# When did Majority become the stablest?

The fireworks happening over at Ryan O’Donnell’s blog reminded me of something that always puzzles me: Is Majority the Stablest? Or the Unstablest?

Consider the class of all monotone, balanced Boolean functions $f:\{-1,1\}^n \rightarrow \{-1,1\}$. We define the noise sensitivity $NS(f)$ of a function $f$ as $NS(f) = P_{x,e}[f(x) \neq f( x \cdot e)]$, where $x \in \{-1,1\}^n$ is chosen uniformly whereas the noise $e \in \{-1,1\}^n$ is sampled from a noise distribution $N$ of our choice. Think of $e$ as telling us which bits of $x$ to flip. Two natural choices are:

1. The $\epsilon$-noise distribution $N(\epsilon)$ where each bit is flipped independently with probability $\epsilon$.
2. The “bit-flip” distribution $N'$ where we flip a single random bit. This behaves a lot like  $N(\epsilon)$ where $\epsilon = 1/n$.

Say we’d like to find the monotone, balanced, Boolean function $f$ which is least sensitive to noise. A bit of thought shows that the answer is any co-ordinate/dictator function $x_i$. So to rule out trivial answers, let us only allow functions where all variables are equally important. More precisely, functions should be invariant under a transitive group of  permutations on the variables. This is a bit of overkill, the right way to do this is to insist that no variable has too much influence.

The Majority function $Maj(x_1,\ldots,x_n) = sgn(\sum_{i=1}^nx_i )$ is pretty insensitive/stable under $\epsilon$-noise, one can show that $NS(Maj) \leq \sqrt{\epsilon}$.  In contrast, for constant $\epsilon$, many other functions will have sensitivity approaching $0.5$.The Majority is Stablest Theorem due Mossel, O’Donnell and Oleskeiwicz shows Majority is indeed about as good as it gets, functions which are more stable need to have a distinguished variable (which we disallow).

Now consider the “bit-flip” distribution $N'$. Using the $\epsilon = 1/n$ rule of thumb, one would expect the noise-stability of Majority to be around $1/\sqrt{n}$. This is indeed the case, no surprises there. What is surprising is that this is as unstable as it gets for monotone functions! The proof of this is simple: we start with the observation that since $f$ is monotone, $E_x[f(x)x_i]$ captures the probability that flipping $x_i$ changes $f$ for a random $x$. To see why this is so, fix all the other bits except $x_i$. Now flipping $x_i$ from $-1$ to $+1$ can result in two possible outcomes:

1. It leaves $f$ unchanged, in which case the contribution to $E_x[f(x)x_i]$ is 0.
2. It causes $f$ to change from $-1$ to $+1$, in which case the contribution to $E_x[f(x)x_i]$ is 2. By monotonicity, the reverse flip from $+1$ to $-1$ is not possible.

So now we have

$NS(f) = \frac{1}{n}\sum_{i=1}^nE_x[f(x)x_i] = \frac{1}{n}E_x\left[f(x)\left(\sum_{i=1}^nx_i\right)\right]$

Since $f(x) \in \{-1,1\}$, we have

$f(x)\left(\sum_{i=1}^nx_i\right) \leq \left|\sum_{i=1}^nx_i\right|$

with equality iff

$f(x) = sgn\left(\sum_{i=1}^nx_i\right) = Maj(x_1,...,x_n)$

So  among balanced, transitive, monotone functions, Majority is maximally unstable under $N'$ and maximally stable under $N(\epsilon)$. And in an intriguing twist, one way to prove the stability under $N(\epsilon)$ is to reduce it to bounding the stability under $N'$ as is done in the proof of Peres’ Theorem, which tells us that the $\sqrt{\epsilon}$ bound is true for all LTFs.

Coming soon:

1. If not Majority, then who?
2. What lies behind Majority’s rapid descent into instability?