  Gradient descent-based optimization methods underpin the parameter training
of neural networks, and hence comprise a significant component in the
impressive test results found in a number of applications. Introducing
stochasticity is key to their success in practical problems, and there is some
understanding of the role of stochastic gradient descent in this context.
Momentum modifications of gradient descent such as Polyak's Heavy Ball method
(HB) and Nesterov's method of accelerated gradients (NAG), are also widely
adopted. In this work our focus is on understanding the role of momentum in the
training of neural networks, concentrating on the common situation in which the
momentum contribution is fixed at each step of the algorithm. To expose the
ideas simply we work in the deterministic setting.
  Our approach is to derive continuous time approximations of the discrete
algorithms; these continuous time approximations provide insights into the
mechanisms at play within the discrete algorithms. We prove three such
approximations. Firstly we show that standard implementations of fixed momentum
methods approximate a time-rescaled gradient descent flow, asymptotically as
the learning rate shrinks to zero; this result does not distinguish momentum
methods from pure gradient descent, in the limit of vanishing learning rate. We
then proceed to prove two results aimed at understanding the observed practical
advantages of fixed momentum methods over gradient descent. We achieve this by
proving approximations to continuous time limits in which the small but fixed
learning rate appears as a parameter. Furthermore in a third result we show
that the momentum methods admit an exponentially attractive invariant manifold
on which the dynamics reduces, approximately, to a gradient flow with respect
to a modified loss function.
