Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

We investigate three related and important problems connected to machine learning, namely approximating a submodular function everywhere, learning a submodular function (in a PAC like setting [26]), and constrained minimization of submodular functions. In all three problems, we provide improved bounds which depend on the “curvature” of a submodular function and improve on the previously known best results for these problems [9, 3, 7, 25] when the function is not too curved – a property which is true of many real-world submodular functions. In the former two problems, we obtain these bounds through a generic black-box transformation (which can potentially work for any algorithm), while in the case of submodular minimization, we propose a framework of algorithms which depend on choosing an appropriate surrogate for the submodular function. In all these cases, we provide almost matching lower bounds. While improved curvature-dependent bounds were shown for monotone submodular maximization [4, 27], the existence of similar improved bounds for the aforementioned problems has been open. We resolve this question in this paper by showing that the same notion of curvature provides these improved results. Empirical experiments add further support to our claims.