On the hardness of learning intersections of two halfspaces

Authors:
Subhash Khot;Rishi Saket
Affiliations:
New York University, United States;Georgia Institute of Technology, United States
Venue:
Journal of Computer and System Sciences
Year:
2011

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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 @? halfspaces (linear threshold functions) for any integer @?. Specifically, we show that for every integer @? and an arbitrarily small constant @e0, 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 @? halfspaces can correctly classify at most 12+@e fraction of the points.