Given a finite ground set N and a value vector a ∈ RN , 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 utility function appears frequently in combinatorial optimization problems whenmodeling risk aversion and decreasing marginal preferen...

In many multi-robot applications such as target search, environmental monitoring and reconnaissance, the multi-robot system operates semi-autonomously, but under the supervision of a remote human who monitors task progress. In these applications, each robot collects a large amount of task-specific data that must be sent to the human periodically to keep the human aware of task progress. It is o...

In this paper, we consider an online optimization process, where the objective functions are not convex (nor concave) but instead belong to a broad class of continuous submodular functions. We first propose a variant of the Frank-Wolfe algorithm that has access to the full gradient of the objective functions. We show that it achieves a regret bound of O( √ T ) (where T is the horizon of the onl...

We consider robust optimization problems, where the goal is to optimize in the worst case over a class of objective functions. We develop a reduction from robust improper optimization to Bayesian optimization: given an oracle that returns αapproximate solutions for distributions over objectives, we compute a distribution over solutions that is α-approximate in the worst case. We show that deran...

We study the problem of summarizing DAG-structured topic hierarchies over a given set of documents. Example applications include automatically generating Wikipedia disambiguation pages for a set of articles, and generating candidate multi-labels for preparing machine learning datasets (e.g., for text classification, functional genomics, and image classification). Unlike previous work, which foc...

Traditional optimization techniques often rely upon functions that are convex or at least locally convex. Such diverse methods as gradient descent, loopy belief propagation, and linear programming all rely upon convex functions. However, many natural functions are not convex, yet optimizing over them is both possible and necessary. The class of submodular functions is particularly well-behaved ...

Discrete convex analysis [18, 40, 43, 47] aims to establish a general theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. The framework of convex analysis is adapted to discrete settings and the mathematical results in matroid/submodular function theory are generalized. Viewed from th...

In recent years, a fundamental problem structure has emerged as very useful in a variety of machine learning applications: Submodularity is an intuitive diminishing returns property, stating that adding an element to a smaller set helps more than adding it to a larger set. Similarly to convexity, submodularity allows one to efficiently find provably (near-) optimal solutions for large problems....

Submodular functions are the discrete ananlogue of convexity and generalize many known and well studied settings. They capture several economic principles such as decreasing marginal cost and correlated utilities. Multi-agent allocation problems over submodular functions arise in a variety of natural scenarios and have been the subject of extensive study over the last decade. In this proposal I...