
Continuous Submodular Function Maximization
Continuous submodular functions are a category of generally nonconvex/n...
read it

Parallel Quasiconcave set optimization: A new frontier that scales without needing submodularity
Classes of set functions along with a choice of ground set are a bedrock...
read it

Continuous Profit Maximization: A Study of Unconstrained Drsubmodular Maximization
Profit maximization (PM) is to select a subset of users as seeds for vir...
read it

Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an M^Concave Set Function
Submodularity is an important concept in combinatorial optimization, and...
read it

An optimization problem for continuous submodular functions
Real continuous submodular functions, as a generalization of the corresp...
read it

Streaming Methods for Restricted Strongly Convex Functions with Applications to Prototype Selection
In this paper, we show that if the optimization function is restricteds...
read it

Shaping Level Sets with Submodular Functions
We consider a class of sparsityinducing regularization terms based on s...
read it
Submodular + Concave
It has been well established that first order optimization methods can converge to the maximal objective value of concave functions and provide constant factor approximation guarantees for (nonconvex/nonconcave) continuous submodular functions. In this work, we initiate the study of the maximization of functions of the form F(x) = G(x) +C(x) over a solvable convex body P, where G is a smooth DRsubmodular function and C is a smooth concave function. This class of functions is a strict extension of both concave and continuous DRsubmodular functions for which no theoretical guarantee is known. We provide a suite of FrankWolfe style algorithms, which, depending on the nature of the objective function (i.e., if G and C are monotone or not, and nonnegative or not) and on the nature of the set P (i.e., whether it is downward closed or not), provide 11/e, 1/e, or 1/2 approximation guarantees. We then use our algorithms to get a framework to smoothly interpolate between choosing a diverse set of elements from a given ground set (corresponding to the mode of a determinantal point process) and choosing a clustered set of elements (corresponding to the maxima of a suitable concave function). Additionally, we apply our algorithms to various functions in the above class (DRsubmodular + concave) in both constrained and unconstrained settings, and show that our algorithms consistently outperform natural baselines.
READ FULL TEXT
Comments
There are no comments yet.