The complexity of properly learning simple concept classes

  • Authors:

  • Misha Alekhnovich;Mark Braverman;Vitaly Feldman;Adam R. Klivans;Toniann Pitassi

  • Affiliations:

  • IAS, Princeton, USA;University of Toronto, Canada;Harvard University, USA;University of Texas at Austin, USA;IAS, Princeton, USA and University of Toronto, Canada

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2008

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Abstract

We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on well-known concept classes:*We show that unless NP=RP, there is no polynomial-time PAC learning algorithm for DNF formulas where the hypothesis is an OR-of-thresholds. Note that as special cases, we show that neither DNF nor OR-of-thresholds are properly learnable unless NP=RP. Previous hardness results have required strong restrictions on the size of the output DNF formula. We also prove that it is NP-hard to learn the intersection of @?=2 halfspaces by the intersection of k halfspaces for any constant k=0. Previous work held for the case when k=@?. *Assuming that NP@?DTIME(2^n^^^@e) for a certain constant @e=0. Previous hardness results for learning decision trees held for k=