Minimizing Ratio of Monotone Non-submodular Functions

Abstract

In this paper, we investigate the problem of minimizing the ratio of normalized non-negative monotone non-submodular set function f and normalized non-negative monotone set function g. We take advantage of the greedy technique and get a performance guarantee depending on the generalized curvature and inverse generalized curvature of f, as well as the submodularity ratio of g. Our results generalize the works of Bai et al. (Algorithms for optimizing the ratio of submodular functions. In: Proceedings of the 33rd International Conference on Machine Learning, 2016) and Qian et al. (Optimizing ratio of monotone set functions. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence, 2017).