  In this paper, we study the behavior of solutions of the ODE associated to
Nesterov acceleration. It is well-known since the pioneering work of Nesterov
that the rate of convergence $O(1/t^2)$ is optimal for the class of convex
functions with Lipschitz gradient. In this work, we show that better
convergence rates can be obtained with some additional geometrical conditions,
such as \L ojasiewicz property. More precisely, we prove the optimal
convergence rates that can be obtained depending on the geometry of the
function $F$ to minimize. The convergence rates are new, and they shed new
light on the behavior of Nesterov acceleration schemes. We prove in particular
that the classical Nesterov scheme may provide convergence rates that are worse
than the classical gradient descent scheme on sharp functions: for instance,
the convergence rate for strongly convex functions is not geometric for the
classical Nesterov scheme (while it is the case for the gradient descent
algorithm). This shows that applying the classical Nesterov acceleration on
convex functions without looking more at the geometrical properties of the
objective functions may lead to sub-optimal algorithms.

Optimization and Control