Hardness of Learning Noisy Halfspaces using Polynomial Thresholds
[edit]
Proceedings of the 31st Conference On Learning Theory, PMLR 75:876917, 2018.
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 $\eps > 0$, it is NPhard to decide: given a set of $\{1,1\}$labeled points in $\mathbb{R}^n$ whether (YES Case) there exists a halfspace that classifies $(1\eps)$fraction of the points correctly, or (NO Case) any degree$d$ PTF classifies at most $(1/2 + \eps)$fraction of the points correctly. This strengthens to all constant degrees the previous NPhardness 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.
Related Material


