  Momentum has become a crucial component in deep learning optimizers,
necessitating a comprehensive understanding of when and why it accelerates
stochastic gradient descent (SGD). To address the question of ''when'', we
establish a meaningful comparison framework that examines the performance of
SGD with Momentum (SGDM) under the \emph{effective learning rates} $\eta_{ef}$,
a notion unifying the influence of momentum coefficient $\mu$ and batch size
$b$ over learning rate $\eta$. In the comparison of SGDM and SGD with the same
effective learning rate and the same batch size, we observe a consistent
pattern: when $\eta_{ef}$ is small, SGDM and SGD experience almost the same
empirical training losses; when $\eta_{ef}$ surpasses a certain threshold, SGDM
begins to perform better than SGD. Furthermore, we observe that the advantage
of SGDM over SGD becomes more pronounced with a larger batch size. For the
question of ``why'', we find that the momentum acceleration is closely related
to \emph{abrupt sharpening} which is to describe a sudden jump of the
directional Hessian along the update direction. Specifically, the misalignment
between SGD and SGDM happens at the same moment that SGD experiences abrupt
sharpening and converges slower. Momentum improves the performance of SGDM by
preventing or deferring the occurrence of abrupt sharpening. Together, this
study unveils the interplay between momentum, learning rates, and batch sizes,
thus improving our understanding of momentum acceleration.
