We investigate the performance of the standardGREEDYalgorithm for cardinality constrainedmaximization of non-submodular nondecreasingset functions. While there are strong theoreticalguarantees on the performance of GREEDYformaximizing submodular functions, there are fewguarantees for non-submodular ones. However,GREEDYenjoys strong empirical performancefor many important non-submodular functions,e.g., the Bayesian A-optimality objective in ex-perimental design. We prove theoretical guaran-tees supporting the empirical performance. Ourguarantees are characterized by a combinationof the (generalized)curvatureαand thesub-modularity ratioγ. In particular, we prove thatGREEDYenjoys atightapproximation guaranteeof1α(1−e−γα)for cardinality constrained max-imization. In addition, we bound the submod-ularity ratio and curvature for several importantreal-world objectives, including the Bayesian A-optimality objective, the determinantal functionof a square submatrix and certain linear programswith combinatorial constraints. We experimen-tally validate our theoretical findings for bothsynthetic and real-world applications.

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