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arXiv:1901.00988

Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0

Alexander A. Sherstov, Pei Wu

Published Jan 4, 2019• 4 citations
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Abstract

The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a real matrix $M$ with $\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}$. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits ($\text{AC}^{0}$) 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 $\text{AC}^{0}$ circuit in $n$ variables that has threshold degree $Ω(n^{1-ε})$ and sign-rank $\exp(Ω(n^{1-ε})),$ improving on the previous best lower bounds of $Ω(\sqrt{n})$ and $\exp(\tildeΩ(\sqrt{n}))$, respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of $\text{AC}^{0}$ 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 $\text{AC}^{0}$, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of $\text{AC}^{0}$.

DOI
10.48550/arxiv.1901.00988

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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.
We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial error. First, we prove a tight Omega~(n^{1/2}) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC^0 function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS 2015). Our result also extends to a 2^{Omega~(n^{1/2})} lower bound on the sign-rank of an AC^0 function, improving on the previous best bound of 2^{Omega(n^{2/5})} (Bun and Thaler, ICALP 2016). Second, for any delta>0, we exhibit a function f : {-1,1}^n -> {-1,1} that is computed by a circuit of depth O(1/delta) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error epsilon=1-2^{-Omega(n^{1-delta})}, even by polynomials of degree n^{1-delta}. Our recent prior work (Bun and Thaler, FOCS 2017) proved a similar lower bound, but which held only for error epsilon=1/3. Our result implies 2^{Omega(n^{1-delta})} lower bounds on the complexity of AC^0 under a variety of basic measures such as discrepancy, margin complexity, and threshold weight. This nearly matches the trivial upper bound of 2^{O(n)} that holds for every function. The previous best lower bound on AC^0 for these measures was 2^{Omega(n^{1/2})} (Sherstov, FOCS 2015). Additional applications in learning theory, communication complexity, and cryptography are described.
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The approximate degree of a Boolean function f ⁣:{−1,1}n→{−1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}f:{−1,1}n→{−1,1} is the least degree of a real polynomial that approximates fff 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 fff with approximate degree ddd into a function FFF on O(n⋅polylog(n))O(n \cdot \mathrm{polylog}(n))O(n⋅polylog(n)) variables with approximate degree at least D=Ω(n1/3⋅d2/3)D = \Omega(n^{1/3} \cdot d^{2/3})D=Ω(n1/3⋅d2/3). In particular, if d=n1−Ω(1)d= n^{1-\Omega(1)}d=n1−Ω(1), then DDD is polynomially larger than ddd. Moreover, if fff is computed by a polynomial-size Boolean circuit of constant depth, then so is FFF. By recursively applying our transformation, for any constant δ>0\delta > 0δ>0 we exhibit an AC0^00 function of approximate degree Ω(n1−δ)\Omega(n^{1-\delta})Ω(n1−δ). This improves upon the best previous lower bound of Ω(n2/3)\Omega(n^{2/3})Ω(n2/3) due to Aaronson and Shi [ J. ACM, 51 (2004), pp. 595--605] and nearly matches the trivial upper bound of nnn 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\delta > 0δ>0, an Ω(n1−δ)\Omega(n^{1-\delta})Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC0^00; (ii) a Boolean function fff with approximate degree at least C(f)2−o(1)C(f)^{2-o(1)}C(f)2−o(1), where C(f)C(f)C(f) is the certificate complexity of fff; this separation is optimal up to the o(1)o(1)o(1) term in the exponent; (iii) improved secret sharing schemes with reconstruction procedures in AC0^00.
In this work, we prove the strongest known lower bounds for QAC0^00, allowing polynomially many gates and ancillae. Our main results show that: (1) Depth-3 QAC0^00 circuits cannot compute PARITY, and require Ω(exp⁡(n))Ω(\exp(\sqrt{n}))Ω(exp(n​)) gates to compute MAJORITY. (2) Depth-2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage, regardless of size. We develop new classical simulation techniques for QAC0^00 to obtain our depth-3 bounds. In these results, we relax the output requirement of the quantum circuit to a single bit, making our depth 222 approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC0^00 circuits, which can compute PARITY exactly in depth 222 (exp size). Our techniques further suggest that, for boolean total functions, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth 222 QAC0^00 circuits, regardless of size, cannot exactly synthesize an nnn-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth 222 exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound). Finally, we argue that approximating PARITY in QAC0, with significantly better than 1/poly(n) advantage on average, is just as hard as computing it exactly. Thus, extending our techniques to higher depths would also rule out approximate circuits for PARITY and related problems
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.