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arXiv:2403.07227 · IEEE Transactions on Information Theory

Noisy Computing of the Threshold Function

Ziao Wang, Nadim Ghaddar, Banghua Zhu, Lele Wang

Published Aug 6, 2026• 0 citations
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Abstract

Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability $p \in (0,1/2)$. As our main result, we show that it is sufficient to use $(1+o(1)) \frac{n\log \frac{m}δ}{D_{\mathsf{KL}}(p \| 1-p)}$ queries in expectation to compute the $\mathsf{TH}_k$ function with a vanishing error probability $δ= o(1)$, where $m\triangleq \min\{k,n-k+1\}$ and $D_{\mathsf{KL}}(p \| 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. Conversely, we show that any algorithm achieving an error probability of $δ= o(1)$ necessitates at least $(1-o(1))\frac{(n-m)\log\frac{m}δ}{D_{\mathsf{KL}}(p \| 1-p)}$ queries in expectation. The upper and lower bounds are tight when $m=o(n)$, and are within a multiplicative factor of $\frac{n}{n-m}$ when $m=Θ(n)$. In particular, when $k=n/2$, the $\mathsf{TH}_k$ function corresponds to the $\mathsf{MAJORITY}$ function, in which case the upper and lower bounds are tight up to a multiplicative factor of two. Compared to previous work, our result tightens the dependence on $p$ in both the upper and lower bounds.

DOI
10.1109/tit.2026.3721399

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Let THk\mathsf{TH}_kTHk​ denote the kkk-out-of-nnn threshold function: given nnn input Boolean variables, the output is 111 if and only if at least kkk of the inputs are 111. We consider the problem of computing the THk\mathsf{TH}_kTHk​ function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability p∈(0,1/2)p \in (0,1/2)p∈(0,1/2). As our main result, we show that it is sufficient to use (1+o(1))nlog⁡mδDKL(p∥1−p)(1+o(1)) \frac{n\log \frac{m}δ}{D_{\mathsf{KL}}(p \| 1-p)}(1+o(1))DKL​(p∥1−p)nlogδm​​ queries in expectation to compute the THk\mathsf{TH}_kTHk​ function with a vanishing error probability δ=o(1)δ= o(1)δ=o(1), where m≜min⁡{k,n−k+1}m\triangleq \min\{k,n-k+1\}m≜min{k,n−k+1} and DKL(p∥1−p)D_{\mathsf{KL}}(p \| 1-p)DKL​(p∥1−p) denotes the Kullback-Leibler divergence between Bern(p)\mathsf{Bern}(p)Bern(p) and Bern(1−p)\mathsf{Bern}(1-p)Bern(1−p) distributions. Conversely, we show that any algorithm achieving an error probability of δ=o(1)δ= o(1)δ=o(1) necessitates at least (1−o(1))(n−m)log⁡mδDKL(p∥1−p)(1-o(1))\frac{(n-m)\log\frac{m}δ}{D_{\mathsf{KL}}(p \| 1-p)}(1−o(1))DKL​(p∥1−p)(n−m)logδm​​ queries in expectation. The upper and lower bounds are tight when m=o(n)m=o(n)m=o(n), and are within a multiplicative factor of nn−m\frac{n}{n-m}n−mn​ when m=Θ(n)m=Θ(n)m=Θ(n). In particular, when k=n/2k=n/2k=n/2, the THk\mathsf{TH}_kTHk​ function corresponds to the MAJORITY\mathsf{MAJORITY}MAJORITY function, in which case the upper and lower bounds are tight up to a multiplicative factor of two. Compared to previous work, our result tightens the dependence on ppp in both the upper and lower bounds.
Noisy Computing of the OR\mathsf{OR}OR and MAX\mathsf{MAX}MAX Functions
We consider the problem of computing a function of nnn variables using noisy queries, where each query is incorrect with some fixed and known probability p∈(0,1/2)p \in (0,1/2)p∈(0,1/2). Specifically, we consider the computation of the OR\mathsf{OR}OR function of nnn bits (where queries correspond to noisy readings of the bits) and the MAX\mathsf{MAX}MAX function of nnn real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of (1±o(1))nlog⁡1δDKL(p∥1−p) (1 \pm o(1)) \frac{n\log \frac{1}δ}{D_{\mathsf{KL}}(p \| 1-p)} (1±o(1))DKL​(p∥1−p)nlogδ1​​ is both sufficient and necessary to compute both functions with a vanishing error probability δ=o(1)δ= o(1)δ=o(1), where DKL(p∥1−p)D_{\mathsf{KL}}(p \| 1-p)DKL​(p∥1−p) denotes the Kullback-Leibler divergence between Bern(p)\mathsf{Bern}(p)Bern(p) and Bern(1−p)\mathsf{Bern}(1-p)Bern(1−p) distributions. Compared to previous work, our results tighten the dependence on ppp in both the upper and lower bounds for the two functions.
We consider the problem of computing a function of n variables using noisy queries, where each query is incorrect with some fixed and known probability p(0,1/2). Specifically, we consider the computation of the OR function of n bits (where queries correspond to noisy readings of the bits) and the MAX function of n real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of (1±o(1))nlog1/δDKL(p||1-p) is both sufficient and necessary to compute both functions with a vanishing error probability δ=o(1), where DKL(p||1-p) denotes the Kullback-Leibler divergence between Bern(p) and Bern(1-p) distributions. Compared to previous work, our results tighten the dependence on p in both the upper and lower bounds for the two functions.
In the noisy query model, the (binary) return value of every query (possibly repeated) is independently flipped with some fixed probability p∈(0,1/2)p \in (0, 1/2)p∈(0,1/2). In this paper, we obtain tight bounds on the noisy query complexity of several fundamental problems. Our first contribution is to show that any Boolean function with total influence Ω(n)Ω(n)Ω(n) has noisy query complexity Θ(nlog⁡n)Θ(n\log n)Θ(nlogn). Previous works often focus on specific problems, and it is of great interest to have a characterization of noisy query complexity for general functions. Our result is the first noisy query complexity lower bound of this generality, beyond what was known for random Boolean functions [Reischuk and Schmeltz, FOCS 1991]. Our second contribution is to prove that Graph Connectivity has noisy query complexity Θ(n2log⁡n)Θ(n^2 \log n)Θ(n2logn). In this problem, the goal is to determine whether an undirected graph is connected using noisy edge queries. While the upper bound can be achieved by a simple algorithm, no non-trivial lower bounds were known prior to this work. Last but not least, we determine the exact number of noisy queries (up to lower order terms) needed to solve the kkk-Threshold problem and the Counting problem. The kkk-Threshold problem asks to decide whether there are at least kkk ones among nnn bits, given noisy query access to the bits. We prove that (1±o(1))nlog⁡(min⁡{k,n−k+1}/δ)(1−2p)log⁡1−pp(1\pm o(1)) \frac{n\log (\min\{k,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}(1±o(1))(1−2p)logp1−p​nlog(min{k,n−k+1}/δ)​ queries are both sufficient and necessary to achieve error probability δ=o(1)δ= o(1)δ=o(1). Previously, such a result was only known when min⁡{k,n−k+1}=o(n)\min\{k,n-k+1\}=o(n)min{k,n−k+1}=o(n) [Wang, Ghaddar, Zhu and Wang, arXiv 2024]. We also show a similar (1±o(1))nlog⁡(min⁡{k+1,n−k+1}/δ)(1−2p)log⁡1−pp(1\pm o(1)) \frac{n\log (\min\{k+1,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}(1±o(1))(1−2p)logp1−p​nlog(min{k+1,n−k+1}/δ)​ bound for the Counting problem, where one needs to count the number of ones among nnn bits given noisy query access and kkk denotes the answer.
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.
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.
Given a Boolean function f:{0,1}n→{0,1}f:\{0,1\}^n\to\{0,1\}f:{0,1}n→{0,1}, the goal in the usual query model is to compute fff on an unknown input x∈{0,1}nx \in \{0,1\}^nx∈{0,1}n while minimizing the number of queries to xxx. One can also consider a "distinguishing" problem denoted by fsabf_{\mathsf{sab}}fsab​: given an input x∈f−1(0)x \in f^{-1}(0)x∈f−1(0) and an input y∈f−1(1)y \in f^{-1}(1)y∈f−1(1), either all differing locations are replaced by a ∗*∗, or all differing locations are replaced by †\dagger†, and an algorithm's goal is to identify which of these is the case while minimizing the number of queries. Ben-David and Kothari [ToC'18] introduced the notion of randomized sabotage complexity of a Boolean function to be the zero-error randomized query complexity of fsabf_{\mathsf{sab}}fsab​. A natural follow-up question is to understand Q(fsab)\mathsf{Q}(f_{\mathsf{sab}})Q(fsab​), the quantum query complexity of fsabf_{\mathsf{sab}}fsab​. In this paper, we initiate a systematic study of this. The following are our main results: ∙    \bullet\;\;∙ If we have additional query access to xxx and yyy, then Q(fsab)=O(min⁡{Q(f),n})\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f),\sqrt{n}\})Q(fsab​)=O(min{Q(f),n​}). ∙    \bullet\;\;∙ If an algorithm is also required to output a differing index of a 0-input and a 1-input, then Q(fsab)=O(min⁡{Q(f)1.5,n})\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f)^{1.5},\sqrt{n}\})Q(fsab​)=O(min{Q(f)1.5,n​}). ∙    \bullet\;\;∙ Q(fsab)=Ω(fbs(f))\mathsf{Q}(f_{\mathsf{sab}}) = Ω(\sqrt{\mathsf{fbs}(f)})Q(fsab​)=Ω(fbs(f)​), where fbs(f)\mathsf{fbs}(f)fbs(f) denotes the fractional block sensitivity of fff. By known results, along with the results in the previous bullets, this implies that Q(fsab)\mathsf{Q}(f_{\mathsf{sab}})Q(fsab​) is polynomially related to Q(f)\mathsf{Q}(f)Q(f). ∙    \bullet\;\;∙ The bound above is easily seen to be tight for standard functions such as And, Or, Majority and Parity. We show that when fff is the Indexing function, Q(fsab)=Θ(fbs(f))\mathsf{Q}(f_{\mathsf{sab}})=Θ(\mathsf{fbs}(f))Q(fsab​)=Θ(fbs(f)), ruling out the possibility that Q(fsab)=Θ(fbs(f))\mathsf{Q}(f_{\mathsf{sab}})=Θ(\sqrt{\mathsf{fbs}(f)})Q(fsab​)=Θ(fbs(f)​) for all fff.