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.