The Average Sensitivity of an Intersection of Half Spaces

TL;DR

This paper establishes optimal bounds on the average sensitivity of indicator functions for intersections of multiple halfspaces, improving theoretical understanding and implications for learning algorithms.

Contribution

It provides the first optimal bound of O(√(n log k)) for the average sensitivity of intersections of halfspaces, generalizing previous Gaussian results.

Findings

Proves the optimal bound of O(√(n log k)) for average sensitivity.

Extends Nazarov's Gaussian case result to the Boolean case.

Implications for the runtime of learning algorithms for intersections of halfspaces.

Abstract

We prove new bounds on the average sensitivity of the indicator function of an intersection of $k$ halfspaces. In particular, we prove the optimal bound of $O(\sqrt{n\log(k)})$. This generalizes a result of Nazarov, who proved the analogous result in the Gaussian case, and improves upon a result of Harsha, Klivans and Meka. Furthermore, our result has implications for the runtime required to learn intersections of halfspaces.