On Improved Degree Lower Bounds for Polynomial Approximation

A polynomial P in F[X_1,...,X_n] is said to epsilon-approximate a boolean function F:{0,1}^n -> {0,1} under distribution D over {0,1}^n if for a random x chosen according to distribution D, the probability that P(x) is not equal to F(x) is at most epsilon. Smolensky (1987) showed that for any constant distinct primes p and q, any polynomial P in F_p[x_1,...,x_n] that (1/2q - Omega(1))-approximates the boolean function MOD_q:{0,1}^n->{0,1} -- which accepts its input iff the number of ones is non-zero modulo q -- under the uniform distribution must have degree Omega(n^{1/2}). We consider the problem of finding an explicit function f:{0,1}^n->{0,1} that has no epsilon-approximating polynomial of degree less than n^{1/2 + Omega(1)} under *some distribution*, for some constant epsilon>0. We show a number of negative results in this direction: specifically, we show that many interesting classes of functions including symmetric functions and linear threshold functions do have approximating polynomials of degree O(n^{1/2+o(1)}) under every distribution. This demonstrates the power of this model of computation. The above results, in turn, provide further motivation for this lower bound question. Using the upper bounds obtained above, we show that finding such a function f would have applications to: lower bounds for AC^0 o F where F is the class of symmetric and threshold gates; stronger lower bounds for 1-round compression by ACC^0[p] circuits; improved correlation lower bounds against low degree polynomials; and (under further conditions) showing that the Inner Product (over F_2) function does not have small AC^0 o MOD_2 circuits