Differential Privacy Dynamics of Langevin Diffusion and Noisy Gradient Descent

What is the information leakage of an iterative learning algorithm about its training data, when the internal state of the algorithm is \emph{not} observable? How much is the contribution of each specific training epoch to the final leakage? We study this problem for noisy gradient descent algorithms, and model the \emph{dynamics} of R\'enyi differential privacy loss throughout the training process. Our analysis traces a provably tight bound on the R\'enyi divergence between the pair of probability distributions over parameters of models with neighboring datasets. We prove that the privacy loss converges exponentially fast, for smooth and strongly convex loss functions, which is a significant improvement over composition theorems. For Lipschitz, smooth, and strongly convex loss functions, we prove optimal utility for differential privacy algorithms with a small gradient complexity.