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IEEE Journal on Selected Areas in Information Theory

Noisy Computing of the OR and MAX Functions

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

Published Jan 1, 2024• 0 citations
Publisher

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(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.

DOI
10.1109/jsait.2024.3396787

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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.
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.
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.
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.
We present a novel technique, based on the Jensen-Shannon divergence from information theory, to prove lower bounds on the query complexity of sampling algorithms that approximate functions over arbitrary domain and range. Unlike previous methods, our technique does not use a reduction from a decision promise problem. As a result, it gives stronger bounds for functions that possess a large set of inputs, each two of which exhibit a gap in the function value.We demonstrate the technique with new query complexity lower bounds for three fundamental problems: (1) the election problem, for which we obtain a quadratic improvement over previous bounds, (2) low rank matrix approximation, for which we prove the first lower bounds, showing that the algorithms given for this problem are almost optimal, and (3) matrix reconstruction.In addition, we introduce a new method for proving lower bounds on the expected query complexity of functions, using the Kullback-Leibler divergence. We demonstrate its use by a simple query complexity lower bound for the mean.
This paper considers the quantum query complexity of e-biased oracles that return the correct value with probability only 1/2 + e. In particular, we show a quantum algorithm to compute N-bit OR functions with O(√N/e) queries to e-biased oracles. This improves the known upper bound of O(√N/e2) and matches the known lower bound; we answer the conjecture raised by the paper yamashita affirmatively. We also show a quantum algorithm to cope with the situation in which we have no knowledge about the value of e. This contrasts with the corresponding classical situation, where it is almost hopeless to construct a bounded error algorithm without knowing the value of e.