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Stochastic gradient descent procedures have gained popularity for parameterestimation from large data sets. However, their statistical properties are notwell understood, in theory. And in practice, avoiding numerical instabilityrequires careful tuning of key parameters. Here, we introduce implicitstochastic gradient descent procedures, which involve parameter updates thatare implicitly defined. Intuitively, implicit updates shrink standardstochastic gradient descent updates. The amount of shrinkage depends on theobserved Fisher information matrix, which does not need to be explicitlycomputed; thus, implicit procedures increase stability without increasing thecomputational burden. Our theoretical analysis provides the first fullcharacterization of the asymptotic behavior of both standard and implicitstochastic gradient descent-based estimators, including finite-sample errorbounds. Importantly, analytical expressions for the variances of thesestochastic gradient-based estimators reveal their exact loss of efficiency. Wealso develop new algorithms to compute implicit stochastic gradientdescent-based estimators for generalized linear models, Cox proportionalhazards, M-estimators, in practice, and perform extensive experiments. Ourresults suggest that implicit stochastic gradient descent procedures are poisedto become a workhorse for approximate inference from large data sets

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