We propose a general framework to incorporate rst-order logic (FOL) clauses, that are thought of as an abstract and partial representation of the environment, into kernel machines that learn within a semi-supervised scheme. We rely on a multi-task learning scheme where each task is associated with a unary predicate dened on the feature space, while higher level abstract representations consist of FOL clauses made of those predicates. We re-use the kernel machine mathematical apparatus to solve the problem as primal optimization of a function composed of the loss on the supervised examples, the regularization term, and a penalty term deriving from forcing real-valued constraints deriving from the predicates. Unlike for classic kernel machines, however, depending on the logic clauses, the overall function to be optimized is not convex anymore. An important contribution is to show that while tackling the optimization by classic numerical schemes is likely to be hopeless, a stage-based learning scheme, in which we start learning the supervised examples until convergence is reached, and then continue by forcing the logic clauses is a viable direction to attack the problem. Some promising experimental results are given on artificial learning tasks and on the automatic tagging of bibtex entries to emphasize the comparison with plain kernel machines.

Diligenti, M., Maggini, M., Gori, M., Rigutini, L. (2012). Bridging Logic and Kernel Machines. MACHINE LEARNING, 86(1), 57-88 [10.1007/s10994-011-5243-x].

Bridging Logic and Kernel Machines

DILIGENTI, MICHELANGELO;MAGGINI, MARCO;GORI, MARCO;RIGUTINI, LEONARDO
2012-01-01

Abstract

We propose a general framework to incorporate rst-order logic (FOL) clauses, that are thought of as an abstract and partial representation of the environment, into kernel machines that learn within a semi-supervised scheme. We rely on a multi-task learning scheme where each task is associated with a unary predicate dened on the feature space, while higher level abstract representations consist of FOL clauses made of those predicates. We re-use the kernel machine mathematical apparatus to solve the problem as primal optimization of a function composed of the loss on the supervised examples, the regularization term, and a penalty term deriving from forcing real-valued constraints deriving from the predicates. Unlike for classic kernel machines, however, depending on the logic clauses, the overall function to be optimized is not convex anymore. An important contribution is to show that while tackling the optimization by classic numerical schemes is likely to be hopeless, a stage-based learning scheme, in which we start learning the supervised examples until convergence is reached, and then continue by forcing the logic clauses is a viable direction to attack the problem. Some promising experimental results are given on artificial learning tasks and on the automatic tagging of bibtex entries to emphasize the comparison with plain kernel machines.
2012
Diligenti, M., Maggini, M., Gori, M., Rigutini, L. (2012). Bridging Logic and Kernel Machines. MACHINE LEARNING, 86(1), 57-88 [10.1007/s10994-011-5243-x].
File in questo prodotto:
File Dimensione Formato  
ML12.pdf

non disponibili

Tipologia: Post-print
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 1.58 MB
Formato Adobe PDF
1.58 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/25716
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo