Before revision:  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 .
After revision:  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 We also identify distinct generalizations of the notion of margin from hyperplanes to polytopes and investigate how they relate geometrically; this result may be of interest beyond the learning setting .