Tight Bounds on Low-Degree Spectral Concentration of Submodular and XOS Functions

Submodular and fractionally subadditive (or equivalently XOS) functions play a fundamental role in combinatorial optimization, algorithmic game theory and machine learning. Motivated by learnability of these classes of functions from random examples, we consider the question of how well such functions can be approximated by low-degree polynomials in ℓ₂ norm over the uniform distribution. This question is equivalent to understanding the concentration of Fourier weight on low-degree coefficients, a central concept in Fourier analysis. Denoting the smallest degree sufficient to approximate f in ℓ₂ norm within ∈ by deg_∈(ℓ₂)(f), we show that : For any submodular function f : {0, 1}ⁿ → [0, 1], deg_∈(ℓ₂)(f) = O(log(1/∈)/∈^4/5) and there is a submodular function that requires degree Ω(1/∈^4/5). : For any XOS function f : {0, 1} → [0, 1], deg_∈(ℓ₂) (f) = O(1/∈) and there exists an XOS function that requires degree Ω(1/∈). This improves on previous approaches that all showed an upper bound of O(1/∈²) for submodular [CKKL12], [FKV13], [FV13] and XOS [FV13] functions. The best previous lower bound was Ω(1/∈^2/3) for monotone submodular functions [FKV13]. Our techniques reveal new structural properties of submodular and XOS functions and the upper bounds lead to nearly optimal PAC learning algorithms for these classes of functions.