Learning Functions of Halfspaces

TL;DR

This paper introduces a novel algorithm capable of PAC learning Boolean functions formed by intersections of halfspaces over real space, with subexponential runtime, advancing the understanding of learning complex geometric functions.

Contribution

It presents the first subexponential-time PAC learning algorithm for intersections of two halfspaces, a significant step in geometric function learning.

Findings

First subexponential-time PAC learning algorithm for intersections of two halfspaces.

Algorithm works in the distribution-free PAC model.

Runs in time $2^{ ext{poly}( ext{sqrt}(n), ext{log} n)}$.

Abstract

We give an algorithm that learns arbitrary Boolean functions of $k$ arbitrary halfspaces over $\mathbb{R}^n$, in the challenging distribution-free Probably Approximately Correct (PAC) learning model, running in time $2^{\sqrt{n} \cdot (\log n)^{O(k)}}$. This is the first algorithm that can PAC learn even intersections of two halfspaces in time $2^{o(n)}.$