Non-monotone Submodular Maximization with Nearly Optimal Adaptivity and Query Complexity

As a general optimization problem, submodular maximization has a wide range of applications in machine learning (e.g., active learning, clustering, and feature selection). In large-scale optimization, the parallel running time of an algorithm is governed by its adaptivity, which measures the number of sequential rounds needed if the algorithm can execute polynomially-many independent oracle queries in parallel. While low adaptivity is ideal, it is not sufficient for an algorithm to be efficient in practice---there are many applications of distributed submodular optimization where the number of function evaluations becomes prohibitively expensive. Motivated by these applications, we study the adaptivity and query complexity of submodular maximization. In this paper, we give the first constant-approximation algorithm for maximizing a non-monotone submodular function subject to a cardinality constraint~$k$ that runs in $O(\log(n))$ adaptive rounds. Additionally, our algorithm makes only $O(n \log(k))$ oracle queries in expectation. In our empirical study, we use three real-world applications to compare our algorithm with several benchmarks for non-monotone submodular maximization, and the results show that our algorithm finds competitive solutions using \emph{significantly fewer rounds and queries}.