Technical Report CS0907

Title: The Query Complexity of Finding Local Minima in the Lattice
Authors: Felix Geller and Eyal Kushilevitz
Abstract: In this paper we study the query complexity of finding local minimum points of a Boolean function. This task occurs frequently in learning many natural classes, such as monotone DNF, $O(\log n)$-term DNF, unate DNF and decision trees, in the model of exact learning. We prove that any algorithm that produces a local minimum of a function $f$ (chosen from a sufficiently "rich" concept class), given an access to a membership oracle for $f$, must ask $\Omega(n^2)$ membership queries in the worst case. In particular, this bound is attained on the class of decision trees. A simple algorithm is known that achieves this lower bound.
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