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