Tight Bounds for Learning Polyhedra with a Margin

• URL: http://arxiv.org/abs/2604.14614v1
• Date: Thu, 16 Apr 2026 04:43:12 GMT
• Title: Tight Bounds for Learning Polyhedra with a Margin
• Abstract summary: We give an algorithm for PAC learning intersections of $k$ halfspaces with a $$ margin to within error.<n>Our algorithm extends to the more general setting when we are only promised that most points have distance at least $$ from the boundary of the polyhedron.
• Score: 0.2836066255205732
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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