Title: Robust Guarantees of Stochastic Greedy Algorithms

Authors: Avinatan Hassidim, Yaron Singer

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 submod- ular function under a cardinality constraint, itera- tively selecting an element whose marginal con- tribution is approximately maximal in expecta- tion is a sufficient condition to obtain the opti- mal approximation guarantee with exponentially high probability, assuming the cardinality is suf- ficiently large. One consequence of our result is that the linear-time STOCHASTIC-GREEDY algo- rithm recently proposed in (Mirzasoleiman et al., 2015) achieves the optimal running time while maintaining an optimal approximation guaran- tee. We also show that high probability guaran- tees cannot be obtained for stochastic greedy al- gorithms under matroid constraints, and prove an approximation guarantee which holds in expec- tation. 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.

Full Text: [PDF]

Accessibility at Yale   Inference, Information, and Decision Group at Yale