  We analyze the convergence rate of various momentum-based optimization
algorithms from a dynamical systems point of view. Our analysis exploits
fundamental topological properties, such as the continuous dependence of
iterates on their initial conditions, to provide a simple characterization of
convergence rates. In many cases, closed-form expressions are obtained that
relate algorithm parameters to the convergence rate. The analysis encompasses
discrete time and continuous time, as well as time-invariant and time-variant
formulations, and is not limited to a convex or Euclidean setting. In addition,
the article rigorously establishes why symplectic discretization schemes are
important for momentum-based optimization algorithms, and provides a
characterization of algorithms that exhibit accelerated convergence.
