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

Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates