  Stochastic gradient methods are among the most important algorithms in
training machine learning problems. While classical assumptions such as strong
convexity allow a simple analysis they are rarely satisfied in applications. In
recent years, global and local gradient domination properties have shown to be
a more realistic replacement of strong convexity. They were proved to hold in
diverse settings such as (simple) policy gradient methods in reinforcement
learning and training of deep neural networks with analytic activation
functions. We prove almost sure convergence rates $f(X_n)-f^*\in o\big(
n^{-\frac{1}{4\beta-1}+\epsilon}\big)$ of the last iterate for stochastic
gradient descent (with and without momentum) under global and local
$\beta$-gradient domination assumptions. The almost sure rates get arbitrarily
close to recent rates in expectation. Finally, we demonstrate how to apply our
results to the training task in both supervised and reinforcement learning.
