Bounding the average sensitivity and noise sensitivity of polynomial threshold functions

Ilias Diakonikolas, Prahladh Harsha, Adam Klivans, Raghu Meka, Prasad Raghavendra, Rocco A. Servedio, Li-Yang Tan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish
Title of host publicationProceedings of the 42nd ACM Symposium on Theory of Computing (STOC '10)
Place of PublicationNew York, NY, USA
PublisherACM
Pages533-542
Number of pages10
ISBN (Print)978-1-4503-0050-6
DOIs
Publication statusPublished - 2010

Keywords

  • average sensitivity
  • boolean function
  • fourier analysis
  • noise sensitivity
  • polynomial threshold function

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