Degree and Sensitivity: tails of two distributions

TL;DR

This paper proves that Boolean functions with sensitivity s can be approximated by low-degree polynomials within any epsilon in L2 norm, providing near-optimal bounds and linking to the sensitivity conjecture.

Contribution

It introduces an approximate version of the sensitivity conjecture, showing polynomial approximation bounds and proposing a robust analogue related to Fourier concentration.

Findings

Boolean functions with sensitivity s can be epsilon-approximated by degree O(s log(1/epsilon)) polynomials.

Improving the approximation bound to O(s^c log(1/epsilon)^d) with d<1 would imply the sensitivity conjecture.

A conjecture that functions with mostly low sensitivity inputs have Fourier mass on small subsets.

Abstract

The sensitivity of a Boolean function f is the maximum over all inputs x, of the number of sensitive coordinates of x. The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean function can be computed by a polynomial over the reals of degree poly(s). The best known upper bounds on degree, however, are exponential rather than polynomial in s. Our main result is an approximate version of the conjecture: every Boolean function with sensitivity s can be epsilon-approximated (in L_2) by a polynomial whose degree is O(s log(1/epsilon)). This is the first improvement on the folklore bound of s/epsilon. Further, we show that improving the bound to O(s^c log(1/epsilon)^d)$ for any d < 1 and any c > 0 will imply the sensitivity conjecture. Thus our result is essentially the best one can hope for without proving the conjecture. We postulate a robust analogue of the sensitivity conjecture: if most inputs to a Boolean function f have low sensitivity, then most of the Fourier mass of f is concentrated on small subsets, and present an approach towards proving this conjecture.