TY - GEN

T1 - Proper learning algorithm for functions of κ terms under smooth distributions

AU - Sakai, Yoshifumi

AU - Takimoto, Eiji

AU - Maruoka, Akira

PY - 1995/7/5

Y1 - 1995/7/5

N2 - Algorithms for learning feasibly Boolean functions from examples are explored. A class of functions we deal with is written as F1 oF2 k = {g(f1(v),...fk(v)) g ∈ F1, f1...,fk ∈ F2} for classes F1 and F2 given by somewhat "simple" description. Letting Γ = {0,1}, we denote by F1 and F2 a class of functions from Γk to Γ and that of functions from Γn to Γ, respectively. For exa.mple, let FOr consist of an OR function of k variables, and let Fn be the class of all monomials of n variables. In the distribution free setting, it is known that FORo Fn k, denoted usually k-term DNF, is not learnable unless P≠NP In this paper, we first introduce a probabilistic distribution, called a smooth distribution, which is a generalization of both q-bounded distribution and product distribution, and define the learnability under this distribution. Then, we give an algorithm that properly learns FkoTn k under smooth distribution in polynomial time for constant k, where Fk is the class of all Boolean functions of k variables. The class FkoTn k is called the functions of k terms and although it was shown by Blum and Singh to be learned using DNF as a hypothesis class, it remains open whether it is properly learnable under distribution free setting.

AB - Algorithms for learning feasibly Boolean functions from examples are explored. A class of functions we deal with is written as F1 oF2 k = {g(f1(v),...fk(v)) g ∈ F1, f1...,fk ∈ F2} for classes F1 and F2 given by somewhat "simple" description. Letting Γ = {0,1}, we denote by F1 and F2 a class of functions from Γk to Γ and that of functions from Γn to Γ, respectively. For exa.mple, let FOr consist of an OR function of k variables, and let Fn be the class of all monomials of n variables. In the distribution free setting, it is known that FORo Fn k, denoted usually k-term DNF, is not learnable unless P≠NP In this paper, we first introduce a probabilistic distribution, called a smooth distribution, which is a generalization of both q-bounded distribution and product distribution, and define the learnability under this distribution. Then, we give an algorithm that properly learns FkoTn k under smooth distribution in polynomial time for constant k, where Fk is the class of all Boolean functions of k variables. The class FkoTn k is called the functions of k terms and although it was shown by Blum and Singh to be learned using DNF as a hypothesis class, it remains open whether it is properly learnable under distribution free setting.

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M3 - Conference contribution

AN - SCOPUS:33646907847

T3 - Proceedings of the 8th Annual Conference on Computational Learning Theory, COLT 1995

SP - 206

EP - 213

BT - Proceedings of the 8th Annual Conference on Computational Learning Theory, COLT 1995

PB - Association for Computing Machinery, Inc

T2 - 8th Annual Conference on Computational Learning Theory, COLT 1995

Y2 - 5 July 1995 through 8 July 1995

ER -