Guarantees for Greedy Maximization of Non-submodular Functions with Applications

arXiv1703.02100MaRDI QIDQ6283974

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Publication date: 6 March 2017

Abstract: We investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of the (generalized) curvature $alpha$ and the submodularity ratio $gamma$. In particular, we prove that Greedy enjoys a tight approximation guarantee of $frac1alpha(1-e^{-gammaalpha})$ for cardinality constrained maximization. In addition, we bound the submodularity ratio and curvature for several important real-world objectives, including the Bayesian A-optimality objective, the determinantal function of a square submatrix and certain linear programs with combinatorial constraints. We experimentally validate our theoretical findings for both synthetic and real-world applications.

Has companion code repository: https://github.com/bianan/non-submodular-max

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