  We give quantitative estimates for the rate of convergence of Riemannian
stochastic gradient descent (RSGD) to Riemannian gradient flow and to a
diffusion process, the so-called Riemannian stochastic modified flow (RSMF).
Using tools from stochastic differential geometry we show that, in the small
learning rate regime, RSGD can be approximated by the solution to the RSMF
driven by an infinite-dimensional Wiener process. The RSMF accounts for the
random fluctuations of RSGD and, thereby, increases the order of approximation
compared to the deterministic Riemannian gradient flow. The RSGD is build using
the concept of a retraction map, that is, a cost efficient approximation of the
exponential map, and we prove quantitative bounds for the weak error of the
diffusion approximation under assumptions on the retraction map, the geometry
of the manifold, and the random estimators of the gradient.
