Query complexity of finding local minima in the lattice

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.