Query complexity of finding local minima in the lattice

Amos Beimel; Felix Geller; Eyal Kushilevitz

doi:10.1145/279943.280000

Query complexity of finding local minima in the lattice

Amos Beimel, Felix Geller, Eyal Kushilevitz

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

1 Scopus citations

Abstract

In this paper we study the query complexity of finding local minimum points of a boolean function. This task occurs frequently in exact learning algorithms for many natural classes, such as monotone DNF, O(log n)-term DNF, unate DNF and decision trees. On the negative side, we prove that any (possibly randomized) algorithm that produces a local minimum of a function f chosen from a sufficiently 'rich' concept class, using a membership oracle for f, must ask Ω(n²) membership queries in the worst case. In particular, this lower bound applies to the class of decision trees. A simple algorithm is known that achieves this lower bound. On the positive side, we show that for the class O(log n)-term DNF finding local minimum points requires only Θ(n log n) membership queries (and more generally Θ(nt) membership queries for t-term DNF with t ≤ n). This efficient procedure improves the time and query complexity of known learning algorithms for the class O(log n)-term DNF.

Original language	English GB
Title of host publication	Proceedings of the 1998 11th Annual Conference on Computational Learning Theory, Madison, WI, USA,
Pages	294-302
Number of pages	9
DOIs	https://doi.org/10.1145/279943.280000
State	Published - 1 Jan 1998
Externally published	Yes
Event	Proceedings of the 1998 11th Annual Conference on Computational Learning Theory - Madison, WI, USA Duration: 24 Jul 1998 → 26 Jul 1998

Conference

Conference	Proceedings of the 1998 11th Annual Conference on Computational Learning Theory
City	Madison, WI, USA
Period	24/07/98 → 26/07/98

ASJC Scopus subject areas

Computational Mathematics

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10.1145/279943.280000

Cite this

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title = "Query complexity of finding local minima in the lattice",

abstract = "In this paper we study the query complexity of finding local minimum points of a boolean function. This task occurs frequently in exact learning algorithms for many natural classes, such as monotone DNF, O(log n)-term DNF, unate DNF and decision trees. On the negative side, we prove that any (possibly randomized) algorithm that produces a local minimum of a function f chosen from a sufficiently 'rich' concept class, using a membership oracle for f, must ask Ω(n2) membership queries in the worst case. In particular, this lower bound applies to the class of decision trees. A simple algorithm is known that achieves this lower bound. On the positive side, we show that for the class O(log n)-term DNF finding local minimum points requires only Θ(n log n) membership queries (and more generally Θ(nt) membership queries for t-term DNF with t ≤ n). This efficient procedure improves the time and query complexity of known learning algorithms for the class O(log n)-term DNF.",

author = "Amos Beimel and Felix Geller and Eyal Kushilevitz",

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N2 - In this paper we study the query complexity of finding local minimum points of a boolean function. This task occurs frequently in exact learning algorithms for many natural classes, such as monotone DNF, O(log n)-term DNF, unate DNF and decision trees. On the negative side, we prove that any (possibly randomized) algorithm that produces a local minimum of a function f chosen from a sufficiently 'rich' concept class, using a membership oracle for f, must ask Ω(n2) membership queries in the worst case. In particular, this lower bound applies to the class of decision trees. A simple algorithm is known that achieves this lower bound. On the positive side, we show that for the class O(log n)-term DNF finding local minimum points requires only Θ(n log n) membership queries (and more generally Θ(nt) membership queries for t-term DNF with t ≤ n). This efficient procedure improves the time and query complexity of known learning algorithms for the class O(log n)-term DNF.

AB - In this paper we study the query complexity of finding local minimum points of a boolean function. This task occurs frequently in exact learning algorithms for many natural classes, such as monotone DNF, O(log n)-term DNF, unate DNF and decision trees. On the negative side, we prove that any (possibly randomized) algorithm that produces a local minimum of a function f chosen from a sufficiently 'rich' concept class, using a membership oracle for f, must ask Ω(n2) membership queries in the worst case. In particular, this lower bound applies to the class of decision trees. A simple algorithm is known that achieves this lower bound. On the positive side, we show that for the class O(log n)-term DNF finding local minimum points requires only Θ(n log n) membership queries (and more generally Θ(nt) membership queries for t-term DNF with t ≤ n). This efficient procedure improves the time and query complexity of known learning algorithms for the class O(log n)-term DNF.

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Query complexity of finding local minima in the lattice

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