Learning Boolean Halfspaces with Small Weights from Membership Queries

TL;DR

This paper presents new algorithms for learning Boolean Halfspaces with small integer weights using membership queries, significantly reducing the number of queries needed compared to previous methods.

Contribution

It introduces an adaptive two-round algorithm with $n^{O(t)}$ queries and a non-adaptive algorithm with $n^{O(t^3)}$ queries, closing the query complexity gap.

Findings

Adaptive algorithm asks $n^{O(t)}$ queries.

Non-adaptive algorithm asks $n^{O(t^3)}$ queries.

Results improve the efficiency of learning Boolean Halfspaces with small weights.

Abstract

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 $n^{O(t^5)}$ membership queries where the best lower bound for the number of membership queries is $n^t$ [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 $n^{O(t)}$ membership queries. We also give a non-adaptive proper learning algorithm that asks $n^{O(t^3)}$ membership queries.