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arXiv:2309.03986

Noisy Computing of the $\mathsf{OR}$ and $\mathsf{MAX}$ Functions

Banghua Zhu, Wang, Ziao, Nadim Ghaddar, Jiantao Jiao, Lele Wang

Published Sep 7, 2023• 0 citations
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Abstract

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 $p \in (0,1/2)$. Specifically, we consider the computation of the $\mathsf{OR}$ function of $n$ bits (where queries correspond to noisy readings of the bits) and the $\mathsf{MAX}$ function of $n$ real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of \[ (1 \pm o(1)) \frac{n\log \frac{1}δ}{D_{\mathsf{KL}}(p \| 1-p)} \] is both sufficient and necessary to compute both functions with a vanishing error probability $δ= o(1)$, where $D_{\mathsf{KL}}(p \| 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. Compared to previous work, our results tighten the dependence on $p$ in both the upper and lower bounds for the two functions.

DOI
10.48550/arxiv.2309.03986

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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 \pm o(1)) \frac{n\log \frac{1}δ}{D_{\mathsf{KL}}(p | 1-p)} ] 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.
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.
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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.
This paper studies the depth of noisy decision trees in which each node gives the wrong answer with some constant probability. In the noisy Boolean decision tree model, tight bounds are given on the number of queries to input variables required to compute threshold functions, the parity function and symmetric functions. In the noisy comparison tree model, tight bounds are given on the number of noisy comparisons for searching, sorting, selection and merging. The paper also studies parallel selection and sorting with noisy comparisons, giving tight bounds for several problems.
We present a new method for deriving lower bounds to the expected number of queries made by noisy decision trees computing Boolean functions. The new method has the feature that expectations are taken with respect to a uniformly distributed random input, as well as with respect to the random noise, thus yielding stronger lower bounds. It also applies to many more functions than do previous results. The method yields a simple proof of the result (previously established by Reischuck and Schmeltz) that almost all Boolean functions of n arguments require Ω(n log n) queries and strengthens this bound from the worst-case over inputs to the average over inputs. The method also yields bounds for specific Boolean functions in terms of their spectra (their Fourier transforms). The simplest instance of this spectral bound yields the result (previously established by Feige, Peleg, Raghavan and Upfal) that the parity function of n arguments requires Ω(n log n) queries, and again strengthens this bound from the worst-case over inputs to the average over inputs. In its full generality, the spectral bound applies to the "highly resilient" functions introduced by Chor, Friedman, Goldreigh, Hastad, Rudich and Smolensky, and it yields non-linear lower bounds whenever the resiliency is asymptotic to the number of arguments.