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    Asymptotic and finite-sample properties of estimators based on stochastic gradients

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    Genre
    Journal Article
    Date
    2017-08-01
    Author
    Toulis, P
    Airoldi, EM
    Subject
    Stochastic approximation
    implicit updates
    asymptotic variance
    generalized linear models
    Cox proportional hazards
    M-estimation
    maximum likelihood
    exponential family
    statistical efficiency
    numerical stability
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/5691
    
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    DOI
    10.1214/16-AOS1506
    Abstract
    © 2017 Institute of Mathematical Statistics. Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradient descent procedures, which involve parameter updates that are implicitly defined. Intuitively, implicit updates shrink standard stochastic gradient descent updates. The amount of shrinkage depends on the observed Fisher information matrix, which does not need to be explicitly computed; thus, implicit procedures increase stability without increasing the computational burden. Our theoretical analysis provides the first full characterization of the asymptotic behavior of both standard and implicit stochastic gradient descent-based estimators, including finite-sample error bounds. Importantly, analytical expressions for the variances of these stochastic gradient-based estimators reveal their exact loss of efficiency. We also develop new algorithms to compute implicit stochastic gradient descent-based estimators for generalized linear models, Cox proportional hazards, M-estimators, in practice, and perform extensive experiments. Our results suggest that implicit stochastic gradient descent procedures are poised to become a workhorse for approximate inference from large data sets.
    Citation to related work
    Institute of Mathematical Statistics
    Has part
    Annals of Statistics
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    ae974a485f413a2113503eed53cd6c53
    http://dx.doi.org/10.34944/dspace/5673
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