Learning convex polytopes with margin
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We present a near-optimal algorithm for properly learning convex polytopes in the realizable PAC setting from data with a margin. Our first contribution is to identify distinct generalizations of the notion of margin from hyperplanes to polytopes and to understand how they relate geometrically; this result may be of interest beyond the learning setting. Our novel learning algorithm constructs a consistent polytope as an intersection of about t t halfspaces in time polynomial in t (where t is the number of halfspaces forming an optimal polytope). This is an exponential improvement over the state of the art [Arriaga and Vempala, 2006]. We also improve over the super-polynomial-in-t algorithm of Klivans and Servedio [2008], while achieving a better sample complexity. Finally, we provide the first nearly matching hardness-of-approximation lower bound, whence our claim of near optimality.
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