  The implicit stochastic gradient descent (ISGD), a proximal version of SGD,
is gaining interest in the literature due to its stability over (explicit) SGD.
In this paper, we conduct an in-depth analysis of the two modes of ISGD for
smooth convex functions, namely proximal Robbins-Monro (proxRM) and proximal
Poylak-Ruppert (proxPR) procedures, for their use in statistical inference on
model parameters. Specifically, we derive non-asymptotic point estimation error
bounds of both proxRM and proxPR iterates and their limiting distributions, and
propose on-line estimators of their asymptotic covariance matrices that require
only a single run of ISGD. The latter estimators are used to construct valid
confidence intervals for the model parameters. Our analysis is free of the
generalized linear model assumption that has limited the preceding analyses,
and employs feasible procedures. Our on-line covariance matrix estimators
appear to be the first of this kind in the ISGD literature.
