Abstract:
The approximate degree of a Boolean function f: {-1, 1}^n --> {-1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits.
Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n * polylog(n)) variables with approximate degree at least D = \Omega(n^{1/3} * d^{2/3}). In particular, if d= n^{1-\Omega(1)}, then D is polynomially larger than d. Moreover, if f is computed by a polynomial-size Boolean circuit of constant depth, then so is F.
By recursively applying our transformation, for any constant delta > 0 we exhibit an AC^0 function of approximate degree \Omega(n^{1-\delta}). This improves over the best previous lower bound of \Omega(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width.
We describe several applications of these results. We give:
*For any constant delta > 0, an \Omega(n^{1-\delta}) lower bound on the quantum
communication complexity of a function in AC^0.
*A Boolean function f with approximate degree at least C(f)^{2-o(1)},
where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent.
*Improved secret sharing schemes with reconstruction procedures in AC^0.
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