A Nearly Optimal Lower Bound on the Approximate Degree of AC$^0$

Abstract: The approximate degree of a Boolean function $f \colon {-1, 1}ⁿ \rightarrow {-1, 1}$ 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 $O(n \cdot \operatorname{polylog}(n))$ variables with approximate degree at least $D = \Omega(n^{1/3} \cdot d<sup>{2/3})$. In particular, if $d= n<sup>{1-\Omega(1)}$, 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 $\delta &gt; 0$ we exhibit an AC$<sup>0$ function of approximate degree $\Omega(n<sup>{1-\delta})$. This improves over the best previous lower bound of $\Omega(n<sup>{2/3})$ 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 $\delta &gt; 0$, an $\Omega(n<sup>{1-\delta})$ lower bound on the quantum communication complexity of a function in AC$<sup>0$. * A Boolean function $f$ with approximate degree at least $C(f)<sup>{2-o(1)}$, 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$<sup>0$.