Weakly Submodular Maximization Beyond Cardinality Constraints: Does Randomization Help Greedy?

Submodular functions are a broad class of set functions, which naturally arise in diverse areas such as economics, operations research and game theory. Many algorithms have been suggested for the maximization of these functions, achieving both strong theoretical guarantees and good practical performance. Unfortunately, once the function deviates from submodularity (even slightly), the known algorithms for submodular maximization may perform arbitrarily poorly. Amending this issue, by obtaining approximation results for classes of set functions generalizing submoduolar functions, has been the focus of several recent works. One such class, known as the class of weakly submodular functions, has received a lot of attention from the machine learning community due to its strong connections to restricted strong convexity and sparse reconstruction. A key result proved by Das and Kempe (2011) showed that the approximation ratio of the standard greedy algorithm for the problem of maximizing a weakly submodular function subject to a cardinality constraint degrades smoothly with the distance of the function from submodularity. However, no results have been obtained so far for the maximization of weakly submodular functions subject to constraints beyond cardinality. In particular, it is not known whether the greedy algorithm achieves any non-trivial approximation ratio for such constraints. In this paper, we prove that a randomized version of the greedy algorithm (previously used by Buchbinder et al. (2014) for a different problem) achieves an approximation ratio of (1 + 1/γ)−2 for the maximization of a weakly submodular function subject to a general matroid constraint, where γ is a parameter measuring the distance of the function from submodularity. Moreover, we also experimentally compare the performance of this version of the greedy algorithm on real world problems such as gene splice site detection and video summarization against natural benchmarks, and show that the algorithm we study performs well also in practice. To the best of our knowledge, this is the first algorithm with a non-trivial approximation guarantee for maximizing a weakly submodular function subject to a constraint other than the simple cardinality constraint. In particular, it is the first algorithm with such a guarantee for the important and broad class of matroid constraints. keywords: weakly submodular functions, optimization, matroid constraint ∗Yale Institute for Network Science, Yale University. E-mail: [email protected]. Authors are listed in alphabetical order. †Depart. of Mathematics and Computer Science, The Open University of Israel. E-mail: [email protected]. ‡Yale Institute for Network Science, Yale University. E-mail: [email protected]. ar X iv :1 70 7. 04 34 7v 1 [ cs .D M ] 1 3 Ju l 2 01 7