  We consider the optimization of a smooth and strongly convex objective using
constant step-size stochastic gradient descent (SGD) and study its properties
through the prism of Markov chains. We show that, for unbiased gradient
estimates with mildly controlled variance, the iteration converges to an
invariant distribution in total variation distance. We also establish this
convergence in Wasserstein-2 distance in a more general setting compared to
previous work. Thanks to the invariance property of the limit distribution, our
analysis shows that the latter inherits sub-Gaussian or sub-exponential
concentration properties when these hold true for the gradient. This allows the
derivation of high-confidence bounds for the final estimate. Finally, under
such conditions in the linear case, we obtain a dimension-free deviation bound
for the Polyak-Ruppert average of a tail sequence. All our results are
non-asymptotic and their consequences are discussed through a few applications.
