  We present two classes of differentially private optimization algorithms
derived from the well-known accelerated first-order methods. The first
algorithm is inspired by Polyak's heavy ball method and employs a smoothing
approach to decrease the accumulated noise on the gradient steps required for
differential privacy. The second class of algorithms are based on Nesterov's
accelerated gradient method and its recent multi-stage variant. We propose a
noise dividing mechanism for the iterations of Nesterov's method in order to
improve the error behavior of the algorithm. The convergence rate analyses are
provided for both the heavy ball and the Nesterov's accelerated gradient method
with the help of the dynamical system analysis techniques. Finally, we conclude
with our numerical experiments showing that the presented algorithms have
advantages over the well-known differentially private algorithms.
