From Sets to Multisets: Provable Variational Inference for Probabilistic Integer Submodular Models

ICML, pp. 8388-8397, 2020.

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We considered the problem of variational inference in Probabilistic Integer Submodular Models and introduced a novel continuous extension for integer submodular functions

Abstract:

Submodular functions have been studied extensively in machine learning and data mining. In particular, the optimization of submodular functions over the integer lattice (integer submodular functions) has recently attracted much interest, because this domain relates naturally to many practical problem settings, such as multilabel graph c...More

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Introduction
  • Submodular functions have many applications in machine learning, which include data summarization (Tschiatschek et al, 2014; Lin & Bilmes, 2012), sensor placement (Krause et al, 2006) and computer vision (Boykov et al, 1999)
  • They are defined on the subsets of the ground set V which contains n elements.
  • The multilinear extension (Calinescu et al, 2007) is a continuous extension of a submodular function to the full hypercube [0, 1]n and one can get the best approximation guarantee for matroid constraints by optimizing it using the continuous greedy algorithm (Calinescu et al, 2011)
Highlights
  • Submodular functions have many applications in machine learning, which include data summarization (Tschiatschek et al, 2014; Lin & Bilmes, 2012), sensor placement (Krause et al, 2006) and computer vision (Boykov et al, 1999)
  • We present the Generalized Multilinear Extension (GME) for integer submodular functions and continuous submodular functions
  • We introduce Probabilistic Integer Submodular Models, which are characterized through integer submodular functions
  • The proposed Generalized Multilinear Extension is smooth and corresponds to the one part of the Evidence Lower Bound for variational inference for Probabilistic Submodular Models with integer submodular functions as we show below
  • We considered the problem of variational inference in Probabilistic Integer Submodular Models and introduced a novel continuous extension for integer submodular functions
Methods
  • The authors illustrate the efficacy of the proposed generalized multilinear extension and algorithms on two classes of applications: Revenue maximization with discrete assignments and facility location with discrete levels.
  • Both of these two applications have integer submodular objectives.
  • The authors use Monte Carlo sampling to estimate F (ρ) by sampling the integer function values according to the categorical distribution.
  • Thanks to the Hoeffding bound (Hoeffding, 1963), one can show that the estimated function value can be arbitrarily close to true F (ρ) with polynomial number of samples
Results
  • 3 N6umbe9r of Ep12ochs15 18 21
  • In this experiment, the authors use an integer facility location function with n = 50 facilities and customers and k = 5 levels.
  • For each level the authors sum up the previous utility values in order to get a monotone function.
  • The authors run Shrunken FW, Two-Phase FW and variants of Block CA for 20 epochs and the variants of Block.
  • CA achieve similar ELBO values whereas Shrunken.
  • FW and Two-Phase FW obtain a lower ELBO value.
Conclusion
  • The authors considered the problem of variational inference in Probabilistic Integer Submodular Models and introduced a novel continuous extension for integer submodular functions.
  • It can be viewed as an expectation under fully-factorized marginals and the authors proved that it is DR-submodular even if the integer function is just submodular.
  • The authors presented an efficient block coordinate ascent algorithm to optimize the ELBO and showed the effectiveness of the method in real world graph mining applications
Summary
  • Introduction:

    Submodular functions have many applications in machine learning, which include data summarization (Tschiatschek et al, 2014; Lin & Bilmes, 2012), sensor placement (Krause et al, 2006) and computer vision (Boykov et al, 1999)
  • They are defined on the subsets of the ground set V which contains n elements.
  • The multilinear extension (Calinescu et al, 2007) is a continuous extension of a submodular function to the full hypercube [0, 1]n and one can get the best approximation guarantee for matroid constraints by optimizing it using the continuous greedy algorithm (Calinescu et al, 2011)
  • Methods:

    The authors illustrate the efficacy of the proposed generalized multilinear extension and algorithms on two classes of applications: Revenue maximization with discrete assignments and facility location with discrete levels.
  • Both of these two applications have integer submodular objectives.
  • The authors use Monte Carlo sampling to estimate F (ρ) by sampling the integer function values according to the categorical distribution.
  • Thanks to the Hoeffding bound (Hoeffding, 1963), one can show that the estimated function value can be arbitrarily close to true F (ρ) with polynomial number of samples
  • Results:

    3 N6umbe9r of Ep12ochs15 18 21
  • In this experiment, the authors use an integer facility location function with n = 50 facilities and customers and k = 5 levels.
  • For each level the authors sum up the previous utility values in order to get a monotone function.
  • The authors run Shrunken FW, Two-Phase FW and variants of Block CA for 20 epochs and the variants of Block.
  • CA achieve similar ELBO values whereas Shrunken.
  • FW and Two-Phase FW obtain a lower ELBO value.
  • Conclusion:

    The authors considered the problem of variational inference in Probabilistic Integer Submodular Models and introduced a novel continuous extension for integer submodular functions.
  • It can be viewed as an expectation under fully-factorized marginals and the authors proved that it is DR-submodular even if the integer function is just submodular.
  • The authors presented an efficient block coordinate ascent algorithm to optimize the ELBO and showed the effectiveness of the method in real world graph mining applications
Tables
  • Table1: Graph datasets and corresponding experimental parameters
Download tables as Excel
Related work
  • The first systematic study for probabilistic models defined through submodular set functions is presented by Djolonga & Krause (2014). Djolonga et al (2016) develop an efficient approximate inference algorithm for more general models with both submodular and supermodular functions. Gotovos et al (2015) analyze MCMC sampling to perform approximate inference in PSMs and Gotovos et al (2018) introduce a new sampling strategy for accelerating mixing in PSMs.

    Integer and continuous submodular optimization problems attract considerable attention recently. Gottschalk & Peis (2015) present a deterministic algorithm to maximize non-monotone submodular functions on a bounded lattice; Soma et al (2014) consider maximizing monotone integer submodular functions with a knapsack constraint; and Soma & Yoshida (2017) study the integer submodular cover problem with applications on sensor placement with discrete energy levels. Qian et al (2018) consider submodular maximization problems subject to size constraints while relaxing the submodularity assumption. Bian et al (2017b) characterize the notion of continuous submodularity and present an optimal algorithm for monotone DR-submodular maximization. Hassani et al (2017) show that the projected gradient ascent algorithm achieves a 1/2 approximation for maximizing monotone DR-submodular functions. Bian et al (2017a) present the local-global relation and guaranteed algorithms for non-monotone DR-submodular maximization. Recently, Bian et al (2019); Niazadeh et al (2018) propose optimal algorithms for non-monotone DR-submodular maximization with a box constraint. Continuous submodular maximization is also well studied in the stochastic setting (Hassani et al, 2017; Mokhtari et al, 2018) and online setting (Chen et al, 2018).
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