In this paper, we showcase the interplay between discrete and continuous optimization in network-structured settings. We propose the first fully decentralized optimization method for a wide class of non-convex objective functions that possess a diminishing returns property. More specifically, given an arbitrary connected network and a global continuous submodular function, formed by a sum of local functions, we develop Decentralized Continuous Greedy (DCG), a message passing algorithm that converges to the tight $(1-1/e)$ approximation factor of the optimum global solution using only local computation and communication. We also provide strong convergence bounds as a function of network size and spectral characteristics of the underlying topology. Interestingly, DCG readily provides a simple recipe for decentralized discrete submodular maximization through the means of continuous relaxations. Formally, we demonstrate that by lifting the local discrete functions to continuous domains and using DCG as an interface we can develop a consensus algorithm that also achieves the tight $(1-1/e)$ approximation guarantee of the global discrete solution once a proper rounding scheme is applied.

In this paper we analyze an adaptive sampling approach for submodular maximization. Adaptive sampling is a technique that has recently been shown to achieve a constant factor approximation guarantee for submodular maximization under a cardinality constraint with exponentially fewer adaptive rounds than any previously studied constant factor approximation algorithm for this problem. Adaptivity quantifies the number of sequential rounds that an algorithm makes when function evaluations can be executed in parallel and is the parallel running time of an algorithm, up to low order terms. Adaptive sampling achieves its exponential speedup at the expense of approximation. In theory, it is guaranteed to produce a solutionthat is a 1/3 approximation to the optimum. Nevertheless, experiments show that adaptive sampling techniques achieve far better values in practice.In this paper we provide theoretical justification for this phenomenon. In particular, we showthat under very mild conditions of curvature of a function, adaptive sampling techniques achieve an approximation arbitrarily close to 1/2 while maintaining their low adaptivity. Furthermore, we show that the approximation ratio approaches 1 in direct relationship to a homogeneity property of the submodular function. In addition, we conductexperiments on real data sets in which the curvature and homogeneity properties can be easilymanipulated and demonstrate the relationship between approximation and curvature, as well asthe effectiveness of adaptive sampling in practice.

We analyze the performance of the greedy algorithm, and also a discrete semi-gradient based algorithm, for maximizing the sum of a suBmodular and suPermodular (BP) function (both of which are non-negative monotone non-decreasing) under two types of constraints, either a cardinality constraint or $p\geq 1$ matroid independence constraints. These problems occur naturally in several real-world applications in data science, machine learning, and artificial intelligence. The problems are ordinarily inapproximable to any factor. Using the curvature $\curv_f$ of the submodular term, and introducing $\curv^g$ for the supermodular term (a natural dual curvature for supermodular functions), however, both of which are computable in linear time, we show that BP maximization can be efficiently approximated by both the greedy and the semi-gradient based algorithm. The algorithms yield multiplicative guarantees of $\frac{1}{\curv_f}\left[1-e^{-(1-\curv^g)\curv_f}\right]$ and $\frac{1-\curv^g}{(1-\curv^g)\curv_f + p}$ for the two types of constraints respectively. For pure monotone supermodular constrained maximization, these yield $1-\curvg$ and $(1-\curvg)/p$ for the two types of constraints respectively. We also analyze the hardness of BP maximization and show that our guarantees match hardness by a constant factor and by $O(\ln(p))$ respectively. Computational experiments are also provided supporting our analysis.

We introduce the problem of grouping a finite ground set into blocks where each block is a subset of the ground set and where: (i) the blocks are individually highly valued by a submodular function (both robustly and in the average case) while satisfying block-specific matroid constraints; and (ii) block scores interact where blocks are jointly scored highly, thus making the blocks mutually non-redundant. Submodular functions are good models of information and diversity; thus, the above can be seen as grouping the ground set into matroid constrained blocks that are both intra- and inter-diverse. Potential applications include forming ensembles of classification/regression models, partitioning data for parallel processing, and summarization. In the non-robust case, we reduce the problem to non-monotone submodular maximization subject to multiple matroid constraints. In the mixed robust/average case, we offer a bi-criterion guarantee for a polynomial time deterministic algorithm and a probabilistic guarantee for randomized algorithm, as long as the involved submodular functions (including the inter-block interaction terms) are monotone. We close with a case study in which we use these algorithms to find high quality diverse ensembles of classifiers, showing good results.

The optimization of submodular functions on the integer lattice has received much attention recently, but the objective functions of many applications are non-submodular. We provide two approximation algorithms for maximizing a non-submodular function on the integer lattice subject to a cardinality constraint; these are the first algorithms for this purpose that have polynomial query complexity. We propose a general framework for influence maximization on the integer lattice that generalizes prior works on this topic, and we demonstrate the efficiency of our algorithms in this context.