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Exception-enrcihed Rule Learning from Knowledge Graphs

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Mohamed Gad-elrab
Mohamed Gad-elrab

Presentation in the International Semantic Web Conference (ISWC2016), Kobe Japan. October 2016

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Exception-enriched Rule Learning from Knowledge Graphs Mohamed Gad-Elrab1, Daria Stepanova1, Jacopo Urbani 2, Gerhard Weikum1 1Max-Planck-Institut fΓΌr Informatik, Saarland Informatics Campus, Germany 2 Vrije Universiteit Amsterdam, Amsterdam, The Netherlands 21st October 2016
Knowledge Graphs (KGs) 2 β€’ Huge collection of < 𝑠𝑒𝑏𝑗𝑒𝑐𝑑, π‘π‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘‘π‘’, π‘œπ‘π‘—π‘’π‘π‘‘ > triples β€’ Positive facts under Open World Assumption (OWA) β€’ Possibly incomplete and/or inaccurate
Mining Rules from KGs 3 Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football
Mining Rules from KGs 4 π‘Ÿ: 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 𝑋, 𝑍 ← π‘–π‘ π‘€π‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘π‘‡π‘œ 𝑋, π‘Œ , 𝑙𝑖𝑣𝑒𝑠𝐼𝑛(π‘Œ, 𝑍) Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football [GalΓ‘rraga et al., 2015]
Mining Rules from KGs 5 π‘Ÿ: 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 𝑋, 𝑍 ← π‘–π‘ π‘€π‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘π‘‡π‘œ 𝑋, π‘Œ , 𝑙𝑖𝑣𝑒𝑠𝐼𝑛(π‘Œ, 𝑍) Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football 1 2
Our Goal 6 π‘Ÿ: 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 𝑋, 𝑍 ← π‘–π‘ π‘€π‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘π‘‡π‘œ 𝑋, π‘Œ , 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 π‘Œ, 𝑍 , π‘›π‘œπ‘‘ 𝑖𝑠𝐴(𝑋, π‘Ÿπ‘’π‘ ) Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football 1 2
Problem Statement β€’ Quality-based theory revision problem β€’ Given β€’ Knowledge graph 𝐾𝐺 β€’ Set of Horn rules 𝑅 𝐻 β€’ Find the nonmonotonic revision 𝑅 𝑁𝑀 of 𝑅 𝐻 β€’ Maximize top-k avg. confidence β€’ Minimize conflicting prediction 7
Problem Statement β€’ Quality-based theory revision problem β€’ Given β€’ Knowledge graph 𝐾𝐺 β€’ Set of Horn rules 𝑅 𝐻 β€’ Find the nonmonotonic revision 𝑅 𝑁𝑀 of 𝑅 𝐻 β€’ Maximize top-k avg. confidence β€’ Minimize conflicting prediction 8 Unknown
Problem Statement: Conflicting Predictions β€’ Defining conflicts β€’ Measuring conflicts (auxiliary rules) 9 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) 𝑅 = π‘Ÿ1: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← π‘–π‘ πΊπ‘Žπ‘šπ‘’ 𝑋 , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ (𝑋) π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) {π‘–π‘ πΊπ‘Žπ‘šπ‘’ π‘Ž , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ π‘Ž , β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ π‘Ž , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄ π‘Ž } Output: {(π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž , π‘Ÿ2: π‘›π‘œπ‘‘ π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž )} π‘Ž =
Problem Statement: Conflicting Predictions β€’ Defining conflicts β€’ Measuring conflicts (auxiliary rules) 10 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) 𝑅 = π‘Ÿ1: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← π‘–π‘ πΊπ‘Žπ‘šπ‘’ 𝑋 , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ (𝑋) π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) {π‘–π‘ πΊπ‘Žπ‘šπ‘’ π‘Ž , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ π‘Ž , β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ π‘Ž , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄ π‘Ž } Output: {(π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž , π‘Ÿ2: π‘›π‘œπ‘‘ π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž )} π‘Ž =
Problem Statement: Conflicting Predictions β€’ Defining conflicts β€’ Measuring conflicts (auxiliary rules) 11 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) 𝑅 = π‘Ÿ1: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← π‘–π‘ πΊπ‘Žπ‘šπ‘’ 𝑋 , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ (𝑋) π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) {π‘–π‘ πΊπ‘Žπ‘šπ‘’ π‘Ž , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ π‘Ž , β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ π‘Ž , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄ π‘Ž } (π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž , π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž ) π‘Ž =
Approach Overview Step 1 β€’ Mining Horn Rules Step 2 β€’ Extracting Exception Witness Set (EWS) Step 3 β€’ Constructing Candidate Revisions Step 4 β€’ Selecting the Best Revision 12
bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ p6 βœ“ βœ“ p7 βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ Step 1: Mining Horn Rules 13 π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normal
bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ p6 βœ“ βœ“ p7 βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normalStep 2: Extracting Exception Witness Set (EWS) 14 πΈπ‘Šπ‘†π‘– = {π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 , π‘π‘œπ‘’π‘‘ 𝑋 }
Step 3: Constructing Candidate Revisions β€’ Horn rules β€’ Rule revisions 15 π‘Ÿ1: π‘™π‘–π‘£π‘’π‘ πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 , π‘›π‘œπ‘‘ π‘π‘œπ‘’π‘‘ 𝑋 π‘Ÿ1: π‘™π‘–π‘£π‘’π‘ πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 , π‘›π‘œπ‘‘ π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 … 𝑅 = π‘Ÿ1: π‘™π‘–π‘£π‘’π‘ πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄(𝑋) π‘Ÿ2: π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 ← π‘ π‘‘π‘Žπ‘‘π‘’π‘™π‘’π‘ π‘ (𝑋) πΈπ‘Šπ‘† = {π‘π‘œπ‘’π‘‘ 𝑋 , π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 , … } πΈπ‘Šπ‘† = {𝑒1 𝑋 , 𝑒2(𝑋), … }
Step 4: Selecting the Best Revision Finding globally best revision is expensive! β€’ NaΓ―ve ranker β€’ For each rule, pick the revision that maximizes confidence β€’ Works in isolation from other rules β€’ Partial materialization ranker β€’ 𝐾𝐺s are incomplete! β€’ Augment the original 𝐾𝐺 with predictions of other rules β€’ Rank revisions on avg. confidence of the rule and its auxiliary. 16
β€’ Partial materialization Ranking Rule’s Revisions 17 bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ p6 βœ“ βœ“ p7 βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normal
β€’ Partial materialization bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ βœ“ p6 βœ“ βœ“ βœ“ p7 βœ“ βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ βœ“ π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normalRanking Rule’s Revisions 18
Ranking Rule’s Revisions β€’ Ordered partial materialization ranker β€’ Only rules with higher quality β€’ Ordered weighted partial materialization ranker β€’ KG fact weight = 1 β€’ Predicted facts inherit their weights from the rules 19
Experiments β€’ Ruleset quality 20 *Higher is better 0.60 0.65 0.70 0.75 0.80 20 40 60 80 100 Avg.Confidence Top-K (%) Rules YAGO3 Horn Naive Ordered & Weighted PM 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 20 40 60 80 100Avg.Confidence Top-K (%) Rules IMDB Horn Naive Ordered & Weighted PM Facts 10M 2M Rules 10K 25K
Experiments β€’ Predictions consistency 21*Lower is better 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 200 400 600 800 1000 ConflictRatio Top-K Rules YAGO3 Naive Ordered & Weighted PM 0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000ConflictRatio Top-K Rules IMDB Naive Ordered & Weighted PM
Experiments β€’ Examples 22 π‘–π‘ π‘€π‘œπ‘’π‘›π‘‘π‘Žπ‘–π‘›(𝑋) ← π‘–π‘ πΌπ‘›π΄π‘’π‘ π‘‘π‘Ÿπ‘–π‘Ž(𝑋), π‘–π‘ πΌπ‘›πΌπ‘‘π‘Žπ‘™π‘¦ 𝑋 , π‘›π‘œπ‘‘ π‘–π‘ π‘…π‘–π‘£π‘’π‘Ÿ (𝑋) π‘–π‘ π‘ƒπ‘œπ‘™π‘–π‘‘π‘‚π‘“π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 , π‘–π‘ πΊπ‘œπ‘£ 𝑋 , π‘›π‘œπ‘‘ π‘–π‘ π‘ƒπ‘œπ‘™π‘–π‘‘π‘ƒπ‘’π‘’π‘Ÿπ‘‘π‘œπ‘…π‘–π‘π‘œ(𝑋) π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘Žπ‘π‘‘π‘’π‘‘πΌπ‘›π‘€π‘œπ‘£π‘–π‘’ 𝑋 , π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’π‘‘π‘€π‘œπ‘£π‘–π‘’ 𝑋 , π‘›π‘œπ‘‘ π‘€π‘œπ‘›πΉπ‘–π‘™π‘šπ‘“π‘Žπ‘Ÿπ‘’(𝑋)
Summary β€’ Conclusion β€’ Quality-based theory revision under OWA β€’ Partial materialization for ranking revisions β€’ Comparison of ranking methods on real life KGs β€’ Outlook β€’ Extending to higher arity predicates β€’ Binary predicates [Tran et al., to appear ILP2016] β€’ Evidence from text corpora β€’ Exploiting partial completeness 23
References β€’ [Angiulli and Fassetti, 2014] Fabrizio Angiulli and Fabio Fassetti. Exploiting domain knowledge to detect outliers. Data Min. Knowl. Discov., 28(2):519–568, 2014. β€’ [Dimopoulos and Kakas, 1995] Yannis Dimopoulos and Antonis C. Kakas. Learning non-monotonic logic programs: Learning exceptions. In Machine Learning: ECML-95, 8th European Conference on Machine Learning, Heraclion, Crete, Greece, April 25-27, 1995, Proceedings, pages 122–137, 1995. β€’ [Galarraga et al., 2015] Luis Galarraga, Christina Teflioudi, Katja Hose, and Fabian M. Suchanek. Fast Rule Mining in Ontological Knowledge Bases with AMIE+. In VLDB Journal, 2015. β€’ [Law et al., 2015] Mark Law, Alessandra Russo, and Krysia Broda. The ILASP system for learning answer set programs, 2015. β€’ [Leone et al., 2006] Nicola Leone, Gerald Pfeifer, Wolfgang Faber, Thomas Eiter, Georg Gottlob, Simona Perri, and Francesco Scarcello. 2006. The DLV system for knowledge representation and reasoning. ACM Trans. Comput. Logic 7, 3 (July 2006), 499-562. β€’ [Suzuki, 2006] Einoshin Suzuki. Data mining methods for discovering interesting exceptions from an unsupervised table. J. UCS, 12(6):627–653, 2006. β€’ [Tran et al., 2016] Hai Dang Tran, Daria Stepanova, Mohamed H. Gad-Elrab, Francesca A. Lisi, Gerhard Weikum. Towards Nonmonotonic Relational Learning from Knowledge Graphs. ILP2016, London, UK, to appear. β€’ [Katzouris et al., 2015] Nikos Katzouris, Alexander Artikis, and Georgios Paliouras. Incremental learning of event definitions with inductive logic programming. Machine Learning, 100(2-3):555– 585, 2015. 24
Related Work β€’ Learning nonmonotonic programs β€’ E.g., [Dimopoulos and Kakas, 1995], ILASP [Law et al., 2015], ILED [Katzouris et al., 2015], etc. β€’ Outlier detection in logic programs β€’ E.g., [Angiulli and Fassetti, 2014], etc. β€’ Mining exception rules β€’ E.g., [Suzuki, 2006], etc. 25
Problem Statement: Ruleset Quality β€’ Independent Rule Measure (π‘Ÿπ‘š) β€’ Support: 𝑠𝑒𝑝𝑝 𝐻 ← 𝐡 = 𝑠𝑒𝑝𝑝(𝐻 βˆͺ 𝐡) β€’ Coverage: cπ‘œπ‘£ 𝐻 ← 𝐡 = 𝑠𝑒𝑝𝑝(𝐡) β€’ Confidence: cπ‘œπ‘›π‘“ 𝐻 ← 𝐡 = 𝑠𝑒𝑝𝑝(𝐻βˆͺ𝐡) 𝑠𝑒𝑝𝑝(𝐡) β€’ Lift: 𝑙𝑖𝑓𝑑 𝐻 ← 𝐡 = π‘π‘œπ‘›π‘“(𝐻←𝐡) 𝑠𝑒𝑝𝑝(𝐻) β€’ … β€’ Average Ruleset Quality 26 π‘ž π‘Ÿπ‘š 𝑅 𝑁𝑀, 𝐺 = π‘Ÿ βˆˆπ‘… 𝑁𝑀 π‘Ÿπ‘š(π‘Ÿ, 𝐺) 𝑅 𝑁𝑀
Problem Statement: Conflicting Predictions β€’ Measuring conflicts (auxiliary rules) 27 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘ž π‘π‘œπ‘›π‘“π‘™π‘–π‘π‘‘ 𝑅 𝑁𝑀, 𝐺 = |{(𝑝 π‘Ž , π‘›π‘œπ‘‘_𝑝 π‘Ž ), … }| {π‘›π‘œπ‘‘_𝑝 π‘Ž , … }
Propositionalization β€’ Unary predicates 28 Binary Predicates β€’ hasType(enstein, scientist) β€’ isMarriedTo(elsa, einstein) β€’ bornIn(einstein, um) Unary β€’ isAScientist(einstein) β€’ isMarriedToEinstein(elsa) β€’ bornInUlm(einstein) Abstraction β€’ isAScientist(einstein) β€’ isMarriedToScientist(elsa) β€’ bornInGermany(einstein)
Experiments β€’ Input β€’ Experiment statistics 29 YAGO3 IMDB Input Facts 10M 2M Horn Rules 10K 25K Revised Rules 6K 22K General-purpose KG Domain-specific KG (Movies)
Experiments β€’ Predictions assessment β€’ Run DLV on YAGO and 𝑅 𝐻 then 𝑅 𝑁𝑀 seperately β€’ Sample facts such that fact 𝑓 ∈ π‘Œπ΄πΊπ‘‚ π»π‘Œπ΄πΊπ‘‚ 𝑁𝑀 β€’ 73% of the sampled facts were found to be erroneous 30 checked predictions
Ranking Rule’s Revisions β€’ Partial materialization ranker β€’ Augment the original 𝐾𝐺 with predictions of other rules β€’ Rank revisions on Avg. confidence of the π‘Ÿ and π‘Ÿ π‘Žπ‘’π‘₯ 31 π‘ π‘π‘œπ‘Ÿπ‘’ π‘Ÿπ‘’, πΎπΊβˆ— = π‘π‘œπ‘›π‘“ π‘Ÿπ‘’, πΎπΊβˆ— + π‘π‘œπ‘›π‘“(π‘Ÿπ‘’ π‘Žπ‘’π‘₯, πΎπΊβˆ—) 2 where π‘Ÿπ‘’ is the rule π‘Ÿ with exception 𝑒 & πΎπΊβˆ— is the augmented 𝐾𝐺.

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Exception-enrcihed Rule Learning from Knowledge Graphs

  • 1. Exception-enriched Rule Learning from Knowledge Graphs Mohamed Gad-Elrab1, Daria Stepanova1, Jacopo Urbani 2, Gerhard Weikum1 1Max-Planck-Institut fΓΌr Informatik, Saarland Informatics Campus, Germany 2 Vrije Universiteit Amsterdam, Amsterdam, The Netherlands 21st October 2016
  • 2. Knowledge Graphs (KGs) 2 β€’ Huge collection of < 𝑠𝑒𝑏𝑗𝑒𝑐𝑑, π‘π‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘‘π‘’, π‘œπ‘π‘—π‘’π‘π‘‘ > triples β€’ Positive facts under Open World Assumption (OWA) β€’ Possibly incomplete and/or inaccurate
  • 3. Mining Rules from KGs 3 Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football
  • 4. Mining Rules from KGs 4 π‘Ÿ: 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 𝑋, 𝑍 ← π‘–π‘ π‘€π‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘π‘‡π‘œ 𝑋, π‘Œ , 𝑙𝑖𝑣𝑒𝑠𝐼𝑛(π‘Œ, 𝑍) Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football [GalΓ‘rraga et al., 2015]
  • 5. Mining Rules from KGs 5 π‘Ÿ: 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 𝑋, 𝑍 ← π‘–π‘ π‘€π‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘π‘‡π‘œ 𝑋, π‘Œ , 𝑙𝑖𝑣𝑒𝑠𝐼𝑛(π‘Œ, 𝑍) Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football 1 2
  • 6. Our Goal 6 π‘Ÿ: 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 𝑋, 𝑍 ← π‘–π‘ π‘€π‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘π‘‡π‘œ 𝑋, π‘Œ , 𝑙𝑖𝑣𝑒𝑠𝐼𝑛 π‘Œ, 𝑍 , π‘›π‘œπ‘‘ 𝑖𝑠𝐴(𝑋, π‘Ÿπ‘’π‘ ) Amsterdam isMarriedToJohn Kate Chicago isMarriedToBrad Anna Berlin hasBrother isMarriedToDave ClaraisMarriedToBob Alice Researcher Berlin Football 1 2
  • 7. Problem Statement β€’ Quality-based theory revision problem β€’ Given β€’ Knowledge graph 𝐾𝐺 β€’ Set of Horn rules 𝑅 𝐻 β€’ Find the nonmonotonic revision 𝑅 𝑁𝑀 of 𝑅 𝐻 β€’ Maximize top-k avg. confidence β€’ Minimize conflicting prediction 7
  • 8. Problem Statement β€’ Quality-based theory revision problem β€’ Given β€’ Knowledge graph 𝐾𝐺 β€’ Set of Horn rules 𝑅 𝐻 β€’ Find the nonmonotonic revision 𝑅 𝑁𝑀 of 𝑅 𝐻 β€’ Maximize top-k avg. confidence β€’ Minimize conflicting prediction 8 Unknown
  • 9. Problem Statement: Conflicting Predictions β€’ Defining conflicts β€’ Measuring conflicts (auxiliary rules) 9 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) 𝑅 = π‘Ÿ1: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← π‘–π‘ πΊπ‘Žπ‘šπ‘’ 𝑋 , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ (𝑋) π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) {π‘–π‘ πΊπ‘Žπ‘šπ‘’ π‘Ž , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ π‘Ž , β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ π‘Ž , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄ π‘Ž } Output: {(π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž , π‘Ÿ2: π‘›π‘œπ‘‘ π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž )} π‘Ž =
  • 10. Problem Statement: Conflicting Predictions β€’ Defining conflicts β€’ Measuring conflicts (auxiliary rules) 10 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) 𝑅 = π‘Ÿ1: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← π‘–π‘ πΊπ‘Žπ‘šπ‘’ 𝑋 , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ (𝑋) π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) {π‘–π‘ πΊπ‘Žπ‘šπ‘’ π‘Ž , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ π‘Ž , β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ π‘Ž , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄ π‘Ž } Output: {(π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž , π‘Ÿ2: π‘›π‘œπ‘‘ π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž )} π‘Ž =
  • 11. Problem Statement: Conflicting Predictions β€’ Defining conflicts β€’ Measuring conflicts (auxiliary rules) 11 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) 𝑅 = π‘Ÿ1: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← π‘–π‘ πΊπ‘Žπ‘šπ‘’ 𝑋 , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ (𝑋) π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) {π‘–π‘ πΊπ‘Žπ‘šπ‘’ π‘Ž , π‘–π‘ π΅π‘Žπ‘ π‘’π‘‘π‘‚π‘›π½π‘ƒπ΄π‘›π‘–π‘šπ‘’ π‘Ž , β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ π‘Ž , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄ π‘Ž } (π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž , π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ π‘Ž ) π‘Ž =
  • 12. Approach Overview Step 1 β€’ Mining Horn Rules Step 2 β€’ Extracting Exception Witness Set (EWS) Step 3 β€’ Constructing Candidate Revisions Step 4 β€’ Selecting the Best Revision 12
  • 13. bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ p6 βœ“ βœ“ p7 βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ Step 1: Mining Horn Rules 13 π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normal
  • 14. bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ p6 βœ“ βœ“ p7 βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normalStep 2: Extracting Exception Witness Set (EWS) 14 πΈπ‘Šπ‘†π‘– = {π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 , π‘π‘œπ‘’π‘‘ 𝑋 }
  • 15. Step 3: Constructing Candidate Revisions β€’ Horn rules β€’ Rule revisions 15 π‘Ÿ1: π‘™π‘–π‘£π‘’π‘ πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 , π‘›π‘œπ‘‘ π‘π‘œπ‘’π‘‘ 𝑋 π‘Ÿ1: π‘™π‘–π‘£π‘’π‘ πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 , π‘›π‘œπ‘‘ π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 … 𝑅 = π‘Ÿ1: π‘™π‘–π‘£π‘’π‘ πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄(𝑋) π‘Ÿ2: π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 ← π‘ π‘‘π‘Žπ‘‘π‘’π‘™π‘’π‘ π‘ (𝑋) πΈπ‘Šπ‘† = {π‘π‘œπ‘’π‘‘ 𝑋 , π‘’π‘šπ‘–π‘”π‘Ÿπ‘Žπ‘›π‘‘ 𝑋 , … } πΈπ‘Šπ‘† = {𝑒1 𝑋 , 𝑒2(𝑋), … }
  • 16. Step 4: Selecting the Best Revision Finding globally best revision is expensive! β€’ NaΓ―ve ranker β€’ For each rule, pick the revision that maximizes confidence β€’ Works in isolation from other rules β€’ Partial materialization ranker β€’ 𝐾𝐺s are incomplete! β€’ Augment the original 𝐾𝐺 with predictions of other rules β€’ Rank revisions on avg. confidence of the rule and its auxiliary. 16
  • 17. β€’ Partial materialization Ranking Rule’s Revisions 17 bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ p6 βœ“ βœ“ p7 βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normal
  • 18. β€’ Partial materialization bornInUSA livesInUSA stateless emigrant singer poet p1 βœ“ βœ“ p2 βœ“ βœ“ p3 βœ“ βœ“ p4 βœ“ βœ“ βœ“ βœ“ p5 βœ“ βœ“ βœ“ βœ“ p6 βœ“ βœ“ βœ“ p7 βœ“ βœ“ βœ“ p8 βœ“ βœ“ βœ“ βœ“ p9 βœ“ βœ“ βœ“ βœ“ βœ“ p10 βœ“ βœ“ βœ“ βœ“ p11 βœ“ βœ“ βœ“ βœ“ π’“π’Š: π’π’Šπ’—π’†π’”π‘°π’π‘Όπ‘Ίπ‘¨ 𝑿 ← 𝒃𝒐𝒓𝒏𝑰𝒏𝑼𝑺𝑨(𝑿) NormalAb-normalRanking Rule’s Revisions 18
  • 19. Ranking Rule’s Revisions β€’ Ordered partial materialization ranker β€’ Only rules with higher quality β€’ Ordered weighted partial materialization ranker β€’ KG fact weight = 1 β€’ Predicted facts inherit their weights from the rules 19
  • 20. Experiments β€’ Ruleset quality 20 *Higher is better 0.60 0.65 0.70 0.75 0.80 20 40 60 80 100 Avg.Confidence Top-K (%) Rules YAGO3 Horn Naive Ordered & Weighted PM 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 20 40 60 80 100Avg.Confidence Top-K (%) Rules IMDB Horn Naive Ordered & Weighted PM Facts 10M 2M Rules 10K 25K
  • 21. Experiments β€’ Predictions consistency 21*Lower is better 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 200 400 600 800 1000 ConflictRatio Top-K Rules YAGO3 Naive Ordered & Weighted PM 0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000ConflictRatio Top-K Rules IMDB Naive Ordered & Weighted PM
  • 22. Experiments β€’ Examples 22 π‘–π‘ π‘€π‘œπ‘’π‘›π‘‘π‘Žπ‘–π‘›(𝑋) ← π‘–π‘ πΌπ‘›π΄π‘’π‘ π‘‘π‘Ÿπ‘–π‘Ž(𝑋), π‘–π‘ πΌπ‘›πΌπ‘‘π‘Žπ‘™π‘¦ 𝑋 , π‘›π‘œπ‘‘ π‘–π‘ π‘…π‘–π‘£π‘’π‘Ÿ (𝑋) π‘–π‘ π‘ƒπ‘œπ‘™π‘–π‘‘π‘‚π‘“π‘ˆπ‘†π΄ 𝑋 ← π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 , π‘–π‘ πΊπ‘œπ‘£ 𝑋 , π‘›π‘œπ‘‘ π‘–π‘ π‘ƒπ‘œπ‘™π‘–π‘‘π‘ƒπ‘’π‘’π‘Ÿπ‘‘π‘œπ‘…π‘–π‘π‘œ(𝑋) π‘π‘œπ‘Ÿπ‘›πΌπ‘›π‘ˆπ‘†π΄ 𝑋 ← π‘Žπ‘π‘‘π‘’π‘‘πΌπ‘›π‘€π‘œπ‘£π‘–π‘’ 𝑋 , π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’π‘‘π‘€π‘œπ‘£π‘–π‘’ 𝑋 , π‘›π‘œπ‘‘ π‘€π‘œπ‘›πΉπ‘–π‘™π‘šπ‘“π‘Žπ‘Ÿπ‘’(𝑋)
  • 23. Summary β€’ Conclusion β€’ Quality-based theory revision under OWA β€’ Partial materialization for ranking revisions β€’ Comparison of ranking methods on real life KGs β€’ Outlook β€’ Extending to higher arity predicates β€’ Binary predicates [Tran et al., to appear ILP2016] β€’ Evidence from text corpora β€’ Exploiting partial completeness 23
  • 24. References β€’ [Angiulli and Fassetti, 2014] Fabrizio Angiulli and Fabio Fassetti. Exploiting domain knowledge to detect outliers. Data Min. Knowl. Discov., 28(2):519–568, 2014. β€’ [Dimopoulos and Kakas, 1995] Yannis Dimopoulos and Antonis C. Kakas. Learning non-monotonic logic programs: Learning exceptions. In Machine Learning: ECML-95, 8th European Conference on Machine Learning, Heraclion, Crete, Greece, April 25-27, 1995, Proceedings, pages 122–137, 1995. β€’ [Galarraga et al., 2015] Luis Galarraga, Christina Teflioudi, Katja Hose, and Fabian M. Suchanek. Fast Rule Mining in Ontological Knowledge Bases with AMIE+. In VLDB Journal, 2015. β€’ [Law et al., 2015] Mark Law, Alessandra Russo, and Krysia Broda. The ILASP system for learning answer set programs, 2015. β€’ [Leone et al., 2006] Nicola Leone, Gerald Pfeifer, Wolfgang Faber, Thomas Eiter, Georg Gottlob, Simona Perri, and Francesco Scarcello. 2006. The DLV system for knowledge representation and reasoning. ACM Trans. Comput. Logic 7, 3 (July 2006), 499-562. β€’ [Suzuki, 2006] Einoshin Suzuki. Data mining methods for discovering interesting exceptions from an unsupervised table. J. UCS, 12(6):627–653, 2006. β€’ [Tran et al., 2016] Hai Dang Tran, Daria Stepanova, Mohamed H. Gad-Elrab, Francesca A. Lisi, Gerhard Weikum. Towards Nonmonotonic Relational Learning from Knowledge Graphs. ILP2016, London, UK, to appear. β€’ [Katzouris et al., 2015] Nikos Katzouris, Alexander Artikis, and Georgios Paliouras. Incremental learning of event definitions with inductive logic programming. Machine Learning, 100(2-3):555– 585, 2015. 24
  • 25. Related Work β€’ Learning nonmonotonic programs β€’ E.g., [Dimopoulos and Kakas, 1995], ILASP [Law et al., 2015], ILED [Katzouris et al., 2015], etc. β€’ Outlier detection in logic programs β€’ E.g., [Angiulli and Fassetti, 2014], etc. β€’ Mining exception rules β€’ E.g., [Suzuki, 2006], etc. 25
  • 26. Problem Statement: Ruleset Quality β€’ Independent Rule Measure (π‘Ÿπ‘š) β€’ Support: 𝑠𝑒𝑝𝑝 𝐻 ← 𝐡 = 𝑠𝑒𝑝𝑝(𝐻 βˆͺ 𝐡) β€’ Coverage: cπ‘œπ‘£ 𝐻 ← 𝐡 = 𝑠𝑒𝑝𝑝(𝐡) β€’ Confidence: cπ‘œπ‘›π‘“ 𝐻 ← 𝐡 = 𝑠𝑒𝑝𝑝(𝐻βˆͺ𝐡) 𝑠𝑒𝑝𝑝(𝐡) β€’ Lift: 𝑙𝑖𝑓𝑑 𝐻 ← 𝐡 = π‘π‘œπ‘›π‘“(𝐻←𝐡) 𝑠𝑒𝑝𝑝(𝐻) β€’ … β€’ Average Ruleset Quality 26 π‘ž π‘Ÿπ‘š 𝑅 𝑁𝑀, 𝐺 = π‘Ÿ βˆˆπ‘… 𝑁𝑀 π‘Ÿπ‘š(π‘Ÿ, 𝐺) 𝑅 𝑁𝑀
  • 27. Problem Statement: Conflicting Predictions β€’ Measuring conflicts (auxiliary rules) 27 π‘Ÿ2: π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘›π‘œπ‘‘ π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘Ÿ2 π‘Žπ‘’π‘₯ : π‘›π‘œπ‘‘_π‘Ÿπ‘’π‘™π‘’π‘Žπ‘ π‘’π‘‘πΌπ‘›π½π‘ƒ 𝑋 ← β„Žπ‘Žπ‘ π½π‘ƒπΆπ‘œπ‘šπ‘π‘œπ‘ π‘’π‘Ÿ 𝑋 , π‘π‘’π‘π‘™π‘–π‘ β„Žπ‘’π‘Ÿπ‘ˆπ‘†π΄(𝑋) π‘ž π‘π‘œπ‘›π‘“π‘™π‘–π‘π‘‘ 𝑅 𝑁𝑀, 𝐺 = |{(𝑝 π‘Ž , π‘›π‘œπ‘‘_𝑝 π‘Ž ), … }| {π‘›π‘œπ‘‘_𝑝 π‘Ž , … }
  • 28. Propositionalization β€’ Unary predicates 28 Binary Predicates β€’ hasType(enstein, scientist) β€’ isMarriedTo(elsa, einstein) β€’ bornIn(einstein, um) Unary β€’ isAScientist(einstein) β€’ isMarriedToEinstein(elsa) β€’ bornInUlm(einstein) Abstraction β€’ isAScientist(einstein) β€’ isMarriedToScientist(elsa) β€’ bornInGermany(einstein)
  • 29. Experiments β€’ Input β€’ Experiment statistics 29 YAGO3 IMDB Input Facts 10M 2M Horn Rules 10K 25K Revised Rules 6K 22K General-purpose KG Domain-specific KG (Movies)
  • 30. Experiments β€’ Predictions assessment β€’ Run DLV on YAGO and 𝑅 𝐻 then 𝑅 𝑁𝑀 seperately β€’ Sample facts such that fact 𝑓 ∈ π‘Œπ΄πΊπ‘‚ π»π‘Œπ΄πΊπ‘‚ 𝑁𝑀 β€’ 73% of the sampled facts were found to be erroneous 30 checked predictions
  • 31. Ranking Rule’s Revisions β€’ Partial materialization ranker β€’ Augment the original 𝐾𝐺 with predictions of other rules β€’ Rank revisions on Avg. confidence of the π‘Ÿ and π‘Ÿ π‘Žπ‘’π‘₯ 31 π‘ π‘π‘œπ‘Ÿπ‘’ π‘Ÿπ‘’, πΎπΊβˆ— = π‘π‘œπ‘›π‘“ π‘Ÿπ‘’, πΎπΊβˆ— + π‘π‘œπ‘›π‘“(π‘Ÿπ‘’ π‘Žπ‘’π‘₯, πΎπΊβˆ—) 2 where π‘Ÿπ‘’ is the rule π‘Ÿ with exception 𝑒 & πΎπΊβˆ— is the augmented 𝐾𝐺.