  We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged
stochastic gradient descent (SGD) to a normal random vector for a class of
twice-differentiable test functions. A crucial intermediate step is proving a
non-asymptotic martingale central limit theorem (CLT), i.e., establishing the
rates of convergence of a multivariate martingale difference sequence to a
normal random vector, which might be of independent interest. We obtain the
explicit rates for the multivariate martingale CLT using a combination of
Stein's method and Lindeberg's argument, which is then used in conjunction with
a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have
potentially interesting consequences for computing confidence intervals for
parameter estimation with SGD and constructing hypothesis tests with SGD that
are valid in a non-asymptotic sense.
