Hardness of learning noisy halfspaces using polynomial thresholds

TL;DR

This paper establishes the NP-hardness of weakly learning noisy halfspaces with polynomial threshold functions, extending previous results to all constant degrees and highlighting the computational difficulty of this problem.

Contribution

It proves NP-hardness for learning noisy halfspaces with degree-$d$ polynomial threshold functions for all constant degrees, strengthening prior hardness results.

Findings

NP-hardness of weakly learning noisy halfspaces with degree-$d$ PTFs

Extends previous degree-2 hardness to all constant degrees

Highlights computational challenges in learning noisy halfspaces

Abstract

We prove the hardness of weakly learning halfspaces in the presence of adversarial noise using polynomial threshold functions (PTFs). In particular, we prove that for any constants $d \in \mathbb{Z}^+$ and $\varepsilon > 0$, it is NP-hard to decide: given a set of $\{-1,1\}$-labeled points in $\mathbb{R}^n$ whether (YES Case) there exists a halfspace that classifies $(1-\varepsilon)$-fraction of the points correctly, or (NO Case) any degree-$d$ PTF classifies at most $(1/2 + \varepsilon)$-fraction of the points correctly. This strengthens to all constant degrees the previous NP-hardness of learning using degree-$2$ PTFs shown by Diakonikolas et al. (2011). The latter result had remained the only progress over the works of Feldman et al. (2006) and Guruswami et al. (2006) ruling out weakly proper learning adversarially noisy halfspaces.