Local Optimality of Dictator Functions with Applications to Courtade--Kumar and Li--M\'edard Conjectures

TL;DR

This paper proves dictator functions are locally optimal for certain stability measures of Boolean functions, confirming key conjectures for specific parameters using majorization and hypercontractivity.

Contribution

It establishes local optimality of dictator functions for $\

Findings

Dictator functions maximize $\

contribution

null

Abstract

Given a convex function $\Phi:[0,1]\to\mathbb{R}$, the $\Phi$-stability of a Boolean function $f$ is defined as $\mathbb{E}[\Phi(T_{\rho}f(\mathbf{X}))]$, where $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{\pm1\}^{n}$ and $T_{\rho}$ is the Bonami-Beckner operator. In this paper, we prove that dictator functions are locally optimal in maximizing the $\Phi$-stability of $f$ over all balanced Boolean functions. When focusing on the symmetric $q$-stability, combining this result with our previous bound, we use computer-assisted methods to prove that dictator functions maximize the symmetric $q$-stability for $q=1$ and $\rho\in[0,0.914]$ or for $q\in[1.36,2)$ and all $\rho\in[0,1]$. In other words, we confirm the (balanced) Courtade--Kumar conjecture with the correlation coefficient $\rho\in[0,0.914]$ and the (symmetrized) Li--M\'edard conjecture with $q\in[1.36,2)$. We conjecture that dictator functions maximize both the symmetric and asymmetric $\frac{1}{2}$-stability over all balanced Boolean functions. Our proofs are based on majorization of noise operators and hypercontractivity inequalities.