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    Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm

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    Type
    Preprint
    Authors
    Salim, Adil
    Richtarik, Peter cc
    KAUST Department
    Visual Computing Center (VCC)
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Computer Science Program
    Date
    2020-06-16
    Permanent link to this record
    http://hdl.handle.net/10754/666021
    
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    Abstract
    We consider the task of sampling with respect to a log concave probability distribution. The potential of the target distribution is assumed to be composite, i.e., written as the sum of a smooth convex term, and a nonsmooth convex term possibly taking infinite values. The target distribution can be seen as a minimizer of the Kullback-Leibler divergence defined on the Wasserstein space (i.e., the space of probability measures). In the first part of this paper, we establish a strong duality result for this minimization problem. In the second part of this paper, we use the duality gap arising from the first part to study the complexity of the Proximal Stochastic Gradient Langevin Algorithm (PSGLA), which can be seen as a generalization of the Projected Langevin Algorithm. Our approach relies on viewing PSGLA as a primal dual algorithm and covers many cases where the target distribution is not fully supported. In particular, we show that if the potential is strongly convex, the complexity of PSGLA is $\cO(1/\varepsilon^2)$ in terms of the 2-Wasserstein distance. In contrast, the complexity of the Projected Langevin Algorithm is $\cO(1/\varepsilon^{12})$ in terms of total variation when the potential is convex.
    Publisher
    arXiv
    arXiv
    2006.09270
    Additional Links
    https://arxiv.org/pdf/2006.09270
    Collections
    Preprints; Computer Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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