This paper presents a unified approach for maximizing continuous DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in three cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining four cases. Notably, our approach for the stochastic function value-based oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions.

Neural Information Processing Systems http://nips.cc/

In this paper, we develop \scg~(\text{SCG}{$++$}), the first efficient variant of a conditional gradient method for maximizing a continuous submodular function subject to a convex constraint. Concretely, for a monotone and continuous DR-submodular function, \SCGPP achieves a tight $[(1-1/e)\OPT -\epsilon]$ solution while using $O(1/\epsilon^2)$ stochastic gradients and $O(1/\epsilon)$ calls to the linear optimization oracle. The best previously known algorithms either achieve a suboptimal $[(1/2)\OPT -\epsilon]$ solution with $O(1/\epsilon^2)$ stochastic gradients or the tight $[(1-1/e)\OPT -\epsilon]$ solution with suboptimal $O(1/\epsilon^3)$ stochastic gradients. We further provide an information-theoretic lower bound to showcase the necessity of $\OM({1}/{\epsilon^2})$ stochastic oracle queries in order to achieve $[(1-1/e)\OPT -\epsilon]$ for monotone and DR-submodular functions. This result shows that our proposed \SCGPP enjoys optimality in terms of both approximation guarantee, i.e., $(1-1/e)$ approximation factor, and stochastic gradient evaluations, i.e., $O(1/\epsilon^2)$ calls to the stochastic oracle. By using stochastic continuous optimization as an interface, we also show that it is possible to obtain the $[(1-1/e)\OPT-\epsilon]$ tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a general matroid constraint after at most $\mathcal{O}(n^2/\epsilon^2)$ calls to the stochastic function value, where $n$ is the number of elements in the ground set.

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 (non-convex/non-concave) 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 DR-submodular function and $C$ is a smooth concave function. This class of functions is a strict extension of both concave and continuous DR-submodular functions for which no theoretical guarantee is known. We provide a suite of Frank-Wolfe style algorithms, which, depending on the nature of the objective function (i.e., if $G$ and $C$ are monotone or not, and non-negative or not) and on the nature of the set $P$ (i.e., whether it is downward closed or not), provide $1-1/e$, $1/e$, 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 (DR-submodular + concave) in both constrained and unconstrained settings, and show that our algorithms consistently outperform natural baselines.

In this work we study the problem of maximizing non-monotone continuous DR-submodular functions under general down-closed convex constraints.

In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains.

This work studies the non-monotone DR-submodular Maximization over a ground set of $n$ subject to a size constraint $k$. We propose two approximation algorithms for solving this problem named FastDrSub and FastDrSub++. FastDrSub offers an approximation ratio of $0.044$ with query complexity of $O(n \log(k))$. The second one, FastDrSub++, improves upon it with a ratio of $1/4-ε$ within query complexity of $(n \log k)$ for an input parameter $ε>0$. Therefore, our proposed algorithms are the first constant-ratio approximation algorithms for the problem with the low complexity of $O(n \log(k))$. Additionally, both algorithms are experimentally evaluated and compared against existing state-of-the-art methods, demonstrating their effectiveness in solving the Revenue Maximization problem with DR-submodular objective function. The experimental results show that our proposed algorithms significantly outperform existing approaches in terms of both query complexity and solution quality.