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This paper presents a new PAC learning algorithm for intersections of $k$ halfspaces with margin $\rho$, achieving a runtime of $\textsf{poly}(k, \varepsilon^{-1}, \rho^{-1}) \cdot \exp \left(O(\sqrt{n \log(1/\rho) \log k})\right)$. This significantly improves upon existing algorithms with exponential dependencies on $k$ or $\rho^{-1}$, and tightly matches known lower bounds up to logarithmic factors. The algorithm's extension to settings with noisy margins makes it applicable to learning from continuous distributions.
Learning polyhedra just got a whole lot faster: this algorithm shaves off exponential dependencies on key parameters, bringing us closer to the theoretical limits.
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.