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

Submodular maximization is a general optimization problem with 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-factor approximation algorithm for maximizing a non-monotone submodular function subject to a cardinality constraint $k$ that runs in $O(\log(n))$ adaptive rounds and makes $O(n \log(k))$ 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. The results demonstrate that our algorithm finds competitive solutions using significantly fewer rounds and queries.Comment: 12 pages, 8 figure

Nearly Linear-Time, Parallelizable Algorithms for Non-Monotone Submodular Maximization

We study parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a cardinality constraint $k$. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and query complexity, up to logarithmic factors in the size $n$ of the ground set, from $0.039 - \epsilon$ to $0.193 - \epsilon$. We provide two algorithms; the first has approximation ratio $1/6 - \epsilon$, adaptivity $O( \log n )$, and query complexity $O( n \log k )$, while the second has approximation ratio $0.193 - \epsilon$, adaptivity $O( \log^2 n )$, and query complexity $O(n \log k)$. Heuristic versions of our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with highly adaptive approximation algorithms.Comment: 24 pages, 2 figure

Submodular Maximization with Matroid and Packing Constraints in Parallel

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-\epsilon$ approximation for monotone functions and a $1/e-\epsilon$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/\epsilon^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-\epsilon$ approximation using $O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-\epsilon$ approximation in $O(\log(n/\epsilon)\log(m)/\epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function