  We prove the following strong hardness result for learning: Given a
distribution of labeled examples from the hypercube such that there exists a
monomial consistent with $(1-\eps)$ of the examples, it is NP-hard to find a
halfspace that is correct on $(1/2+\eps)$ of the examples, for arbitrary
constants $\eps > 0$. In learning theory terms, weak agnostic learning of
monomials is hard, even if one is allowed to output a hypothesis from the much
bigger concept class of halfspaces. This hardness result subsumes a long line
of previous results, including two recent hardness results for the proper
learning of monomials and halfspaces. As an immediate corollary of our result
we show that weak agnostic learning of decision lists is NP-hard.
  Our techniques are quite different from previous hardness proofs for
learning. We define distributions on positive and negative examples for
monomials whose first few moments match. We use the invariance principle to
argue that regular halfspaces (all of whose coefficients have small absolute
value relative to the total $\ell_2$ norm) cannot distinguish between
distributions whose first few moments match. For highly non-regular subspaces,
we use a structural lemma from recent work on fooling halfspaces to argue that
they are ``junta-like'' and one can zero out all but the top few coefficients
without affecting the performance of the halfspace. The top few coefficients
form the natural list decoding of a halfspace in the context of dictatorship
tests/Label Cover reductions.
  We note that unlike previous invariance principle based proofs which are only
known to give Unique-Games hardness, we are able to reduce from a version of
Label Cover problem that is known to be NP-hard. This has inspired follow-up
work on bypassing the Unique Games conjecture in some optimal geometric
inapproximability results.
