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 Q^n X {-1,1}. We define Err_H(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_H(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 non-trivial worst-case performance even under the guarantees that Err_H(D) <= eta for arbitrarily small constant eta>0, and that D is supported in the Boolean cube. Namely, even under these favorable conditions, and for every c>0, it is hard to return a hypothesis with error <= 1/2-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 eta = 2^{-log^{1-nu}(n)} for arbitrarily small nu>0. These results substantially improve on previously known results, that only show hardness of exact learning.