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

2025-07-27

AI Summary

This paper addresses the exact learning of $k$-term DNF formulas, aiming to break the $mathrm{poly}(n, 2^k)$ time barrier established by Blum and Rudich (1992). We propose a novel framework integrating dynamic feature-space augmentation with adaptive query strategies, unifying techniques from Winnow2, attribute-efficient learning, learning low-weight linear threshold functions, and junta variable identification. Our theoretical analysis leverages extremal polynomials and noise operators to control approximation error and query complexity. As our main contribution, we achieve the first subexponential-in-$k$ exact learning algorithm for $k$-term DNF, running in $mathrm{poly}(n) cdot 2^{ ilde{O}(sqrt{k})}$ time under the membership and equivalence query model. This constitutes the first substantial speedup over prior bounds and introduces a new paradigm—along with key analytical tools—for learning Boolean functions.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning Theory
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 $ extsf{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 $ extsf{poly}(n,2^k)$. We give an algorithm that uses membership and equivalence queries to learn $k$-term DNF formulas in time $ extsf{poly}(n) cdot 2^{ ilde{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.

Problem

• Improving runtime for learning k-term DNF formulas
• Using membership and equivalence queries efficiently
• Combining novel algorithmic tools and analytic techniques

Innovation

• Enhanced feature space with Winnow2 algorithm
• Strengthened DNF term reduction technique
• Novel junta testing and noise operators