Submodular functions are noise stable

We show that all non-negative submodular functions have high 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 e). 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). Additionally we give simple algorithms that efficiently release differentially private answers to all Boolean conjunctions and to all halfspaces with constant average error, subsuming and improving recent work due to Gupta, Hardt, Roth and Ullman (STOC 2011).