  The stochastic gradient descent (SGD) optimization algorithm plays a central
role in a series of machine learning applications. The scientific literature
provides a vast amount of upper error bounds for the SGD method. Much less
attention as been paid to proving lower error bounds for the SGD method. It is
the key contribution of this paper to make a step in this direction. More
precisely, in this article we establish for every $\gamma, \nu \in (0,\infty)$
essentially matching lower and upper bounds for the mean square error of the
SGD process with learning rates $(\frac{\gamma}{n^\nu})_{n \in \mathbb{N}}$
associated to a simple quadratic stochastic optimization problem. This allows
us to precisely quantify the mean square convergence rate of the SGD method in
dependence on the asymptotic behavior of the learning rates.
