On the hardness of learning intersections of two halfspaces
Journal of Computer and System Sciences
Hardness of Reconstructing Multivariate Polynomials over Finite Fields
SIAM Journal on Computing
Bypassing UGC from some optimal geometric inapproximability results
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Hardness results for agnostically learning low-degree polynomial threshold functions
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
2log1-ε n hardness for the closest vector problem with preprocessing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Nearly optimal solutions for the chow parameters problem and low-weight approximation of halfspaces
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
An invariance principle for polytopes
Journal of the ACM (JACM)
Approximation resistance from pairwise independent subgroups
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Approximation resistance on satisfiable instances for predicates with few accepting inputs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We prove the following strong hardness result for learning: Given a distribution on labeled examples from the hypercube such that there exists a monomial (or conjunction) consistent with (1-ε)-fraction of the examples, it is NP-hard to find a halfspace that is correct on ( 1/2 +ε)-fraction of the examples, for arbitrary constant ε 0. In learning theory terms, weak agnostic learning of monomials by halfspaces is NP-hard. This hardness result bridges between and subsumes two previous results which showed similar hardness results for the proper learning of monomials and halfspaces. As immediate corollaries of our result, we give the first optimal hardness results for weak agnostic learning of decision lists and majorities. Our techniques are quite different from previous hardness proofs for learning. We use an invariance principle and sparse approximation of halfspaces from recent work on fooling halfspaces to give a new natural list decoding of a halfspace in the context of dictatorship tests/label cover reductions. In addition, unlike previous invariance principle based proofs which are only known to give Unique Games hardness, we give a reduction from a smooth version of Label Cover that is known to be NP-hard.