  We model the dynamics of privacy loss in Langevin diffusion and extend it to
the noisy gradient descent algorithm: we compute a tight bound on R\'enyi
differential privacy and the rate of its change throughout the learning
process. We prove that the privacy loss converges exponentially fast. This
significantly improves the prior privacy analysis of differentially private
(stochastic) gradient descent algorithms, where (R\'enyi) privacy loss
constantly increases over the training iterations. Unlike composition-based
methods in differential privacy, our privacy analysis does not assume that the
noisy gradients (or parameters) during the training could be revealed to the
adversary. Our analysis tracks the dynamics of privacy loss through the
algorithm's intermediate parameter distributions, thus allowing us to account
for privacy amplification due to convergence. We prove that our privacy
analysis is tight, and also provide a utility analysis for strongly convex,
smooth and Lipshitz loss functions.
