Noisy Computing of the Threshold Function
Ziao Wang, Nadim Ghaddar, Banghua Zhu, Lele Wang
TL;DR
This work characterizes the query complexity of computing the threshold function $\mathsf{TH}_k$ under noisy bit readings through a Binary Symmetric Channel with fixed noise $p\in(0,1/2)$. It delivers asymptotically tight results: an achievability bound of $\Theta\left(\frac{n\log(m/\delta)}{D_{\mathrm{KL}}(p\|1-p)}\right)$ and a matching (up to smaller factors) converse bound, where $m=\min\{k,n-k+1\}$ and $D_{\mathrm{KL}}(p\|1-p)$ is the KL divergence between $\mathrm{Bern}(p)$ and $\mathrm{Bern}(1-p)$. The paper sharpens previous results by tightening dependence on $p$ and establishing tightness for $m=o(n)$ and near-tightness for general $m$, with MAJORITY ($k=n/2$) achieving exact-throughput up to a factor of 2. The authors develop two complementary analyses: a Le Cam-based lower bound and a genie-aided, balls-in-bins lower-bound for large $m$, and two regimes for the upper bound using per-bit checking or a filtering-plus-heap strategy. These findings advance understanding of robust threshold computation under noise and point to extensions when $p$ is unknown or varies with $n$.
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.