  This paper is concerned with convergence of stochastic gradient algorithms
with momentum terms in the nonconvex setting. A class of stochastic momentum
methods, including stochastic gradient descent, heavy ball, and Nesterov's
accelerated gradient, is analyzed in a general framework under mild
assumptions. Based on the convergence result of expected gradients, we prove
the almost sure convergence by a detailed discussion of the effects of momentum
and the number of upcrossings. It is worth noting that there are not additional
restrictions imposed on the objective function and stepsize. Another
improvement over previous results is that the existing Lipschitz condition of
the gradient is relaxed into the condition of Holder continuity. As a
byproduct, we apply a localization procedure to extend our results to
stochastic stepsizes.
