A Counterexample to the "Majority is Least Stable" Conjecture

TL;DR

This paper presents a specific counterexample in the form of a 5-variable linear threshold function that has lower noise stability than the majority function, disproving a longstanding conjecture.

Contribution

It provides the first explicit counterexample to the 'Majority is Least Stable' Conjecture, challenging previous assumptions about noise stability in Boolean functions.

Findings

Counterexample with 5 variables showing lower noise stability than majority

Disproves the 'Majority is Least Stable' Conjecture

Highlights limitations of previous stability conjectures

Abstract

We exhibit a linear threshold function in 5 variables with strictly smaller noise stability (for small values of the correlation parameter) than the majority function on 5 variables, thereby providing a counterexample to the "Majority is Least Stable" Conjecture of Benjamini, Kalai, and Schramm.

Equations

$$f(x_1,x_2,x_3,x_4,x_5)=\text{sgn}(2x_1+2x_2+x_3+x_4+x_5)$$

$${\text{Inf}}_1[{\text{Maj}}_5]=\frac{\left(\genfrac{}{}{0ex}{}{4}{2}\right)}{2^4}=\frac{3}{8}$$

$${\text{Inf}}_1[f]=\frac{2+2\times \left(\genfrac{}{}{0ex}{}{3}{1}\right)}{16}=\frac{1}{2}$$

$${\text{Inf}}_3[f]=\frac{2\times 2}{16}=\frac{1}{4}$$

$$W^1[{\text{Maj}}_5]$$

$$W^1[f]$$

Full text

In their document Real Analysis in Computer Science: A Collection of Open Problems, Filmus et al. [FHHMOSWW14] present the following “Majority is Least Stable” conjecture due to Benjamini, Kalai and Schramm [H17]:

In this note, we provide a simple counterexample to this conjecture, even when $n=5$. We begin by noting that for an unbiased linear threshold function $f:\{-1,1\}^{n}\rightarrow\{-1,1\}$, the statement $W^{1}[f]<W^{1}[\text{Maj}_{n}]$ would disprove the conjecture. An example of such an LTF is provided by $f\colon\{-1,1\}^{5}\rightarrow\{-1,1\}$ given by

[TABLE]

To show this, we explicitly compute the degree-1 Fourier coefficients of $f$ and of $\text{Maj}_{5}$. Since both these functions are monotone, computing $\hat{f}(i)$ is the same as computing the influence $\text{Inf}_{i}[f]$. Moreover, by symmetry, it suffices to compute $\text{Inf}_{1}[\text{Maj}_{5}],\text{Inf}_{1}[f]$ and $\text{Inf}_{3}[f]$.

Since coordinate $1$ is influential for $\text{Maj}_{5}$ iff $x_{2}+x_{3}+x_{4}+x_{5}=0$, it follows that

[TABLE]

Since coordinate $1$ is influential for $f$ iff $|2x_{2}+x_{3}+x_{4}+x_{5}|=1$, and this happens precisely either if $x_{2}=\pm 1$ and $x_{3}=x_{4}=x_{5}=\mp 1$, or $x_{2}=x_{i}=\pm 1$, $x_{k},x_{l}=\mp 1$ where $(i,j,k)$ is some permutation of $(3,4,5)$, we get

[TABLE]

Finally, coordinate $3$ is influential for $f$ iff $2x_{1}+2x_{2}+x_{4}+x_{5}=0$. This can happen iff $x_{1}=-x_{2}$ and $x_{4}=-x_{5}$. Hence

[TABLE]

It follows that

[TABLE]

whereas

[TABLE]

which completes the verification.

Acknowledgements

The author would like to thank Elchanan Mossel for help with the presentation of the argument, as well as introducing him to [FHHMOSWW14].