Learning Convex Polyhedra With Margin

We present an improved algorithm for
quasi-properly
learning convex polyhedra in the realizable PAC setting from data with a margin. Our learning algorithm constructs a consistent polyhedron as an intersection of about
$t \log t$
halfspaces with constant-size margins in time polynomial in
$t$
(where
$t$
is the number of halfspaces forming an optimal polyhedron). We also identify distinct generalizations of the notion of margin from hyperplanes to polyhedra and investigate how they relate geometrically; this result may have ramifications beyond the learning setting.