The Power of Asymmetry in Constant-Depth Circuits

Abstract

The threshold degree of a Boolean function $f$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv {sgn}~ p(x)$. Introduced in the seminal work of Minsky and Papert [Perceptrons: An Introduction to Computational Geometry, MIT Press, 1969], this notion is central to some of the strongest algorithmic and complexity-theoretic results for constant-depth circuits. One problem that has remained open for several decades, with applications to computational learning and communication complexity, is to determine the maximum threshold degree of a polynomial-size constant-depth circuit in $n$ variables. The best lower bound prior to our work was $\Omega(n^{(d-1)/(2d-1)})$ for circuits of depth $d$. We obtain a polynomial improvement for every depth $d,$ with a lower bound of $\Omega(n^{3/7})$ for depth 3 and $\Omega(\sqrt{n})$ for depth $d\geq4.$ The proof contributes an approximation-theoretic technique of independent interest, which exploits asymmetry in circuits to prove their hardness for polynomials.