Learning random monotone DNF

We give an algorithm that with high probability properly learns random monotone DNF with t(n) terms of length ≈ logt(n) under the uniform distribution on the Boolean cube {0,1}n. For any function t(n) ≤ poly(n) the algorithm runs in time poly(n,1/i¾?) and with high probability outputs an i¾?-accurate monotone DNF hypothesis. This is the first algorithm that can learn monotone DNF of arbitrary polynomial size in a reasonable average-case model of learning from random examples only.

[1]  Avrim Blum,et al.  Machine learning: my favorite results, directions, and open problems , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[2]  Ryan O'Donnell,et al.  Learning Monotone Decision Trees in Polynomial Time , 2007, SIAM J. Comput..

[3]  Avrim Blum.  Learning a Function of r Relevant Variables , 2003, COLT.

[4]  Karsten A. Verbeurgt.  Learning Sub-classes of Monotone DNF on the Uniform Distribution , 1998, ALT.

[5]  Yishay Mansour,et al.  An O(nlog log n) learning algorithm for DNF under the uniform distribution , 1992, COLT '92.

[6]  Eyal Kushilevitz,et al.  Learning decision trees using the Fourier spectrum , 1991, STOC '91.

[7]  Yishay Mansour,et al.  An O(n^(log log n)) Learning Algorithm for DNT under the Uniform Distribution , 1995, J. Comput. Syst. Sci..

[8]  Rocco A. Servedio,et al.  On learning monotone DNF under product distributions , 2001, Inf. Comput..

[9]  Noam Nisan,et al.  Constant depth circuits, Fourier transform, and learnability , 1993, JACM.

[10]  Linda Sellie.  Learning Random Monotone DNF Under the Uniform Distribution , 2008, COLT.

[11]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[12]  Rocco A. Servedio,et al.  On Learning Random DNF Formulas Under the Uniform Distribution , 2005, Theory of Computing.

[13]  Mark Braverman,et al.  The complexity of properly learning simple concept classes , 2008, J. Comput. Syst. Sci..

[14]  Noam Nisan,et al.  Constant depth circuits, Fourier transform, and learnability , 1989, 30th Annual Symposium on Foundations of Computer Science.

[15]  Marco Protasi,et al.  On Learning Monotone DNF Formulae under Uniform Distributions , 1994, Inf. Comput..

[16]  Nader H. Bshouty,et al.  On the Fourier spectrum of monotone functions , 1995, STOC '95.

[17]  Rocco A. Servedio,et al.  Learnability beyond AC0 , 2002, STOC '02.

[18]  John H. Reif,et al.  Proceedings of the thiry-fourth annual ACM symposium on Theory of computing , 2002, STOC 2002.

[19]  Karsten A. Verbeurgt.  Learning DNF under the uniform distribution in quasi-polynomial time , 1990, COLT '90.

[20]  Elchanan Mossel,et al.  On the noise sensitivity of monotone functions , 2003, Random Struct. Algorithms.

[21]  Vitaly Feldman.  Hardness of approximate two-level logic minimization and PAC learning with membership queries , 2009, J. Comput. Syst. Sci..

[22]  Umesh V. Vazirani,et al.  An Introduction to Computational Learning Theory , 1994 .

[23]  Yishay Mansour,et al.  Learning Boolean Functions via the Fourier Transform , 1994 .

[24]  Akira Maruoka,et al.  Learning Monotone Log-Term DNF Formulas under the Uniform Distribution , 2000, Theory of Computing Systems.

[25]  Yishay Mansour,et al.  Learning monotone ku DNF formulas on product distributions , 1991, COLT '91.

[26]  Akira Maruoka,et al.  On Learning Monotone Boolean Functions under the Uniform Distribution , 2002, ALT.

[27]  Leslie G. Valiant,et al.  Learning Boolean formulas , 1994, JACM.

[28]  Leonard Pitt,et al.  On the learnability of disjunctive normal form formulas , 2004, Machine Learning.

[29]  Yishay Mansour,et al.  Weakly learning DNF and characterizing statistical query learning using Fourier analysis , 1994, STOC '94.

[30]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, CACM.

[31]  Jeffrey C. Jackson,et al.  An efficient membership-query algorithm for learning DNF with respect to the uniform distribution , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[32]  Linda Sellie,et al.  Exact learning of random DNF over the uniform distribution , 2009, STOC '09.

[33]  Rocco A. Servedio,et al.  Learning Random Log-Depth Decision Trees under Uniform Distribution , 2005, SIAM J. Comput..

[34]  John Langford,et al.  On learning monotone Boolean functions , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[35]  Leslie G. Valiant,et al.  Computational limitations on learning from examples , 1988, JACM.