Mansour's Conjecture is True for Random DNF Formulas

In 1994, Y. Mansour conjectured that for every DNF formula on n variables with t terms there exists a polynomial p with tO(log(1/)) non-zero coefficients such that Ex{0,1}n [(p(x) - f(x))2 ] . We make the first progress on this conjecture and show that it is true for several natural subclasses of DNF formulas including randomly chosen DNF formulas and read-k DNF formulas for constant k. Our result yields the first polynomial-time query algorithm for agnostically learning these subclasses of DNF formulas with respect to the uniform distribution on {0, 1} n (for any constant error parameter). Applying recent work on sandwiching polynomials, our results imply that a t-O(log 1/) -biased distribution fools the above subclasses of DNF formulas. This gives pseudorandom generators for these subclasses with shorter seed length than all previous work.

• Type: Journal
• Year: 2010
• Where: COLT
• Authors: Adam R. Klivans, Homin K. Lee, Andrew Wan
• Keywords: COLT 2010, DNF Formulas, Formulas, Machine Learning, Read-k Dnf Formulas