Bounding the average sensitivity and noise sensitivity of polynomial threshold functions

Abstract

We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-d polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube {-1,1}n and for PTFs over Rn under the standard n-dimensional Gaussian distribution N(0,In). Our bound on the Boolean average sensitivity of PTFs represents progress towards the resolution of a conjecture of Gotsman and Linial [17], which states that the symmetric function slicing the middle d layers of the Boolean hypercube has the highest average sensitivity of all degree-d PTFs. Via the L1 polynomial regression algorithm of Kalai et al. [22], our bounds on Gaussian and Boolean noise sensitivity yield polynomial-time agnostic learning algorithms for the broad class of constant-degree PTFs under these input distributions.The main ingredients used to obtain our bounds on both average and noise sensitivity of PTFs in the Gaussian setting are tail bounds and anti-concentration bounds on low-degree polynomials in Gaussian random variables [20, 7]. To obtain our bound on the Boolean average sensitivity of PTFs, we generalize the critical-index machinery of [37] (which in that work applies to halfspaces, i.e. degree-1 PTFs) to general PTFs. Together with the invariance principle of [30], this lets us extend our techniques from the Gaussian setting to the Boolean setting. Our bound on Boolean noise sensitivity is achieved via a simple reduction from upper bounds on average sensitivity of Boolean PTFs to corresponding bounds on noise sensitivity.