# Tight Bounds on Low-degree Spectral Concentration of Submodular and XOS functions

arXiv: https://arxiv.org/abs/1504.03391 · HTML: https://mlsanity.com/paper/1504.03391/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Apr 2015.

## Built on by (most cited first, 2 of 2)

- [Submodularity In Machine Learning and Artificial Intelligence](https://mlsanity.com/paper/2202.00132/) (2022)
- [Testing distributional assumptions of learning algorithms](https://mlsanity.com/paper/2204.07196/) (2022)

## Abstract (arXiv)

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 of the concentration of Fourier weight on low-degree coefficients, a central concept in Fourier analysis. We show that 1. For any submodular function f:0,1n rightarrow [0,1], there is a polynomial of degree O(log (1/ε) / ε4/5) approximating f within ε in ℓ2, and there is a submodular function that requires degree Omega(1/ε4/5). 2. For any XOS function f:0,1n rightarrow [0,1], there is a polynomial of degree O(1/ε) and there exists an XOS function that requires degree Omega(1/ε). This improves on previous approaches that all showed an upper bound of O(1/ε2) for submodular and XOS functions. The best previous lower bound was Omega(1/ε2/3) for monotone submodular functions. 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.