  This paper provides a framework to analyze stochastic gradient algorithms in
a mean squared error (MSE) sense using the asymptotic normality result of the
stochastic gradient descent (SGD) iterates. We perform this analysis by taking
the asymptotic normality result and applying it to the finite iteration case.
Specifically, we look at problems where the gradient estimators are biased and
have reduced variance and compare the iterates generated by these gradient
estimators to the iterates generated by the SGD algorithm. We use the work of
Fabian to characterize the mean and the variance of the distribution of the
iterates in terms of the bias and the covariance matrix of the gradient
estimators. We introduce the sliding window SGD (SW-SGD) algorithm, with its
proof of convergence, which incurs a lower MSE than the SGD algorithm on
quadratic and convex problems. Lastly, we present some numerical results to
show the effectiveness of this framework and the superiority of SW-SGD
algorithm over the SGD algorithm.
