Learning convex polyhedra with margin

TL;DR

This paper introduces an efficient algorithm for learning convex polyhedra with margin in the PAC setting, constructing a consistent polyhedron from data with theoretical guarantees.

Contribution

It presents an improved polynomial-time algorithm for learning convex polyhedra with margin, extending margin concepts from hyperplanes to polyhedra.

Findings

Algorithm constructs polyhedra as intersection of about t log t halfspaces

Time complexity is polynomial in the number of halfspaces t

Explores geometric generalizations of margin for polyhedra

Abstract

We present an improved algorithm for {\em 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.