Breaking the minsky-papert barrier for constant-depth circuits

The threshold degree of a Boolean function f is the minimum degree of a real polynomial p that represents f in sign: f(x) ≡ sgn p(x). In a seminal 1969 monograph, Minsky and Papert constructed a polynomial-size constant-depth {∧, ∨)-circuit in n variables with threshold degree Ω(n1/3). This bound underlies some of today's strongest results on constant-depth circuits. It has been an open problem (O'Donnell and Servedio, STOC 2003) to improve Minsky and Papert's bound to nΩ(1)+1/3. We give a detailed solution to this problem. For any fixed k ≥ 1, we construct an {∧, ∨)-formula of size n and depth k with threshold degree Ω(n k-1/2k-1). This lower bound nearly matches a known O(√n) bound for arbitrary formulas, and is exactly tight for regular formulas. Our result proves a conjecture due to O'Donnell and Servedio (STOC 2003) and a different conjecture due to Bun and Thaler (2013). Applications to communication complexity and computational learning are given.