  We study the algorithmic stability of Nesterov's accelerated gradient method.
For convex quadratic objectives, \citet{chen2018stability} proved that the
uniform stability of the method grows quadratically with the number of
optimization steps, and conjectured that the same is true for the general
convex and smooth case. We disprove this conjecture and show, for two notions
of stability, that the stability of Nesterov's accelerated method in fact
deteriorates \emph{exponentially fast} with the number of gradient steps. This
stands in sharp contrast to the bounds in the quadratic case, but also to known
results for non-accelerated gradient methods where stability typically grows
linearly with the number of steps.
