Minimizing Ratio of Monotone Non-submodular Functions

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).

[1]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[2]  Rishabh K. Iyer,et al.  Submodularity in Data Subset Selection and Active Learning , 2015, ICML.

[3]  Yang Yu,et al.  Optimizing Ratio of Monotone Set Functions , 2017, IJCAI.

[4]  Yoshio Okamoto,et al.  Submodular fractional programming for balanced clustering , 2011, Pattern Recognit. Lett..

[5]  C. J. van Rijsbergen,et al.  FOUNDATION OF EVALUATION , 1974 .

[6]  Rishabh K. Iyer,et al.  Algorithms for Optimizing the Ratio of Submodular Functions , 2016, ICML.

[7]  G. McLachlan.  Discriminant Analysis and Statistical Pattern Recognition , 1992 .

[8]  Alper Atamtürk,et al.  The submodular knapsack polytope , 2009, Discret. Optim..

[9]  Éva Tardos,et al.  “The quickest transshipment problem” , 1995, SODA '95.

[10]  Roy Schwartz,et al.  Online Submodular Maximization with Preemption , 2015, SODA.

[11]  Neil D. Lawrence,et al.  Fast Sparse Gaussian Process Methods: The Informative Vector Machine , 2002, NIPS.

[12]  Abhimanyu Das,et al.  Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection , 2011, ICML.

[13]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[14]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[15]  Joseph Naor,et al.  Submodular Maximization with Cardinality Constraints , 2014, SODA.

[16]  Jan Vondr.  Submodularity and Curvature: The Optimal Algorithm , 2010 .

[17]  Laurence A. Wolsey,et al.  Best Algorithms for Approximating the Maximum of a Submodular Set Function , 1978, Math. Oper. Res..

[18]  Rishabh K. Iyer,et al.  Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions , 2013, NIPS.

[19]  Andreas Krause,et al.  Guarantees for Greedy Maximization of Non-submodular Functions with Applications , 2017, ICML.

[20]  Naoyuki Kamiyama.  A Note on Submodular Function Minimization with Covering Type Linear Constraints , 2017, Algorithmica.

[21]  Zhi-Hua Zhou,et al.  Constrained Monotone $k$ -Submodular Function Maximization Using Multiobjective Evolutionary Algorithms With Theoretical Guarantee , 2018, IEEE Transactions on Evolutionary Computation.

[22]  Gérard Cornuéjols,et al.  Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem , 1984, Discret. Appl. Math..

[23]  Laurence A. Wolsey,et al.  An analysis of the greedy algorithm for the submodular set covering problem , 1982, Comb..

[24]  L. Shapley.  Cores of convex games , 1971 .

[25]  Satoru Fujishige,et al.  Polymatroidal Dependence Structure of a Set of Random Variables , 1978, Inf. Control..

[26]  Andreas Krause,et al.  Continuous DR-submodular Maximization: Structure and Algorithms , 2017, NIPS 2017.

[27]  Volkan Cevher,et al.  Robust Maximization of Non-Submodular Objectives , 2018, AISTATS.

[28]  Andreas Krause,et al.  Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies , 2008, J. Mach. Learn. Res..

[29]  Jitendra Malik,et al.  Normalized Cuts and Image Segmentation , 2000, IEEE Trans. Pattern Anal. Mach. Intell..