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    SGD: General Analysis and Improved Rates

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    Name:
    1901.09401.pdf
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    1.192Mb
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    PDF
    Description:
    Preprint
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    Type
    Preprint
    Authors
    Gower, Robert Mansel
    Loizou, Nicolas
    Qian, Xun cc
    Sailanbayev, Alibek
    Shulgin, Egor
    Richtarik, Peter cc
    KAUST Department
    Applied Mathematics and Computational Science
    Applied Mathematics and Computational Science Program
    Computer Science
    Computer Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Visual Computing Center
    Visual Computing Center (VCC)
    Date
    2019-01-27
    Permanent link to this record
    http://hdl.handle.net/10754/653123
    
    Metadata
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    Abstract
    We propose a general yet simple theorem describing the convergence of SGDunder the arbitrary sampling paradigm. Our theorem describes the convergence ofan infinite array of variants of SGD, each of which is associated with aspecific probability law governing the data selection rule used to formmini-batches. This is the first time such an analysis is performed, and most ofour variants of SGD were never explicitly considered in the literature before.Our analysis relies on the recently introduced notion of expected smoothnessand does not rely on a uniform bound on the variance of the stochasticgradients. By specializing our theorem to different mini-batching strategies,such as sampling with replacement and independent sampling, we derive exactexpressions for the stepsize as a function of the mini-batch size. With this wecan also determine the mini-batch size that optimizes the total complexity, andshow explicitly that as the variance of the stochastic gradient evaluated atthe minimum grows, so does the optimal mini-batch size. For zero variance, theoptimal mini-batch size is one. Moreover, we prove insightfulstepsize-switching rules which describe when one should switch from a constantto a decreasing stepsize regime.
    Sponsors
    RMG acknowledges the support by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Program for optimization, operations research and their interactions with data sciences.
    Publisher
    arXiv
    arXiv
    1901.09401
    Additional Links
    https://arxiv.org/pdf/1901.09401
    Collections
    Preprints; Applied Mathematics and Computational Science Program; Computer Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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