Randomized Composable Core-sets for Distributed Submodular Maximization

An effective technique for solving optimization problems over massive data sets is to partition the data into smaller pieces, solve the problem on each piece and compute a representative solution from it, and finally obtain a solution inside the union of the representative solutions for all pieces. This technique can be captured via the concept of {\em composable core-sets}, and has been recently applied to solve diversity maximization problems as well as several clustering problems. However, for coverage and submodular maximization problems, impossibility bounds are known for this technique \cite{IMMM14}. In this paper, we focus on efficient construction of a randomized variant of composable core-sets where the above idea is applied on a {\em random clustering} of the data. We employ this technique for the coverage, monotone and non-monotone submodular maximization problems. Our results significantly improve upon the hardness results for non-randomized core-sets, and imply improved results for submodular maximization in a distributed and streaming settings. In summary, we show that a simple greedy algorithm results in a $1/3$-approximate randomized composable core-set for submodular maximization under a cardinality constraint. This is in contrast to a known $O({\log k\over \sqrt{k}})$ impossibility result for (non-randomized) composable core-set. Our result also extends to non-monotone submodular functions, and leads to the first 2-round MapReduce-based constant-factor approximation algorithm with $O(n)$ total communication complexity for either monotone or non-monotone functions. Finally, using an improved analysis technique and a new algorithm $\mathsf{PseudoGreedy}$, we present an improved $0.545$-approximation algorithm for monotone submodular maximization, which is in turn the first MapReduce-based algorithm beating factor $1/2$ in a constant number of rounds

Differentially Private Decomposable Submodular Maximization

We study the problem of differentially private constrained maximization of decomposable submodular functions. A submodular function is decomposable if it takes the form of a sum of submodular functions. The special case of maximizing a monotone, decomposable submodular function under cardinality constraints is known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al., 2008]. Previous work by Gupta et al. [2010] gave a differentially private algorithm for the CPP problem. We extend this work by designing differentially private algorithms for both monotone and non-monotone decomposable submodular maximization under general matroid constraints, with competitive utility guarantees. We complement our theoretical bounds with experiments demonstrating empirical performance, which improves over the differentially private algorithms for the general case of submodular maximization and is close to the performance of non-private algorithms

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

Constrained Submodular Maximization: Beyond 1/e

In this work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over a down-closed polytope. The approximation guarantee is 0.372 and it is the first improvement over the 1/e approximation achieved by the unified Continuous Greedy algorithm [Feldman et al., FOCS 2011]