Learning convex bodies is hard

TL;DR

This paper proves that learning convex bodies in high-dimensional space from random samples is computationally hard, requiring exponentially many samples relative to the dimension and accuracy, by constructing a simple hard family using error correcting codes.

Contribution

It establishes a fundamental lower bound on the sample complexity of learning convex bodies, introducing a simple construction based on error correcting codes.

Findings

Learning convex bodies requires exponential samples in dimension and accuracy.

A simple family of convex bodies is constructed to prove the lower bound.

The result highlights inherent computational difficulty in high-dimensional convex body learning.

Abstract

We show that learning a convex body in $\RR^d$, given random samples from the body, requires $2^{\Omega(\sqrt{d/\eps})}$ samples. By learning a convex body we mean finding a set having at most $\eps$ relative symmetric difference with the input body. To prove the lower bound we construct a hard to learn family of convex bodies. Our construction of this family is very simple and based on error correcting codes.