The Power of Subsampling in Submodular Maximization

• URL: http://arxiv.org/abs/2104.02772v1
• Date: Tue, 6 Apr 2021 20:25:57 GMT
• Title: The Power of Subsampling in Submodular Maximization
• Authors: Christopher Harshaw, Ehsan Kazemi, Moran Feldman, Amin Karbasi
• Abstract summary: We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.
• Score: 51.629656762796564
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set, and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual offline setting, we present SampleGreedy, which obtains a $(p + 2 + o(1))$-approximation for maximizing a submodular function subject to a $p$-extendible system using $O(n + nk/p)$ evaluation and feasibility queries, where $k$ is the size of the largest feasible set. The approximation ratio improves to $p+1$ and $p$ for monotone submodular and linear objectives, respectively. In the streaming setting, we present SampleStreaming, which obtains a $(4p +2 - o(1))$-approximation for maximizing a submodular function subject to a $p$-matchoid using $O(k)$ memory and $O(km/p)$ evaluation and feasibility queries per element, where $m$ is the number of matroids defining the $p$-matchoid. The approximation ratio improves to $4p$ for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.

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