Black Box Submodular Maximization: Discrete and Continuous Settings

Lin Chen, Mingrui Zhang, Hamed Hassani, Amin Karbasi
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:1058-1070, 2020.

Abstract

In this paper, we consider the problem of black box continuous submodular maximization where we only have access to the function values and no information about the derivatives is provided. For a monotone and continuous DR-submodular function, and subject to a bounded convex body constraint, we propose Black-box Continuous Greedy, a derivative-free algorithm that provably achieves the tight $[(1-1/e)OPT-\epsilon]$ approximation guarantee with $O(d/\epsilon^3)$ function evaluations. We then extend our result to the stochastic setting where function values are subject to stochastic zero-mean noise. It is through this stochastic generalization that we revisit the discrete submodular maximization problem and use the multi-linear extension as a bridge between discrete and continuous settings. Finally, we extensively evaluate the performance of our algorithm on continuous and discrete submodular objective functions using both synthetic and real data.

Cite this Paper


BibTeX
@InProceedings{pmlr-v108-chen20c, title = {Black Box Submodular Maximization: Discrete and Continuous Settings}, author = {Chen, Lin and Zhang, Mingrui and Hassani, Hamed and Karbasi, Amin}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {1058--1070}, year = {2020}, editor = {Silvia Chiappa and Roberto Calandra}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/chen20c/chen20c.pdf}, url = { http://proceedings.mlr.press/v108/chen20c.html }, abstract = {In this paper, we consider the problem of black box continuous submodular maximization where we only have access to the function values and no information about the derivatives is provided. For a monotone and continuous DR-submodular function, and subject to a bounded convex body constraint, we propose Black-box Continuous Greedy, a derivative-free algorithm that provably achieves the tight $[(1-1/e)OPT-\epsilon]$ approximation guarantee with $O(d/\epsilon^3)$ function evaluations. We then extend our result to the stochastic setting where function values are subject to stochastic zero-mean noise. It is through this stochastic generalization that we revisit the discrete submodular maximization problem and use the multi-linear extension as a bridge between discrete and continuous settings. Finally, we extensively evaluate the performance of our algorithm on continuous and discrete submodular objective functions using both synthetic and real data.} }
Endnote
%0 Conference Paper %T Black Box Submodular Maximization: Discrete and Continuous Settings %A Lin Chen %A Mingrui Zhang %A Hamed Hassani %A Amin Karbasi %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-chen20c %I PMLR %P 1058--1070 %U http://proceedings.mlr.press/v108/chen20c.html %V 108 %X In this paper, we consider the problem of black box continuous submodular maximization where we only have access to the function values and no information about the derivatives is provided. For a monotone and continuous DR-submodular function, and subject to a bounded convex body constraint, we propose Black-box Continuous Greedy, a derivative-free algorithm that provably achieves the tight $[(1-1/e)OPT-\epsilon]$ approximation guarantee with $O(d/\epsilon^3)$ function evaluations. We then extend our result to the stochastic setting where function values are subject to stochastic zero-mean noise. It is through this stochastic generalization that we revisit the discrete submodular maximization problem and use the multi-linear extension as a bridge between discrete and continuous settings. Finally, we extensively evaluate the performance of our algorithm on continuous and discrete submodular objective functions using both synthetic and real data.
APA
Chen, L., Zhang, M., Hassani, H. & Karbasi, A.. (2020). Black Box Submodular Maximization: Discrete and Continuous Settings. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:1058-1070 Available from http://proceedings.mlr.press/v108/chen20c.html .

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