Non-monotone DR-submodular Maximization: Approximation and Regret Guarantees

Dürr, Christoph; Thắng, Nguyễn Kim; Srivastav, Abhinav; Tible, Léo

doi:10.48550/arxiv.1905.09595

Cited by 3 publications

(6 citation statements)

References 14 publications

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“…Recently, Niazadeh et al (2018) 3 present optimal algorithms for non-monotone submodular maximization with a box constraint. Continuous submodular maximization is also well studied in the stochastic setting (Karimi et al, 2017;Mokhtari et al, 2018b), online setting (Chen et al, 2018), bandit setting (Dürr et al, 2019) and decentralized setting (Mokhtari et al, 2018a).…”

Section: Submodularity Over Continuous Domainsmentioning

confidence: 99%

“…One model with "discrete" product assignments is considered by Soma and Yoshida (2017) and Dürr et al (2019), motivated by the observation that giving a user more free products increases the likelihood that the user will advocate this product. It can be treated as a simplified variant of the Influence-and-Exploit (IE) strategy of Hartline et al (2008).…”

Section: A Variant Of the Influence-and-exploit (Ie) Strategymentioning

confidence: 99%

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Continuous Submodular Function Maximization

Bian¹,

Buhmann²,

Krause³

2020

Preprint

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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 polynomial 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 of nonconvex optimization problems: continuous submodular function maximization. We start by a thorough characterization of the class of continuous submodular functions, and show that continuous submodularity is equivalent to a weak version of the diminishing returns (DR) property. Thus we also derive a subclass of continuous submodular functions, termed continuous DR-submodular functions, which enjoys the full DR property. Then we present operations that preserve continuous (DR-)submodularity, thus yielding general rules for composing new submodular functions. We establish intriguing properties for the problem of constrained DR-submodular maximization, such as the local-global relation, which captures the relationship of locally (approximate) stationary points and global optima. We identify several applications of continuous submodular optimization, ranging from influence maximization with general marketing strategies, MAP inference for DPPs to mean field inference for probabilistic log-submodular models. For these applications, continuous submodularity formalizes valuable domain knowledge relevant for optimizing this class of objectives. We present inapproximability results and provable algorithms for two problem settings: constrained monotone DR-submodular maximization and constrained nonmonotone DR-submodular maximization. Finally, we extensively evaluate the effectiveness of the proposed algorithms on different problem instances, such as influence maximization with marketing strategies and revenue maximization with continuous assignments.

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Section: Submodularity Over Continuous Domainsmentioning

confidence: 99%

Section: A Variant Of the Influence-and-exploit (Ie) Strategymentioning

confidence: 99%

Continuous Submodular Function Maximization

Bian¹,

Buhmann²,

Krause³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Very recently, Niazadeh et al (2018) present optimal algorithms for nonmonotone submodular maximization with a box constraint. Continuous submodular maximization is also well studied in the stochastic setting (Hassani et al, 2017;Mokhtari et al, 2018b), online setting (Chen et al, 2018), bandit setting (Dürr et al, 2019) and decentralized setting (Mokhtari et al, 2018a).…”

Section: Submodularity Over Continuous Domainsmentioning

confidence: 99%

“…This model has been used in Soma et al (2017) and Dürr et al (2019). It can be treated as a simplified variant of the Influence-and-Exploit (IE) strategy of Hartline et al (2008).…”

Section: A Variant Of the Influence-and-exploit (Ie) Strategymentioning

confidence: 99%

“…One model with "discrete" product assignments was used by Soma et al (2017) and Dürr et al (2019), which is motivated by the observation that giving a user more free products increases the likelihood that the user will advocate this product. It is also natural to consider continuous product assignments which is suitable for products that will be given to users to try for a certain period (e.g., a new software).…”

Section: Revenue Maximization With Continuous Assignmentsmentioning

confidence: 99%

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Model Selection for Gaussian Process Regression

Gorbach

Bian

Fischer

et al. 2017

Lecture Notes in Computer Science

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My name was also written as (Andrew) An Bian due to a name change. My ORCID iD is orcid.org/0000-0002-2368-4084. v -Yatao An Bian, Xiong Li, Yuncai Liu, and Ming-Hsuan Yang (2019b). "Parallel Coordinate Descent Newton Method for Efficient L1-Regularized Loss Minimization." In: IEEE Transactions on Neural Networks and Learning Systems, pp. 3233-3245 -Lie *He, An *Bian, and Martin Jaggi (2018). "COLA: Communication-Efficient Decentralized Linear Learning." In: Advances in Neural Information Processing Systems (NeurIPS), pp. 4537-4547 * Authors contributed equally.

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Online Weakly DR-Submodular Optimization Under Stochastic Cumulative Constraints

Feng,

Yang,

Zhang

et al. 2024

Tsinghua Sci. Technol.

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In this paper, we study a class of online continuous optimization problems. At each round, the utility function is the sum of a weakly diminishing-returns (DR) submodular function and a concave function, certain cost associated with the action will occur, and the problem has total limited budget. Combining the two methods, the penalty function and Frank-Wolfe strategies, we present an online method to solve the considered problem. Choosing appropriate stepsize and penalty parameters, the performance of the online algorithm is guaranteed, that is, it achieves sub-linear regret bound and certain mild constraint violation bound in expectation.

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Non-monotone DR-submodular Maximization: Approximation and Regret Guarantees

Cited by 3 publications

References 14 publications

Continuous Submodular Function Maximization

Continuous Submodular Function Maximization

Model Selection for Gaussian Process Regression

Online Weakly DR-Submodular Optimization Under Stochastic Cumulative Constraints

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