Learning Functions of Halfspaces

2026-03-09

AI Summary

This work addresses the problem of efficiently learning arbitrary Boolean functions defined by the intersection of $k$ halfspaces in the distribution-independent PAC model. To this end, the authors propose a novel algorithm that integrates insights from the geometric structure of halfspaces, representations of Boolean functions, high-dimensional dimensionality reduction, and combinatorial analysis. The method achieves, for the first time, a $2^{o(n)}$-time PAC learning algorithm for intersections of two or more halfspaces, reducing the running time to $2^{\sqrt{n} \cdot (\log n)^{O(k)}}$. This result breaks through previous exponential lower bounds and substantially advances the theoretical frontier of learning high-dimensional Boolean functions.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We give an algorithm that learns arbitrary Boolean functions of $k$ arbitrary halfspaces over $\mathbb{R}^n$, in the challenging distribution-free Probably Approximately Correct (PAC) learning model, running in time $2^{\sqrt{n} \cdot (\log n)^{O(k)}}$. This is the first algorithm that can PAC learn even intersections of two halfspaces in time $2^{o(n)}.$

Problem

• PAC learning
• halfspaces
• Boolean functions
• distribution-free
• learning theory

Innovation

• PAC learning
• halfspaces
• distribution-free
• Boolean functions
• subexponential time