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.

本文研究了凸函数和 Lipschitz 目标的随机镜像下降法的优化误差，提供了新的尾部界限，并将其扩展到更重尾噪声的情况。研究了最后一次迭代和迭代平均值的优化误差，并在指数尾和多项式尾两种重要情况下实例化了结果。我们的结果的一个显著特点是不需要对定义域的直径设置上界。最后，我们通过示例实验证明在重尾噪声情况下迭代平均值与最后一次迭代的行为的比较。