  Ordinary differential equation (ODE) models of gradient-based optimization
methods can provide insights into the dynamics of learning and inspire the
design of new algorithms. Unfortunately, this thought-provoking perspective is
weakened by the fact that, in the worst case, the error between the algorithm
steps and its ODE approximation grows exponentially with the number of
iterations. In an attempt to encourage the use of continuous-time methods in
optimization, we show that, if some additional regularity on the objective is
assumed, the ODE representations of Gradient Descent and Heavy-ball do not
suffer from the aforementioned problem, once we allow for a small perturbation
on the algorithm initial condition. In the dynamical systems literature, this
phenomenon is called shadowing. Our analysis relies on the concept of
hyperbolicity, as well as on tools from numerical analysis.
