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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, when implemented in the
non-asymptotic regime with fixed small, but non-zero, learning rate. We achieve this by
proving approximations to continuous time limits in which the small but fixed learning rate
appears as a parameter; this is known as the method of modified equations in the numerical
analysis literature, recently rediscovered as the high resolution ODE approximation in the
machine learning context. In our second result we show that the momentum method is approximated by a continuous time gradient flow, with an additional momentum-dependent
second order time-derivative correction, proportional to the learning rate; this may be used
to explain the stabilizing effect of momentum algorithms in their transient phase. 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, equal to the original loss function plus a small
perturbation proportional to the learning rate; this small correction provides convexification of the loss function and encodes additional robustness present in momentum methods,
beyond the transient phase.

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