Abstract
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.
| Original language | English |
|---|---|
| Title of host publication | Proceedings of the 8th Annual Conference on Computational Learning Theory, COLT 1995 |
| Publisher | Association for Computing Machinery, Inc |
| Pages | 206-213 |
| Number of pages | 8 |
| ISBN (Electronic) | 0897917235, 9780897917230 |
| Publication status | Published - Jul 5 1995 |
| Externally published | Yes |
| Event | 8th Annual Conference on Computational Learning Theory, COLT 1995 - Santa Cruz, United States Duration: Jul 5 1995 → Jul 8 1995 |
Publication series
| Name | Proceedings of the 8th Annual Conference on Computational Learning Theory, COLT 1995 |
|---|---|
| Volume | 1995-January |
Other
| Other | 8th Annual Conference on Computational Learning Theory, COLT 1995 |
|---|---|
| Country | United States |
| City | Santa Cruz |
| Period | 7/5/95 → 7/8/95 |
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All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Artificial Intelligence
- Software
Cite this
Proper learning algorithm for functions of κ terms under smooth distributions. / Sakai, Yoshifumi; Takimoto, Eiji; Maruoka, Akira.
Proceedings of the 8th Annual Conference on Computational Learning Theory, COLT 1995. Association for Computing Machinery, Inc, 1995. p. 206-213 (Proceedings of the 8th Annual Conference on Computational Learning Theory, COLT 1995; Vol. 1995-January).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
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
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
ER -