  The stochastic momentum method is a commonly used acceleration technique for
solving large-scale stochastic optimization problems in artificial neural
networks. Current convergence results of stochastic momentum methods under
non-convex stochastic settings mostly discuss convergence in terms of the
random output and minimum output. To this end, we address the convergence of
the last iterate output (called last-iterate convergence) of the stochastic
momentum methods for non-convex stochastic optimization problems, in a way
conformal with traditional optimization theory. We prove the last-iterate
convergence of the stochastic momentum methods under a unified framework,
covering both stochastic heavy ball momentum and stochastic Nesterov
accelerated gradient momentum. The momentum factors can be fixed to be
constant, rather than time-varying coefficients in existing analyses. Finally,
the last-iterate convergence of the stochastic momentum methods is verified on
the benchmark MNIST and CIFAR-10 datasets.
