NDLI: On Agnostic Learning of Parities, Monomials, and Halfspaces

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Khot, Subhash Feldman, Vitaly Gopalan, Parikshit Ponnuswami, Ashok Kumar
Copyright Year	2009
Abstract	We study the learnability of several fundamental concept classes in the agnostic learning framework of [D. Haussler, Inform. and Comput., 100 (1992), pp. 78150] and [M. Kearns, R. Schapire, and L. Sellie, Machine Learning, 17 (1994), pp. 115141]. We show that under the uniform distribution, agnostically learning parities reduce to learning parities with random classification noise, commonly referred to as the noisy parity problem. Together with the parity learning algorithm of [A. Blum, A. Kalai, and H. Wasserman, J. ACM, 50 (2003), pp. 506519], this gives the first nontrivial algorithm for agnostic learning of parities. We use similar techniques to reduce learning of two other fundamental concept classes under the uniform distribution to learning of noisy parities. Namely, we show that learning of disjunctive normal form (DNF) expressions reduces to learning noisy parities of just logarithmic number of variables, and learning of k-juntas reduces to learning noisy parities of k variables. We give essentially optimal hardness results for agnostic learning of monomials over $\{0,1\}^n$ and halfspaces over $\mathbb{Q}^n$. We show that for any constant $\epsilon$ finding a monomial (halfspace) that agrees with an unknown function on $1/2+\epsilon$ fraction of the examples is NP-hard even when there exists a monomial (halfspace) that agrees with the unknown function on $1-\epsilon$ fraction of the examples. This resolves an open question due to Blum and significantly improves on a number of previous hardness results for these problems. We extend these results to $\epsilon=2^{-\log^{1-\lambda}n}$ ($\epsilon=2^{-\sqrt{\log n}}$ in the case of halfspaces) for any constant $\lambda>0$ under stronger complexity assumptions.
Starting Page	606
Ending Page	645
Page Count	40
File Format	PDF
ISSN	00975397
DOI	10.1137/070684914
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2 (Special Section on Foundations of Computer Science)
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-07-23
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems halfspaces monomials random noise Computational learning theory agnostic learning parities
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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