On Polynomial Approximations to AC

We make progress on some questions related to polynomial approximations of AC0. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that any AC0 circuit of size s and depth d has an ε-error probabilistic polynomial over the reals of degree (log(s/ε))O(d). We improve this upper bound to (log s)O(d) · log(1/ε), which is much better for small values of ε. We give an application of this result by using it to resolve a question posed by Tal (ECCC 2014): we show that (log s)O(d) · log(1/ε)-wise independence fools AC0, improving on Tal’s strengthening of Braverman’s theorem (J. ACM 2010) that (log(s/ε))O(d)-wise independence fools AC0. Up to the constant implicit in the O(d), our result is tight. As far as we know, this is the first PRG construction for AC0 that achieves optimal dependence on the error ε. We also prove lower bounds on the best polynomial approximations to AC0. We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least Ω̃( √ log n). This result improves exponentially on a recent lower bound demonstrated by Meka, Nguyen, and Vu (arXiv 2015). 1 Motivation and Results We use AC0(s, d) to denote AC0 circuits of size s and depth d. Polynomial approximations to AC0. In his breakthrough work on proving lower bounds for the class AC0[⊕], Razborov [14] studied how well small circuits can be approximated by low-degree polynomials. We recall (an equivalent version of) his notion of polynomial approximation over the reals. An ε-error probabilistic polynomial (over the reals) for a circuit C(x1, . . . , xn) is a random polynomial P(x1, . . . , xn) ∈ R[x1, . . . , xn] such that for any a ∈ {0, 1}n, we have PrP[C(a) 6= P(a)] ≤ ε. Further, we say that P has degree D and ‖P‖∞ ≤ L if P is supported on polynomials P of degree at most D and L∞ norm at most L (i.e. polynomials P such that maxa∈{0,1}n |P(a)| ≤ L). If there is such a P for C, we say that C has ε-error probabilistic degree at most D. It is well-known [19, 18, 2] that any circuit C ∈ AC0(s, d) has an ε-error probabilistic polynomial P of degree (log(s/ε))O(d) and satisfying ‖P‖∞ < exp((log s/ε)O(d)). This can be used to ∗TIFR, Mumbai, India. [email protected] †Department of Mathematics, IIT Bombay, Mumbai, India. [email protected]