Noisy Computing of the Threshold Function

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, when $k = o(n)$, it is both sufficient and necessary to use $$(1 \pm o(1)) \frac{n\log \frac{k}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$$ queries in expectation to compute the $\mathsf{TH}_k$ function with a vanishing error probability $\delta = o(1)$, where $D_{\mathsf{KL}}(p || 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. In particular, this says that $(1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$ queries in expectation are both sufficient and necessary to compute the $\mathsf{OR}$ and $\mathsf{AND}$ functions of $n$ Boolean variables. Compared to previous work, our result tightens the dependence on $p$ in both the upper and lower bounds.