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2017 | OriginalPaper | Chapter

Learning Łukasiewicz Logic Fragments by Quadratic Programming

Authors : Francesco Giannini, Michelangelo Diligenti, Marco Gori, Marco Maggini

Published in: Machine Learning and Knowledge Discovery in Databases

Publisher: Springer International Publishing

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Abstract

In this paper we provide a framework to embed logical constraints into the classical learning scheme of kernel machines, that gives rise to a learning algorithm based on a quadratic programming problem. In particular, we show that, once the constraints are expressed using a specific fragment from the Łukasiewicz logic, the learning objective turns out to be convex. We formulate the primal and dual forms of a general multi–task learning problem, where the functions to be determined are predicates (of any arity) defined on the feature space. The learning set contains both supervised examples for each predicate and unsupervised examples exploited to enforce the constraints. We give some properties of the solutions constructed by the framework along with promising experimental results.

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previous chapter Kernel Sequential Monte Carlo

next chapter Nyström Sketches

A variety is a class of algebras closed under the taking of homomorphic images, subalgebras and direct products.

Through the paper we use the same symbols for connectives and algebraic operations.

The difference can be explained as follows: \(\alpha \wedge \beta \) gives us the availability of both \(\alpha \) and \(\beta \) but we can pick just one of them; \(\alpha \otimes \beta \) forces us to pick both formulas in the pair \((\alpha ,\beta )\).

Where, for every \(j\in \mathbb {N},\,\mathbb {N}_j=\{n\in \mathbb {N}:\,n\le j\}\).

If we consider the usual 0–1 logic values as labels, we can consider the condition \((2y_l-1)p_j(\mathbf{x}_l)\ge y_l\).

For every \(j=1,\ldots ,J\), \(\phi _j\) is determined by the j-th kernel function \(k_j\) of the RKHS \(\mathcal {H}_j\) and it is such that \(k_j(\mathbf{x},\mathbf{y})=\langle \phi _j(\mathbf{x}),\phi _j(\mathbf{y})\rangle _{\mathbb {R}^{N_j}}\).

http://www.image-net.org.

The dataset and code to reproduce the results can be downloaded from https://github.com/diligmic/ECML2017_1 .

Bach, S.H., Broecheler, M., Huang, B., Getoor, L.: Hinge-loss markov random fields and probabilistic soft logic. arXiv preprint arXiv:1505.04406 (2015)

Bofill, M., Manya, F., Vidal, A., Villaret, M.: Finding hard instances of satisfiability in lukasiewicz logics. In: 2015 IEEE International Symposium on Multiple-Valued Logic (ISMVL), pp. 30–35. IEEE (2015)

Cumby, C.M., Roth, D.: On kernel methods for relational learning. In: Proceedings of the 20th International Conference on Machine Learning (ICML-03), pp. 107–114 (2003)

Diligenti, M., Gori, M., Maggini, M., Rigutini, L.: Bridging logic and kernel machines. Mach. Learn. 86(1), 57–88 (2012)MathSciNetCrossRefMATH

Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artif. Intell. (2015). https://doi.org/10.1016/j.artint.2015.08.011

Diligenti, M., Gori, M., Scoca, V.: Learning efficiently in semantic based regularization. In: Frasconi, P., Landwehr, N., Manco, G., Vreeken, J. (eds.) ECML PKDD 2016. LNCS (LNAI), vol. 9852, pp. 33–46. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-46227-1_3 CrossRef

Gnecco, G., Gori, M., Melacci, S., Sanguineti, M.: Foundations of support constraint machines. Neural comput. 27(2), 388–480 (2015)CrossRefMATH

Hu, Z., Ma, X., Liu, Z., Hovy, E., Xing, E.: Harnessing deep neural networks with logic rules. arXiv preprint arXiv:1603.06318 (2016)

Lowe, D.G.: Object recognition from local scale-invariant features. In: The Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999, vol. 2, pp. 1150–1157. IEEE (1999)

10.

Marchioni, E., Wooldridge, M.: Łukasiewicz games: a logic-based approach to quantitative strategic interactions. ACM Trans. Comput. Logic (TOCL) 16(4), 33 (2015)CrossRefMATH

11.

Muggleton, S., Lodhi, H., Amini, A., Sternberg, M.J.E.: Support vector inductive logic programming. In: Hoffmann, A., Motoda, H., Scheffer, T. (eds.) DS 2005. LNCS (LNAI), vol. 3735, pp. 163–175. Springer, Heidelberg (2005). https://doi.org/10.1007/11563983_15 CrossRef

12.

Novák, V., Perlieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic, vol. 517. Springer Science & Business Media, Heidelberg (2012)

13.

Richardson, M., Domingos, P.: Markov logic networks. Mach. Learn. 62(1), 107–136 (2006)CrossRef

14.

Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, vol. 317. Springer Science and Business Media, Berlin (2009)MATH

15.

Serafini, L., Garcez, A.D.: Logic tensor networks: deep learning and logical reasoning from data and knowledge. arXiv preprint arXiv:1606.04422 (2016)

16.

Winston, P.H., Horn, B.K.: Lisp. Addison Wesley Pub., Reading (1986)MATH

Title: Learning Łukasiewicz Logic Fragments by Quadratic Programming
Authors: Francesco Giannini
Michelangelo Diligenti
Marco Gori
Marco Maggini
Publisher: Springer International Publishing
Book: Machine Learning and Knowledge Discovery in Databases
Print ISBN: 978-3-319-71248-2

Electronic ISBN: 978-3-319-71249-9

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-71249-9_25

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