  Accelerated gradient descent iterations are widely used in optimization. It
is known that, in the continuous-time limit, these iterations converge to a
second-order differential equation which we refer to as the accelerated
gradient flow. Using geometric singular perturbation theory, we show that,
under certain conditions, the accelerated gradient flow possesses an attracting
invariant slow manifold to which the trajectories of the flow converge
asymptotically. We obtain a general explicit expression in the form of
functional series expansions that approximates the slow manifold to any
arbitrary order of accuracy. To the leading order, the accelerated gradient
flow reduced to this slow manifold coincides with the usual gradient descent.
We illustrate the implications of our results on three examples.
