Learning Boolean Halfspaces with Small Weights from Membership Queries

We consider the problem of proper learning a Boolean Halfspace with integer weights 0,1,ldots,t from membership queries only. The best known algorithm for this problem is an adaptive algorithm that asks nO(t5) membership queries where the best lower bound for the number of membership queries is nt [Learning Threshold Functions with Small Weights Using Membership Queries. COLT 1999] In this paper we close this gap and give an adaptive proper learning algorithm with two rounds that asks nO(t) membership queries. We also give a non-adaptive proper learning algorithm that asks nO(t3) membership queries.