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.
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.
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). Introduced in the seminal work of Minsky and Papert (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 Ω(n(d-1)/(2d-1)) for circuits of depth d. We obtain a polynomial improvement for every depth d, with a lower bound of Ω(n3/7) for depth 3 and Ω(√n) for depth d ≥4. The proof contributes a novel approximation-theoretic technique of independent interest, which exploits asymmetry in circuits to prove their hardness for polynomials.
The threshold degree of a Boolean function f:{0,1}n→{0,1} is the minimum degree of a real polynomial p that represents f in sign: sgnp(x)=(−1)f(x). A related notion is sign-rank, defined for a Boolean matrix F=[Fij] as the minimum rank of a real matrix M with sgnMij=(−1)Fij. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (AC0) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications. We give an essentially optimal solution to this problem. For any ε>0, we construct an AC0 circuit in n variables that has threshold degree Ω(n1−ε) and sign-rank exp(Ω(n1−ε)), improving on the previous best lower bounds of Ω(n) and exp(Ω~(n)), respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of AC0 circuits of any given depth, with a strict improvement starting at depth 4. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of AC0, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of AC0.
The approximate degree of a Boolean function f∶{0,1}n→{0,1} is the minimum degree of a real polynomial p that approximates f pointwise: |f(x)−p(x)|≤1/3 for all x∈{0,1}n. For any δ>0, we construct DNF and CNF formulas of polynomial size with approximate degree Ω(n1−δ), essentially matching the trivial upper bound of n. This fully resolves the approximate degree of constant-depth circuits (AC0), a question that has seen extensive research over the past 10 years. Prior to our work, an Ω(n1−δ) lower bound was known only for AC0 circuits of depth that grows with 1/δ (Bun and Thaler, FOCS 2017). Furthermore, the DNF and CNF formulas that we construct are the simplest possible in that they have constant width.
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 = Ω(n1/3· d2/3). In particular, if d = n1-Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a constant-depth polynomial-size Boolean circuit, then so is F. By recursively applying our transformation, for any constant δ > 0 we exhibit an AC0function of approximate degree Ω(n1-δ). This improves over the best previous lower bound of Ω(n2/3) 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 δ > 0, an Ω(n1-δ) lower bound on the quantum communication complexity of a function in AC0. · 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 AC0.
A Nearly Optimal Lower Bound on the Approximate Degree of AC0
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=Ω(n1/3⋅d2/3). In particular, if d=n1−Ω(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 δ>0 we exhibit an AC0 function of approximate degree Ω(n1−δ). This improves upon the best previous lower bound of Ω(n2/3) due to Aaronson and Shi [ J. ACM, 51 (2004), pp. 595--605] and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) disjunctive normal forms of polylogarithmic width. We describe several applications of these results and provide the following: (i) for any constant δ>0, an Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC0; (ii) 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; (iii) improved secret sharing schemes with reconstruction procedures in AC0.