Algorithms for optimizing the ratio of submodular functions
Wenruo Bai, Rishabh Iyer, Kai Wei, Jeff Bilmes · International Conference on Machine Learning · 2016
We investigate a new optimization problem involving minimizing the Ratio of two Submodular (RS) functions. We argue that this problem occurs naturally in several real world applications. We then show the connection between this problem and several related problems including minimizing the difference between submodular functions (Iyer & Bilmes, 2012b; Narasimhan & Bilmes, 2005), and to submodular optimization subject to submodular constraints (Iyer & Bilmes, 2013). We show that RS optimization can be solved with bounded approximation factors. We also provide a hardness bound and show that our tightest algorithm matches the lower bound up to a log factor. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.