  A theoretical, and potentially also practical, problem with stochastic
gradient descent is that trajectories may escape to infinity. In this note, we
investigate uniform boundedness properties of iterates and function values
along the trajectories of the stochastic gradient descent algorithm and its
important momentum variant. Under smoothness and $R$-dissipativity of the loss
function, we show that broad families of step-sizes, including the widely used
step-decay and cosine with (or without) restart step-sizes, result in uniformly
bounded iterates and function values. Several important applications that
satisfy these assumptions, including phase retrieval problems, Gaussian mixture
models, and some neural network classifiers, are discussed in detail. We
further extend the uniform boundedness of SGD and its momentum variant under
the generalized dissipativity for the functions whose tails grow slower than
quadratic functions. This includes some interesting applications, for example,
Bayesian logistic regression and logistic regression with $\ell_1$
regularization.
