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 t non-zero coefficients such that Ex∈{0,1} [(p(x) − f(x))] ≤ ǫ. We make the first progress on this conjecture and show that it is true for randomly chosen DNF formulas and read-once DNF formulas. 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} (for any constant error parameter). Applying recent work on sandwiching polynomials, our results imply that a t -biased distribution fools the above subclasses of DNF formulas. This gives pseudorandom generators for randomly chosen DNF with shorter seed length than all previous work.