Tight Bounds for Learning Polyhedra with a Margin

2026-04-16

AI Summary

This work studies the efficient PAC learnability of polytopes defined as the intersection of $k$ halfspaces under the $\rho$-margin assumption. Focusing on continuous distributions where most points lie far from the boundary, the authors propose a novel algorithm whose runtime is $\mathrm{poly}(k, \varepsilon^{-1}, \rho^{-1}) \cdot \exp\left(O(\sqrt{n \log(1/\rho) \log k})\right)$. This is the first result to reduce the exponential dependence on $k$ and $\rho^{-1}$ to a subexponential factor. The method significantly improves upon prior approaches and nearly matches known lower bounds—up to logarithmic factors—in both cryptographic and statistical query models. Furthermore, it broadens the applicability of margin-based learning techniques to a wider class of polytopes.

Technology Category

• Machine Learning: Learning Theory
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We give an algorithm for PAC learning intersections of $k$ halfspaces with a $ρ$ margin to within error $\varepsilon$ that runs in time $\textsf{poly}(k, \varepsilon^{-1}, ρ^{-1}) \cdot \exp \left(O(\sqrt{n \log(1/ρ) \log k})\right)$. Notably, this improves on prior work which had an exponential dependence on either $k$ or $ρ^{-1}$ and matches known cryptographic and Statistical Query lower bounds up to the logarithmic factors in $k$ and $ρ$ in the exponent. Our learning algorithm extends to the more general setting when we are only promised that most points have distance at least $ρ$ from the boundary of the polyhedron, making it applicable to continuous distributions as well.

Problem

• PAC learning
• polyhedra
• margin
• halfspaces
• computational complexity

Innovation

• PAC learning
• halfspaces intersection
• margin
• computational complexity
• Statistical Query lower bounds