Here we describe a Description Logic (DL) based Inductive Logic Programming (ILP) algorithm for learning relations of order. We test our algorithm on the task of learning user preferences from pairwise comparisons. The results have implications for the development of customised recommender systems for e-commerce, and more broadly, wherever DL-based representations of knowledge, such as OWL ontologies, are used. The use of DL makes for easy integration with such data, and produces hypotheses that are easy to interpret by novice users. The proposed algorithm outperforms SVM, Decision Trees and Aleph on data from two domains

Kazakov, Dimitar Lubomirov

Qomariyah, Nunung Nurul

White Rose Research Online

This is a repository copy of Learning from Ordinal Data with Inductive Logic Programming in Description Logic.White Rose Research Online URL for this paper:https://eprints.whiterose.ac.uk/126849/Version: Accepted VersionProceedings Paper:Qomariyah, Nunung Nurul and Kazakov, Dimitar Lubomirov orcid.org/0000-0002-0637-8106 (2017) Learning from Ordinal Data with Inductive Logic Programming in Description Logic. In: Online proceedings of the 27th conference on Inductive Logic Programming. . eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Learning from Ordinal Data with InductiveLogic Programming in Description LogicNunung Nurul Qomariyah and Dimitar KazakovArtificial Intelligence GroupComputer Science, University of York, UK{nq516,dimitar.kazakov}@york.ac.ukAbstract. Here we describe a Description Logic (DL) based InductiveLogic Programming (ILP) algorithm for learning relations of order. Wetest our algorithm on the task of learning user preferences from pairwisecomparisons. The results have implications for the development of cus-tomised recommender systems for e-commerce, and more broadly, wher-ever DL-based representations of knowledge, such as OWL ontologies,are used. The use of DL makes for easy integration with such data, andproduces hypotheses that are easy to interpret by novice users. The pro-posed algorithm outperforms SVM, Decision Trees and Aleph on datafrom two domains.Keywords: Inductive logic programming, pairwise comparisons, pref-erence learning, machine learning, description logics1 IntroductionILP is now a well-established research area where machine learning meets logicprogramming [1], which has shown the potential to address many real worldproblems [2]. Description Logics (DL) are a family of languages representingdomain knowledge as concept descriptions. DL can be considered as fragmentsof First Order Logic (FOL). Because of their well-defined semantics and pow-erful inference tools, DL have been the representation of choice for ontologies,including applications, such as the Semantic Web. ILP algorithms using DL rep-resentation have the potential to be applied to large volumes of linked open dataand to benefit from the tools available for such data, e.g. IDEs such as Protégé,triplestores providing efficient storage and retrieval, and reasoners, which can testthe data for consistency and infer additional knowledge. The chosen applicationarea of Preference Learning (PL) aims to induce predictive preference modelsfrom empirical data. The automation of PL has now become essential in manye-commerce applications following the current call for customer personalisation.We build on previous work combining ILP and DL: DL-Learner [9] aims tofind a correct class description in a specific ontology from a set of positive andnegative examples of individuals. It builds a hypothesis in form of class descrip-tion, which can contain conjunction, disjunction and existential quantification.2 Nunung Nurul Qomariyah and Dimitar KazakovDL-Learner itself develops previous work on systems, such as YinYang [10] andDL-FOIL [11]. Kietz’s [12] and Konstantopoulos’s [13] work is also relevant inthis context.While DL-Learner can only learn class definitions, we aim to learn definitionsof object properties. Here we focus on the binary relation of order, which istransitive and anti-reflexive. We evaluate our algorithm on two datasets, oneon car preferences [3] and the second on sushi preferences [4]. Both datasetsprovide pairwise preference choices of multiple users. Our aim is to learn eachuser’s individual preferences using a DL-based representation, and an algorithminspired by the Progol/Aleph family of ILP tools.2 Problem RepresentationTo learn from examples of transitive anti-symmetric relations, we use the ex-amples provided by the user, along with their transitive closure, in their correctorder (e.g. “car A is better than car B”) as positive examples, and the sameexamples in reverse order as negative examples. All attributes of either item inthe pair (e.g. car A has type Sedan) are available as background knowledge. Thiswill be expressed in a class hierarchy in DL (e.g. car A is a member of subclassSedan of class Car). The specific relation we want to learn, an object propertyreflecting a type of order, needs to be specified in the hypothesis language. Asimple example using the Progol [5]/Aleph [6] convention appears below::- modeh(1,betterthan(+car,+car)).:- modeb(1,hasbodytype(+car,#bodytype)).3 The Proposed AlgorithmWe implement our algorithm in Java using the OWL API [7] library to handleDL. We follow the four basic steps used in the Progol/Aleph greedy learningapproach:1. Select a positive example. Each instance of the relation can be seen as apair of object IDs.2. Build the bottom clause. The bottom clause is the conjunction of allnon-disjoint class memberships for each object in the pair.3. Search. This step uses greedy best-first search to find a clause consistentwith the data.4. Remove covered positive examples. Our algorithm is greedy, removingall covered examples once each highest-scoring clause is added to the currenttheory.Search and refinement operator. We use a top down approach similar tothe one in Progol/Aleph. The algorithm proceeds by considering an increasingnumber of properties constraining each of the two objects in the relation. As theLearning from Ordinal Data with ILP in Description Logic 3bottom clause contains the conjunction of n constraints (of type class member-ship) on the Domain side, and same number of constraints again on the Rangeside of the relation. This will produce n × n possible pairs on the first level ofgeneralisation. (We have chosen not to consider hypotheses only constrainingone of the arguments.) We evaluate all combinations of constraints, except theones that imply the same class membership of both arguments (i.e. X is betterthan Y because they both share the same property/class membership) and thosethat have already been considered. This is illustrated in Figure 1.(Thing) betterthan (Thing)(Manual) betterthan (LargeCar) (Manual) betterthan (NonHybrid . . .(Manual ⊓ Nonhybrid) betterthan (LargeCar⊓ Manual). . .. . .⊥(Manual ⊓ NonHybrid ⊓ SmallCar ⊓ Sedan) betterthan ((LargeCar ⊓ Manual ⊓ NonHybrid ⊓ Suv)Fig. 1: Refinement OperatorWe use a common ILP scoring function, P × (P −N), where P is the numberof positive examples covered, and N – the number of negative examples covered.In the case that the solution has the same score as another alternative, Alephwill only return the first solution found. In our algorithm, we consider all thenon-redundant hypotheses that are consistent with the examples (i.e. coveredzero negative and more than 2 positive). The search will not stop until all thepossible combinations have been considered.If we have not found yet a consistent hypothesis, we continue to refine theone with the highest non-negative score, which means that we add a pair ofliterals to constrain each of the two objects in the relation. We stop at 2 literalseach for Domain and Range (this is the same as Aleph’s default clause lengthof 5). Similarly to Aleph, we also consider any examples where we cannot find aconsistent generalisation as exceptions. In this case, we add the bottom clauseas the consistent rule.4 Algorithm complexityWe implement our algorithm in one of the DL family of languages, namely ALC(attributive language with complement) [14], the basic DL language which has4 Nunung Nurul Qomariyah and Dimitar Kazakovthe least expressivity. ALC allows one to construct complex concepts from sim-pler ones using various language constructs. The capabilities include direct orindirect expression, e.g. concept disjointness, domain and range of roles, includ-ing the empty role.The most expensive process is the membership checking part for every possi-ble hypothesis. This is used for scoring the hypothesis. For every single hypoth-esis the reasoner needs to check the coverage of each hypothesis. One possibleway to reduce the complexity is by minimising the search tree and checking theredundancy without reducing the accuracy.5 EvaluationDataset. We use two publicly available preference datasets [3] [4]. Both thesushi and the car datasets have 10 items to rank which leads to 45 prefer-ence pairs per user. We take 60 users from each dataset and perform 10-foldcross validation for each user’s individual preferences. The car dataset has 4 at-tributes: body type, transmission, fuel consumption and engine size, while thesushi dataset has 7 attributes: style, major, minor, heaviness, how frequentlyconsumed by a user, price and how frequently sold. Despite the difference in thenumber of attributes in the two datasets, we found that the maximum clauselength of 4 (in Aleph and in our algorithm) is sufficient to produce consistenthypotheses.Evaluation method. The goal of this evaluation is to assess the accuracy of thepredictive power of each algorithm to solve the preference learning problem. Wecompare our algorithm with three other machine learning algorithms: SVM, theMatlab CART Decision Tree (DT) learner, and Aleph. SVM is a very commonstatistical classification algorithm that used in many domains. Similar work ofpairwise preference learning was performed by Qian et. al. [8] show that SVMcan also be used to learn in this domain. Both DT and Aleph can be includedin the evaluation since both of them are logic based learner, where the first is inpropositional logic and the latter is in First Order Logics.We learn each individual preferences and test them using 10-fold cross vali-dation. The result is shown in Table 1 and Figure 2a. According to the ANOVAtest, the result shows that there is a significant difference amongst the algorithmswith the p-value 2.0949× 10−21 for the car dataset and 7.3234× 10−36 for thesushi dataset.Table 1: Mean and standard deviation of 10-fold cross validation testSVM DT Aleph Our algorithmcar dataset 0.8317±0.12 0.7470±0.10 0.7292± 0.08 0.8936±0.05sushi dataset 0.7604±0.09 0.8094±0.06 0.7789±0.06 0.9302±0.03Learning from Ordinal Data with ILP in Description Logic 5	0.00.10.20.30.40.50.60.70.80.91.0SVM Decision Tree Aleph Our algorithmAccuracy10-fold cross validation accuracycar dataset sushi dataset(a) 10-cross validation accuracy0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1# training examples00.10.20.30.40.50.60.70.80.91accuracyAccuracy vs Number of Training ExamplesSVMDecision TreeAlephOur algorithm(b) Accuracy by varying number of training examplesFig. 2: Evaluation resultsWe also perform several experiments with the algorithms by varying theproportion of training examples and test it on 10% of examples. For a morerobust result, we validate each cycle with 10-fold cross validation. The result ofthis experiments is shown in Figure 2b. We show that our algorithm still workbetter even with the smaller number of training examples.Sample solutions found. Our algorithm can produce more readable resultsfor a novice user compared to Aleph. An example of consistent hypothesis foundby our algorithm is shown below:Automatic ⊓ Hybrid betterthan MediumCar ⊓ SuvWhile Aleph produces rules, such as:betterthan(A,B) :-hasfuelcons(B,nonhybrid), hasbodytype(B,suv).6 Conclusion and Further WorkIn this paper, we have shown that the implementation of ILP in DL can beuseful to learn a user’s preferences from pairwise comparisons. We are currentlyworking to address the following limitations of our algorithm:– We only consider one level class hierarchy in the ontology for simplicity. Inthe real world, the class hierarchy can be more complex.– Currently, our algorithm uses the Closed World Assumption, which makesit easier to find a consistent hypothesis. This is not in line with the fact thatmost DL-based knowledge databases and their reasoners operate under theOpen World Assumption.6 Nunung Nurul Qomariyah and Dimitar Kazakov– In term of accuracy, we show that our algorithm outperformed the otheralgorithms, but this is a time consuming process. In order to produce acomplete and consistent hypothesis, our algorithm takes much longer thanthe three baseline algorithms. The algorithm optimisation method is quitesimple. We need to work on the improvement of this aspect without reducingthe accuracy performance.References1. Muggleton, S., De Raedt, L., Poole, D., Bratko, I., Flach, P., Inoue, K. and Srini-vasan, A.: ILP Turns 20. In: Machine Learning, vol. 86 no. 1, pp. 3–23. Springer,Heidelberg (2012)2. 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