Submodular Functions Are Noise Stable

TL;DR

This paper proves that all non-negative submodular functions are highly noise-stable, enabling a polynomial-time learning algorithm that works under any product distribution and in the agnostic setting, improving upon prior methods.

Contribution

It introduces a universal noise-stability property for non-negative submodular functions and develops a new efficient learning algorithm applicable in broad settings.

Findings

All non-negative submodular functions are noise-stable.

The proposed algorithm learns submodular functions in polynomial time.

The algorithm works under any product distribution and in the agnostic setting.

Abstract

We show that all non-negative submodular functions have high {\em noise-stability}. As a consequence, we obtain a polynomial-time learning algorithm for this class with respect to any product distribution on $\{-1,1\}^n$ (for any constant accuracy parameter $\epsilon$). Our algorithm also succeeds in the agnostic setting. Previous work on learning submodular functions required either query access or strong assumptions about the types of submodular functions to be learned (and did not hold in the agnostic setting).