Hardness Amplification and the Approximate Degree of Constant-Depth Circuits

Abstract

We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to within constant error in a certain one-sided sense, then an OR of disjoint copies of that circuit cannot be pointwise approximated even with very high error. As our main application, we show that for every sequence of degrees $d(n)$, there is an explicit depth-three circuit $F: \{-1,1\}^n \rightarrow \{-1,1\}$ of polynomial-size such that any degree-$d$ polynomial cannot pointwise approximate $F$ to error better than $1-\exp (-\tilde{\Omega }(nd^{-3/2}))$. As a consequence of our main result, we obtain an $\exp (-\tilde{\Omega }(n^{2/5}))$ upper bound on the the discrepancy of a function in AC$^{\text{0 }}$, and an $\exp (\tilde{\Omega }(n^{2/5}))$ lower bound on the threshold weight of AC$^{\text{0 }}$, improving over the previous best results of $\exp (-\Omega (n^{1/3}))$ and $\exp (\Omega (n^{1/3}))$ respectively.

Our techniques also yield a new lower bound of $\Omega (n^{1/2}/\log ^{(d-2)/2}(n))$ on the approximate degree of the AND-OR tree of depth $d$, which is tight up to polylogarithmic factors for any constant $d$, as well as new bounds for read-once DNF formulas. In turn, these results imply new lower bounds on the communication and circuit complexity of these classes, and demonstrate strong limitations on existing PAC learning algorithms.

The full version of this paper is available at http://arxiv.org/abs/1311.1616.

Supported by an NDSEG Fellowship and NSF grant CNS-1237235.

Parts of this work were done while the author was a graduate student at Harvard University, and a Research Fellow at the Simons Institute for the Theory of Computing. This work was supported by an NSF Graduate Research Fellowship, NSF grants CNS-1011840 and CCF-0915922, and a Research Fellowship from the Simons Institute for the Theory of Computing.