TY - JOUR
T1 - The query complexity of finding local minima in the lattice
AU - Beimel, Amos
AU - Geller, Felix
AU - Kushilevitz, Eyal
N1 - Funding Information:
1An early version of this paper appeared in the proceedings of the 11th Conference on Computational Learning Theory (COLT), pp. 294-302, July 1998. 2 This work was done while the author was with the division of engineering & applied sciences at Harvard University, supported by grants ONR-N00014-96-1-0550 and ARO-DAAL-03-92-G0115. 3Supported by Technion V.P.R. Fund 120-872, by Japan Technion Society Research Fund, by the fund for the promotion of research at the Technion, and by the German-Israeli Foundation for scientific research and development (GIF). 69
PY - 2001/11/25
Y1 - 2001/11/25
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(tn) 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(tn) 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.
UR - http://www.scopus.com/inward/record.url?scp=0035951021&partnerID=8YFLogxK
U2 - 10.1006/inco.2001.3065
DO - 10.1006/inco.2001.3065
M3 - Article
AN - SCOPUS:0035951021
SN - 0890-5401
VL - 171
SP - 69
EP - 83
JO - Information and Computation
JF - Information and Computation
IS - 1
ER -