I. Diakonikolas and R. O'Donnell and R. Servedio and Y. Wu.

In

Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low degree polynomial threshold function (PTF) which has the maximum possible agreement with a given set of labeled examples in $\R^n \times \{-1,1\}.$ We prove that for any constants $d\geq 1, \eps > 0$,

These results immediately imply the following hardness of learning results: (i) Assuming the Unique Games Conjecture, there is no better-than-trivial proper learning algorithm that agnostically learns degree-$d$ PTFs under arbitrary distributions; (ii) There is no better-than-trivial learning algorithm that outputs degree-$2$ PTFs and agnostically learns halfspaces (i.e. degree-$1$ PTFs) under arbitrary distributions.