We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $τ$ from the returned set may be removed. We focus on the worst-case setting considered in (Orlin et al., 2016), in which a constant-factor approximation guarantee was given for $τ= o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $τ= o(k)$. Our algorithm constructs partitions consisting of buckets with exponentially increasing sizes, and applies standard submodular optimization subroutines on the buckets in order to construct the robust solution. We numerically demonstrate the performance of PRo in data summarization and influence maximization, demonstrating gains over both the greedy algorithm and the algorithm of (Orlin et al., 2016).
Continuous submodular functions are a category of generally non-convex/non-concave functions with a wide spectrum of applications. The celebrated property of this class of functions - continuous submodularity - enables both exact minimization and approximate maximization in poly. time. Continuous submodularity is obtained by generalizing the notion of submodularity from discrete domains to continuous domains. It intuitively captures a repulsive effect amongst different dimensions of the defined multivariate function. In this paper, we systematically study continuous submodularity and a class o…
Yale University（耶鲁大学）；University of Pennsylvania（宾夕法尼亚大学）
In this paper, we propose three online algorithms for submodular maximisation. The first one, Mono-Frank-Wolfe, reduces the number of per-function gradient evaluations from $T^{1/2}$ [Chen2018Online] and $T^{3/2}$ [chen2018projection] to 1, and achieves a $(1-1/e)$-regret bound of $O(T^{4/5})$. The second one, Bandit-Frank-Wolfe, is the first bandit algorithm for continuous DR-submodular maximization, which achieves a $(1-1/e)$-regret bound of $O(T^{8/9})$. Finally, we extend Bandit-Frank-Wolfe to a bandit algorithm for discrete submodular maximization, Responsive-Frank-Wolfe, which attains a …
Naveen Jindal School of Management, The University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金德尔管理学院）
In this paper, we develop fast algorithms for two stochastic submodular maximization problems. We start with the well-studied adaptive submodular maximization problem subject to a cardinality constraint. We develop the first linear-time algorithm which achieves a $(1-1/e-ε)$ approximation ratio. Notably, the time complexity of our algorithm is $O(n\log\frac{1}ε)$ (number of function evaluations) which is independent of the cardinality constraint, where $n$ is the size of the ground set. Then we introduce the concept of fully adaptive submodularity, and develop a linear-time algorithm for maxim…
Mean field inference in probabilistic models is generally a highly nonconvex problem. Existing optimization methods, e.g., coordinate ascent algorithms, can only generate local optima. In this work we propose provable mean filed methods for probabilistic log-submodular models and its posterior agreement (PA) with strong approximation guarantees. The main algorithmic technique is a new Double Greedy scheme, termed DR-DoubleGreedy, for continuous DR-submodular maximization with box-constraints. It is a one-pass algorithm with linear time complexity, reaching the optimal 1/2 approximation ratio, …
In this paper, we introduce a novel technique for constrained submodular maximization, inspired by barrier functions in continuous optimization. This connection not only improves the running time for constrained submodular maximization but also provides the state of the art guarantee. More precisely, for maximizing a monotone submodular function subject to the combination of a $k$-matchoid and $\ell$-knapsack constraint (for $\ell\leq k$), we propose a potential function that can be approximately minimized. Once we minimize the potential function up to an $ε$ error it is guaranteed that we hav…
Naveen Jindal School of Management, The University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）
In this paper, we study the non-monotone adaptive submodular maximization problem subject to a cardinality constraint. We first revisit the adaptive random greedy algorithm proposed in \citep{gotovos2015non}, where they show that this algorithm achieves a $1/e$ approximation ratio if the objective function is adaptive submodular and pointwise submodular. It is not clear whether the same guarantee holds under adaptive submodularity (without resorting to pointwise submodularity) or not. Our first contribution is to show that the adaptive random greedy algorithm achieves a $1/e$ approximation rat…
Microsoft Research Cambridge（微软研究院剑桥）；ETH Zurich（苏黎世联邦理工学院）
We consider learning of submodular functions from data. These functions are important in machine learning and have a wide range of applications, e.g. data summarization, feature selection and active learning. Despite their combinatorial nature, submodular functions can be maximized approximately with strong theoretical guarantees in polynomial time. Typically, learning the submodular function and optimization of that function are treated separately, i.e. the function is first learned using a proxy objective and subsequently maximized. In contrast, we show how to perform learning and optimizati…
Inference for latent feature models is inherently difficult as the inference space grows exponentially with the size of the input data and number of latent features. In this work, we use Kurihara & Welling (2008)'s maximization-expectation framework to perform approximate MAP inference for linear-Gaussian latent feature models with an Indian Buffet Process (IBP) prior. This formulation yields a submodular function of the features that corresponds to a lower bound on the model evidence. By adding a constant to this function, we obtain a nonnegative submodular function that can be maximized via …
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）；Expedia Group（亿客行集团）；University of Haifa（海法大学）
In this paper, we propose scalable methods for maximizing a regularized submodular function $f = g - \ell$ expressed as the difference between a monotone submodular function $g$ and a modular function $\ell$. Indeed, submodularity is inherently related to the notions of diversity, coverage, and representativeness. In particular, finding the mode of many popular probabilistic models of diversity, such as determinantal point processes, submodular probabilistic models, and strongly log-concave distributions, involves maximization of (regularized) submodular functions. Since a regularized function…
Khoury College of Computer Sciences, Northeastern University（东北大学胡里计算机科学学院）
We study the problem of differentially private constrained maximization of decomposable submodular functions. A submodular function is decomposable if it takes the form of a sum of submodular functions. The special case of maximizing a monotone, decomposable submodular function under cardinality constraints is known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al., 2008]. Previous work by Gupta et al. [2010] gave a differentially private algorithm for the CPP problem. We extend this work by designing differentially private algorithms for both monotone and non-monotone d…
A variety of large-scale machine learning problems can be cast as instances of constrained submodular maximization. Existing approaches for distributed submodular maximization have a critical drawback: The capacity - number of instances that can fit in memory - must grow with the data set size. In practice, while one can provision many machines, the capacity of each machine is limited by physical constraints. We propose a truly scalable approach for distributed submodular maximization under fixed capacity. The proposed framework applies to a broad class of algorithms and constraints and provid…
University of Washington（华盛顿大学）；Yahoo Research（雅虎研究院）
We propose a new random pruning method (called "submodular sparsification (SS)") to reduce the cost of submodular maximization. The pruning is applied via a "submodularity graph" over the $n$ ground elements, where each directed edge is associated with a pairwise dependency defined by the submodular function. In each step, SS prunes a $1-1/\sqrt{c}$ (for $c>1$) fraction of the nodes using weights on edges computed based on only a small number ($O(\log n)$) of randomly sampled nodes. The algorithm requires $\log_{\sqrt{c}}n$ steps with a small and highly parallelizable per-step computation. An …
Naveen Jindal School of Management, The University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）；Department of Computer Science, The University of Texas at Dallas（德克萨斯大学达拉斯分校计算机科学系）
In this paper, we propose and study the cascade submodular maximization problem under the adaptive setting. The input of our problem is a set of items, each item is in a particular state (i.e., the marginal contribution of an item) which is drawn from a known probability distribution. However, we can not know its actual state before selecting it. As compared with existing studies on stochastic submodular maximization, one unique setting of our problem is that each item is associated with a continuation probability which represents the probability that one is allowed to continue to select the n…
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 objective…
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）；University of Pennsylvania（宾夕法尼亚大学）
In this paper, we consider the problem of black box continuous submodular maximization where we only have access to the function values and no information about the derivatives is provided. For a monotone and continuous DR-submodular function, and subject to a bounded convex body constraint, we propose Black-box Continuous Greedy, a derivative-free algorithm that provably achieves the tight $[(1-1/e)OPT-ε]$ approximation guarantee with $O(d/ε^3)$ function evaluations. We then extend our result to the stochastic setting where function values are subject to stochastic zero-mean noise. It is thro…
Submodular functions have applications throughout machine learning, but in many settings, we do not have direct access to the underlying function $f$. We focus on stochastic functions that are given as an expectation of functions over a distribution $P$. In practice, we often have only a limited set of samples $f_i$ from $P$. The standard approach indirectly optimizes $f$ by maximizing the sum of $f_i$. However, this ignores generalization to the true (unknown) distribution. In this paper, we achieve better performance on the actual underlying function $f$ by directly optimizing a combination …
In this paper, we consider online continuous DR-submodular maximization with linear stochastic long-term constraints. Compared to the prior work on online submodular maximization, our setting introduces the extra complication of stochastic linear constraint functions that are i.i.d. generated at each round. To be precise, at step $t\in\{1,\dots,T\}$, a DR-submodular utility function $f_t(\cdot)$ and a constraint vector $p_t$, i.i.d. generated from an unknown distribution with mean $p$, are revealed after committing to an action $x_t$ and we aim to maximize the overall utility while the expecte…
Over the last two decades, submodular function maximization has been the workhorse of many discrete optimization problems in machine learning applications. Traditionally, the study of submodular functions was based on binary function properties. However, such properties have an inherit weakness, namely, if an algorithm assumes functions that have a particular property, then it provides no guarantee for functions that violate this property, even when the violation is very slight. Therefore, recent works began to consider continuous versions of function properties. Probably the most significant …
ETH Zurich（苏黎世联邦理工学院）；University of Pennsylvania（宾夕法尼亚大学）
Stochastic optimization of continuous objectives is at the heart of modern machine learning. However, many important problems are of discrete nature and often involve submodular objectives. We seek to unleash the power of stochastic continuous optimization, namely stochastic gradient descent and its variants, to such discrete problems. We first introduce the problem of stochastic submodular optimization, where one needs to optimize a submodular objective which is given as an expectation. Our model captures situations where the discrete objective arises as an empirical risk (e.g., in the case o…
We address the problem of maximizing an unknown submodular function that can only be accessed via noisy evaluations. Our work is motivated by the task of summarizing content, e.g., image collections, by leveraging users' feedback in form of clicks or ratings. For summarization tasks with the goal of maximizing coverage and diversity, submodular set functions are a natural choice. When the underlying submodular function is unknown, users' feedback can provide noisy evaluations of the function that we seek to maximize. We provide a generic algorithm -- \submM{} -- for maximizing an unknown submo…
Continuous DR-submodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing works have studied monotone continuous DR-submodular maximization subject to a convex constraint and have proposed efficient algorithms with approximation guarantees. However, in many applications, e.g., computing the stability number of a graph and mean-field inference for probabilistic log-submodular models, the DR-submodular function has the additional property of being \emph{strongly} concave along non-ne…
In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $\frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $\frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular m…
Data summarization has become a valuable tool in understanding even terabytes of data. Due to their compelling theoretical properties, submodular functions have been in the focus of summarization algorithms. These algorithms offer worst-case approximations guarantees to the expense of higher computation and memory requirements. However, many practical applications do not fall under this worst-case, but are usually much more well-behaved. In this paper, we propose a new submodular function maximization algorithm called ThreeSieves, which ignores the worst-case, but delivers a good solution in h…
In this paper, we study fundamental problems of maximizing DR-submodular continuous functions that have real-world applications in the domain of machine learning, economics, operations research and communication systems. It captures a subclass of non-convex optimization that provides both theoretical and practical guarantees. Here, we focus on minimizing regret for online arriving non-monotone DR-submodular functions over different types of convex sets: hypercube, down-closed and general convex sets. First, we present an online algorithm that achieves a $1/e$-approximation ratio with the regre…
In this paper we describe a new algorithm called Fast Adaptive Sequencing Technique (FAST) for maximizing a monotone submodular function under a cardinality constraint $k$ whose approximation ratio is arbitrarily close to $1-1/e$, is $O(\log(n) \log^2(\log k))$ adaptive, and uses a total of $O(n \log\log(k))$ queries. Recent algorithms have comparable guarantees in terms of asymptotic worst case analysis, but their actual number of rounds and query complexity depend on very large constants and polynomials in terms of precision and confidence, making them impractical for large data sets. Our ma…
Yale University（耶鲁大学）；The Open University of Israel（以色列开放大学）
Submodular functions are a broad class of set functions, which naturally arise in diverse areas. Many algorithms have been suggested for the maximization of these functions. Unfortunately, once the function deviates from submodularity, the known algorithms may perform arbitrarily poorly. Amending this issue, by obtaining approximation results for set functions generalizing submodular functions, has been the focus of recent works. One such class, known as weakly submodular functions, has received a lot of attention. A key result proved by Das and Kempe (2011) showed that the approximation ratio…
Khoury College of Computer Sciences, Northeastern University（东北大学胡里计算机科学学院）
In this work, we study the problem of privately maximizing a submodular function in the streaming setting. Extensive work has been done on privately maximizing submodular functions in the general case when the function depends upon the private data of individuals. However, when the size of the data stream drawn from the domain of the objective function is large or arrives very fast, one must privately optimize the objective within the constraints of the streaming setting. We establish fundamental differentially private baselines for this problem and then derive better trade-offs between privac…
We study the problem of maximizing a monotone set function subject to a cardinality constraint $k$ in the setting where some number of elements $τ$ is deleted from the returned set. The focus of this work is on the worst-case adversarial setting. While there exist constant-factor guarantees when the function is submodular, there are no guarantees for non-submodular objectives. In this work, we present a new algorithm Oblivious-Greedy and prove the first constant-factor approximation guarantees for a wider class of non-submodular objectives. The obtained theoretical bounds are the first constan…
Diminishing-returns (DR) submodular optimization is an important field with many real-world applications in machine learning, economics and communication systems. It captures a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we study the fundamental problem of maximizing non-monotone DR-submodular functions over down-closed and general convex sets in both offline and online settings. First, we show that for offline maximizing non-monotone DR-submodular functions over a general convex set, the Frank-Wolfe algorithm achieves an approxim…
Sharif University（谢里夫大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）
The standard greedy algorithm has been recently shown to enjoy approximation guarantees for constrained non-submodular nondecreasing set function maximization. While these recent results allow to better characterize the empirical success of the greedy algorithm, they are only applicable to simple cardinality constraints. In this paper, we study the problem of maximizing a non-submodular nondecreasing set function subject to a general matroid constraint. We first show that the standard greedy algorithm offers an approximation factor of $\frac{0.4 γ^{2}}{\sqrt{γr} + 1}$, where $γ$ is the submodu…
University of Cambridge（剑桥大学）；Georgia Institute of Technology（佐治亚理工学院）
We study the worst-case adaptive optimization problem with budget constraint that is useful for modeling various practical applications in artificial intelligence and machine learning. We investigate the near-optimality of greedy algorithms for this problem with both modular and non-modular cost functions. In both cases, we prove that two simple greedy algorithms are not near-optimal but the best between them is near-optimal if the utility function satisfies pointwise submodularity and pointwise cost-sensitive submodularity respectively. This implies a combined algorithm that is near-optimal w…
