# Approximating the noise sensitivity of a monotone Boolean function

## TL;DR

This paper presents a near-optimal randomized algorithm for approximating the noise sensitivity of monotone Boolean functions within a specified range of parameters, introducing new analytical tools and lower bound techniques.

## Contribution

It introduces a novel descending-ascending view of noise sensitivity and develops new lower bound techniques based on combinatorial and probabilistic methods.

## Key findings

- The algorithm achieves query complexity close to the theoretical lower bound.
- A new descending-ascending framework for analyzing noise sensitivity is proposed.
- Lower bounds are established for approximating noise sensitivity, influence, and bias.

## Abstract

The noise sensitivity of a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is one of its fundamental properties. A function of a positive noise parameter $\delta$, it is denoted as $NS_{\delta}[f]$. Here we study the algorithmic problem of approximating it for monotone $f$, such that $NS_{\delta}[f] \geq 1/n^{C}$ for constant $C$, and where $\delta$ satisfies $1/n \leq \delta \leq 1/2$. For such $f$ and $\delta$, we give a randomized algorithm performing $O\left(\frac{\min(1,\sqrt{n} \delta \log^{1.5} n) }{NS_{\delta}[f]} \text{poly}\left(\frac{1}{\epsilon}\right)\right)$ queries and approximating $NS_{\delta}[f]$ to within a multiplicative factor of $(1\pm \epsilon)$. Given the same constraints on $f$ and $\delta$, we also prove a lower bound of $\Omega\left(\frac{\min(1,\sqrt{n} \delta)}{NS_{\delta}[f] \cdot n^{\xi}}\right)$ on the query complexity of any algorithm that approximates $NS_{\delta}[f]$ to within any constant factor, where $\xi$ can be any positive constant. Thus, our algorithm's query complexity is close to optimal in terms of its dependence on $n$. We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield previously unknown lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias.