  In this paper, we provide novel tail bounds on the optimization error of
Stochastic Mirror Descent for convex and Lipschitz objectives. Our analysis
extends the existing tail bounds from the classical light-tailed Sub-Gaussian
noise case to heavier-tailed noise regimes. We study the optimization error of
the last iterate as well as the average of the iterates. We instantiate our
results in two important cases: a class of noise with exponential tails and one
with polynomial tails. A remarkable feature of our results is that they do not
require an upper bound on the diameter of the domain. Finally, we support our
theory with illustrative experiments that compare the behavior of the average
of the iterates with that of the last iterate in heavy-tailed noise regimes.
