  The article examines in some detail the convergence rate and
mean-square-error performance of momentum stochastic gradient methods in the
constant step-size and slow adaptation regime. The results establish that
momentum methods are equivalent to the standard stochastic gradient method with
a re-scaled (larger) step-size value. The size of the re-scaling is determined
by the value of the momentum parameter. The equivalence result is established
for all time instants and not only in steady-state. The analysis is carried out
for general strongly convex and smooth risk functions, and is not limited to
quadratic risks. One notable conclusion is that the well-known bene ts of
momentum constructions for deterministic optimization problems do not
necessarily carry over to the adaptive online setting when small constant
step-sizes are used to enable continuous adaptation and learn- ing in the
presence of persistent gradient noise. From simulations, the equivalence
between momentum and standard stochastic gradient methods is also observed for
non-differentiable and non-convex problems.
