@MISC{Khot_nonnumericalalgorithms, author = {Subhash Khot and Rishi Saket}, title = {Nonnumerical Algorithms and Problems—Geometrical}, year = {} }

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Abstract

We show that unless NP = RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in R n using a hypothesis which is a function of up to ℓ linear threshold functions for any integer ℓ. Specifically, we show that for every integer ℓ and an arbitrarily small constant ε> 0, unless NP = RP, no polynomial time algorithm can distinguish whether there is an intersection of two halfspaces that correctly classifies a given set of labeled points in R n, or whether any function of ℓ linear threshold functions can correctly classify at most 1 2 + ε fraction of the points. Categories and Subject Descriptors