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 alg...

We introduce submodular optimization to the problem of training data subset selection for statistical machine translation (SMT). By explicitly formulating data selection as a submodular program, we obtain fast scalable selection algorithms with mathematical performance guarantees, resulting in a unified framework that clarifies existing approaches and also makes both new and many previous appro...

We consider the problem of optimization from samples of monotone submodular functions with bounded curvature. In numerous applications, the function optimized is not known a priori, but instead learned from data. What are the guarantees we have when optimizing functions from sampled data? In this paper we show that for any monotone submodular function with curvature c there is a (1 c)/(1 + c c2...

We study the problem of maximizing a function that is approximately submodular under a cardinality constraint. Approximate submodularity implicitly appears in a wide range of applications as in many cases errors in evaluation of a submodular function break submodularity. Say that F is ε-approximately submodular if there exists a submodular function f such that (1−ε)f(S) ≤ F (S) ≤ (1+ε)f(S) for ...

In this paper, we present fast polynomial-time algorithms for solving classes of submodular constraints over Boolean domains. We pose the identified classes of problems within the general framework of Weighted Constraint Satisfaction Problems (WCSPs), reformulated as minimum weighted vertex cover problems. We examine the Constraint Composite Graphs (CCGs) associated with these WCSPs and provide...

Motivated by extremely large-scale machine learning problems, we introduce a new multistage algorithmic framework for submodular maximization (called MultGreed), where at each stage we apply an approximate greedy procedure to maximize surrogate submodular functions. The surrogates serve as proxies for a target submodular function but require less memory and are easy to evaluate. We theoreticall...

Generative moment matching network (GMMN), which is based on the maximum mean discrepancy (MMD) measure, is a generative model for unsupervised learning, where the mini-batch stochastic gradient descent is applied for the update of parameters. In this work, instead of obtaining a mini-batch randomly, each mini-batch in the iterations is selected in a submodular way such that the most informativ...

We design a class of submodular functions meant for document summarization tasks. These functions each combine two terms, one which encourages the summary to be representative of the corpus, and the other which positively rewards diversity. Critically, our functions are monotone nondecreasing and submodular, which means that an efficient scalable greedy optimization scheme has a constant factor...

We call a market competitive if increasing the endowment of one buyer does not increase the equilibrium utility of another. We show every competitive uniform utility allocation market is a submodular utility allocation market, answering a question of Jain and Vazirani, (2007). Our proof proceeds via characterizing non-submodular fractionally sub-additive functions.