Robust Guarantees of Stochastic Greedy Algorithms

Avinatan Hassidim, Yaron Singer
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:1424-1432, 2017.

Abstract

In this paper we analyze the robustness of stochastic variants of the greedy algorithm for submodular maximization. Our main result shows that for maximizing a monotone submodular function under a cardinality constraint, iteratively selecting an element whose marginal contribution is approximately maximal in expectation is a sufficient condition to obtain the optimal approximation guarantee with exponentially high probability, assuming the cardinality is sufficiently large. One consequence of our result is that the linear-time STOCHASTIC-GREEDY algorithm recently proposed in (Mirzasoleiman et al.,2015) achieves the optimal running time while maintaining an optimal approximation guarantee. We also show that high probability guarantees cannot be obtained for stochastic greedy algorithms under matroid constraints, and prove an approximation guarantee which holds in expectation. In contrast to the guarantees of the greedy algorithm, we show that the approximation ratio of stochastic local search is arbitrarily bad, with high probability, as well as in expectation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v70-hassidim17a, title = {Robust Guarantees of Stochastic Greedy Algorithms}, author = {Avinatan Hassidim and Yaron Singer}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {1424--1432}, year = {2017}, editor = {Precup, Doina and Teh, Yee Whye}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v70/hassidim17a/hassidim17a.pdf}, url = {https://proceedings.mlr.press/v70/hassidim17a.html}, abstract = {In this paper we analyze the robustness of stochastic variants of the greedy algorithm for submodular maximization. Our main result shows that for maximizing a monotone submodular function under a cardinality constraint, iteratively selecting an element whose marginal contribution is approximately maximal in expectation is a sufficient condition to obtain the optimal approximation guarantee with exponentially high probability, assuming the cardinality is sufficiently large. One consequence of our result is that the linear-time STOCHASTIC-GREEDY algorithm recently proposed in (Mirzasoleiman et al.,2015) achieves the optimal running time while maintaining an optimal approximation guarantee. We also show that high probability guarantees cannot be obtained for stochastic greedy algorithms under matroid constraints, and prove an approximation guarantee which holds in expectation. In contrast to the guarantees of the greedy algorithm, we show that the approximation ratio of stochastic local search is arbitrarily bad, with high probability, as well as in expectation.} }
Endnote
%0 Conference Paper %T Robust Guarantees of Stochastic Greedy Algorithms %A Avinatan Hassidim %A Yaron Singer %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-hassidim17a %I PMLR %P 1424--1432 %U https://proceedings.mlr.press/v70/hassidim17a.html %V 70 %X In this paper we analyze the robustness of stochastic variants of the greedy algorithm for submodular maximization. Our main result shows that for maximizing a monotone submodular function under a cardinality constraint, iteratively selecting an element whose marginal contribution is approximately maximal in expectation is a sufficient condition to obtain the optimal approximation guarantee with exponentially high probability, assuming the cardinality is sufficiently large. One consequence of our result is that the linear-time STOCHASTIC-GREEDY algorithm recently proposed in (Mirzasoleiman et al.,2015) achieves the optimal running time while maintaining an optimal approximation guarantee. We also show that high probability guarantees cannot be obtained for stochastic greedy algorithms under matroid constraints, and prove an approximation guarantee which holds in expectation. In contrast to the guarantees of the greedy algorithm, we show that the approximation ratio of stochastic local search is arbitrarily bad, with high probability, as well as in expectation.
APA
Hassidim, A. & Singer, Y.. (2017). Robust Guarantees of Stochastic Greedy Algorithms. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:1424-1432 Available from https://proceedings.mlr.press/v70/hassidim17a.html.

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