In this paper we investigate the problem of training data partitioning for parallel learning of statistical models. Motivated by [10], we utilize submodular functions to model the utility of data subsets for training machine learning classifiers and formulate this problem mathematically as submodular partitioning. We introduce a simple and scalable greedy algorithm that near-optimally solves th...

Given a finite ground set N and a value vector a ∈ R , we consider optimization problems involving maximization of a submodular set utility function of the form h(S) = f (∑ i∈S ai ) , S ⊆ N , where f is a strictly concave, increasing, differentiable function. This function appears frequently in combinatorial optimization problems when modeling risk aversion and decreasing marginal preferences, ...

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call proportionally submodular functions. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to ...

"Real continuous submodular functions, as a generalization of the corresponding discrete notion to domain, gained considerable attention recently. The analog for entropy functions requires additional properties: real function defined on non-negative orthant $\R^n$ is entropy-like (EL) if it submodular, takes zero at zero, non-decreasing, and has Diminishing Returns property. Motivated by proble...

A set function f defined on the subsets of a finite set V is said to be submodular if it satisfies f(X) + f(Y ) ≥ f(X ∪ Y ) + f(X ∩ Y ), ∀X, Y ⊆ V. Submodular functions are discrete analogues of convex functions. They arise in various branches of applied mathematics such as game theory, information theory, and queueing theory. Examples include the matroid rank functions, the cut capacity functi...

Recent years have seen many algorithmic advances in the area of submodular optimization: (SO) min /max f(S) : S ∈ F , where F is a given family of feasible sets over a ground set V and f : 2 → R is submodular. This progress has been coupled with a wealth of new applications for these models. Our focus is on a more general class of multi-agent submodular optimization (MASO) which was introduced ...

Submodular optimization has found many applications in machine learning andbeyond. We carry out the first systematic investigation of inference in probabilis-tic models defined through submodular functions, generalizing regular pairwiseMRFs and Determinantal Point Processes. In particular, we present L-FIELD, avariational approach to general log-submodular and log-supermodul...

In this work we present a quadratic programming approximation of the Semi-Supervised Support Vector Machine (S3VM) problem, namely approximate QP-S3VM, that can be efficiently solved using off the shelf optimization packages. We prove that this approximate formulation establishes a relation between the low density separation and the graph-based models of semi-supervised learning (SSL) which is ...

We consider the problem of maximizing a monotone submodular function under noise, which to the best of our knowledge has not been studied in the past. There has been a great deal of work on optimization of submodular functions under various constraints, with many algorithms that provide desirable approximation guarantees. However, in many applications we do not have access to the submodular fun...

Many important problems in discrete optimization require maximization of a monotonic submodular function subject to matroid constraints. For these problems, a simple greedy algorithm is guaranteed to obtain near-optimal solutions. In this article, we extend this classic result to a general class of adaptive optimization problems under partial observability, where each choice can depend on obser...