On additive approximate submodularity

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