It has been well established that first order optimization methods can converge to the maximal objective value of concave functions and provide constant factor approximation guarantees for (non-convex/non-concave) continuous submodular functions. In this work, we initiate the study of the maximization of functions of the form F(x) = G(x) + C(x) over a solvable convex body P, where G is a smooth DR-submodular function and C is a smooth concave function. This class of functions is a strict extension of both concave and continuous DR-submodular functions for which no theoretical guarantee is known. We provide a suite of Frank-Wolfe style algorithms, which, depending on the nature of the objective function (i.e., if G and C are monotone or not, and non-negative or not) and on the nature of the set P (i.e., whether it is downward closed or not), provide 1−1/e, 1/e, or 1/2 approximation guarantees. We then use our algorithms to get a framework to smoothly interpolate between choosing a diverse set of elements from a given ground set (corresponding to the mode of a determinantal point process) and choosing a clustered set of elements (corresponding to the maxima of a suitable concave function). Additionally, we apply our algorithms to various functions in the above class (DR-submodular + concave) in both constrained and unconstrained settings, and show that our algorithms consistently outperform natural baselines.
|Title of host publication||Advances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021|
|Editors||Marc'Aurelio Ranzato, Alina Beygelzimer, Yann Dauphin, Percy S. Liang, Jenn Wortman Vaughan|
|Publisher||Neural information processing systems foundation|
|Number of pages||15|
|State||Published - 2021|
|Event||35th Conference on Neural Information Processing Systems, NeurIPS 2021 - Virtual, Online|
Duration: 6 Dec 2021 → 14 Dec 2021
|Name||Advances in Neural Information Processing Systems|
|Conference||35th Conference on Neural Information Processing Systems, NeurIPS 2021|
|Period||6/12/21 → 14/12/21|
Bibliographical noteFunding Information:
Funding in direct support of this work: The work of Moran Feldman was supported in part by Israel Science Foundation (ISF) grant no. 459/20. Amin Karbasi acknowledges funding in direct support of this work from NSF (IIS-1845032) and ONR (N00014-19-1-2406).
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