  The classical convergence analysis of SGD is carried out under the assumption
that the norm of the stochastic gradient is uniformly bounded. While this might
hold for some loss functions, it is violated for cases where the objective
function is strongly convex. In Bottou et al. (2018), a new analysis of
convergence of SGD is performed under the assumption that stochastic gradients
are bounded with respect to the true gradient norm. We show that for stochastic
problems arising in machine learning such bound always holds; and we also
propose an alternative convergence analysis of SGD with diminishing learning
rate regime. We then move on to the asynchronous parallel setting, and prove
convergence of Hogwild! algorithm in the same regime in the case of diminished
learning rate. It is well-known that SGD converges if a sequence of learning
rates $\{\eta_t\}$ satisfies $\sum_{t=0}^\infty \eta_t \rightarrow \infty$ and
$\sum_{t=0}^\infty \eta^2_t < \infty$. We show the convergence of SGD for
strongly convex objective function without using bounded gradient assumption
when $\{\eta_t\}$ is a diminishing sequence and $\sum_{t=0}^\infty \eta_t
\rightarrow \infty$. In other words, we extend the current state-of-the-art
class of learning rates satisfying the convergence of SGD.
