Faster exact learning of k-term DNFs with membership and equivalence queries

TL;DR

This paper presents a significantly faster algorithm for learning k-term DNF formulas using membership and equivalence queries, improving the runtime from exponential in k to polynomial in n times an exponential in the square root of k.

Contribution

It introduces a new algorithm that reduces the runtime for learning k-term DNF formulas to extsf{poly}(n) imes 2^{ ilde{O}(\sqrt{k})}, marking the first improvement since 1992.

Findings

Achieves runtime extsf{poly}(n) imes 2^{ ilde{O}(\sqrt{k})} for learning k-term DNF.

Employs Winnow2 algorithm with adaptive feature space construction.

Combines novel techniques from extremal polynomials and junta testing.

Abstract

In 1992 Blum and Rudich [BR92] gave an algorithm that uses membership and equivalence queries to learn $k$-term DNF formulas over $\{0,1\}^n$ in time $\textsf{poly}(n,2^k)$, improving on the naive $O(n^k)$ running time that can be achieved without membership queries [Val84]. Since then, many alternative algorithms [Bsh95, Kus97, Bsh97, BBB+00] have been given which also achieve runtime $\textsf{poly}(n,2^k)$. We give an algorithm that uses membership and equivalence queries to learn $k$-term DNF formulas in time $\textsf{poly}(n) \cdot 2^{\tilde{O}(\sqrt{k})}$. This is the first improvement for this problem since the original work of Blum and Rudich [BR92]. Our approach employs the Winnow2 algorithm for learning linear threshold functions over an enhanced feature space which is adaptively constructed using membership queries. It combines a strengthened version of a technique that effectively reduces the length of DNF terms from the original work of [BR92] with a range of additional algorithmic tools (attribute-efficient learning algorithms for low-weight linear threshold functions and techniques for finding relevant variables from junta testing) and analytic ingredients (extremal polynomials and noise operators) that are novel in the context of query-based DNF learning.