Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions

Abstract

We give the first nontrivial upper bounds on the Boolean average sensitivity and noise sensitivity of degree-$d$ polynomial threshold functions (PTFs). Our bound on the Boolean average sensitivity of PTFs represents the first progress toward the resolution of a conjecture of Gotsman and Linial [Combinatorica, 14 (1994), pp. 35--50], 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 $L_1$ polynomial regression algorithm of Kalai et al. [SIAM J. Comput., 37 (2008), pp. 1777--1805], our bound on Boolean noise sensitivity yields the first polynomial-time agnostic learning algorithm for the broad class of constant-degree PTFs under the uniform distribution. To obtain our bound on the Boolean average sensitivity of PTFs, we generalize the “critical-index” machinery of [R. Servedio, Comput. Complexity, 16 (2007), pp. 180--209] (which in that work applies to halfspaces, i.e., degree-1 PTFs) to general...