Do Less, Get More: Streaming Submodular Maximization with Subsampling
In this paper, we develop the first one-pass streaming algorithm for submodular maximization that does not evaluate the entire stream even once. By carefully subsampling each element of the data stream, our algorithm enjoys the tightest approximation guarantees in various settings while having the smallest memory footprint and requiring the lowest number of function evaluations. More specifically, for a monotone submodular function and a p-matchoid constraint, our randomized algorithm achieves a 4p approximation ratio (in expectation) with O(k) memory and O(km/p) queries per element (k is the size of the largest feasible solution and m is the number of matroids used to define the constraint). For the non-monotone case, our approximation ratio increases only slightly to 4p + 2 o(1). To the best or our knowledge, our algorithm is the first that combines the benefits of streaming and subsampling in a novel way in order to truly scale submodular maximization to massive machine learning problems. To showcase its practicality, we empirically evaluated the performance of our algorithm on a video summarization application and observed that it outperforms the state-of-the-art algorithm by up to fifty-fold while maintaining practically the same utility. We also evaluated the scalability of our algorithm on a large dataset of Uber pick up locations.
This is a clearly written and well argued paper that improves the known state of the art algorithms for the specific problem of streaming submodular optimization subject to multiple matroid (or matchoid) constraints. The authors improve the running time and either match or improve the approximation factors of known algorithms.  The experimental results can be significantly strengthened.   One downside of the work is that the theoretical improvement is relatively small, however in my mind this is outweighed by the fact that the algorithm is clean, seemingly universal (subsuming previous work), and practical.  On the technical side, the main contribution is Theorem 4, which gives a parametrized approximation guarantee for the Sample-Streaming algorithm. This is then combined with known results, and tuned settings for the parameters to obtain the stated results for monotone and non-monotone functions.  The proof is involved, it would have been good to provide a high level intuition picture in the Appendix (or if possible in the main body) of why this approach succeeds where previous methods have failed.   More notes: - Spell check -- l176 (independence), l248 (diverse),  - In the experimental section (line 265), it is worth noting explicitly that the proposed algorithm performs _worse_ than the SeqDPP baseline on OVP videos, and on half of the YouTube videos. A sentence is also warranted to describe the performance versus local search, which has a strictly worse approximation ratio, but performs comparably here.  - While it's nice to see absolute running time comparisons in Figure 1c, a much more meaningful metric would be the number of oracle calls, since this is precisely what the proposed method is trying to optimized. Moreover, such a metric would be independent of machine speed, or programming prowess, and can be used as a basis for comparison in the future. I strongly urge the authors to provide such a plot. Rebuttal Question: how does the number of evaluations scale between different methods?  - In my mind Section 4.2 should either be cut, or augmented with a stronger baseline. As mentioned above, simply reporting running time numbers without a baseline is not very meaningful, and future researchers would not be able to compare their methods against them.