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### Continuous DR-submodular Maximization: Structure and Algorithms

DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints. We start by investigating geometric properties that underlie such objectives, e.g., a strong relation between (approximately) stationary points and global optimum is proved. These properties are then used to devise two optimization algorithms with provable guarantees. Concretely, we first devise a "two-phase'' algorithm with 1/4 approximation guarantee. This algorithm allows the use of existing methods for finding (approximately) stationary points as a subroutine, thus, harnessing recent progress in non-convex optimization. Then we present a non-monotone Frank-Wolfe variant with 1/e approximation guarantee and sublinear convergence rate. Finally, we extend our approach to a broader class of generalized DR-submodular continuous functions, which captures a wider spectrum of applications. Our theoretical findings are validated on synthetic and real-world problem instances.

### Continuous DR-submodular Maximization: Structure and Algorithms

DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints. We start by investigating geometric properties that underlie such objectives, e.g., a strong relation between (approximately) stationary points and global optimum is proved. These properties are then used to devise two optimization algorithms with provable guarantees. Concretely, we first devise a "two-phase" algorithm with $1/4$ approximation guarantee. This algorithm allows the use of existing methods for finding (approximately) stationary points as a subroutine, thus, harnessing recent progress in non-convex optimization. Then we present a non-monotone Frank-Wolfe variant with $1/e$ approximation guarantee and sublinear convergence rate. Finally, we extend our approach to a broader class of generalized DR-submodular continuous functions, which captures a wider spectrum of applications. Our theoretical findings are validated on synthetic and real-world problem instances.

### Continuous DR-submodular Maximization: Structure and Algorithms

DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints. We start by investigating geometric properties that underlie such objectives, e.g., a strong relation between (approximately) stationary points and global optimum is proved. These properties are then used to devise two optimization algorithms with provable guarantees.

### Stochastic Conditional Gradient Methods: From Convex Minimization to Submodular Maximization

This paper considers stochastic optimization problems for a large class of objective functions, including convex and continuous submodular. Stochastic proximal gradient methods have been widely used to solve such problems; however, their applicability remains limited when the problem dimension is large and the projection onto a convex set is costly. Instead, stochastic conditional gradient methods are proposed as an alternative solution relying on (i) Approximating gradients via a simple averaging technique requiring a single stochastic gradient evaluation per iteration; (ii) Solving a linear program to compute the descent/ascent direction. The averaging technique reduces the noise of gradient approximations as time progresses, and replacing projection step in proximal methods by a linear program lowers the computational complexity of each iteration. We show that under convexity and smoothness assumptions, our proposed method converges to the optimal objective function value at a sublinear rate of $O(1/t^{1/3})$. Further, for a monotone and continuous DR-submodular function and subject to a general convex body constraint, we prove that our proposed method achieves a $((1-1/e)OPT-\eps)$ guarantee with $O(1/\eps^3)$ stochastic gradient computations. This guarantee matches the known hardness results and closes the gap between deterministic and stochastic continuous submodular maximization. Additionally, we obtain $((1/e)OPT -\eps)$ guarantee after using $O(1/\eps^3)$ stochastic gradients for the case that the objective function is continuous DR-submodular but non-monotone and the constraint set is down-closed. By using stochastic continuous optimization as an interface, we provide the first $(1-1/e)$ tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a matroid constraint and $(1/e)$ approximation guarantee for the non-monotone case.

### Online Non-Monotone DR-submodular Maximization

In this paper, we study problems at the interface of two important fields: \emph{submodular optimization} and \emph{online learning}. Submodular functions play a vital role in modelling cost functions that naturally arise in many areas of discrete optimization. These functions have been studied under various models of computation. Independently, submodularity has been considered in continuous domains. In fact, many problems arising in machine learning and statistics have been modelled using continuous DR-submodular functions. In this work, we are study the problem of maximizing \textit{non-monotone} continuous DR-submodular functions within the framework of online learning. We provide three main results. First, we present an online algorithm (in full-information setting) that achieves an approximation guarantee (depending on the search space) for the problem of maximizing non-monotone continuous DR-submodular functions over a \emph{general} convex domain. To best of our knowledge, no prior approximation algorithm in full-information setting was known for the non-monotone continuous DR submodular functions even for the \emph{down-closed} convex domain. Second, we show that the online stochastic mirror ascent algorithm (in full information setting) achieves an improved approximation ratio of $(1/4)$ for maximizing the non-monotone continuous DR-submodular functions over a \emph{down-closed} convex domain. At last, we extend our second result to the bandit setting where we present the first approximation guarantee of $(1/4)$. To best of our knowledge, no approximation algorithm for non-monotone submodular maximization was known in the bandit setting.