  The vast majority of convergence rates analysis for stochastic gradient
methods in the literature focus on convergence in expectation, whereas
trajectory-wise almost sure convergence is clearly important to ensure that any
instantiation of the stochastic algorithms would converge with probability one.
Here we provide a unified almost sure convergence rates analysis for stochastic
gradient descent (SGD), stochastic heavy-ball (SHB), and stochastic Nesterov's
accelerated gradient (SNAG) methods. We show, for the first time, that the
almost sure convergence rates obtained for these stochastic gradient methods on
strongly convex functions, are arbitrarily close to their optimal convergence
rates possible. For non-convex objective functions, we not only show that a
weighted average of the squared gradient norms converges to zero almost surely,
but also the last iterates of the algorithms. We further provide last-iterate
almost sure convergence rates analysis for stochastic gradient methods on
weakly convex smooth functions, in contrast with most existing results in the
literature that only provide convergence in expectation for a weighted average
of the iterates.
