Robust Budget Allocation Via Continuous Submodular Functions

Matthew Staib, Stefanie Jegelka
2019 Applied Mathematics and Optimization  
A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT Abstract The optimal allocation of resources for maximizing influence, spread of information or coverage, has gained attention in the past years, in particular in machine learning and data mining. But in applications, the parameters of the problem are rarely known exactly, and using wrong parameters can lead to undesirable outcomes. We hence
more » ... t a continuous version of the Budget Allocation or Bipartite Influence Maximization problem introduced by Alon et al. [3] from a robust optimization perspective, where an adversary may choose the least favorable parameters within a confidence set. The resulting problem is a nonconvex-concave saddle point problem (or game). We show that this nonconvex problem can be solved exactly by leveraging connections to continuous submodular functions, and by solving a constrained submodular minimization problem. Although constrained submodular minimization is hard in general, here, we establish conditions under which such a problem can be solved to arbitrary precision ε. Introduction The optimal allocation of resources for maximizing influence, spread of information or coverage, has gained attention in the past few years, in particular in machine learning and data mining [25, 44, 22, 37, 16] . Formally, in the Budget Allocation Problem, one is given a bipartite influence graph between channels S and people T , and the task is to assign a budget y(s) to each channel s in S with the goal of maximizing the expected number of influenced people I(y). Each edge (s,t) ∈ E between channel s and person t is weighted with a probability p st that, e.g., an advertisement on radio station s will influence person t to buy some product. The budget y(s) controls how many independent attempts are M. Staib A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT Corollary 1 The influence function I(y; x) defined in Section 2 is continuous submodular in x over the nonnegative orthant, for each y ≥ 0. A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT A u t h o r a c c e p t e d m a n u s c r i p t AUTHOR ACCEPTED MANUSCRIPT
doi:10.1007/s00245-019-09567-0 fatcat:lxbezcnc7ffslcrote6hzy7ckm