Noisy Computing of the Threshold Function

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