Complexity theoretic limitations on learning halfspaces

We study the problem of agnostically learning halfspaces which is defined by a fixed but unknown distribution D on ℚⁿ × {±1}. We define Err_HALF(D) as the least error of a halfspace classifier for D. A learner who can access D has to return a hypothesis whose error is small compared to Err_HALF(D). Using the recently developed method of Daniely, Linial and Shalev-Shwartz we prove hardness of learning results assuming that random K-XOR formulas are hard to (strongly) refute. We show that no efficient learning algorithm has nontrivial worst-case performance even under the guarantees that ErrHALp(D) ≤ η for arbitrarily small constant η > 0, and that D is supported in {±1 }ⁿ × {±1}. Namely, even under these favorable conditions, and for every c > 0, it is hard to return a hypothesis with error ≤ 1/2/1/n^c. In particular, no efficient algorithm can achieve a constant approximation ratio. Under a stronger version of the assumption (where K can be poly-logarithmic in n), we can take η = 2-^log1-v(n) for arbitrarily small ? > 0. These results substantially improve on previously known results, that only show hardness of exact learning.