# On Additive Approximate Submodularity

arXiv: https://arxiv.org/abs/2010.02912 · HTML: https://mlsanity.com/paper/2010.02912/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2020.

## Built on by (most cited first, 3 of 3)

- [A Unified Framework for Task-Driven Data Quality Management](https://mlsanity.com/paper/2106.05484/) (2021)
- [A Unified Approach to Submodular Maximization Under Noise](https://mlsanity.com/paper/2510.21128/) (2025)
- [Parallel Quasi-concave set optimization: A new frontier that scales without needing submodularity](https://mlsanity.com/paper/2108.08758/) (2021)

## Abstract (arXiv)

A real-valued set function is (additively) approximately submodular if it satisfies the submodularity conditions with an additive error. Approximate submodularity arises in many settings, especially in machine learning, where the function evaluation might not be exact. In this paper we study how close such approximately submodular functions are to truly submodular functions. We show that an approximately submodular function defined on a ground set of n elements is O(n2) pointwise-close to a submodular function. This result also provides an algorithmic tool that can be used to adapt existing submodular optimization algorithms to approximately submodular functions. To complement, we show an Omega(√n) lower bound on the distance to submodularity. These results stand in contrast to the case of approximate modularity, where the distance to modularity is a constant, and approximate convexity, where the distance to convexity is logarithmic.