, where each row represents a single data instance, and each
4
Note that simplest relational features may correspond even to single propositional
features of the form table-attribute-value, which are evaluated true if the given at-
tribute has a particular value for a given training instance.
Towards Explainable Relational Boosting via Propositionalization
column represents a feature (a simple attribute-value, a relational query pj , or
a conjunction of such queries). This transformation into symbolic vector space
(i.e. a symbolic data table format), referred to as propositionalization, is defined
below.
Definition 1 (Propositionalization). Consider the input of a given data type
and format and heterogeneous background knowledge of various data types and
formats. Propositionalization corresponds to the process of constructing a tabu-
lar (matrix) representation of the data enriched with the background knowledge,
where each row represents a single data instance and each column represents a
feature in a d-dimensional binary vector space Bd .
Propositionalization thus transforms a complex data structure such as a re-
lational database to a simpler, binary vector space, where each feature reflects
the presence (or absence) of the relational property modelled by the constructed
relational feature for the given training instance. Selected propositionalization
algorithms used in this study are briefly described below.
RSD [23] is a relational subgroup discovery algorithm composed of two main
steps: the propositionalization step and the (optional) subgroup discovery
step. The output of the propositionalization step can also be used as input
to other propositional learners. RSD effectively produces an exhaustive list
of first-order features that comply with the user-defined mode constraints,
similar to those of Progol [16] and Aleph [20]. Furthermore, RSD features
satisfy the connectivity requirement, which imposes that no feature can be
decomposed into a conjunction of two or more features. Mode declarations
define the algorithm’s syntactic bias, i.e. the space of possible features.
Tertius (Treeliker) [7] is a top-down rule discovery system, incorporating first-
order clausal logic. As no particular prediction target is specified beforehand,
Tertius can be seen as an ILP system that learns rules in an unsupervised
manner. Its relevance lies in the fact that Tertius encompasses 1BC, i.e.
relational data is handled through 1BC transformation [6].
Aleph [20] is the most popular ILP algorithm, which is actually an ILP toolkit
with many modes of functionality: learning of theories, feature construction,
incremental learning, etc. It includes a feature construction functionality,
which makes it also act as a propositionalization approach. Aleph uses mode
declarations to define the syntactic bias. Input relations are Prolog clauses,
defined either extensionally or intensionally.
2.2 Learning by Boosting
In this section, we discuss the notion of boosting machine learning, focusing on
the aspects used in this work. We begin with an overview of this methodology.
Boosting refers to a group of learning algorithms. The key idea of boosting
revolves around the fact that a series of weak learners can, when joined, form a
strong learner [9]. In general, boosting algorithms iteratively construct a strong
Škrlj and Lavrač.
classifier, where weak learners’ predictions are used to weight the following mod-
els, emphasizing misclassifications: misclassified input data gain a higher weight
and examples that are classified correctly lose weight. Thus, future weak learn-
ers focus more on the examples that previous weak learners have mis-classified.
Boosting algorithms are described, for example, in the AnyBoost framework [12].
Furthermore, it was shown theoretically that boosting performs gradient descent
in a function space using a convex cost function. Commonly, boosting-based en-
sembles are not directly explainable. We next discuss how SHAP [11], a game
theory-based approach, can be used to overcome this issue partially.
2.3 Explaining black-box models
Recent trends in machine learning attempt to solve hard problems using black-
box, non-interpretable models. Even though such models are not explainable per
se, individual predictions can be approximated using existing, symbolic learners
in order to obtain feature relevances for a given classifier. As the final part of
the learning workflow proposed in this work we exploit the recently introduced
SHAP tool [11] to identify which features, constructed using propositionalization
approaches, were the most relevant for a given classification problem.
Shapley regression values can be interpreted as feature importances for linear
models in the presence of multicolinearity. The method begins by training the
model on all subsets S ⊆ F , where F is the set of all features. Each feature is
assigned a numeric score denoting the impact of this feature on a given model’s
performance. This impact is computed by training two models; one with a given
feature and one without. Predictions of the two models are compared on a given
input. As the effect of withholding a single feature from the model depends on all
other features, the performance differences are computed for all possible subsets
S ⊆ F \ fi ; where i denotes fi -th feature. Shapley values ρ can be thus (for the
fi -th feature) defined as:
X |S|!(|F | − |S| − 1)!
ρi = mS∪fi (xS∪fi ) − mS (xS ) ;
|F |!
S⊆F \fi
where mS∪fi (xS∪fi ) corresponds to the model’s performance when the fi -th
feature is considered and mS (xS ) when it is not considered. The xS corresponds
to feature values of the feature set S. The used SHAP methodology employs
efficient sampling schemes for computing feature relevance. In this work, we use
the “tree-explainer” module of SHAP. We refer the interested reader to [11] for
a detailed overview of SHAP. The empirical setup is discussed next.
3 Propositionalization and semantic data mining
Recall the definition of propositionalization. (Definition 1) that involves hetero-
geneous background knowledge of various data types and formats with the aim to
construct a tabular (matrix) representation of the data enriched with the back-
ground knowledge, where each column represents a feature in a d-dimensional
Towards Explainable Relational Boosting via Propositionalization
binary vector space Bd . Let τ represent a mapping from a relational database
(RDB) to a d-dimensional binary vector space B. Let n represent the number
of instances from the target table, for which relational features are constructed.
Each propositionalization algorithm thus attempts to identify relational features
representing the initial RDB, and can be described as the mapping:
τ : RDB → B n×d . (1)
An example b ∈ B 5×5 can be represented as
p5
p1 ∧
p2
p2
p3
p2
p1 ∧
p3 ∧
p1 ∧
p5 ∧
p4 ∧
1 1 1 1 0
0 1 0 0 1
0 0 1 0 0
0 1 0 0 1
1 0 0 0 0
Features pj describing the data can either be low-level features describing the
actual experimental data or higher-level semantic features representing higher-
level concepts in ontologies or taxonomies encoded as the background knowl-
edge. The latter refers to the semantic data mining setting. Higher-level features
have higher generalization potential, given that the use of these features ensures
higher coverage of induced rules.
4 Proposed approach and experimental setting
In this section we describe the experimental setting used to evaluate the per-
formance of various classifiers trained using some of the mentioned proposition-
alization approaches. The two-step approach described next is summarized in
Figure 1. Here, a relational database is used as input, where only a single en-
tity (table — green circle) is considered for learning, and all others are used to
construct features related to this target table. The algorithm yields a proposi-
tionalized database of dimension n × d, where n corresponds to the number of
examples and d to a number of (selected) relational features. Note that values
of feature vectors are boolean {0, 1}.
4.1 Propositionalization algorithms implementation
We tested the following propositionalization methods, which represent a selection
of well-established approaches developed in the Inductive Logic Programming
(ILP) community: RSD [23], Tertius [7] and Aleph [20], presented in Section 2.1.
Škrlj and Lavrač.
d
Boosting
Propositionalized data
Relational data
Fig. 1. Schematic representation of propositionalization combined with boosting. A
d-dimensional representation of the input relational structure is used as input to a
boosting-based learner.
As individual propositionalization algorithms are implemented in different
programming languages using different frameworks, executing all of the ap-
proaches in the same learning setting can be very challenging. The RSD and
Aleph algorithms are written in Prolog, while Tertius is in Java. We built on
a recent effort to unify propositionalization approaches under a joint Python
framework called PyRDM [22]5 , which is freely accessible, and was updated as
part of this work.6 A contribution of this work is also a significant extension of
the PyRDM library to include the newest libraries for data pre-processing and
manipulation, including Orange 3 [5], Pandas [13], and Scikit-learn [17].
4.2 The classifiers used
In this subsection, we describe the classifiers that learned from the transformed,
propositionalized databases. The classifiers were tested on all the propositional-
ization methods described in the previous section.
Support vector machines. We use the implementation, available as part of
the libSVM library [2]. We used C = 20 for the regularization value.
Gradient boosting machines. The GBM implementation [17] used is sup-
ported in Scikit-learn. We use the default settings for learning. We varied
the number of estimators from the range: 10,20,50,100.
Logistic regression. This classifier is also implemented in Scikit-learn [17]. It
uses a logistic function to model a binary distributed (dependent) random
variable. The regularization parameter was set to 1 (default setting).
Extra-randomized trees. Extremely randomized trees are an algorithm that
attempts to capture relevant patterns by constructing larger, random trees.
We use the implementation supported in Scikit-learn [17]. We set the number
of estimators to 40. We set the number of samples needed for a split to 2.
We varied the number of estimators from the range: 10,20,50,100.
5
https://github.com/xflows/rdm
6
https://github.com/xflows/rdm
Towards Explainable Relational Boosting via Propositionalization
Fig. 2. Carcinogenesis database scheme. Propositionalization algorithms attempt to
traverse the database by foreign keys and capture the patterns of relevance to the task.
Extreme Gradient Boosting [3]. The known XGB classifier is one of the
state-of-the-art learners commonly used in contests. We varied the number
of estimators from the range: 10,20,50,100.
4.3 Data sets used
In this section, we describe the data sets considered for experimental evaluation.
The data sets were obtained from the CTU repository [15]. The data sets were
selected as they include diverse schemas and come from different domains. An
example scheme for the Carcinogenesis data is shown in Figure 2.
Carcinogenesis [21] task is to predict the carcinogenicity of a diverse set of
chemical compounds. The data set was obtained by testing different chem-
icals on rodents, where each trial would take several years and hundreds of
animals. The data set consists of 329 compounds, of which One hundred
eighty-two are carcinogens.
Mutagenesis [4] task addresses the problem of predicting mutagenicity of aro-
matic and heteroaromatic nitro compounds. Predicting mutagenicity is an
important task as it is very relevant to the prediction of carcinogenesis. The
compounds from the data are known to be more structurally heterogeneous
than in any other ILP data set of chemical structures. The database contains
230 compounds, of which 138 have positive levels of mutagenicity and are
labelled as ‘active’. Others have class value ‘inactive’ and are considered to
be negative examples. We took the data sets of the original paper [4], where
the data was split into two subsets: 188 compound data set and a smaller
data set with 42 compounds.
Škrlj and Lavrač.
Trains [14] the task addresses the prediction of a given train’s direction based
on the cargo and other trains’ properties. This is one of the canonical data
sets for relational learning.
Facebook [10] the data set, consisting of 10,137 rows and 265 columns, repre-
sents Facebook friendships, where the task is to predict the gender of users.
4.4 The scoring scheme
We validated individual propositionalization-learner pairs using 10-fold stratified
cross-validation, as implemented in PyRDM. For each fold split, we run first a
given propositionalization algorithm, followed by a learner. This way, folds re-
main consistent; hence the algorithms’ results can be compared. We computed
Accuracy and AUCs for individual classifiers (all are binary classification prob-
lems). If the propositionalization-learner pair was not able to finish in 3 hours,
we marked such runs as unsuccessful. We present the results as average classifier
ranks, plotted in the same space as vertical lines along a horizontal line corre-
sponding to all possible algorithm combinations. Such representation is sufficient,
as we were mainly interested in whether Extreme Gradient Boosting (XGB) per-
forms well when combined with the considered transformation methods, as this
relationship was not explored before. Further, good performance of XGB would
indicate explanations are sensible, as if the model performed badly, explanations
can be less reliable. We use the SHAP algorithm as follows. For a given data
set, we train the XGB model on all instances. We compute the Shapley matrix,
which contains feature relevance information. Feature importances are finally
visualized for selected (correctly classified) examples.
5 Results
We present the results obtained using the proposed benchmark. We begin by
showing bar diagrams where differences between classifiers are shown with re-
spect to a given propositionalization algorithm. Next, we present the critical
distance diagrams showing classifier-propositionalization algorithm performance.
Finally, we show how XGB’s predictions can be interpreted using SHAP.
5.1 Results - classification
We observe the best overall classifier performances were obtained when RSD was
used as the propositionalization method. As can be observed in the following
results corresponding to individual classifier performances, the Aleph proposi-
tionalization performed the worst on average. The average rank diagram showing
the performance of individual classifier-propositionalization pairs (or just propo-
sitionalization) is shown in Figures 3, 4 and 5. Note that we omit computation
of critical distances due to reasons stated in [1].
The classification results further indicate that propositionalization represents
the key aspect of the overall classification performance. This observation would
Towards Explainable Relational Boosting via Propositionalization
SVM treeliker
LR treeliker DT aleph
XT aleph LR aleph
DT RSD SVM aleph
DT treeliker MF treeliker
XT treeliker XGB treeliker
SVM RSD MF RSD
LR RSD GBM treeliker
GBM RSD GBM aleph
XT RSD MF aleph
XGB RSD XGB aleph
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Fig. 3. Propositionalization + learner Accuracy comparison across data sets. Average
ranks are shown. LR treeliker
XT aleph SVM aleph
SVM treeliker XGB treeliker
XGB RSD GBM treeliker
XT treeliker GBM aleph
DT treeliker MF treeliker
DT RSD LR aleph
XT RSD DT aleph
GBM RSD MF RSD
SVM RSD MF aleph
LR RSD XGB aleph
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Fig. 4. Propositionalization + learner AUC comparison across data sets. Average ranks
are shown.
treeliker
RSD aleph
1 2 3
Fig. 5. Comparison of propositionalization approaches (classifier accuracy was aver-
aged).
imply that no matter how well the classifier can learn if the input space is not
representative of the modelled problem, the classifier will not learn well, which is
Škrlj and Lavrač.
indeed a sensible assumption in the relational setting, as the considered propo-
sitionalization algorithms consider different aspects of the relational database,
and do not output equal representations.
5.2 Results - explanation example
In this section, we look at an example where we analysed feature importances
via SHAP. The computed importance-based diagram is shown in Figure 6.
Fig. 6. Feature importances using Shapley values (Mutagenesis). Here, an XGB clas-
sifier was trained on the feature space, constructed with RSD.
Conjuncts of atom types and charges emerged as relevant for the classi-
fication performance for the mutagenesis (42) data set (Figure 6), indicating
that the two attributes are highly relevant for correctly predicting the muta-
genicity. Note, however, that the propositionalization is not able to perform
discretization, meaning that the explanations are potentially too specific for non-
categorical/ordinal variables. Further, it can be observed that there are at least
three relational features that impact the model’s prediction the most, indicating
that diverse relational features were used by the XGBoost learner.
6 Discussion and conclusions
In this section, we discuss the obtained results and present some of the relevant
future prospects regarding boosting in a relational setting.
Towards Explainable Relational Boosting via Propositionalization
Overall, we tested a set of different learners trained on feature matrices ob-
tained by using three different propositionalization algorithms. We show that
boosting based learners outperform many other classifiers. Further, we believe
the conducted experiments show a direction for future research, where scalability
could be an issue (boosting is very scalable). One of the results of this study also
shows that the RSD algorithm is one of the top-performing propositionalization
algorithms. Although it performs well, we believe that RSD and other proposi-
tionalization algorithms do not scale well, which is an issue that can be addressed
using ideas proposed in [18, 19], which will be addressed in future work. To the
best of our knowledge, this is one of the first works which utilize SHAP to ex-
plain relational features. The obtained results indicate that Shapley values are
as such one of the possible ways to explain how the black box learners learn in
a relational setting. We have yet to explore the different discretization schemes,
which would yield more relevant explanations. We believe the presented results
demonstrate the relevance of such approaches and are to our knowledge the first
of such kind.
7 Availability
The code is freely available as part of the PyRDM library7 .
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