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 $(1+o(1)) \frac{n\log \frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ queries in expectation to compute the $\mathsf{TH}_k$ function with a vanishing error probability $\delta = o(1)$, where $m\triangleq \min\{k,n-k\}$ 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 $\delta = o(1)$ necessitates at least $(1-o(1))\frac{(n-m)\log\frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ 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 $m=\Theta(n)$. 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.

翻译：令 $\mathsf{TH}_k$ 表示 $k$-out-of-$n$ 阈值函数：给定 $n$ 个布尔输入变量，当且仅当至少 $k$ 个输入为 $1$ 时输出为 $1$。我们研究在布尔变量存在噪声读数的情形下计算 $\mathsf{TH}_k$ 函数的问题，其中每次读数以固定已知概率 $p \in (0,1/2)$ 发生错误。作为主要结果，我们证明：为以趋于零的错误概率 $\delta = o(1)$ 计算 $\mathsf{TH}_k$ 函数，期望查询次数只需 $(1+o(1)) \frac{n\log \frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ 即可，其中 $m\triangleq \min\{k,n-k\}$，$D_{\mathsf{KL}}(p \| 1-p)$ 表示 $\mathsf{Bern}(p)$ 与 $\mathsf{Bern}(1-p)$ 分布间的 Kullback-Leibler 散度。反之，我们证明任何达到错误概率 $\delta = o(1)$ 的算法至少需要 $(1-o(1))\frac{(n-m)\log\frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ 次期望查询。当 $m=o(n)$ 时上下界紧致，当 $m=\Theta(n)$ 时二者相差 $\frac{n}{n-m}$ 倍乘因子。特别地，当 $k=n/2$ 时，$\mathsf{TH}_k$ 函数对应于 $\mathsf{MAJORITY}$ 函数，此时上下界在二倍乘因子内紧致。与已有工作相比，我们的结果强化了上下界中对 $p$ 的依赖关系。