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arXiv:2107.03171 · Leibniz international proceedings in informatics

On the Probabilistic Degree of an nnn-variate Boolean Function

Srikanth Srinivasan, S. Venkitesh

Published Jan 1, 2021• 0 citations
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Abstract

Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log n - O(\log \log n)$. This was improved to a tight $(\log n - O(1))$ bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for \emph{Probabilistic degree}. The function $f$ has probabilistic degree at most $d$ if there is a random real-valued polynomial of degree at most $d$ that agrees with $f$ at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between $(\log n)^{1/2-o(1)}$ and $O(\log n)$ (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of $n$-variate functions $f$, \emph{modulo} our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is $(\log n)^c$, then the minimum possible probabilistic degree of such an $f$ is at least $(\log n)^{c/(c+1)-o(1)}$, and we show this is tight up to $(\log n)^{o(1)}$ factors.

DOI
10.4230/lipics.approx/random.2021.42

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On the probabilistic degree of an n-variate boolean function

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SIAM Journal on Computing•Oct 22, 2019•6 citations
Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}n→{0,1}f:\{0,1\}^n\rightarrow \{0,1\}f:{0,1}n→{0,1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x1,…,xn)P(x_1,\ldots,x_n)P(x1​,…,xn​), has degree at least log⁡n−O(log⁡log⁡n)\log n - O(\log \log n)logn−O(loglogn). This was improved to a tight (log⁡n−O(1))(\log n - O(1))(logn−O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for \emph{Probabilistic degree}. The function fff has probabilistic degree at most ddd if there is a random real-valued polynomial of degree at most ddd that agrees with fff at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log⁡n)1/2−o(1)(\log n)^{1/2-o(1)}(logn)1/2−o(1) and O(log⁡n)O(\log n)O(logn) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of nnn-variate functions fff, \emph{modulo} our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log⁡n)c(\log n)^c(logn)c, then the minimum possible probabilistic degree of such an fff is at least (log⁡n)c/(c+1)−o(1)(\log n)^{c/(c+1)-o(1)}(logn)c/(c+1)−o(1), and we show this is tight up to (log⁡n)o(1)(\log n)^{o(1)}(logn)o(1) factors.
Nisan and Szegedy (CC 1994) showed that any Boolean function f : {0, 1}n → {0, 1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x1,..., xn), has degree at least log n − O(log log n). This was improved to a tight (log n − O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log n)1/2−o(1) and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)c, then the minimum possible probabilistic degree of such an f is at least (log n)c/(c+1)−o(1), and we show this is tight up to (log n)o(1) factors.
The approximate degree of a Boolean function f(x1,x2,…,xn)f(x_{1},x_{2},\ldots,x_{n})f(x1​,x2​,…,xn​) is the minimum degree of a real polynomial that approximates fff pointwise within 1/31/31/3. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a novel, first-principles approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: O(n34−14(2k−1))O(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}})O(n43​−4(2k−1)1​) for the kkk-element distinctness problem; O(n1−1k+1)O(n^{1-\frac{1}{k+1}})O(n1−k+11​) for the kkk-subset sum problem; O(n1−1k+1)O(n^{1-\frac{1}{k+1}})O(n1−k+11​) for any kkk-DNF or kkk-CNF formula; and O(n3/4)O(n^{3/4})O(n3/4) for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the Θ(n)\Theta(n)Θ(n) quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree Ω(n)\Omega(n)Ω(n). In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity.
The approximate non-deterministic degree of a Boolean function fff, denoted ndegε(f)\mathsf{ndeg}_ε(f)ndegε​(f) (written Nε(f)\mathsf{N}_ε(f)Nε​(f) for brevity), is the minimum degree of a real polynomial ppp such that 0≤∣p(x)∣≤ε0 \le |p(x)| \le ε0≤∣p(x)∣≤ε whenever f(x)=0f(x) = 0f(x)=0, and ∣p(x)∣≥1|p(x)| \ge 1∣p(x)∣≥1 whenever f(x)=1f(x) = 1f(x)=1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on 111-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all 000-inputs and bounded away from zero on all 111-inputs. The rational degree conjecture, open for over three decades, was recently resolved by Kothari, Kovacs-Deak, Wang, and Yang, who showed that for every total Boolean function fff, deg(f)≤O~ ⁣(rdeg⁡(f)3). deg(f) \le \widetilde O\!\left(\operatorname{rdeg}(f)^3\right). deg(f)≤O(rdeg(f)3). In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by Nε(f)\mathsf{N}_ε(f)Nε​(f) and Nε(f‾)\mathsf{N}_ε(\overline{f})Nε​(f​) jointly, i.e., for every total Boolean function fff and every constant 0<ε<10<ε<10<ε<1, deg~(f)≤poly⁡(Nε(f),Nε(f‾)). \widetilde{deg}(f) \le \operatorname{poly}(\mathsf N_ε(f), \mathsf N_ε(\overline f)). deg​(f)≤poly(Nε​(f),Nε​(f​)). This conjecture, if true, would imply a polynomial version of the rational degree result and bring us closer to resolving de Wolf's longstanding non-deterministic degree conjecture. In this work, we make the first systematic progress on this problem, establishing the conjecture for several broad and natural function classes: monotone and unate functions, functions of bounded alternation number, symmetric functions, kkk-uniform hypergraph properties, and read-kkk Disjunctive Normal Form (DNF) formulas.
We study the query complexity of Boolean functions f:{0,1}n→{0,1}f: \{0, 1\}^n \rightarrow \{0, 1\}f:{0,1}n→{0,1} in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability p∈(0,1/2)p \in (0, 1/2)p∈(0,1/2); repeated queries are allowed. The noisy query complexity Np(f)\mathsf{N}_p(f)Np​(f) of a function fff is defined as the minimum expected number of queries needed to compute f(x)f(x)f(x) with error probability at most 1/31/31/3, for the worst case input xxx. We prove a general lower bound on Np(f)\mathsf{N}_p(f)Np​(f) based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that Np(f)\mathsf{N}_p(f)Np​(f) is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that Np(f)=Ω(I(f)log⁡I(f))\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))Np​(f)=Ω(I(f)logI(f)), where I(f)\mathsf{I}(f)I(f) denotes the total influence of fff. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.
Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.
A Nearly Optimal Lower Bound on the Approximate Degree of AC0^00
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.