Tight Bounds for Learning Polyhedra with a Margin

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

Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG)

arXiv:2604.14614 [cs.DS]

(or arXiv:2604.14614v2 [cs.DS] for this version)

https://doi.org/10.48550/arXiv.2604.14614 Focus to learn more arXiv-issued DOI via DataCite

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